Digital Modulation 1
– Lecture Notes –
Ingmar Land and Bernard H. Fleury
Navigation and Communications (NavCom)
Department of Electronic Systems
Aalborg University, DK
Version: February 5, 2007
i
Contents
I
Basic Concepts
1
1 The Basic Constituents of a Digital Communication System
1.1 Discrete Information Sources . . . . . . . . . . . .
4
1.2 Considered System Model . . . . . . . . . . . . . .
6
1.3 The Digital Transmitter . . . . . . . . . . . . . . . 10
1.3.1
Waveform Look-up Table . . . . . . . . . . 10
1.3.2
Waveform Synthesis . . . . . . . . . . . . . 12
1.3.3
Canonical Decomposition . . . . . . . . . . 13
1.4 The Additive White Gaussian-Noise Channel . . . . 15
1.5 The Digital Receiver . . . . . . . . . . . . . . . . . 17
1.5.1
Bank of Correlators . . . . . . . . . . . . . 17
1.5.2
Canonical Decomposition . . . . . . . . . . 18
1.5.3
Analysis of the correlator outputs . . . . . . 20
1.5.4
Statistical Properties of the Noise Vector . . 23
1.6 The AWGN Vector Channel . . . . . . . . . . . . . 27
1.7 Vector Representation of a Digital Communication System 28
1.8 Signal-to-Noise Ratio for the AWGN Channel . . . 31
2 Optimum Decoding for the AWGN Channel
33
2.1 Optimality Criterion . . . . . . . . . . . . . . . . . 33
2.2 Decision Rules . . . . . . . . . . . . . . . . . . . . 34
2.3 Decision Regions . . . . . . . . . . . . . . . . . . . 36
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2.4 Optimum Decision Rules . . . . . . . . . . . . . . 37
2.5 MAP Symbol Decision Rule . . . . . . . . . . . . . 39
2.6 ML Symbol Decision Rule . . . . . . . . . . . . . . 41
2.7 ML Decision for the AWGN Channel . . . . . . . . 42
2.8 Symbol-Error Probability . . . . . . . . . . . . . . 44
2.9 Bit-Error Probability . . . . . . . . . . . . . . . . . 46
2.10 Gray Encoding . . . . . . . . . . . . . . . . . . . . 48
II
Digital Communication Techniques
3 Pulse Amplitude Modulation (PAM)
50
51
3.1 BPAM . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1
Signaling Waveforms . . . . . . . . . . . . . 51
3.1.2
Decomposition of the Transmitter . . . . . . 52
3.1.3
ML Decoding . . . . . . . . . . . . . . . . . 53
3.1.4
Vector Representation of the Transceiver . . 54
3.1.5
Symbol-Error Probability . . . . . . . . . . 55
3.1.6
Bit-Error Probability . . . . . . . . . . . . . 57
3.2 MPAM . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1
Signaling Waveforms . . . . . . . . . . . . . 58
3.2.2
Decomposition of the Transmitter . . . . . . 59
3.2.3
ML Decoding . . . . . . . . . . . . . . . . . 62
3.2.4
Vector Representation of the Transceiver . . 63
3.2.5
Symbol-Error Probability . . . . . . . . . . 64
3.2.6
Bit-Error Probability . . . . . . . . . . . . . 66
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4 Pulse Position Modulation (PPM)
67
4.1 BPPM . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1
Signaling Waveforms . . . . . . . . . . . . . 67
4.1.2
Decomposition of the Transmitter . . . . . . 68
4.1.3
ML Decoding . . . . . . . . . . . . . . . . . 69
4.1.4
Vector Representation of the Transceiver . . 71
4.1.5
Symbol-Error Probability . . . . . . . . . . 72
4.1.6
Bit-Error Probability . . . . . . . . . . . . . 73
4.2 MPPM . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1
Signaling Waveforms . . . . . . . . . . . . . 74
4.2.2
Decomposition of the Transmitter . . . . . . 75
4.2.3
ML Decoding . . . . . . . . . . . . . . . . . 77
4.2.4
Symbol-Error Probability . . . . . . . . . . 78
4.2.5
Bit-Error Probability . . . . . . . . . . . . . 80
5 Phase-Shift Keying (PSK)
85
5.1 Signaling Waveforms . . . . . . . . . . . . . . . . . 85
5.2 Signal Constellation . . . . . . . . . . . . . . . . . 87
5.3 ML Decoding . . . . . . . . . . . . . . . . . . . . . 90
5.4 Vector Representation of the MPSK Transceiver . . 93
5.5 Symbol-Error Probability . . . . . . . . . . . . . . 94
5.6 Bit-Error Probability . . . . . . . . . . . . . . . . . 97
6 Offset Quadrature Phase-Shift Keying (OQPSK)100
6.1 Signaling Scheme . . . . . . . . . . . . . . . . . . . 100
6.2 Transceiver . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Phase Transitions for QPSK and OQPSK
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7 Frequency-Shift Keying (FSK)
104
7.1 BFSK with Coherent Demodulation . . . . . . . . . 104
7.1.1
Signal Waveforms . . . . . . . . . . . . . . 104
7.1.2
Signal Constellation . . . . . . . . . . . . . 106
7.1.3
ML Decoding . . . . . . . . . . . . . . . . . 107
7.1.4
Vector Representation of the Transceiver . . 108
7.1.5
Error Probabilities . . . . . . . . . . . . . . 109
7.2 MFSK with Coherent Demodulation . . . . . . . . 110
7.2.1
Signal Waveforms . . . . . . . . . . . . . . 110
7.2.2
Signal Constellation . . . . . . . . . . . . . 111
7.2.3
Vector Representation of the Transceiver . . 112
7.2.4
Error Probabilities . . . . . . . . . . . . . . 113
7.3 BFSK with Noncoherent Demodulation . . . . . . . 114
7.3.1
Signal Waveforms . . . . . . . . . . . . . . 114
7.3.2
Signal Constellation . . . . . . . . . . . . . 115
7.3.3
Noncoherent Demodulator . . . . . . . . . . 116
7.3.4
Conditional Probability of Received Vector . 117
7.3.5
ML Decoding . . . . . . . . . . . . . . . . . 119
7.4 MFSK with Noncoherent Demodulation . . . . . . 121
7.4.1
Signal Waveforms . . . . . . . . . . . . . . 121
7.4.2
Signal Constellation . . . . . . . . . . . . . 122
7.4.3
Conditional Probability of Received Vector . 123
7.4.4
ML Decision Rule . . . . . . . . . . . . . . 123
7.4.5
Optimal Noncoherent Receiver for MFSK . . 124
7.4.6
Symbol-Error Probability . . . . . . . . . . 125
7.4.7
Bit-Error Probability . . . . . . . . . . . . . 129
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8 Minimum-Shift Keying (MSK)
130
8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . 130
8.2 Transmit Signal . . . . . . . . . . . . . . . . . . . 132
8.3 Signal Constellation . . . . . . . . . . . . . . . . . 135
8.4 A Finite-State Machine . . . . . . . . . . . . . . . 138
8.5 Vector Encoder
. . . . . . . . . . . . . . . . . . . 140
8.6 Demodulator . . . . . . . . . . . . . . . . . . . . . 144
8.7 Conditional Probability Density of Received Sequence145
8.8 ML Sequence Estimation . . . . . . . . . . . . . . 146
8.9 Canonical Decomposition of MSK . . . . . . . . . . 149
8.10 The Viterbi Algorithm . . . . . . . . . . . . . . . . 150
8.11 Simplified ML Decoder
. . . . . . . . . . . . . . . 155
8.12 Bit-Error Probability . . . . . . . . . . . . . . . . . 160
8.13 Differential Encoding and Differential Decoding . . 161
8.14 Precoded MSK . . . . . . . . . . . . . . . . . . . . 162
8.15 Gaussian MSK . . . . . . . . . . . . . . . . . . . . 166
9 Quadrature Amplitude Modulation
169
10 BPAM with Bandlimited Waveforms
170
III
Appendix
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A Geometrical Representation of Signals
172
A.1 Sets of Orthonormal Functions . . . . . . . . . . . 172
A.2 Signal Space Spanned by an Orthonormal Set of Functions173
A.3 Vector Representation of Elements in the Vector Space174
A.4 Vector Isomorphism . . . . . . . . . . . . . . . . . 175
A.5 Scalar Product and Isometry . . . . . . . . . . . . 176
A.6 Waveform Synthesis . . . . . . . . . . . . . . . . . 178
A.7 Implementation of the Scalar Product . . . . . . . . 179
A.8 Computation of the Vector Representation . . . . . 181
A.9 Gram-Schmidt Orthonormalization . . . . . . . . . 182
B Orthogonal Transformation of a White Gaussian Noise Vec
B.1 The Two-Dimensional Case . . . . . . . . . . . . . 185
B.2 The General Case . . . . . . . . . . . . . . . . . . 186
C The Shannon Limit
188
C.1 Capacity of the Band-limited AWGN Channel . . . 188
C.2 Capacity of the Band-Unlimited AWGN Channel
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References
[1] J. G. Proakis and M. Salehi, Communication Systems Engineering, 2nd ed. Prentice-Hall, 2002.
[2] B. Rimoldi, “A decomposition approach to CPM,” IEEE
Trans. Inform. Theory, vol. 34, no. 2, pp. 260–270, Mar. 1988.
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Part I
Basic Concepts
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1
The Basic Constituents of a Digital Communication System
ments
Block diagram of a digital communication system:
Digital Transmitter
Binary Information Source
Information
Source
Source
Encoder
Uk
1/Tb
Vector
Encoder
X(t)
Waveform
Modulator
1/T
Waveform
Channel
Digital Receiver
Binary Information Sink
Information
Sink
Source
Decoder
1/Tb
Ûk
Vector
Decoder
Waveform
Demodulator
• Time duration of one binary symbol (bit) Uk : Tb
• Time duration of one waveform X(t): T
• Waveform channel may introduce
- distortion,
- interference,
- noise
Goal: Signal of information source and signal at information sink
should be as similar as possible according to some measure.
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1/T
Y (t)
3
Some properties:
• Information source: may be analog or digital
• Source encoder: may perform sampling, quantization and
compression; generates one binary symbol (bit) Uk ∈ {0, 1}
per time interval Tb
• Waveform modulator: generates one waveform x(t) per time
interval T
• Vector decoder: generates one binary symbol (bit) Ûk ∈
{0, 1} (estimates of Uk ) per time interval Tb
• Source decoder: reconstructs the original source signal
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1.1
Discrete Information Sources
A discrete memoryless source (DMS) is a source that generates
a sequence U1, U2, . . . of independent and identically distributed
(i.i.d.) discrete random variables, called an i.i.d. sequence.
PSfrag replacements
DMS
U1 , U 2 , . . .
Properties of the output sequence:
discrete U1, U2, . . . ∈ {a1, a2, . . . , aQ}.
memoryless U1, U2, . . . are statistically independent.
stationary U1, U2, . . . are identically distributed.
Notice: The symbol alphabet {a1, a2, . . . , aQ} has cardinality Q.
The value of Q may be infinite, the elements of the alphabet only
have to be countable.
Example: Discrete sources
(i) Output of the PC keyboard, SMS
(usually not memoryless).
(ii) Compressed file
(near memoryless).
(iii) The numbers drawn on a roulette table in a casino
(ought to be memoryless, but may not. . . ).
3
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A DMS is statistically completely described by the probabilities
pU (aq ) = Pr(U = aq )
of the symbols aq , q = 1, 2, . . . , Q. Notice that
(i) pU (aq ) ≥ 0 for all q = 1, 2, . . . , Q, and
(ii)
Q
X
pU (aq ) = 1.
q=1
As the sequence is i.i.d., we have Pr(Ui = aq ) = Pr(Uj = aq ).
For convenience, we may write pU (a) shortly as p(a).
Binary Symmetric Source
A binary memoryless source is a DMS with a binary symbol alphabet.
Remark:
Commonly used binary symbol alphabets are F2 := {0, 1} and
B := {−1, +1}. In the following, we will use F2.
A binary symmetric source (BSS) is a binary memoryless source
with equiprobable symbols,
1
pU (0) = pU (1) = .
2
Example: Binary Symmetric Source
All length-K sequences of a BSS have the same probability
pU1U2...UK (u1u2 . . . uK ) =
K
Y
i=1
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pU (ui) =
1 K
2
.
3
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1.2
Considered System Model
This course focuses on the transmitter and the receiver. Therefore,
we replace the information source by a BSS
PSfrag replacements
Information
Source
Source
Encoder
U1 , U2 , . . .
BSS
U1 , U2 , . . .
Binary
Decoder
Sink
and the information sink by a binary (information) sink.
PSfrag replacements
BSS
Information
Sink
Source
Decoder
Û1 , Û2 , . . .
Binary
Sink
Û1 , Û2 , . . .
Encoder
U1 , U2 , . . .
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Some Remarks
(i) There is no loss of generality resulting from these substitutions.
Indeed it can be demonstrated within Shannon’s Information Theory that an efficient source encoder converts the output of an information source into a random binary independent and uniformly
distributed (i.u.d.) sequence. Thus, the output of a perfect source
encoder looks like the output of a BSS.
(ii) Our main concern is the design of communication systems for
reliable transmission of the output symbols of a BSS. We will not
address the various methods for efficient source encoding.
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Digital communication system considered in this course:
BSS
U
bit rate 1/Tb
Binary
Sink
X(t)
Digital
Transmitter
rate 1/T
Digital
Receiver
Û
Waveform
Channel
Y (t)
• Source: U = [U1, . . . , UK ], Ui ∈ {0, 1}
• Transmitter: X(t) = x(t, u)
• Sink: Û = [Û1, . . . , ÛK ], Ûi ∈ {0, 1}
• Bit rate: 1/Tb
• Rate (of waveforms): 1/T = 1/(KTb )
• Bit error probability
K
1 X
Pb =
Pr(Uk 6= Ûk )
K
k=1
Objective: Design of efficient digital communication systems
Efficiency means
(
small bit error probability and
high bit rate 1/Tb.
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Constraints and limitations:
• limited power
• limited bandwidth
• impairments (distortion, interference, noise) of the channel
Design goals:
• “good” waveforms
• low-complexity transmitters
• low-complexity receivers
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1.3
The Digital Transmitter
PSfrag replacements
Waveform
Look-up
PSfrag replacementsTable
PSfrag replacements
Example: 4PPM (Pulse-Position Modulation)
eplacements
Set of four different waveforms, S = {s1(t), s2 (t), s3 (t), s4 (t)}:
1.3.1
s1 (t)
A
0
0
s2 (t)
A
s1 (t)
t 0
0
T
s3 (t)
s1 (t)
A
s2 (t)
t 0
0
T
s1 (t)
s4 (t)
s2 (t)
A
s3 (t)
t 0
0
T
T
t
Each waveform X(t) ∈ S is addressed by vector [U1, U2] ∈ {0, 1}2 :
[U1, U2]
7→
X(t).
The mapping may be implemented by a waveform look-up table.
PSfrag replacements
4PPM
Transmitter
[00] 7→ s1(t)
[01] 7→ s2(t)
[10] 7→ s3(t)
[11] 7→ s4(t)
[U1, U2]
PSfrag replacements
X(t)
Example: [00011100] is transmitted as
x(t)
A
0
0
T
2T
3T
4T
t
3
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rag replacements
For an arbitrary digital transmitter, we have the following:
Digital
Transmitter
[U1, U2, . . . , UK ]
Waveform
Look-up
Table
X(t)
Input Binary vectors of length K from the input set U:
[U1, U2, . . . , UK ] ∈ U := {0, 1}K .
The “duration” of one binary symbol Uk is Tb.
Output Waveforms of duration T from the output set S:
x(t) ∈ S := {s1(t), s2 (t), . . . , sM (t)}.
Waveform duration T means that for m = 1, 2, . . . , M ,
sm(t) = 0 for t ∈
/ [0, T ].
The look-up table maps each input vector [u1, . . . , uK ] ∈ U to one
waveform x(t) ∈ S. Thus the digital transmitter may defined by
a mapping U → S with
[u1, . . . , uK ] 7→ x(t).
The mapping is one-to-one and onto such that
M = 2K .
Relation between signaling interval (waveform duration) T and bit
interval (“bit duration”) Tb:
T = K · Tb .
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1.3.2
Waveform Synthesis
The set of waveforms,
S := {s1(t), s2 (t), . . . , sM (t)},
spans a vector space. Applying the Gram-Schmidt orthogonalization procedure, we can find a set of orthonormal functions
Sψ := {ψ1(t), ψ2 (t), . . . , ψD (t)} with D ≤ M
such that the space spanned by Sψ contains the space spanned
by S.
Hence, each waveform sm(t) can be represented by a linear combination of the orthonormal functions:
sm(t) =
D
X
i=1
sm,i · ψi(t),
m = 1, 2, . . . , M . Each signal sm(t) can thus be geometrically
represented by the D-dimensional vector
sm = [sm,1 , sm,2 , . . . , sm,D ]T ∈ RD
with respect to the set Sψ .
Further details are given in Appendix A.
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1.3.3
Canonical Decomposition
The digital transmitter may be represented by a waveform look-up
table:
u = [u1, . . . , uK ] 7→ x(t),
where u ∈ U and x(t) ∈ S.
From the previous section, we know how to synthesize the waveform sm(t) from sm (see also Appendix A) with respect to a set of
basis functions
Sψ := {ψ1(t), ψ2(t), . . . , ψD (t)}.
Making use of this method, we can split the waveform look-up
table into a vector look-up table and a waveform synthesizer:
u = [u1, . . . , uK ]
where
eplacements
7→
x = [x1, x2, . . . , xD ]
7→
x(t),
u ∈ U = {0, 1}K ,
x ∈ X = {s1, s2, . . . , sD } ⊂ RD ,
x(t) ∈ S = {s1(t), s2 (t), . . . , sM (t)}.
This splitting procedure leads to the sought canonical decomposition of a digital transmitter:
[U1, . . . , UK ]
Vector
Encoder
Vector
Look-up Table
[0 . . . 0] 7→ s1
[1 . . . 1] 7→ sM
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X1
Waveform
Modulator
ψ1(t)
X(t)
XD
ψD (t)
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PSfrag replacements
PSfrag replacements
PSfrag replacements
√
2/ T
√
placements
Example: 4PPM 2/ T
eplacements
√
or EncoderThe four
2/ orthonormal
T
ψ1(t)
basis functions are
Vector
ψ1(t)
ψ2(t)
ψ3(t)
ψ4(t)
ψ1(t)
√
ψ1(t)
ψ2(t)
2/ T
Modulator
ψ2(t)
ψ3(t)
0
t 0
t 0
t 0
0
0
0
0
T
T
T
14
T
t
The canonical
√ decomposition of the receiver is the following, where
√
Es = A · T /2.
Vector
Encoder
Vector
Look-up Table
[U1, U2]
√
X1
X2
Waveform
Modulator
ψ1(t)
T
[00] 7→ [ Es, 0, 0, 0]
√
[01] 7→ [0, Es, 0, 0]T
√
[10] 7→ [0, 0, Es, 0]T
√
[11] 7→ [0, 0, 0, Es]T
X3
X4
ψ2(t)
X(t)
ψ3(t)
ψ4(t)
Remarks:
- The
signal energy of each basis function is equal to one:
R
ψd(t)dt = 1 (due to their construction).
- The basis functions are orthonormal (due to their construction).
√
- The value Es is chosen such that the synthesized signals have
the same energy as the original signals, namely Es (compare
Example 3).
3
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1.4
The Additive White Gaussian-Noise
PSfrag Channel
replacements
W (t)
Y (t)
X(t)
The additive white Gaussian noise (AWGN) channel-model is
widely used in communications. The transmitted signal X(t) is
superimposed by a stochastic noise signal W (t), such that the
transmitted signal reads
Y (t) = X(t) + W (t).
The stochastic process W (t) is a stationary process with the
following properties:
(i) W (t) is a Gaussian process, i.e., for each time t, the
samples v = w(t) are Gaussian distributed with zero mean
and variance σ 2:
v2 exp − 2 .
pV (v) = √
2
2σ
2πσ
1
(ii) W (t) has a flat power spectrum with height N0/2:
SW (f ) =
N0
2
(Therefore it is called “white”.)
(iii) W (t) has the autocorrelation function
RW (τ ) =
Land, Fleury: Digital Modulation 1
N0
δ(τ ).
2
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Remarks
1. Autocorrelation function and power spectrum:
RW (τ ) = E W (t)W (t + τ ) , SW (f ) = F RW (τ ) .
2. For Gaussian processes, weak stationarity implies strong stationarity.
3. An WGN is an idealized process without physical reality:
• The process is so “wild” that its realizations are not
ordinary functions of time.
• Its power is infinite.
However, a WGN is a useful approximation of a noise with a
flat power spectrum in the bandwidth B used by a communication system:
Snoise(f )
PSfrag replacements
N0/2
−f0
f0
B
B
f
4. In satellite communications, W (t) is the thermal noise of the
receiver front-end. In this case, N0/2 is proportional to the
squared temperature.
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1.5
The Digital Receiver
Consider a transmission system using the waveforms
n
o
S = s1(t), s2 (t), . . . , sM (t)
with sm(t) = 0 for t ∈
/ [0, T ], m = 1, 2, . . . , M , i.e., with duration T .
Assume transmission over an AWGN channel, such that
y(t) = x(t) + w(t),
x(t) ∈ S.
1.5.1
Bank of Correlators
The transmitted waveform may be recovered from the received
waveform y(t) by correlating y(t) with all possible waveforms:
eplacements
cm = hy(t), sm(t)i,
m = 1, 2, . . . , M . Based on the correlations [c1, . . . , cM ], the waveform ŝ(t) with the highest correlation is chosen and the corresponding (estimated) source vector û = [û1, . . . , ûK ] is output.
s1 (t)
RT
sM (t)
RT
0
y(t)
0
(.)dt
(.)dt
c1
cM
estimation
of x̂(t)
and
table
look-up
[û1, . . . , ûK ]
Disadvantage: High complexity.
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1.5.2
Canonical Decomposition
Consider a set of orthonormal functions
n
o
Sψ = ψ1(t), ψ2(t), . . . , ψD (t)
obtained by applying the
s1(t), s2 (t), . . . , sM (t). Then,
sm(t) =
D
X
d=1
Gram-Schmidt
procedure
to
sm,d · ψd(t),
m = 1, 2, . . . , M . The vector
sm = [sm,1 , sm,2 , . . . , sm,D ]T
entirely determines sm(t) with respect to the orthonormal set Sψ .
The set Sψ spans the vector space
n
Sψ := s(t) =
D
X
i=1
siψi(t) : [s1, s2, . . . , sD ] ∈ R
D
o
.
Basic Idea
• The received waveform y(t) may contain components outside
of Sψ . However, only the components inside Sψ are relevant,
as x(t) ∈ Sψ .
• Determine the vector representation y of the components
of y(t) that are inside Sψ .
• This vector y is sufficient for estimating the transmitted
waveform.
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Canonical decomposition of the optimal receiver for
the AWGN channel
Appendix A describes two ways to compute the vector representation of a waveform. Accordingly, the demodulator may be implemented in two ways, leading to the following two receiver structures.
replacements
Correlator-based Demodulator and Vector Decoder
RT
(.)dt
RT
(.)dt
0
Y (t)
ψ1(t)
0
ψD (t)
Y1
Vector
Decoder
[Û1, . . . , ÛK ]
YD
replacements
Matched-filter based Demodulator and Vector Decoder
Y (t)
MF
ψ1(T − t)
Y1
MF
ψD (T − t)
YD
Vector
Decoder
[Û1, . . . , ÛK ]
T
The optimality of the receiver structures is shown in the following
sections.
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1.5.3
Analysis of the correlator outputs
Assume that the waveform sm(t) is transmitted, i.e.,
x(t) = sm(t).
The received waveform is thus
y(t) = x(t) + w(t) = sm(t) + w(t).
The correlator outputs are
yd =
ZT
y(t)ψd(t)dt
0
=
ZT
0
=
ZT
|0
i
sm(t) + w(t) ψd(t)dt
sm(t)ψd(t)dt +
{z
sm,d
= sm,d + wd,
}
ZT
w(t)ψd (t)dt
|0
{z
wd
}
d = 1, 2, . . . , D.
Using the notation
y = [y1, . . . , yD ]T,
w = [w1, . . . , wD ]T,
wd =
ZT
w(t)ψd(t)dt,
0
we can recast the correlator outputs in the compact form
y = sm + w.
The waveform channel between x(t) = sm(t) and y(t) is thus transformed into a vector channel between x = sm and y.
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Remark:
Since the vector decoder “sees” noisy observations of the vectors
sm, these vectors are also called signal points with respect to Sψ ,
and the set of signal points is called the signal constellation.
From Appendix A, we know that sm(t) entirely characterizes sm(t).
The vector y, however, does not represent the complete waveform y(t); it represents only the waveform
y 0(t) =
D
X
ydψd(t)
d=1
which is the “part” of y(t) within the vector space spanned by the
set Sψ of orthonormal functions, i.e.,
y 0(t) ∈ Sψ .
The “remaining part” of y(t), i.e., the part outside of Sψ , reads
y ◦(t) = y(t) − y 0(t)
D
X
= y(t) −
ydψd(t)
d=1
= sm(t) + w(t) −
= sm(t) −
D
X
d=1
D
X
[sm,d + wd]ψd(t)
d=1
sm,d ψd(t) +w(t) −
{z
}
=0
D
X
wdψd(t).
= w(t) −
|
D
X
wdψd(t)
d=1
d=1
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Observations
• The waveform y ◦(t) contains no information about sm(t).
• The waveform y 0(t) is completely represented by y.
Therefore, the receiver structures do not lead to a loss of information, and so they are optimal.
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1.5.4
Statistical Properties of the Noise Vector
The components of the noise vector
w = [w1, . . . , wD ]T
are given by
wd =
ZT
w(t)ψd(t)dt,
0
d = 1, 2, . . . , D. The random variables Wd are Gaussian as they
result from a linear transformation of the Gaussian process W (t).
Hence the probability density function of Wd is of the form
1
(wd − µd)2 pWd (wd) = p
.
exp −
2σd2
2πσd2
where the mean value µd and the variance σd2 are given by
2
2
σd = E (Wd − µd) .
µd = E W d ,
These values are now determined.
Mean value
Using the definition of Wd and taking into account that the process W (t) has zero mean, we obtain
E Wd = E
hZT
0
=
ZT
0
= 0.
W (t)ψd (t)dt
i
E W (t) ψd(t)dt
Thus, all random variables Wd have zero mean.
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Variance
For computing the variance, we use µd = 0, the definition of Wd,
and the autocorrelation function of W (t),
RW (τ ) =
N0
δ(τ ).
2
We obtain the following chain of equations:
E (Wd − µd)2 = E Wd2
ZT
ZT
=E
W (t)ψd (t)dt
W (t0)ψd(t0)dt0
0
ZT ZT
0
E W (t)W (t0 ) ψd(t)ψd(t0)dtdt0
{z
}
|
0
0 0
RW (t − t)
ZT ZT
N0
=
δ(t − t0)ψd(t0)dt0 ψd(t)dt
2
0
{z
}
|0
δ(t) ∗ ψd(t) = ψd(t)
ZT
N0
N0
=
ψd2(t)dt =
.
2
2
=
0
Distribution of the noise vector components
Thus we have µd = 0 and σd2 = N0/2, and the distributions read
wd2 1
exp −
.
pWd (wd) = √
N0
πN0
for d = 1, 2, . . . , D. Notice that the components Wd are all identically distributed.
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As the Gaussian distribution is very important, the shorthand
notation
A ∼ N (µ, σ 2)
is often used. This notation means: The random variable A is
Gaussian distributed with mean value µ and variance σ 2.
Using this notation, we have for the components of the noise vector
Wd ∼ N (0, N0/2),
d = 1, 2, . . . , D.
Distribution of the noise vector
Using the same reasoning as before, we can conclude that W is a
Gaussian noise vector, i.e.,
W ∼ N (µ, Σ),
where the expectation µ ∈ RD×1 and the covariance matrix Σ ∈
RD×D are given by
µ=E W ,
Σ = E (W − µ)(W − µ)T .
From the previous considerations, we have immediately
µ = 0.
Using a similar approach as for computing the variances, the statistical independence of the components of W can be shown. That
is,
N0
E Wi Wj =
δi,j ,
2
where δi,j denotes the Kronecker symbol defined as
(
1 for i = j,
δi,j =
0 otherwise.
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As a consequence, the covariance matrix of W is
Σ=
N0
I D,
2
where I D denotes the D × D identity matrix.
To sum up, the noise vector W is distributed as
N
0
W ∼ N 0, I D .
2
As the components of W are statistically independent, the probability density function of W can be written as
pW (w) =
D
Y
pWd (wd)
d=1
D
Y
wd2 1
√
exp −
=
N0
πN
0
d=1
D
1 D
1 X 2
exp −
= √
wd
N0
πN0
d=1
−D/2
1
2
exp − kwk
= πN0
N0
with kwk denoting the Euclidean norm
kwk =
D
X
d=1
wd2
1/2
of the D-dimensional vector w = [w1, . . . , wD ]T.
The independency of the components of W give rise to the AWGN
vector channel.
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1.6
The AWGN Vector Channel
The additive white Gaussian noise vector channel is defined as a
channel with input vector
X = [X1, X2 , . . . , XD ]T,
output vector
Y = [Y1, Y2, . . . , YD ]T,
and the relation
Y = X + W,
where
W = [W1, W2, . . . , WD ]T
is a vector of D independent random variables distributed as
Wd ∼ N (0, N0/2).
PSfrag replacements
W
X
Y
(As nobody would have been expected differently. . . )
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1.7
Vector Representation of a Digital
Communication System
Consider a digital communication system transmitting over
an AWGN channel. Let s1(t), s2 (t), . . . , sM (t) , sm(t) = 0 for
t∈
/ [0, T ], m = 1, 2, . . . , M , denote the waveforms. Let further
Sψ = ψ1(t), ψ2(t), . . . , ψD (t)
denote a set of orthonormal functions for these waveforms.
placements
The canonical decompositions of the digital transmitter and the
digital receiver convert the block formed by the waveform modulator, the AWGN channel, and the waveform demodulator into an
AWGN vector channel.
Modulator, AWGN channel, demodulator
W (t)
X1
ψ1(t)
X(t)
RT
(.)dt
Y1
RT
(.)dt
YD
0
Y (t)
ψ1(t)
XD
0
ψD (t)
ψD (t)
AWGN vector channel
PSfrag replacements
[W1, . . . , WD ]
[X1, . . . , XD ]
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29
placements
Using this equivalence, the communication system operating over a
(waveform) AWGN channel can be represented as a communication
system operating over an AWGN vector channel:
U
BSS
Vector
Encoder
X
W
Binary
Sink
Vector
Decoder
Û
Y
AWGN
Vector
Channel
• Source vector: U = [U1, . . . , UK ], Ui ∈ {0, 1}
• Transmit vector: X = [X1, . . . , XD ]
• Receive vector: Y = [Y1, . . . , YD ]
• Estimated source vector: Û = [Û1, . . . , ÛK ], Ui ∈ {0, 1}
• AWGN vector: W = [W1, . . . , WD ],
N
0
W ∼ N 0, I D
2
This is the vector representation of a digital communication system
transmitting over an AWGN channel.
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For estimating the transmitted vector x, we are interested in the
likelihood of x for the given received vector y:
pY |X (y|x) = pW (y − x)
−D/2
1
2
= πN0
exp − ky − xk .
N0
Symbolically, we may write
Y |X = sm
∼
N0 N sm , I D .
2
Remark:
The likelihood of x is the conditional probability density function
of y given x, and it is regarded as a function of x.
Example: Signal buried in AWGN
Consider D = 3 and x = sm. Then the “Gaussian cloud”
around sm may look like this:
ψ2
sm
PSfrag replacements
ψ1
ψ3
3
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1.8
Signal-to-Noise Ratio for the AWGN
Channel
Consider a digital communication system using the set of waveforms
n
o
S = s1(t), s2 (t), . . . , sM (t) ,
sm(t) = 0 for t ∈
/ [0, T ], m = 1, 2, . . . , M , for transmission over
an AWGN channel with power spectrum
SW (f ) =
N0
.
2
Let Esm denote the energy of waveform sm(t), and let E s denote
the average waveform energy:
E sm =
ZT
0
2
sm(t) dt,
M
1 X
Es =
Es
M m=1 m
Thus, E s is the average energy per waveform x(t) and also
the average energy per vector x (due to the one-to-one correspondence.)
The average signal-to-noise ratio (SNR) per waveform is
then defined as
Es
γs =
.
N0
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On the other hand, the average energy per source bit uk is of
interest. Since K source bits uk are transmitted per waveform, the
average energy per bit is
Eb =
1
E s.
K
Accordingly, the average signal-to-noise ratio (SNR) per
bit is then defined as
Eb
γb =
.
N0
As K bits are transmitted per waveform, we have the relation
γb =
1
γs .
K
Remark:
The energies and signal-to-noise ratios usually denote received values.
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2
Optimum Decoding
AWGN Channel
placements2.1
for
the
Optimality Criterion
In the previous chapter, the vector representation of a digital
transceiver transmitting over an AWGN channel was derived.
U
BSS
Vector
Encoder
X
W
Binary
Sink
Vector
Decoder
Û
Y
AWGN
Vector
Channel
Nomenclature
u = [u1, . . . , uK ] ∈ U,
û = [û1, . . . , ûK ] ∈ U,
U = {0, 1}K ,
x = [x1, . . . , xD ] ∈ X,
y = [y1, . . . , yD ] ∈ RD ,
X = {s1, . . . , sM }.
U : set of all binary sequences of length K,
X : set of vector representations of the waveforms.
The set X is called the signal constellation, and its elements
are called signals, signal points, or modulation symbols.
For convenience, we may refer to uk as bits and to x as symbols.
(But we have to be careful not to cause ambiguity.)
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As the vector representation implies no loss of information, recovering the transmitted block u = [u1, . . . , uK ] from the received waveform y(t) is equivalent to recovering u from the vector y = [y1, . . . , yD ].
Functionality of the vector encoder
u →
7
x = enc(u)
U → X
enc :
Functionality of the vector decoder
decu :
y →
7
û = decu(y)
RD → U
Optimality criterion
A vector decoder is called an optimal vector decoder if it
minimizes the block error probability Pr(U 6= Û ).
2.2
Decision Rules
Since the encoder mapping is one-to-one, the functionality of the
vector decoder can be decomposed into the following two operations:
(i) Modulation-symbol decision:
dec :
y →
7
x̂ = dec(y)
RD → X
(ii) Encoder inverse:
enc−1 :
Land, Fleury: Digital Modulation 1
x̂ 7→ û = enc−1(x̂)
X → U
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placements
Thus, we have the following chain of mappings:
y
dec
7→
x̂
enc−1
7→
û.
Block Diagram
Vector
Encoder
enc
U
U
X
X
W
Encoder
Inverse
enc−1
Û
U
X̂
X
Mod. Symbol
Decision
dec
RD
Y
RD
Vector Decoder
ag replacements
A decision rule is a function which maps each observation y ∈
RD to a modulation symbol x ∈ X, i.e., to an element of the signal
constellation X.
Example: Decision rule
RD
y (1)
y (2)
y (3)
s 1 , s 2 , . . . , sM
=X
3
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2.3
Decision Regions
A decision rule dec : RD → X defines a partition of RD into decision regions
ag replacements
D
Dm := y ∈ R : dec(y) = sm ,
m = 1, 2, . . . , M .
Example: Decision regions
RD
.
..
DM
D1
D2
s 1 , s 2 , . . . , sM
=X
3
Since the decision regions D1, D2, . . . , DM form a partition, they
have the following properties:
(i) They are disjoint:
Di ∩ Dj = ∅ for i 6= j.
(ii) They cover the whole observation space RD :
D1 ∪ D 2 ∪ · · · ∪ D M = R D .
Obviously, any partition of RD defines a decision rule as well.
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2.4
Optimum Decision Rules
Optimality criterion for decision rules
Since the mapping enc : U → X of the vector encoder is one-toone, we have
Û 6= U ⇔ X̂ 6= X.
Therefore, the decision rule of an optimal vector decoder minimizes
the modulation-symbol error probability Pr(X̂ 6= X).
Such a decision rule is called optimal.
Sufficient condition for an optimal decision rule
Let dec : RD → X be a decision rule which minimizes the
a-posteriori probability of making a decision error,
Pr dec(y) 6= X Y = y ,
| {z }
x̂
for any observation y. Then dec is optimal.
Remark:
The a-posteriori probability of a decoding error is
Sfrag replacements
the probability of the event {dec(y) 6= X} conditioned on the
event {Y = y}:
W
X̂ = dec(y)
Mod. Symbol
Decision
dec
Land, Fleury: Digital Modulation 1
Y =y
X
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Proof:
Let dec : RD → X denote any decision rule, and let
decopt : RD → X denote an optimal decision rule. Then we
have
Z
Pr dec(Y ) 6= X = Pr dec(Y ) 6= X Y = y · pY (y)dy
RD
Z
=
RD
(?)
≥
Z
RD
=
Z
RD
Pr dec(y) 6= X Y = y · pY (y)dy
Pr decopt (y) 6= X Y = y · pY (y)dy
Pr decopt(Y ) 6= X Y = y · pY (y)dy
= Pr decopt (Y ) 6= X .
(?): by definition of decopt , we have
Pr dec(y) 6= X Y = y ≥ Pr decopt (y) 6= X Y = y
for all x ∈ X.
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2.5
MAP Symbol Decision Rule
Starting with the necessary condition for an optimal decision rule
given above, we derive now a method to make an optimal decision.
The corresponding decision rule is called the maximum a-posteriori
(MAP) decision rule.
Consider an optimal decision rule
dec : y 7→ x̂ = dec(y),
where the estimated modulation-symbol is denoted by x̂.
As the decision rule is assumed to be optimum, the probability
Pr(x̂ 6= X|Y = y)
is minimal.
For any x ∈ X, we have
Pr(X 6= x|Y = y) = 1 − Pr(X = x|Y = y)
= 1 − pX|Y (x|y)
=1−
pY |X (y|x) pX (x)
,
pY (y)
where Bayes’ rule has been used in the last line. Notice that
Pr(X = x|Y = y) = pX|Y (x|y)
is the a-posteriori probability of {X = x} given {Y = y}.
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Using the above equalities, we can conclude that
⇔
⇔
Pr(X 6= x|Y = y) is minimal
pX|Y (x|y) is maximal
pY |X (y|x) pX (x) is maximal,
since pY (y) does not depend on x.
Thus, the estimated modulation-symbol x̂ = dec(y) resulting from
the optimal decision rule satisfies the two equivalent conditions
pX|Y (x̂|y) ≥ pX|Y (x|y)
for all x ∈ X;
pY |X (y|x̂) pX (x̂) ≥ pY |X (y|x) pX (x) for all x ∈ X.
These two conditions may be used to define the optimal decision
rule.
Maximum a-posteriori (MAP) decision rule:
n
o
x̂ = decMAP (y) := argmax pX|Y (x|y)
x∈X
n
o
:= argmax pY |X (y|x) pX (x) .
x∈X
The decision rule is called the MAP decision rule as the selected
modulation-symbol x̂ maximizes the a-posteriori probability density function pX|Y (x|y). Notice that the two formulations are
equivalent.
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2.6
ML Symbol Decision Rule
If the modulation-symbols at the output of the vector encoder are
equiprobable, i.e., if
Pr(X = sm) =
1
M
for m = 1, 2, . . . , M , the MAP decision rule reduces to the following decision rule.
Maximum likelihood (ML) decision rule:
n
o
x̂ = decML(y) := argmax pY |X (y|x) .
x∈X
In this case, the the selected modulation-symbol x̂ maximizes the
likelihood function pY |X (y|x) of x.
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2.7
ML Decision for the AWGN Channel
Using the canonical decomposition of the transmitter and of the
receiver, we obtain the AWGN vector-channel:
PSfrag replacements
N0
W ∼ N 0, 2 I D
Y ∈ RD
X∈X
The conditional probability density function of Y
X = x ∈ X is
−D/2
1
2
pY |X (y|x) = πN0
exp − ky − xk .
N0
given
Since exp(.) is a monotonically increasing function, we obtain the
following formulation of the ML decision rule
n
o
x̂ = decML(y) = argmax pY |X (y|x)
x∈X
o
1
2
= argmax exp − ky − xk
N0
x∈X
o
n
2
= argmin ky − xk .
n
x∈X
Thus, the ML decision rule for the AWGN vector channel selects
the vector in the signal constellation X that is the closest one to the
observation y, where the distance measure is the squared Euclidean
distance.
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Implementation of the ML decision rule
The squared Euclidean distance can be expanded as
ky − xk2 = kyk2 − 2hy, xi + kxk2 .
The term kyk2 is irrelevant for the minimization with respect to x.
Thus we get the following equivalent formulation of the ML decision
rule:
o
n
2
x̂ = decML(y) = argmin −2hy, xi + kxk
x∈X
n
= argmax 2hy, xi − kxk
x∈X
n
= argmax hy, xi −
x∈X
2
2
1
kxk
2
o
o
.
PSfrag replacements
Block diagram of the ML decision rule for the AWGN vectorchannel:
y
s1
1
2
2 ks1 k
sM
1
2
2 ksM k
..
.
..
.
select
largest
one
x̂
PSfragVector
replacements
correlator:
a∈R
D
ha, bi =
b ∈ RD
Land, Fleury: Digital Modulation 1
PD
d=1 ad bd
∈R
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Remark:
Some signal constellations X have the property that kxk2 has the
same value for all x ∈ X. In these cases (but only then), the ML
decision rule simplifies further to
n
o
x̂ = decML(y) = argmax hy, xi .
x∈X
2.8
Symbol-Error Probability
The error probability for modulation-symbols X, or for short, the
symbol-error probability is defined as
Ps = Pr(X̂ 6= X) = Pr(dec(Y ) 6= X).
The symbol-error probability coincides with the block error probability
Ps = Pr(Û 6= U ) = Pr [Û1, . . . , ÛK ] 6= [U1, . . . , UK ] .
We compute Ps in two steps:
(i) Compute the conditional symbol-error probability given
X = x for any x ∈ X:
Pr(X̂ 6= X|X = x).
(ii) Average over the signal constellation:
X
Pr(X̂ 6= X) =
Pr(X̂ 6= X|X = x) · Pr(X = x).
x∈X
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Remember: Decision regions for the symbol decision rule:
RD
.
..
DM
D1
D2
s 1 , s 2 , . . . , sM
=X
Assume that x = sm is transmitted. Then,
Pr(X̂ 6= X|X = sm) = Pr(X̂ 6= sm|X = sm)
= Pr(Y ∈
/ Dm|X = sm)
= 1 − Pr(Y ∈ Dm|X = sm).
For the AWGN vector channel, the conditional probability density
function of Y given X = sm is
−D/2
1
pY |X (y|sm) = πN0
exp − ky − smk2 .
N0
Hence,
Z
Pr(Y ∈ Dm|X = sm) =
pY |X (y|sm)dy =
Dm
= πN0
−D/2
Z
Dm
exp −
1
ky − smk2 dy.
N0
The symbol-error probability is then obtained by averaging
over the signal constellation:
Pr(X̂ 6= X) = 1 −
M
X
m=1
Pr(Y ∈ Dm|X = sm) · Pr(X = sm).
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If the modulation symbols are equiprobable, the formula simplifies
to
M
1 X
Pr(Y ∈ Dm|X = sm).
Pr(X̂ 6= X) = 1 −
M m=1
2.9
Bit-Error Probability
The error probability for the source bits Uk , or for short, the biterror probability is defined as
K
1 X
Pb =
Pr(Ûk 6= Uk ),
K
k=1
i.e., as the average number of erroneous bits in the block
Û = [Û1, . . . , ÛK ].
Relation between the bit-error probability and the
symbol-error probability
Clearly, the relation between Pb and Ps depends on the encoding
function
enc : u = [u1, . . . , uk ] 7→ x = enc(u).
The analytical derivation of the relation between Pb and Ps is
usually cumbersome. However, we can derive an upper bound
and a lower bound for Pb when Ps is given (and also vice versa).
These bounds are valid for any encoding function.
The lower bound provides a guideline for the design of a “good”
encoding function.
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Lower bound:
If a symbol is erroneous, there is at least one erroneous bit in the
block of K bits. Therefore
Pb ≥
1
K Ps .
Upper bound:
If a symbol is erroneous, there are at most K erroneous bits in the
block of K bits. Therefore
Pb ≤ P s .
Combination:
Combining the lower bound and the upper bound yields
1
K Ps
≤ Pb ≤ Ps .
An efficient encoder should achieve the lower bound for the biterror probability, i.e.,
Pb ≈ K1 Ps.
Remark:
The bounds can also be obtained by considering the union bound
of the events
{Û1 6= U1}, {Û2 6= U2}, . . . , {ÛK 6= UK },
i.e, by relating the probability
Pr {Û1 6= U1} ∪ {Û2 6= U2} ∪ . . . ∪ {ÛK 6= UK }
to the probabilities
Pr {Û1 6= U1} , Pr {Û2 6= U2} , . . . , Pr {ÛK 6= UK } .
Land, Fleury: Digital Modulation 1
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2.10
Gray Encoding
Sfrag replacements
In Gray encoding, blocks of bits, u = [u1, . . . , uK ], which differ in
only one bit uk are assigned to neighboring vectors (modulation
symbols) x in the signal constellation X.
Example: 4PAM (Pulse Amplitude Modulation)
Signal constellation: X = {[−3], [−1], [+1], [+3]}.
00
s1
−2
01
11
s2
s3
10
2
s4
3
Motivation
When a decision error occurs, it is very likely that the transmitted
symbol and the detected symbol are neighbors. When the bit
blocks of neighboring symbols differ only in one bit, there will be
only one bit error per symbol error.
Effect
Gray encoding achieves
Pb ≈
1
K Ps
for high signal-to-noise ratios (SNR).
Land, Fleury: Digital Modulation 1
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49
Sketch of the proof:
We consider the case K = 2, i.e., U = [U1, U2].
Ps = Pr(X̂ 6= X)
= Pr [Û1, Û2] 6= [U1, U2]
= Pr {Û1 6= U1} ∪ {Û2 6= U2}
= Pr(Û1 6= U1) + Pr(Û2 6= U2)
|
{z
}
2 · Pb
− Pr {Û1 6= U1} ∩ {Û2 6= U2} .
{z
}
|
(?)
The term (?) is small compared to the other two terms if the SNR
is high. Hence
Ps ≈ 2 · P b .
Land, Fleury: Digital Modulation 1
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Part II
Digital Communication
Techniques
Land, Fleury: Digital Modulation 1
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51
3
Pulse
Amplitude
(PAM)
Modulation
In pulse amplitude modulation (PAM), the information is conveyed
by the amplitude of the transmitted waveforms.
3.1
BPAM
PAM with only two waveforms is called binary PAM (BPAM).
3.1.1
Signaling Waveforms
PSfrag replacements
Set of two rectangular
waveforms
with amplitudes A and −A over
t ∈ [0, T ], S := s1(t), s2 (t) :
s1 (t)
s2 (t)
A
0
−A
A
T t
0
−A
T t
As |S| = 2, one bit is sufficient to address the waveforms (K = 1).
Thus, the BPAM transmitter performs the mapping
U 7→ x(t)
with
U ∈ U = {0, 1}
and
x(t) ∈ S.
Remark: In the general case, the basic waveform may also have
another shape.
Land, Fleury: Digital Modulation 1
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52
3.1.2
Decomposition of the Transmitter
Gram-Schmidt procedure
1
ψ1(t) = √ s1(t)
E1
with
E1 =
ZT
2
s1(t) dt =
0
Thus
ψ1(t) =
(
ZT
0
√1
T
0
A2dt = A2 · T.
for t ∈ [0, T ] ,
otherwise.
Both s1(t) and s2(t) are a linear combination of ψ(t):
p
√
s1(t) = + Es ψ1(t) = +A T ψ1(t),
p
√
s2(t) = − Es ψ1(t) = −A T ψ1(t),
√
√
where Es = A T and Es = E1 = E2 . (The two waveforms
have the same energy.)
Signal constellation
Vector representation of the two waveforms:
p
s1(t) ↔ s1 = s1 = + Es
p
PSfrag replacements s2(t) ↔ s2 = s2 = − Es
√
√ Thus the signal constellation is X = − Es, + Es .
√
Es
√
Es
ψ1
s2
Land, Fleury: Digital Modulation 1
0
s1
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53
Vector Encoder
The vector encoder is usually defined as
( √
+ Es for u = 0,
x = enc(u) =
√
− Es for u = 1.
The signal constellation comprises only two waveforms, and thus
per symbol and per bit, the average transmit energies and the
SNRs are the same:
Eb = E s ,
3.1.3
γs = γb .
ML Decoding
ML symbol decision rule
( √
+ Es for y ≥ 0,
2
decML(y) =
argmin
ky − xk =
√
√
√
Es for y < 0.
−
x∈{− Es ,+ Es }
PSfrag replacements
Decision regions
D2ψ=
1 {y ∈ R : y < 0},
D1 = {y ∈ R : y ≥ 0}.
D1
D2
√
s2 = − E s
0
√
y
s1 = + E s
Remark: The decision rule
( √
+ Es for y > 0,
decML(y) =
√
− Es for y ≤ 0.
is also a ML decision rule.
Land, Fleury: Digital Modulation 1
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54
placements
The mapping of the encoder inverse reads
(
√
Es ,
0
for
x̂
=
+
û = enc-1 (x̂) =
√
1 for x̂ = − Es.
Combining the (modulation) symbol decision rule and the encoder
inverse yields the ML decoder for BPAM:
(
0 for y ≥ 0,
û =
1 for y < 0.
3.1.4
Vector Representation of the Transceiver
BPAM transceiver:
BSS
U
placements
√1
T
Sink
RT
0
Û
Encoder
√
0 7→ + √Es
1 7→ − Es
X
Decoder
D1 7→ 0
D2 7→ 1
Y
X(t)
W (t)
ψ1(t)
√1
T
RT
0
Y (t)
(.)dt
(.)dt
Vector representation:
BSS
Sink
U
Û
Encoder
√
0 7→ + √Es
1 7→ − Es
X
Decoder
D1 7→ 0
D2 7→ 1
Y
Land, Fleury: Digital Modulation 1
W ∼ N (0, N0 /2)
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55
3.1.5
Symbol-Error Probability
Conditional probability densities
p
p
1
1
pY |X (y| + Es) = √
exp − |y − Es|2
N0
πN0
p
p
1
1
2
exp − |y + Es|
pY |X (y| − Es) = √
N0
πN0
Conditional symbol-error probability
√
Assume that X = x = + Es is transmitted.
Pr(X̂ 6= X|X = +
p
p
Es) = Pr(Y ∈ D2|X = + Es)
p
= Pr(Y < 0|X = + Es)
Z0
p
1
1
2
=√
exp − |y − Es| dy
N0
πN0
−∞
√
− 2Es /N0
Z
1
=√
exp − 12 z 2 dz,
2π
−∞
where the last equality results from substituting
p
p
z = (y − Es)/ N0/2.
The Q-function is defined as
Q(v) :=
Z∞
q(z)dz,
v
Land, Fleury: Digital Modulation 1
1
q(z) = √ exp − 12 z 2 .
2π
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56
The function q(z) denotes a Gaussian pdf with zero mean and variance 1; i.e., it may be interpreted as the pdf of a random variable
Z ∼ N (0, 1).
Using this definition, the conditional symbol-error probabilities
may be written as
r 2E p
s
Pr(X̂ 6= X|X = + Es) = Q
,
N
0
r 2E p
s
Pr(X̂ 6= X|X = − Es) = Q
.
N0
The second equation follows from symmetry.
Symbol-error probability
The symbol-error probability results from averaging over the conditional symbol-error probabilities:
Ps = Pr(X̂ 6= X)
p
p
= Pr(X = + Es) · Pr(X̂ 6= X|X = + Es)
p
p
+ Pr(X = − Es) · Pr(X̂ 6= X|X = − Es)
r 2E p
s
= Q( 2γs)
=Q
N0
√
√
as Pr(X = + Es) = Pr(X = − Es) = 1/2.
Alternative method to derive the conditional symbolerror probability
√
Assume again that X = Es is transmitted. Instead of considering the conditional pdf of Y , we directly use the pdf of the white
Gaussian noise W .
Land, Fleury: Digital Modulation 1
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57
p
p
Pr(X̂ 6= X|X = + Es) = Pr(Y < 0|X = + Es)
p
p
= Pr( Es + W < 0|X = + Es)
p
p
= Pr(W < − Es|X = + Es)
p
= Pr(W < − Es)
p
0
= Pr(W < − 2Es/N0)
p
= Q( 2Es/N0),
p
where we substituted W 0 = W/ N0/2; notice that
W 0 ∼ N (0, 1). A plot of the symbol-error probability can found
in [1, p. 412].
The initial event (here p
{Y < 0}) is transformed into an equivalent
event (here {W 0 < − 2Es/N0}), of which the probability can
easily be calculated. This method is frequently applied to determine symbol-error probabilities.
3.1.6
Bit-Error Probability
In BPAM, each waveform conveys one source bit. Accordingly,
each symbol-error leads to exactly one bit-error, and thus
Pb = Pr(Û 6= U ) = Pr(X̂ 6= X) = Ps.
Due to the same reason, the transmit energy per symbol is equal
to the transmit energy per bit, and so also the SNR per symbol is
equal to the SNR per bit:
Eb = E s ,
γs = γb .
Therefore, the bit-error probability (BEP) of BPAM results as
p
Pb = Q( 2γb).
Notice: Pb ≈ 10−5 is achieved for γb ≈ 9.6 dB.
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58
3.2
MPAM
In the general case of M -ary pulse amplitude modulation (MPAM)
with M = 2K , input blocks of K bits modulate the amplitude of
a unique pulse.
3.2.1
Signaling Waveforms
Waveform encoder:
where
u = [u1, . . . , uK ] →
7
x(t)
K
U = {0, 1} → S = s1(t), . . . , sM (t) ,
sm(t) =
(
(2m − M − 1) A g(t) for t ∈ [0, T ],
0
otherwise,
A ∈ R, and g(t) is a predefined pulse of duration T ,
g(t) = 0 for t ∈
/ [0, T ].
Values of the factors (2m − M − 1)A that are weighting g(t):
−(M − 1)A, −(M − 3)A, . . . , −A, A, . . . , (M − 3)A, (M − 1)A.
Example: 4PAM with rectangular pulse.
Using the rectangular pulse
(
g(t) =
1 for t ∈ [0, T ],
0 otherwise,
the resulting waveforms are
S = −3Ag(t), −Ag(t), Ag(t), 3Ag(t) .
Land, Fleury: Digital Modulation 1
3
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59
3.2.2
Decomposition of the Transmitter
Gram-Schmidt procedure:
Orthonormal function
1
1
ψ(t) = √ s1(t) = p g(t),
E1
Eg
2
2
RT
RT
E1 = 0 s1(t) dt,
Eg = 0 g(t) dt.
Every waveform in S is a linear combination of (only) ψ(t):
p
sm(t) = (2m − M − 1) A Eg ψ(t)
| {z }
d
= (2m − M − 1)d ψ(t).
Signal constellation
s1(t)
s2(t)
sm(t)
sM −1(t)
sM (t)
←→
←→
···
←→
···
←→
←→
s1 = −(M − 1)d
s2 = −(M − 3)d
sm = (2m − M − 1)d
sM −1 = (M − 3)d
sM = (M − 1)d
X = s1 , s 2 , . . . , s M
= −(M − 1)d, −(M − 3)d, . . . , −d, d, . . .
. . . , (M − 3)d, (M − 1)d .
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Example: M = 2, 4, 8
0
1
s1
s2
00
01
11
10
s1
s2
s3
s4
000
001
011
010
110
111
101
100
s1
s2
s3
s4
s5
s6
s7
s8
3
Vector Encoder
The vector encoder
enc :
u = [u1, . . . , uK ] →
7
x = enc(u)
K
U = {0, 1} → X = −(M − 1)d, . . . , (M − 1)d
is any Gray encoding function.
Energy and SNR
The source bits are generated by a BSS, and so the symbols of the
signal constellation X are equiprobable:
Pr(X = sm) =
1
M
for all m = 1, 2, . . . , M .
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61
The average symbol energy results as
Es = E X
2
M
1 X
(sm)2
=
M m=1
M
d2 X
d2 1
2
M (M 2 − 1)
=
(2m − M − 1) =
M m=1
M3
Thus,
M2 − 1 2
Es =
d
3
⇐⇒
d=
s
3E s
M2 − 1
As M = 2K , the average energy per transmitted source bit results
as
r
2
M −1 2
Es
3 log2 M
Eb =
=
d
⇐⇒
d=
E b.
K
3 log2 M
M2 − 1
The SNR per symbol and the SNR per bit are related as
γb =
Land, Fleury: Digital Modulation 1
1
1
γs =
γs .
K
log2 M
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62
3.2.3 ML Decoding
PSfrag replacements
ML symbol decision rule
decML(y) = argmin ky − xk2
x∈X
Decision regions
Example: Decision regions for M = 4.
D1
D2
D3
D4
s2
s3
s4
s1
−2d
0
y
2d
3
Starting from the special cases M = 2, 4, 8, the decision regions
can be inferred:
D1 = {y
D2 = {y
···
Dm = {y
···
DM −1 = {y
DM = {y
∈ R : y ≤ −(M − 2)d},
∈ R : −(M − 2)d < y ≤ −(M − 4)d},
∈ R : (2m − M − 2)d < y ≤ (2m − M )d},
∈ R : (M − 4)d < y ≤ (M − 2)d},
∈ R : (M − 2)d < y}.
Accordingly, the ML symbol decision rule may also be written as
decML(y) = sm for y ∈ Dm.
ML decoder for MPAM
û = enc-1 (decML(y))
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3.2.4
BPAM transceiver:
U
BSS
placements
√1
T
Vector Representation of the Transceiver
Sink
Û
Gray
Encoder
um 7→ sm
X
Decoder
Dm 7→ um
Y
X(t)
W (t)
ψ(t)
√1
T
RT
0
Y (t)
(.)dt
Vector representation:
RT
0
(.)dt
U
BSS
Sink
Û
Gray
Encoder
um 7→ sm
X
Decoder
Y
W ∼ N (0, N0 /2)
Dm 7→ um
Notice: um denotes the block of source bits corresponding to sm,
i.e.,
sm = enc(um),
Land, Fleury: Digital Modulation 1
um = enc-1 (sm).
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64
3.2.5
Symbol-Error Probability
Conditional symbol-error probability
Two cases must be considered:
1. The transmitted symbol is an edge symbol (a symbol at the
edge of the signal constellation):
X ∈ {s1, sM }.
2. The transmitted symbol is not an edge symbol:
X ∈ {s2, s3, . . . , sM −1}.
Case 1: X ∈ {s1, sM }
Assume first X = s1. Then
Pr(X̂ 6= X|X = s1) = Pr(Y ∈
/ D1|X = s1)
= Pr(Y > s1 + d|X = s1)
= Pr(W > d)
W
d >p
= Pr p
N0/2
N0/2
d .
=Q p
N0/2
p
Notice that W/ N0/2 ∼ N (0, 1).
By symmetry,
d
Pr(X̂ 6= X|X = sM ) = Q p
.
N0/2
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65
Case 2: X ∈ {s2, . . . , sM −1 }
Assume X = sm. Then
Pr(X̂ 6= X|X = sm) = Pr(Y ∈
/ Dm|X = sm)
= Pr(W < −d or W > d)
= Pr(W < −d) + Pr(W > d)
W
d < −p
= Pr p
N0/2
N0/2
W
d >p
+ Pr p
N0/2
N0/2
d d =Q p
+Q p
N0/2
N0/2
d .
=2·Q p
N0/2
Symbol-error probability
Averaging over the symbols, one obtains
M
1 X
Pr(X̂ 6= X|X = sm)
Ps =
M m=1
2 d 2(M − 2) d =
Q p
+
Q p
M
M
N0/2
N0/2
d
M −1
=2
Q p
.
M
N0/2
Expressing the symbol-error probability as a function of the SNR
per symbol, γs, or the SNR per bit, γb, yields
r
6
M −1
Ps = 2
Q
γs
M
M2 − 1
r
M −1
6 log2 M =2
Q
γb .
M
M2 − 1
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66
Plots of the symbol-error probabilities can be found in [1, p. 412].
Remark: When comparing BPAM and 4PAM for the same Ps,
the SNRs per bit, γb, differ by 10 log10(5/2) ≈ 4 dB. (Proof?)
3.2.6
Bit-Error Probability
When Gray encoding is used, we have the relation
Ps
log2 M
r 6 log M M −1
2
=2
Q
γb
M log2 M
M2 − 1
Pb ≈
for high SNR γb = E b/N0.
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4
Pulse Position Modulation (PPM)
In pulse position modulation (PPM), the information is conveyed
by the position of the transmitted waveforms.
4.1
BPPM
PPM with only two waveforms is called binary PPM (BPPM).
4.1.1 Signaling Waveforms
PSfrag replacements
Set of two rectangular waveforms
with amplitude
A and dura
tion T /2 over t ∈ [0, T ], S := s1(t), s2 (t) :
s1 (t)
s2 (t)
A
0
−A
A
T
2
T
t
0
−A
T
2
T
t
As the cardinality of the set is |S| = 2, one bit is sufficient to
address each waveforms, i.e., K = 1. Thus, the BPPM transmitter
performs the mapping
U 7→ x(t)
with
U ∈ U = {0, 1}
and
x(t) ∈ S.
Remark: In the general case, the basic waveform may also have
another shape.
Land, Fleury: Digital Modulation 1
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4.1.2
Decomposition of the Transmitter
Gram-Schmidt procedure
The waveforms s1(t) and s2(t) are orthogonal and have the same
energy
ZT
ZT
Es = s21(t)dt = s22(t)dt = A2 · T2 .
0
p
√
Notice that Es = A T /2.
0
The two orthonormal functions and the corresponding waveform
representations are
p
1
ψ1(t) = √ s1(t),
s1(t) = Es · ψ1(t) + 0 · ψ2(t),
Es
p
1
ψ2(t) = √ s2(t),
s2(t) = 0 · ψ1(t) + Es · ψ2(t).
Es
Signal constellation
Vector representation of the two waveforms:
p
s1(t) ←→ s1 = ( Es, 0)T,
p
s2(t) ←→ s2 = (0, Es)T.
√
√
Thus the signal constellation is X = ( Es, 0)T , (0, Es)T .
PSfrag replacements
ψ2
√
Es
s2
√
2Es
s1
0
0
Land, Fleury: Digital Modulation 1
√
Es
ψ1
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69
Vector Encoder
The vector encoder is defined as
(√
( Es, 0)T for u = 0,
x = enc(u) =
√
(0, Es)T for u = 1.
The signal constellation comprises only two waveforms, and thus
per (modulation) symbol and per (source) bit, the average transmit
energies and the SNRs are the same:
Eb = E s ,
4.1.3
γs = γb .
ML Decoding
ML symbol decision rule
(√
( Es, 0)T for y1 ≥ y2,
2
decML(y) = argmin ky − xk =
√
(0, Es)T for y1 < y2.
x∈{s1 ,s2 }
Notice: ky − s1k2 ≤ ky − s2k2 ⇔ y1 ≥ y2.
Decision
regions
PSfrag
replacements
D1 = {y ∈ R2 : y1 ≥ y2},
√
2Es y2
√
Es
s2
D2 = {y ∈ R2 : y1 < y2}.
D2
D1
s1
0
0
Land, Fleury: Digital Modulation 1
√
Es
y1
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70
Combining the (modulation) symbol decision rule and the encoder
inverse yields the ML decoder for BPPM:
(
0 for y1 ≥ y2,
û = enc-1 (decML(y)) =
1 for y1 < y2.
PSfrag replacements
The ML decoder may be implemented by
Y1
Y2
Z
z≥0→0
Û
z<0→1
Notice that Z = Y1 − Y2 is then the decision variable.
Land, Fleury: Digital Modulation 1
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4.1.4
Vector Representation of the Transceiver
BPPM transceiver for the waveform AWGN channel
ψ1(t)
BSS
U
placements
Vector
Encoder
√
0 7→ ( Es, 0)T
√
1 7→ (0, Es)T
X1
X(t)
W (t)
X2
ψ2(t)
Y1
Sink
Û
Vector
Decoder
D1 7→ 0
D2 7→ 1
q R
T /2
2
T
Y2
0
q R
T
2
T
T /2
(.)dt
(.)dt
Y (t)
ψ1(t)
ψ (t)
q R 2
T /2
2
(.)dtVector representation including a AWGN vector channel
T 0
q R
2 T
T T /2 (.)dt
X1
Vector
U
Encoder
√
BSS
0 7→ ( Es, 0)T
√
1 7→ (0, Es)T
X2
W1 ∼ N (0, N0 /2)
Y1
Sink
Û
Vector
Decoder
D1 7→ 0
D2 7→ 1
Land, Fleury: Digital Modulation 1
W2 ∼ N (0, N0 /2)
Y2
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72
4.1.5
Symbol-Error Probability
Conditional symbol-error probability
√
Assume that X = s1 = ( Es, 0)T is transmitted.
Pr(X̂ 6= X|X = s1) = Pr(Y ∈ D2|X = s1)
= Pr(s1 + W ∈ D2|X = s1)
PSfrag replacements
= Pr(s1 + W ∈ D2)
0
p
The noiseEvector
W may be represented in the coordinate system
s /2
(ξ1, ξ2) or in the rotated coordinate system (ξ10 , ξ20 ). The latter is
more suited to computing the above probability, as only the noise
component along the ξ20 -axis is relevant.
ξ2
D2
ξ20
D1
s2
s1 + w
w2
ξ10
√
Es
w10
α = π/4
w20
√
Es
s1
w1
ξ1
The elements of the noise vector in the rotated coordinate system
have the same statistical properties as in the non-rotated coordinate system (see Appendix B), i.e.,
W10 ∼ N (0, N0/2),
Land, Fleury: Digital Modulation 1
W20 ∼ N (0, N0 /2).
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73
Thus, we obtain
p
Pr(X̂ =
6 X|X = s1) =
> Es/2)
r
W20
Es
= Pr( p
>
)
N
N /2
0
p 0
√
= Q( Es/N0) = Q( γs).
Pr(W20
By symmetry,
√
Pr(X̂ 6= X|X = s2) = Q( γs).
Symbol-error probability
Averaging over the conditional symbol-error probabilities, we obtain
√
Ps = Pr(X̂ 6= X) = Q( γs).
4.1.6
Bit-Error Probability
For BPPM, Eb = Es, γs = γb, Pb = Ps (see above), and so the
bit-error probability (BEP) results as
√
Pb = Pr(Û 6= U ) = Q( γb).
A plot of the BEP versus γb may be found in [1, p. 426]
Remark
Due to their signal constellations, BPPM is called orthogonal signaling and BPAM is called antipodal signaling. Their BEPs are
given by
p
√
Pb,BPAM = Q( 2γb).
Pb,BPPM = Q( γb),
When comparing the SNR for a certain BEP, BPPM requires 3 dB
more than BPAM. For a plot, see [1, p. 409]
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4.2
MPPM
In the general case of M -ary pulse position modulation (MPPM)
with M = 2K , input blocks of K bits determine the position of a
unique pulse.
4.2.1
Signaling Waveforms
Waveform encoder
where
u = [u1, . . . , uK ] →
7
x(t)
K
U = {0, 1} → S = s1(t), . . . , sM (t) ,
(m − 1)T ,
sm(t) = g t −
M
m = 1, 2, . . . , M , and g(t) is a predefined pulse of duration T /M ,
g(t) = 0 for t ∈
/ [0, T /M ].
The waveforms are orthogonal and have the same energy as g(t),
Es = E g =
ZT
g 2(t)dt.
0
PSfrag replacements
Example: (M=4)
PSfrag replacements
Consider
a rectangular pulse g(t) with amplitude A and duPSfrag
replacements
ration T /4. Then, we have the set of four waveforms, S =
eplacements
{s1(t), s2 (t), s3 (t), s4 (t)}:
s1 (t)
A
0
0
s2 (t)
A
s1 (t)
t 0
0
T
s3 (t)
s1 (t)
A
s2 (t)
t 0
0
T
s1 (t)
s4 (t)
s2 (t)
A
s3 (t)
t 0
0
T
T
t
3
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4.2.2
Decomposition of the Transmitter
Gram-Schmidt procedure
As all waveforms are orthogonal, we have M orthonormal functions:
1
ψm(t) = √ sm(t),
Es
m = 1, 2, . . . , M .
The waveforms and the orthonormal functions are related as
p
sm(t) = Es · ψm(t),
m = 1, 2, . . . , M .
Signal constellation
Waveforms and their vector representations:
←→
←→
...
←→
s1(t)
s2(t)
sM (t)
M
zp
}|
{
s1 = ( Es, 0, 0, . . . , 0)T
p
s2 = (0, Es, 0, . . . , 0)T
sM
p
= (0, 0, 0, . . . , Es)T
Signal constellation:
X = s1 , s 2 , . . . , s M ∈ R M .
The modulation symbols x ∈ X are orthogonal, and they are
placed on corners of an M -dimensional hypercube.
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Example: M = 3
The signal constellation
p
p
p
T
T
X = ( Es, 0, 0) , (0, Es, 0) , (0, Es, 0)T ∈ R3
√
may be visualized using a cube with side-length Es.
3
Vector Encoder
enc :
u = [u1, . . . , uK ] →
7
x = enc(u)
K
U = {0, 1} → X = s1, s2, . . . , sM
Energy and SNR
The symbol energy is Es, and as M = 2K , the average energy per
transmitted source bit results as
Eb =
Es
K
The SNR per symbol and the SNR per bit are related as
γb =
Land, Fleury: Digital Modulation 1
1
1
γs =
γs .
K
log2 M
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4.2.3
ML Decoding
ML symbol decision rule
decML(y) = argmin ky − xk2
x∈X
= argmin kyk2 − 2hy, xi + kxk2
| {z }
x∈X
Es
= argmax hy, xi
x∈X
ML decoder for MPAM
û = enc-1 (decML(y))
placements
Implementation of the ML detector using vector correlators:
hy, s1i
y
select
s1
..
.
max.
x̂
enc
-1
û
hy, sM i
sM
Notice that û = [û1, . . . , ûK ].
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4.2.4
Symbol-Error Probability
Conditional symbol-error probability
Assume first that X = s1 is transmitted. Then
Pr(X̂ 6= X|X = s1) = 1 − Pr(X̂ = X|X = s1)
= 1 − Pr hy, s1i ≥ hy, smi for all m 6= 1|X = s1 .
√
For X = s1 = ( Es, 0, . . . , 0), the received vector is
p
Y = s1 + W = ( Es, 0, . . . , 0) + (W1 , . . . , WM ),
so that
hy, smi =
(
√
Es + Es · W1 for m = 1,
√
Es · Wm
for m = 2, . . . , M .
Therefore, the symbol-error probability may be written as follows:
Pr(X̂ 6= X|X = s1) =
p
p
= 1 − Pr Es + EsW1 ≥ EsWm for all m 6= 1
s
Es
W1
Wm
= 1 − Pr
+p
≥p
for all m 6= 1
N0/2
N /2
N /2
| {z0 } | {z0 }
W10
Wm0
r
2Es
= 1 − Pr Wm0 ≤
+ W10 for all m 6= 1
N0
with
0 T
W 0 = (W10 , . . . , WM
) ∼ N (0, I),
i.e., the components Wm0 are independent and Gaussian distributed
with zero mean and variance 1.
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The probability is now written using the pdf of W10 :
q
2Es
0
0
Pr Wm ≤ N0 + W1 for all m 6= 1 =
Z+∞
q
0
2Es
0
0
Pr Wm ≤ N0 + W1 for all m 6= 1W1 = v pW10 (v)dv.
=
|
{z
}
−∞
(?)
The events corresponding to different values of m are independent,
and thus
(?) =
=
M
Y
Pr
m=2 |
M
Y
Q
m=2
q
=v
+
{zq
}
s
Pr Wm0 ≤ 2E
N0 + v
Wm0
q
≤
2Es
N0
2Es
N0
+v =Q
W10 W10
M −1
p
( 2γs + v).
Using this result and inserting pW10 (v), we obtain the conditional
symbol-error probability
Pr(X̂ 6= X|X = s1) =
Z+∞h
i 1
p
v2 M −1
=
1−Q
( 2γs + v) √ exp − dv
2
2π
−∞
Symbol-error probability
The above expression is independent of s1 and thus valid for all
x ∈ X. Therefore, the average symbol-error probability results as
Ps =
Z+∞h
−∞
1−Q
M −1
Land, Fleury: Digital Modulation 1
i 1
p
v2 ( 2γs + v) √ exp − dv.
2
2π
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4.2.5
Bit-Error Probability
Hamming distance
The Hamming distance dH(u, u0) between two length-K vectors
u and u0 is the number of positions in which they differ:
dH(u, u0) =
K
X
δ(uk , u0k ),
k=1
where the indicator function is defined as
(
1 for u = u0,
0
δ(u, u ) =
0 for u 6= u0.
Example: 4PPM
Let the blocks of source bits and the modulation symbols be associated as
u1
u2
u3
u4
= enc-1 (s1) = [0, 0]
= enc-1 (s2) = [0, 1]
= enc-1 (s3) = [1, 0]
= enc-1 (s4) = [1, 1].
Then we have for example
dH(u1, u1) = 0,
dH(u2, u3) = 2,
dH(u1, u2) = 1,
dH(u2, u4) = 1.
3
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Conditional bit-error probability
Assume that X = s1, and let
um = [um,1 , . . . , um,K ] = enc-1 (sm).
The conditional bit-error probability is defined as
K
1 X
Pr(U 6= Û |X = s1) =
Pr(Uk =
6 Ûk |X = s1).
K
k=1
The argument of the sum may be expanded as
Pr(Uk 6= Ûk |X = s1) =
M
X
=
Pr(Uk 6= Ûk |X̂ = sm, X = s1) Pr(X̂ = sm|X = s1).
m=1
Since
X = s1
X̂ = sm
we have
⇔
U = u1
⇔
Û = um
⇒
⇒
Uk = u1,k ,
Ûk = um,k ,
Pr(Uk 6= Ûk |X̂ = sm, X = s1) =
= Pr(u1,k 6= um,k |X̂ = sm, X = s1)
= δ(u1,k , um,k ).
Using this result in the expression for the conditional bit-error
probability, we obtain
Pr(U 6= Û |X = s1)
K M
1 XX
δ(u1,k , um,k ) Pr(X̂ = sm|X = s1)
=
K
m=1
=
1
K
k=1
K
M X
X
m=1 k=1
|
δ(u1,k , um,k ) Pr(X̂ = sm|X = s1)
{z
}
dH(u1, um)
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Due to the symmetry of the signal constellation, the probabilities
of making a specific decision error are identical, i.e.,
Pr(X̂ = sm|X = sm0 ) =
Ps
Ps
= K
M −1 2 −1
for all m 6= m0. Using this in the above sum yields
M
1
Ps X
Pr(U =
6 Û |X = s1) =
·
dH(u1, um).
K 2K − 1 m=1
Without loss of generality, we assume u1 = 0. The sum over the
Hamming distances becomes then the overall number of ones in all
bit blocks of length K.
Example:
Consider K = 3 and write the binary vectors of length 3 in the
columns of a matrix:


0
0
0
0
1
1
1
1
T
T
u1 , . . . , u 8 =  0 0 1 1 0 0 1 1  .
0 1 0 1 0 1 0 1
In every row, half of the entries are ones. The overall number of
ones is thus 3 · 23−1 .
3
Generalizing the example, the above sum can shown to be
M
X
m=1
dH(0, um) = K · 2K−1 .
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83
Applying this result, we obtain the conditional bit-error probability
2K−1
· Ps ,
Pr(U =
6 Û |X = s1) = K
2 −1
which is independent of s1 and thus the same for all transmitted
symbols X = sm, m = 1, . . . , M .
Average bit-error probability
From above, we obtain
2K−1
Pr(U =
6 Û ) = K
· Ps .
2 −1
Plots of the BEPs may be found in [1, p. 426]
Notice that for large K,
Pr(U 6= Û ) ≈
1
· Ps .
2
Due to the signal constellation, Gray mapping is not possible.
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Remarks
(i) Provided that the SNR γb = Eb/N0 is larger than ln 2 (corresponding to −1.6 dB), the BEP of MPPM can be
made arbitrarily small by selecting M large enough.
The price to pay is an increasing decoding delay and an increasing required bandwidth.
Let the transmission rate be denoted by
R=
K log2 M
=
T
T
[bit/sec].
Decoding delay:
1
· log2 M.
R
Hence, for a fixed rate R, T → ∞ for M → ∞.
T =
Required bandwidth:
B = Bg ≈
M
M
1
=
=R·
,
T /M
T
log2 M
where Bg denotes the bandwidth of the pulse g(t).
Hence, for a fixed rate R, B → ∞ for M → ∞.
(ii) The power of the MPPM signal is
Eb · K
= Eb · R.
T
Hence, the condition γb > ln 2 is equivalent to the condition
P =
R<
1
P
·
= C∞ .
ln 2 N0
The value C∞ denotes the capacity of the (infinite bandwidth) AWGN channel (see Appendix C).
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85
5
Phase-Shift Keying (PSK)
In phase-shift keying (PSK), the waveforms are sines with the same
frequency, and the information is conveyed by the phase of the
waveforms.
5.1
Signaling Waveforms
Set of sines sm(t) with the same frequency f0, duration T , and
different phases φm:
√
m−1

2P
cos
2πf
t
+
2π
for t ∈ [0, T ],

0

| {zM }
sm(t) =
φm


0
elsewhere,
m = 1, 2, . . . , M . The phase can take the values
φm ∈ 0, 2π M1 , 2π M2 , . . . , 2π MM−1
Hence, the digital information is embedded in the phase of a carrier.
The frequency f0 is called the carrier frequency.
For M = 2, this modulation scheme is called binary phaseshift keying (BPSK), and for M = 4, it is called quadrature
phase-shift keying. (QPSK).
All waveforms have the same energy
Es =
Z∞
s2m(t)dt = P T.
−∞
(Notice: cos2 x = 1 + cos 2x.)
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86
Usually, f0 1/T . For simplifying the theoretical analysis, we
assume that f0 = L/T for some large L ∈ N.
BPSK
For M = 2, we obtain φm ∈ {0, π} and thus the two waveforms
√
s1(t) = 2P cos 2πf0t ,
√
s2(t) = 2P cos 2πf0t + π = −s1(t)
for t ∈ [0, T ].
QPSK
For M = 4, we obtain φm ∈ {0, π2 , π, 3π
2 } and thus the waveforms
are
√
s1(t) = + 2P cos 2πf0t ,
√
s2(t) = − 2P sin 2πf0t ,
√
s3(t) = − 2P cos 2πf0t ,
√
s4(t) = + 2P sin 2πf0t
for t ∈ [0, T ].
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5.2
Signal Constellation
BPSK
Orthonormal functions:
ψ1(t) =
√1 s1 (t)
Es
=
for t ∈ [0, T ].
q
2
T
cos 2πf0 t
Geometrical representation of the waveforms:
p
√
√
s1(t) = + P T ψ1(t) ←→ s1 = + P T = + Es
p
√
√
s2(t) = − P T ψ1(t) ←→ s2 = − P T = − Es
PSfrag replacements
Signal constellation:
√
Es
√
Es
ψ1
s2
s1
0
Vector encoder:
x = enc(u) =
Land, Fleury: Digital Modulation 1
(
s1 for u = 1,
s2 for u = 0.
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88
QPSK
Orthonormal functions:
ψ1(t) =
ψ2(t) =
q
√1 s1 (t)
Es
=+
√1 s2 (t)
Es
=−
q
2
T
cos 2πf0t
2
T sin 2πf0 t
for t ∈ [0, T ].
Geometrical representation of the waveforms:
p
√
s1(t) = + P T ψ1(t) ←→ s1 = ( Es, 0)T
p
p
s2(t) = Esψ2(t) ←→ s2 = (0, Es)T
p
√
s3(t) = − P T ψ1(t) ←→ s3 = (− Es, 0)T
p
p
s4(t) = − Esψ2(t) ←→ s4 = (0, − Es)T
with Es = P T .
Signal constellation:
ψ2
PSfrag replacements
s2
ψ1
s1
s3
s4
Vector encoder: any Gray encoder
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General Case
The expression for sm(t) can be expanded as follows:
p
sm(t) = Es cos 2πf0t + φm
p
p
= Es cos φm cos(2πf0 t) − Es sin φm sin(2πf0t),
where φm = 2π m−1
M , m = 1, 2, . . . , M .
(Notice: cos(x + y) = cos x cos y − sin x sin y.)
Orthonormal functions:
ψ1(t) =
q
ψ2(t) = −
for t ∈ [0, T ].
2
T
q
cos(2πf0 t)
2
T
sin 2πf0t
Geometrical representation of the waveforms:
p
p
sm(t) = Es cos φm · ψ1(t) + Es sin φm · ψ2(t)
p
p
T
Es cos φm, Es sin φm ,
←→ sm =
m = 1, 2, . . . , M . Remember: Es = P T .
Signal constellation:
o
n p
p
m−1
m−1 T
Es cos 2π M , Es sin 2π M
: m = 1, 2, . . . , M
X=
Example: 8PSK
For M = 8, we obtain φm = π m−1
4 , m = 1, 2, . . . , 8. (Figure?)
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5.3
ML Decoding
ML symbol decision rule
decML(y) = argmin ky − xk2
x∈X
ML decoder for MPSK
û = [û1, û2, . . . , ûK ] = enc-1 (decML(y))
PSfrag replacements
BPSK
ψ1
BPSK has the same signal constellation as BPAM. Thus, the ML
detectors for these two modulation schemes are the same.
D1
D2
s2
0
(u = 0)
y
s1
(u = 1)
PSfrag replacements
Implementation of the ML detector:
Y
Land, Fleury: Digital Modulation 1
y≥0→1
y<0→0
Û
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91
PSfrag replacements
QPSK
Decision regions for the symbols sm and corresponding bit blocks
u = [u1, u2] (Gray encoding):
y2
s2
01
D2
11
00
s3
s1
D3
s4
D1
y1
10
D4
Decision rule for the data bits:
(
0 for y ∈ D1 ∪ D2
û1 =
1 for y ∈ D3 ∪ D4
(
PSfrag replacements
0 for y ∈ D1 ∪ D4
û2 =
1 for y ∈ D2 ∪ D3
⇔ y1 + y 2 > 0
⇔ y1 + y 2 < 0
⇔ y2 − y 1 < 0
⇔ y2 − y 1 > 0
Implementation of the ML decoder:
Y1
Y2
Land, Fleury: Digital Modulation 1
Z1
z1 > 0 → 0
z1 < 0 → 1
Û1
Z2
z2 < 0 → 0
z2 > 0 → 1
Û2
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92
General Case
The decision regions for the general case can be determined in a
straight-forward way:
PSfrag replacements
y2
s2
2π/M
π/M
D1
y1
s1
sM
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93
5.4
Vector Representation of the MPSK
Transceiver
placements
q
2
T
cos 2πf0 t
T
X1
or Channel
U
X(t)
Vector
Encoder
X2
q
2
T
T
− sin 2πf0 t
W (t)
cos 2πf0 t
Y1
Û
Vector
Decoder
q R
T
2
T
q R
T
2
T
Y2
0
0
(.)dt
Y (t)
(.)dt
− sin 2πf0 t
AWGN Vector Channel:
PSfrag replacements
W1 ∼ N (0, N0 /2)
X1
Y1
X2
Y2
W2 ∼ N (0, N0 /2)
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5.5
Symbol-Error Probability
BPSK
Since BPSK has the same signal constellation as BPAM, the two
modulation schemes have the same symbol-error probability:
q p
2Es
= Q( 2γs).
Ps = Q
N0
QPSK
Sfrag replacements
0 that X = s is transmitted. Then, the conditional
Assume
1
p
symbol-error probability is given by
Es/2
Pr(X̂ 6= X|X = s1) = Pr(y ∈
/ D1|X = s1)
= 1 − Pr(y ∈ D1|X = s1)
= 1 − Pr(s1 + W ∈ D1).
For computing this probability, the noise vector is projected on the
rotated coordinate system (ξ10 , ξ20 ) (compare BPPM):
y2
o
isi
c
de
bo
un
ξ20
de
cis
ion
D2
s2
da
ry
ξ2
n
w2
D1
s1 + w
bo
ξ10
y
ar
d
un
w10
D3
w20
√
Es
s1
w1
π/4
y1
ξ1
D4
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The probability may then be written as
q q
Es
0
0
Pr(s1 + W ∈ D1) = Pr W1 > − 2 and W2 < E2s
q q = Pr W10 > − E2s · Pr W20 < E2s
q q W10
W20
E
Es
s
= Pr √
> − N0 · Pr √
< N
0
N0 /2
N0 /2
√
√
= 1 − Q( γs) 1 − Q( γs)
√ 2
= 1 − Q( γs) .
Hence,
√
2
Pr(X̂ 6= X|X = s1) = 1 − 1 − Q( γs)
√
√
= 2 Q( γs) − Q2( γs).
By symmetry, the four conditional symbol-error probabilities are
identical, and thus the average symbol-error probability results as
√
√
Ps = 2 Q( γs) − Q2( γs).
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General Case
A closed-form expression for Ps does not exist. It can be upperbounded using an union-bound technique, or it can be numerically
computed using an integral expression. (See literature.) A plot of
Ps versus the SNR can be found in [1, p. 416].
Remark: Union Bound
For two random events A and B, we have
Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B) ≤ Pr(A) + Pr(B).
| {z }
≤0
The expression on the right-hand side is called the union bound for
the probability on the left-hand side. In a similar way, we obtain
the union bound for multiple events:
Pr(A1 ∪ A2 ∪ · · · ∪ AM ) ≤ Pr(A1) + Pr(A2) + · · · + Pr(AM ).
Each of these events may denote the case that the received vector
is within a certain decision region.
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5.6
Bit-Error Probability
BPSK
Since Eb = Es, γs = γb, Pb = Ps, the bit-error probability is given
by
p
Pb = Q( 2γb).
QPSK
For Gray encoding, Pb can directly be computed.
Assume first that X = s1 is transmitted, which corresponds to
[U1, U2] = [0, 0]. Then, the conditional bit-error probability is
given by
Pr(Û 6= U |X = s1) =
= 12 Pr(Û1 6= U1|X = s1) + Pr(Û2 6= U2|X = s1)
= 21 Pr(Û1 6= 0|X = s1) + Pr(Û2 6= 0|X = s1)
= 21 Pr(Û1 = 1|X = s1) + Pr(Û2 = 1|X = s1) .
From the direct decision rule for data bits, we have
Û1 = 1
Û2 = 1
Land, Fleury: Digital Modulation 1
⇔
⇔
Y ∈ D 3 ∪ D4 ,
Y ∈ D 2 ∪ D3 .
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98
Thus,
Pr(Û 6= U |X = s1) =
= 12 Pr(Y ∈ D3 ∪ D4) + Pr(Y ∈ D2 ∪ D3)
q q = 21 Pr W10 < − E2s + Pr W20 > E2s
q q 0
0
W
Es
Es
√W2 > N
= 12 Pr √ 1 < − N
+
Pr
0
0
N0 /2
N0 /2
√
= Q( γs).
By symmetry, all four conditional bit-error probabilities are identical. Since two data bits are transmitted per modulation symbol (waveform) in QPSK, we have the relations Es = 2Eb and
γs = 2γb. Therefore,
p
Pb = Q( 2γb).
(Remember: Pb ≈ 10−5 for γb ≈ 9.6 dB.)
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Comparison of QPSK and BPSK
(i) QPSK has exactly the same bit-error probability as BPSK.
(ii) Relation between symbol duration T and (average) bit duration Tb = T /K:
BPSK: T = Tb,
QPSK: T = 2Tb.
Hence, for a fixed symbol duration T , the bit rate of QPSK
is twice as fast as that of BPSK.
General Case
Since Gray encoding is used, we have
Pb ≈
1
Ps
log2 M
for high SNR.
The symbol-error probability may be determined using one of the
methods mentioned above.
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6
Offset Quadrature
Keying (OQPSK)
Phase-Shift
Offset QPSK is a variant of QPSK. It has the same performance
as QPSK, but the requirements on the transmitter and receiver
hardware are lower.
6.1
Signaling Scheme
Consider QPSK with the following “tilted” signal constellation and Gray encoding:
PSfrag replacements
ψ2
01
11
00
√
Es
p
Es/2
ψ1
10
The phase of QPSK can change by maximal π. This is the case
for instance, when two consecutive 2-bit blocks are [00], [11] or
[01], [10]. These phase changes put high requirements on the dynamic behavior of the RF-part of the QPSK transmitter and receiver.
Phase changes of π can be avoided by staggering the quadrature
branch of QPSK by T /2 = Tb. The resulting modulation scheme
is called staggered QPSK or offset QPSK (OQPSK).
Land, Fleury: Digital Modulation 1
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Without significant loss of generality, we assume
2L
for some L 1, L ∈ N.
f0 =
T
Remark
The ψ1 component of the modulation symbol is called the in-phase
component (I), and the ψ2 component is called the quadrature
component (Q).
placements6.2
Transceiver
We consider communication over an AWGN channel.
q
or Channel
[U1, U2]
2
T
cos 2πf0 t
T
X1
X(t)
Vector
Encoder
X2
q
2
T
T
2
3T
2
− sin 2πf0 t
W (t)
cos 2πf0 t
Y1
[Û1, Û2]
Vector
Decoder
q R
T
2
T
(.)dt
q R
3T /2
2
T
Y2
0
T /2
(.)dt
Y (t)
− sin 2πf0 t
Land, Fleury: Digital Modulation 1
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replacements
102
6.3
Phase Transitions
OQPSK
for
QPSK
and
10
Example: Transmission of three symbols
For the transmission of three symbols, we consider the data bits
u = [u1, u2], the in-phase component (I) and the quadrature component (Q) of the modulation
p symbol, and the phase change ∆φ.
For convenience, let a := Es/2.
QPSK
replacements
u
I
Q
00
11
01
+a
+a
−a
−a
−a
+a
10
−π
+ π2
− π2
+π
∆φ
t
T
2
0
= Tb
2T
T
3T
OQPSK
11
00
u
+a
I
Q
+π ∆φ
−π
0
−a
+a
+a
+a
+ π2
01
−a
−a
+ π2
−a
−a
0
−a
+a
+a
− π2
t
T
2
= Tb
T
2T
3T
3
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QPSK exhibits phase changes {± π2 , ±π} at times that are multiples of the signaling interval T . In contrast to this, OQPSK
exhibits phase changes {± π2 } at times that are multiples of the bit
duration Tb = T /2.
Remarks
(i) OQPSK has the same bit-error probability as QPSK.
(ii) Staggering the quadrature component of QPSK by T /2 does
not prevent phase changes of π if the signaling constellation
of QPSK is not tilted. (Why?)
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7
Frequency-Shift Keying (FSK)
In FSK, the waveforms are sines, and the information is conveyed
by the frequencies of the waveforms. FSK with two waveforms is
called binary FSK (BFSK), and FSK with M waveforms is called
M -ary FSK (MFSK).
7.1
7.1.1
BFSK with Coherent Demodulation
Signal Waveforms
Set S of two sinus waveforms with different frequencies f1, f2 and
(possibly) different but known phases φ1 , φ2 ∈ [0, 2π):
√
s1(t) = 2P cos 2πf1 t + φ1 ,
√
s2(t) = 2P cos 2πf2 t + φ2 .
Both values are limited to the time interval t ∈ [0, T ]. Thus we
have
S = s1(t), s2 (t) .
Usually, f1, f2 1/T . To simplify the theoretical analysis, we
assume that
k1
k2
f1 = ,
f2 = ,
T
T
where k1, k2 ∈ N and k1, k2 1.
Both waveforms have the same energy
Z∞
Z∞
s21(t)dt =
s22(t)dt = P T.
Es =
−∞
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We assume the following BFSK transmitter:
{0, 1} → S
u 7→ x(t)
with the mapping
0 →
7
s1(t)
1 →
7
s2(t).
The frequency separation
∆f = |f1 − f2|
is selected such that s1(t) and s2(t) are orthogonal, i.e.,
hs1(t), s2 (t)i =
ZT
s1(t)s2 (t)dt = 0.
0
The necessary condition for orthogonality is
∆f =
k
T
with k ∈ N. Notice that the minimum frequency separation is
1/T .
The phases are assumed to be known to the receiver. Demodulation with phase information (known phases or estimated phases)
is called coherent demodulation.
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7.1.2
Signal Constellation
Orthonormal functions:
ψ1(t) =
ψ2(t) =
√1 s1 (t)
Es
=
√1 s2 (t)
Es
=
for t ∈ [0, T ].
q
q
2
T
cos 2πf1t + φ1
2
T
cos 2πf2t + φ2
Geometrical representation of the waveforms:
p
p
s1(t) = Es · ψ1(t) + 0 · ψ2(t) ←→ s1 = ( Es, 0)T
p
p
s2(t) = 0 · ψ1(t) + Es · ψ2(t) ←→ s1 = (0, Es)T.
PSfrag replacements
Signal constellation:
ψ2
√
Es
D1
D2
s2
√
2Es
s1
0
0
√
Es
ψ1
BFSK with coherent demodulation has the same signal constellation as BPPM.
Accordingly, these two modulation methods are equivalent.
Land, Fleury: Digital Modulation 1
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7.1.3
ML Decoding
ML symbol decision rule
(√
( Es, 0)T for y1 ≥ y2,
2
decML(y) = argmin ky − xk =
√ T
(0,
Es) for y1 < y2.
x∈{s1 ,s2 }
PSfrag replacements
Decision regions
2
D1 = {y ∈
√R : y1 ≥ y2},
2Es
y2
√
Es
s2
D2 = {y ∈ R2 : y1 < y2}.
(u = 1)
D2
D1
(u = 0)
s1
0
√
Es
0
y1
ML decoder for BFSK
û = enc-1 (decML(y)) =
PSfrag replacements
(
0 for y1 ≥ y2,
1 for y1 < y2.
Implementation of the ML decoder
Y1
Y2
Land, Fleury: Digital Modulation 1
Z
z≥0→0
Û
z<0→1
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108
7.1.4
Vector Representation of the Transceiver
We assume BFSK over the AWGN channel and coherent demodulation.
placements
or Channel
q
2
T
cos(2πf1 t + φ1 )
T
X1
U
X(t)
Vector
Encoder
X2
q
2
T
cos(2πf2 t + φ2 )
T
W (t)
cos(2πf1 t + φ1 )
Y1
Û
Vector
Decoder
q R
T
2
T
q R
T
2
T
Y2
0
0
(.)dt
Y (t)
(.)dt
cos(2πf2 t + φ2 )
The dashed box forms a 2-dimensional AWGN vector channel with
independent noise components
W1, W2 ∼ N (0, N0/2).
Remark: Notice the similarities to BPSK and BPPM.
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7.1.5
Error Probabilities
As the signal constellation of coherently demodulated BFSK is the
same as that of BPPM, the error probabilities are the same:
√
Ps = Q( γs),
√
Pb = Q( γb).
For the derivation, see analysis of BPPM.
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7.2
7.2.1
MFSK with Coherent Demodulation
Signal Waveforms
The signal waveforms are
(√
2P cos 2πfmt + φm for t ∈ [0, T ],
sm(t) =
0
elsewhere,
m = 1, 2, . . . , M .
The parameters of the waveforms are as follows:
(a) frequencies fm = km/T for km ∈ N and km 1;
(b) frequency separations ∆fm,n = |fm − fn| = km,n/T , km,n ∈
N;
(c) phases φm ∈ [0, 2π);
m, n = 1, 2, . . . , M .
The phases are assumed to be known to the receiver. Thus, we
consider coherent demodulation.
As in the binary case, all waveforms have the same energy
Es =
Z∞
s2m(t)dt = P T.
−∞
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7.2.2
Signal Constellation
Orthonormal functions:
ψm(t) =
√1 sm (t)
Es
=
for t ∈ [0, T ], m = 1, 2, . . . , M .
q
2
T
cos 2πfmt + φm
Geometrical representation of the waveforms:
p
s1(t) = Es · ψ1(t)
sM (t) =
p
Es · ψM (t)
M elements
zp
}|
{
←→ s1 = ( Es, 0, . . . , 0)T
...
p
←→ s1 = (0, . . . , 0, Es)T.
MFSK with coherent demodulation has the same signal constellation as MPPM.
Accordingly, these two modulation methods are equivalent and
have the same performance.
Vector Encoder
enc :
u = [u1, . . . , uK ] →
7
x = enc(u)
U = {0, 1}K → X = s1, s2, . . . , sM
with K = log2 M .
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7.2.3
Vector Representation of the Transceiver
We assume MFSK over the AWGN channel and coherent demodulation.
placements
or Channel
q
2
T
cos(2πf1 t + φ1 )
T
X1
U
X(t)
Vector
Encoder
XM
q
2
T
cos(2πfM t + φM )
T
W (t)
cos(2πf1 t + φ1 )
Y1
Û
Vector
Decoder
q R
T
2
T
q R
T
2
T
YM
0
0
(.)dt
Y (t)
(.)dt
cos(2πfM t + φM )
The dashed box forms an M -dimensional AWGN vector channel
with mutually independent components
Wm ∼ N (0, N0/2),
m = 1, 2, . . . , M .
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Remarks
The M waveforms sm(t) ∈ S are orthogonal. The receiver simply
correlates the received waveform y(t) to each of the possibly transmitted waveforms sm(t) ∈ S and selects the one with the highest
correlation as the estimate.
7.2.4
Error Probabilities
The signal constellation of coherently demodulated MFSK is the
same as that of MPPM. Therefor, the symbol-error probability
and the bit-error probability are the same. For the analysis, see
MPPM.
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7.3
7.3.1
BFSK with Noncoherent Demodulation
Signal Waveforms
The signal waveforms are
√
s1(t) = 2P cos 2πf1 t + φ1 ,
√
s2(t) = 2P cos 2πf2 t + φ2 .
for t ∈ [0, T ) with
(i) frequencies fm = km/T for km ∈ N and km 1, m = 1, 2,
and
(ii) frequency separation ∆f = |f1 − f2| = k/T for some k ∈ N.
As opposed to the case of coherent demodulation, the phases φ1
and φ2 are not known to the receiver. Instead, they are assumed
as random variables that are uniformly distributed, i.e.,
pΦ1 (φ1 ) =
1
,
2π
pΦ2 (φ2 ) =
1
2π
for φ1, φ2 ∈ [0, 2π).
Demodulation without use of phase information is called noncoherent demodulation. As the phases need not be estimated,
the noncoherent demodulator has a lower complexity than the coherent demodulator.
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7.3.2
Signal Constellation
Using the same method as for the canonical decomposition of PSK,
the orthonormal functions are obtained as
q
ψ1(t) = T2 cos 2πf1t
q
ψ2(t) = − T2 sin 2πf1t
q
ψ3(t) = T2 cos 2πf2t
q
ψ4(t) = − T2 sin 2πf2t
for t ∈ [0, T ].
(Remember: cos(x + y) = cos x cos y − sin x sin y.)
Geometrical representations of the waveforms:
p
p
s1(t) = Es cos φ1 · ψ1(t) + Es sin φ1 · ψ2(t)
p
p
T
Es cos φ1, Es sin φ1, 0, 0 ,
←→ s1 =
p
p
s2(t) = Es cos φ2 · ψ3(t) + Es sin φ2 · ψ4(t)
p
p
T
←→ s2 = 0, 0, Es cos φ2, Es sin φ2 .
The signal constellation is thus
X = s1 , s 2 .
The transmitted (modulation) symbols are the fourdimensional(!) vectors
T
X = |X1{z
, X}2 , X
,
X
| 3{z }4 ∈ X.
“for s1” “for s2”
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7.3.3
Noncoherent Demodulator
As there are four orthonormal basis functions, the demodulator
has four branches:
cos 2πf1 t
q R
T
2
T
0
q R
T
2
PSfrag replacements
T
Y (t)
0
(.)dt
Y1
(.)dt
Y2
(.)dt
Y3
(.)dt
Y4
− sin 2πf1 t
cos 2πf2 t
q R
T
2
T
0
q R
T
2
T
0
− sin 2πf2 t
Notice the similarity to PSK.
We introduce the vectors
Y1
,
Y1=
Y2
Y3
Y2=
Y4
and
Y = Y T1 , Y T2
T
T
= Y1 , Y 2 , Y 3 , Y 4 .
(Loosly speaking, Y 1 corresponds to s1 and Y 2 corresponds to s2.)
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7.3.4
Conditional Probability of Received Vector
Assume that the transmitted vector is
p
p
T
Es cos φ1, Es sin φ1 , 0, 0 .
X = s1 =
Then, the received vector is
p
p
T
E cos φ1 + W1, Es sin φ1 + W2, W
,W
Y =
| s
{z
} | 3{z }4
Y T2
Y T1
T
= Y T1 , Y T2 .
The conditional probability density of Y given X = s1 and φ1 is
pY |X,Φ1 (y|s1, φ1) =
1 2
= πN0 · exp − N10 ky − s1k2
h
p
2
1
1 2
= πN0 · exp − N0 y1 − Es cos φ1 +
p
2
y32
y42
+ y2 − Es sin φ1 + +
i
h
Es
2
2
1
1 2
= πN0 · exp − N0 · exp − N0 ky 1k + ky 2k ·
√
2 Es
· exp N0 y1 cos φ1 + y2 sin φ1
i
With α1 = tan−1(y2/y1), we may write
y1 cos φ1 + y2 sin φ1 = ky 1k cos α1 cos φ1 + sin α1 sin φ1
= ky 1k cos(φ1 − α1).
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The phase can be removed from the condition by averaging over
it. The phase is uniformly distributed in [0, 2π), and thus
pY |X (y|s1) =
=
Z2π
pY |X,Φ1 (y|s1, φ1) pφ1 (φ1) dφ1
0
=
=
1
2π
1
2π
Z2π
0
Z2π
0
pY |X,Φ1 (y|s1, φ1) dφ1
1 2
πN0
· exp
=
1 2
πN0
·
1
2π
|
exp
· exp
√
2 Es
N0 ky 1 k cos(φ1
· exp
Z2π
0
· exp
Es
−N
0
Es
−N
0
· exp
h
2
ky 1k + ky 2k
2
i
·
− α1) dφ1
− N10
√
2 Es
N0 ky 1 k cos(φ1
{z
I0 2 NE0 s ky 1k
√
− N10
h
2
ky 1k + ky 2k
2
i
·
− α1) dφ1 .
}
The above function is the modified Bessel function of order 0, and
it is defined as
I0(v) :=
1
2π
Z2π
0
exp v cos(φ − α) dφ
for α ∈ [0, 2π). It is monotonically increasing. A plot can be found
in [1, p. 403].
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Combining all the above results yields
Es
1 2
pY |X (y|s1) = πN0 · exp − N0 ·
i
h
2
2
1
· I0
· exp − N0 ky 1k + ky 2k
Using the same reasoning, we obtain
Es
1 2
pY |X (y|s2) = πN0 · exp − N0 ·
i
h
2
2
1
· exp − N0 ky 1k + ky 2k
· I0
7.3.5
√
2 Es
N0 ky 1 k .
√
2 Es
N0 ky 2 k .
ML Decoding
Remember the definitions
T
T T
T
Y = Y
, Y}2, Y
, Y}4 = Y 1 , Y 2 .
| 1{z
| 3{z
Y1 Y2
ML symbol decision rule
x̂ = decML(y) = argmax pY |X (y|x)
x∈X
Let x̂ = sm̂. Then, the index of the ML vector is
√
2 Es
m̂ = argmax I0 N0 ky mk
m∈{1,2}
= argmax ky mk
m∈{1,2}
= argmax ky mk2.
m∈{1,2}
Notice that selecting x̂ and selecting m̂ are equivalent.
Land, Fleury: Digital Modulation 1
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ML decoder
û =
(
0 for x̂ = s1 ⇔ ky 1k ≥ ky 2k,
1 for x̂ = s2 ⇔ ky 1k < ky 2k.
placementsOptimal noncoherent receiver for BFSK
cos 2πf1 t
q R
T
2
T
0
q R
T
2
T
Y (t)
0
Y1
(.)dt
(.)2
(.)dt
(.)2
Y2
− sin 2πf1 t
kY 1k2
Z
z≥0→0
z<0→1
cos 2πf2 t
q R
T
2
T
0
q R
T
2
T
0
Y3
(.)dt
(.)2
(.)dt
(.)2
Y4
kY 2k2
− sin 2πf2 t
Notice:
q
kY 1k = Y12 + Y22,
q
2
kY 2k = Y32 + Y42.
2
Land, Fleury: Digital Modulation 1
Z = kY 1k2 − kY 2k2,
NavCom
Û
121
7.4
7.4.1
MFSK with Noncoherent Demodulation
Signal Waveforms
The signal waveforms are
√
sm(t) = 2P cos 2πfmt + φm
for t ∈ [0, T ), m = 1, 2, . . . , M , with
(i) fm = km/T for km ∈ N and km 1, m = 1, 2, . . . , M ,
(ii) ∆f m, n = |fm − fn| = k/T for some km,n ∈ N, m, n =
1, 2, . . . , M .
The phases φ1, φ2 , . . . , φM are not known to the receiver. They
are assumed to be uniformly distributed in [0, 2π).
Land, Fleury: Digital Modulation 1
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7.4.2
Signal Constellation
Orthonormal functions:
ψ1(t) =
q
2
T
q
for t ∈ [0, T ].
cos 2πf1t
2
T
ψ2(t) = −
sin 2πf1t
...
q
ψ2m−1 (t) = T2 cos 2πfmt
q
2
ψ2m(t) = − T sin 2πfmt
...
q
ψ2M −1 (t) = T2 cos 2πfM t
q
ψ2M (t) = − T2 sin 2πfM t
Signal vectors (of length 2M ):
p
p
T
s1 =
Es cos φ1, Es sin φ1, 0, . . . , 0
p
p
T
s2 = 0, 0, Es cos φ2, Es sin φ2 , 0, . . . , 0
...
p
p
T
sm = 0, . . . , 0, Es cos φm, Es sin φm , 0, . . . , 0
{z
}
|
positions 2m − 1 and 2m
...
p
p
T
sM = 0, . . . , 0, Es cos φM , Es sin φM
Signal constellation:
X = s1 , s 2 , . . . , s M .
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7.4.3
Conditional Probability of Received Vector
Using the same reasoning as for BFSK, we can show that
M
Es
·
· exp − N
pY |X (y|sm) = πN1 0
0
M
√
X
2 Es
2
1
· exp − N0
ky mk · I0 N0 ky mk
m=1
with the definition
Y = |Y1{z
, Y}2, Y
, Y}4, . . . , Y2M −1 , Y2M ,
| 3{z
| {z }
Y1 Y2
YM
7.4.4
T
= Y
T T
T
T
.
1,Y 2,...,Y M
ML Decision Rule
Original ML decision rule:
x̂ = decML(y) = argmax pY |X (y|x)
x∈X
Let x̂ = sm̂. Then, the original decision rule may equivalently be
expressed as
m̂ = argmax ky mk2.
m∈{1,...,M }
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7.4.5
placements
Optimal Noncoherent Receiver for MFSK
Remember: “noncoherent demodulation” means that no phase information is used for demodulation.
The optimal noncoherent receiver for MFSK over the AWGN channel is as follows:
cos 2πf1 t
Y1
q R
T
2
T
0
q R
T
2
T
Y (t)
0
(.)dt
(.)2
(.)dt
(.)2
kY 1k2
select
maximum
Y2
− sin 2πf1 t
and
cos 2πfM t
compute
encoder
inverse
q R
T
2
T
0
q R
T
2
T
0
Y2M −1
(.)dt
(.)2
(.)dt
(.)2
Y2M
Û
kY M k2
− sin 2πfM t
Notice that Û = [Û1, . . . , ÛK ] with K = log2 M .
Land, Fleury: Digital Modulation 1
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7.4.6
Symbol-Error Probability
Assume that X = s1 is transmitted. Then, the probability of a
correct decision is
Pr(X̂ = X|X = s1) =
= Pr kY mk2 ≤ kY 1k2 for all m 6= 1|X = s1 .
Provided that X = s1 is transmitted, we have
p
p
T
Y1=
Es cos φ1 + W1, Es sin φ1 + W2
T
Y 2 = W3 , W 4 = W 2
...
T
Y m = W2m−1 , W2m = W m
...
T
Y M = W2M −1 , W2M = W M .
Consequently,
Pr(X̂ = X|X = s1) =
= Pr kW mk2 ≤ kY 1k2 for all m 6= 1|X = s1
= Pr √ 1 kW mk2 ≤ √ 1 kY 1k2 for all m 6= 1|X = s1
N /2
N /2
| 0 {z
} | 0 {z
}
Rm
R1
= Pr Rm ≤ R1 for all m 6= 1|X = s1
Z∞
= Pr Rm ≤ R1 for all m 6= 1|X = s1, R1 = r1 ·
0
· pR1 (r1) dr1.
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The probability density function of R1 is denoted by pR1 (r1).
The probability of a correct decision can further be written as
Pr(X̂ = X|X = s1) =
=
=
Z∞
0
Z∞
0
Pr Rm ≤ r1 for all m 6= 1|X = s1, R1 = r1 pR1 (r1) dr1
Pr Rm ≤ r1 for all m 6= 1 pR1 (r1) dr1
The noise vectors W2, . . . , WM are mutually statistically independent, and thus also the values R2, . . . , RM are mutually statistically independent. Therefore,
Pr(X̂ = X|X = s1) =
=
Z∞ Y
M
Pr(Rm ≤ r1) pR1 (r1) dr1
M −1
0 m=2
Z∞
0
Pr(R2 ≤ r1)
pR1 (r1) dr1.
Notice that
Pr(Rm ≤ r1) = Pr(R2 ≤ r1)
for m = 2, 3, . . . , M , because R2, . . . , RM are identically distributed.
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The conditional probability of a symbol-error can thus be written
as
Pr(X̂ 6= X|X = s1) = 1 − Pr(X̂ = X|X = s1)
Z∞
M −1
Pr(R2 ≤ r1)
=1−
pR1 (r1) dr1.
0
By symmetry, all conditional symbol-error probabilities are the
same. Thus,
Ps = 1 −
Z∞
0
Pr(R2 ≤ r1)
M −1
pR1 (r1) dr1.
In the above expressions, we have the following distributions (see
literature):
(i) R2 (and also R3, . . . , RM ) is Rayleigh distributed:
2
pR2 (r2) = r2 e−r2 /2.
Accordingly, its cumulative distribution function results as
2
Pr(R2 ≤ r1) = 1 − e−r1 /2
(ii) R1 is Rice distributed:
p
pR1 (r1) = r1 · exp − 21 (r1 + 2γs)2 · I0 2γsr1
Remark
The Rayleigh distribution and the Ricd distribution are often used
to model mobile radio channels.
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Using the first relation, we can expand the first term as
M −1
M −1 −r12 /2
Pr(R2 ≤ r1)
= 1−e
=
M
−1
X
(−1)m
m=0
M −1
m
2
e−mr1 /2
After integrating over r1, we obtain the following closed-form solution for the symbol-error probability of MFSK with
noncoherent demodulation:
M
−1
X
M −1
1
m
m+1
Ps =
(−1)
exp −
γs .
m
m
+
1
m
+
1
m=1
For M = 2, this sum simplifies to
Ps =
Land, Fleury: Digital Modulation 1
1
2
e−γs/2.
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7.4.7
Bit-Error Probability
BFSK
We have γb = γs and Pb = Ps.
Therefore, the bit-error probability is given by
Pb =
1
2
e−γb/2.
MFSK
For M > 2, we have
(i) γb =
1
K γs ,
2K−1
(ii) Pb = K
Ps with K = log2 M (see MPPM).
2 −1
Therefore the bit-error probability results as
M −1
mK
2K−1 X
M
−
1
1
m+1
(−1)
Pb = K
exp −
γs .
2 − 1 m=1
m
m+1
m+1
Plots of the error probability can be found in [1, p. 433].
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8
Minimum-Shift Keying (MSK)
MSK is a binary modulation scheme with coherent detection, and
it is similar to BFSK with coherent detection. The waveforms
are sines, the information is conveyed by the frequencies of the
waveforms, and the phases of the waveforms are chosen such that
there are no phase discontinuities. Thus, the power spectrum of
MSK is more compact than that of BFSK. The description of MSK
follows [2].
8.1
Motivation
Consider BFSK. The signal waveforms of BFSK are
√
U = 0 7→ s1(t) = 2P cos 2πf1t ,
√
U = 1 7→ s2(t) = 2P cos 2πf2t ,
for t ∈ [0, T ]. The minimum frequency separation such that s1(t)
and s2(t) are orthogonal (assuming coherent detection) is
1
∆f = |f1 − f2| =
.
2T
In the following, we assume minimum frequency separation.
Without loss of generality, we assume that
k
k
1
k + 1/2
f1 = ,
f2 = +
=
,
T
T 2T
T
for k ∈ N and k 1, i.e., k is a large integer. Thus, we have the
following properties:
waveform number of periods in t ∈ [0, T )
s1(t)
k
s2(t)
k + 1/2
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Example: Output of BFSK Transmitter
Consider BFSK with f1 = 1/T and f2 = 3/2T such that the frequency separation ∆ = 1/T is minimum. Assume the bit sequence
U [0,4] = [U0, U1, U2, U3, U4] = [0, 1, 1, 0, 0].
The resulting output signal x•(t) of the BFSK transmitter shows
3
discontinuities.
The discontinuities produce high side-lobes in the power spectrum
of X(t), which is either inefficient or problematic. This can be
avoided by slightly modifying the modulation scheme.
We rewrite the signal waveforms as follows:
√
s1(t) = 2P cos 2πf1t ,
√
t s2(t) = 2P cos 2πf1t + 2π
,
2T
for t ∈ [0, T ]. (Remember that f2 = f1 + 1/2T .) Thus, we can
express the output signal as
√
x(t) = 2P cos 2πf1t + φ•(t) ,
t ∈ [0, T ), with the phase function
(
0
φ•(t) =
2π 2Tt
for s1(t),
for s2(t).
Notice that φ•(t) depends directly on the transmitted source bit U .
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Example: Phase function of BFSK
The phase function φ•(t) of the previous example shows discontinuities.
3
The discontinuities of the phase and thus of x(t) can be avoided by
making the phase continuous. This can be achieved by disabling
the “reset to zero” of φ•(t), yielding the new phase function φ(t).
Notice that this introduces memory in the transmitter.
Example: Modified BFSK Transmitter
(We continue the previous example.) The phase function φ(t) of
the modified transmitter is continuous. Accordingly, also the signal x(t) generated by the modified transmitter is continuous. 3
FSK with phase made continuous as described above is called
continuous-phase FSK (CPFSK). Binary CPFSK with minimum
frequency separation is termed minimum-shift keying (MSK).
8.2
Transmit Signal
The continuous-phase output signal of an MSK transmitter consists of the following waveforms (the expressions on the right-hand
side are only used as abbreviations in the following):
√
s1(t) = + 2P cos 2πf1 t
“f1”
√
s2(t) = − 2P cos 2πf1t
“−f1”
√
s3(t) = + 2P cos 2πf2 t
“f2”
√
s4(t) = − 2P cos 2πf2t
“−f2”
t ∈ [0, T ].
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The input bit Uj = 0 generates either s1(t) or s2(t); the input
bit Uj = 1 generates either s3(t) or s4(t), which means a phase
difference of π between the beginning and the end of the waveform.
Consider a bit sequence of length L,
u[0,L−1] = [u0, u1, . . . , uL−1 ].
According to the above considerations (see also examples), the
output signal of the MSK transmitter can be written as
follows:
L−1
X
√
x(t) = 2P cos 2πf1t +
ul · 2π b(t − lT ) ,
|l=0
{z
φ(t)
t ∈ [0, LT ), with the phase pulse


for τ < 0,

0
b(τ ) = τ /2T for 0 ≤ τ < T ,


1/2
for T ≤ τ .
}
Remark
The BFSK signal x(t) can be generated with the same expression
as above, when the phase pulse


for τ < 0,

0
b•(τ ) = τ /2T for 0 ≤ τ < T ,


0
for T ≤ τ
is used.
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Sfrag replacements
Plotting all possible trajectories of φ(t), we obtain the (tilted)
phase tree of MSK.
φ(t)
f2
f1
4π
−f2
f2
f2
−f1
3π
f2
f1
2π
5T
π
0
f1
f1
−f2
−f2
−f2
f2
f2
f2
f2
f1
0
−f1
−f2
−f1
f2
Sfrag replacements
−f2
−f1
f1
f1
2T
T
−f1
−f1
f1
f1
4T
3T
t
The physical phase is obtained by taking φ(t) modulo 2π. Plotting
all possible trajectories, we obtain the (tilted) physical-phase
trellis of MSK.
φ(t) mod 2π
2π
−f2
−f2
−f2
−f2
f2
f2
f2
f2
−f1
π
f2
5T
f1
0
0
f1
T
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−f1
f1
2T
−f1
f1
3T
−f1
f1
4T
t
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8.3
Signal Constellation
Signal waveforms:
√
s1(t) = + 2P cos
√
s2(t) = − 2P cos
√
s3(t) = + 2P cos
√
s4(t) = − 2P cos
t ∈ [0, T ].
2πf1t ,
2πf1t ,
2πf2t ,
2πf2t ,
Orthonormal functions:
p
ψ1(t) = 2/T cos 2πf1t ,
p
ψ2(t) = 2/T cos 2πf2t ,
t ∈ [0, T ].
Vector representation of the signal waveforms:
p
s1(t) = + Es ψ1(t)
p
s2(t) = − Es ψ1(t)
p
s3(t) = + Es ψ2(t)
p
s4(t) = − Es ψ2(t)
with Es = P T .
Land, Fleury: Digital Modulation 1
←→
←→
←→
←→
p
s1 = (+ Es, 0)T
p
s2 = (− Es, 0)T = −s1
p
s3 = (0, + Es)T
p
s4 = (0, − Es)T = −s3
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Remark:
The modulation symbol at discrete time j is written as
X j = (Xj,1 , Xj,2 )T.
√
For example, X j = s1 means Xj,1 = Es and Xj,2 = 0.
Signal constellation:
ψ2
PSfrag replacements
s3
ψ1
s2
s1
s4
Remark:
MSK and QPSK have the same signal constellation. However, the
two modulation schemes are not equivalent because their vector
encoders are different. This is discussed in the following.
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Modulator:
The symbol X j = (Xj,1 , Xj,2 )T is transmitted in the time interval
t ∈ [jT, (j + 1)T ).
This time interval can also be written as
τ ∈ [0, T ).
t = jT + τ,
The modulation for τ = t − jT ∈ [0, T ), j = 0, 1, 2, . . . , is then
given by the mapping
X j 7→ Xj (τ ),
and it can be implemented by
PSfrag replacements
q
2
T
cos(2πf1τ )
T
Xj,1
Xj (τ )
Xj,2
q
2
T
cos(2πf2τ )
T
Remark:
The index j is not necessary for describing the modulator. It is
only introduced to have the same notation as being used for the
vector encoder.
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8.4
A Finite-State Machine
Consider the following device (which is a simple finite-state maPSfrag chine)
replacements
Vj+1
Uj
Vj
T
It consists of the following components:
(i) A modulo-2 adder:
PSfrag replacements
Uj
Vj+1
Vj
It has the functionality
Vj+1 = Uj ⊕ Vj = Uj + Vj mod 2.
PSfrag replacements
Uj
(ii) A shift register clocked at times T (“every T seconds”):
Vj+1
T
Vj
(iii) The sequence {Uj } is a binary sequence with elements
in {0, 1}, and so is the sequence {Vj }.
The output value Vj describes the state of this device.
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As Vj ∈ {0, 1}, this device can be in two different states. All possible transitions can be depicted in a state transition diagram:
0
0
PSfrag replacements1
0
1
1
1
0
The values in the circles denote the states v, and the labels of the
arrows denote the inputs u.
eplacements
The state transitions and the associated input values can also be
depicted in a trellis diagram. The initial state is assumed to be
zero.
1
1
1
1
0
0
2
0
0
1
1
0
0
1
1
0
0
0
1
1
1
0
0
1
1
0
1
1
1
0
3
0
0
4
0
5
The time indices are written below the states.
The emphasized path corresponds to the input sequence
[U0, U1, U2, U3, U4] = [0, 1, 1, 0, 0].
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8.5
Vector Encoder
According to the trellis of the physical phase, the initial phase at
the beginning of each symbol interval can take the two values 0
and π. These values determine the state of the encoder. We use
the following association:
↔
↔
phase 0
phase π
state V = 0,
state V = 1.
The functionality of the encoder can be described by the following
state transition diagram:
0/s1
PSfrag replacements
0
1/s3
1/s4
π
0/s2
The transitions are labeled by u/x. The value u denotes the binary
source symbol at the encoder input, and x denotes the modulation
symbol at the encoder output.
Remember: s2 = −s1 and s4 = −s3.
Remark
The vector encoder is a finite-state machine as considered in the
previous section.
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eplacements
141
The temporal behavior of the encoder can be described by the
trellis diagram:
0/s2
0/s2
0/s2
0/s2
1/
s3
1/
s3
1/
s3
1/
s3
π
1/
s3
π
s4
1/
π
s4
1/
π
s4
1/
π
s4
1/
π
0 0/s
1
0 0/s
1
0 0/s
1
0 0/s
1
0 0/s
1
0
0
1
2
3
4
5
The emphasized path corresponds to the input sequence
[U0, U1, U2, U3, U4] = [0, 1, 1, 0, 0].
General Conventions
(i) The input sequence of length L is written as
u = [u0, u1, . . . , uL−1 ].
(ii) The initial state is V0 = 0.
(iii) The final state is VL = 0.
The last bit UL−1 is selected such that the encoder is driven back
to state 0:
VL−1 = 0
VL−1 = 1
−→
−→
UL−1 = 0,
UL−1 = 1.
Therefore, UL−1 does not carry information.
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Generation of the Encoder Outputs
PSfrag replacements
Consider one trellis section:
0/s2
π
1/
s3
s4
1/
π
0 0/s
1
j
0
j+1
Remember: s2 = −s1 and s4 = −s3.
Observations:
(i) Input symbols → output vectors
Uj = 0
Uj = 1
−→
−→
X j ∈ {s1, s2} = {s1, −s1},
X j ∈ {s3, s4} = {s3, −s3}.
(ii) Encoder state → output vectors
Vj = 0 (“0”)
Vj = 1 (“π”)
−→
−→
X j ∈ {s1, s3}
X j ∈ {s2, s4} = {−s1, −s3}.
Using these observations, the above device can be supplemented
to obtain an implementation of the vector encoder for MSK.
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Block Diagram of the Vector Encoder for MSK
Uj
Uj
Vj+1
Vj
T
s3 10
01
s1
s2
00
Uj Vj
Xj,1
Xj,2
s4 11
At time index j:
• input symbol Uj ,
• state Vj ,
• output symbol X j = (Xj,1 , Xj,2 )T,
• subsequent state Vj+1.
Notice:
(i) Uj controls the frequency of the transmitted waveform.
(ii) Vj controls the polarity of the transmitted waveform.
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8.6
Demodulator
The received symbol Y j = (Yj,1 , Yj,2 )T is determined in the time
interval
t ∈ [jT, (j + 1)T ).
This time interval can also be written as
τ ∈ [0, T ).
t = jT + τ,
The demodulation for τ = t − jT ∈ [0, T ), j = 0, 1, 2, . . . , can
then be implemented as follows:
PSfrag replacements
cos(2πf1 τ )
Yj,1
qT
2
T
Yj,2
q R
T
2
T
0
q R
T
2
T
0
(.)dτ
Y (t)
(.)dτ
cos(2πf2 τ )
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8.7
Conditional Probability Density of Received Sequence
Sequence of (two-dimensional) received symbols
[Y 0, . . . , Y L−1 ] = [X 0 + W 0, . . . , X L−1 + W L−1 ].
The vector [W 0, . . . , W L−1] is a white Gaussian noise sequence
with the properties:
(i) W 0, . . . , W L−1 are independent ;
(ii) each (two-dimensional) noise vector is distributed as
Wj,1
0
N /2 0
Wj =
, 0
∼N
0
0 N0/2
Wj,2
PSfrag replacements
Illustration of the AWGN vector channel:
Yj,1
Xj,1
Wj,1 W
j
Wj,2
Xj
Xj,2
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Yj,2
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Conditional probability of y [0,L−1] = [y 0, . . . , y L−1 ] given that
x[0,L−1] = [x0, . . . , xL−1 ] was transmitted:
pY [0,L−1] |X [0,L−1] y 0, . . . , y L−1 |x0, . . . , xL−1 =
=
=
L−1
Y
pY j |X j y j |xj
j=0
L−1
Yh
j=0
i
1
2
1
· exp − N0 ky j − xj k
πN0
L−1
1 L
X
· exp − N10
=
ky j − xj k2 .
πN0
j=0
Notice that this is the likelihood function of x[0,L−1] for a given
received sequence y [0,L−1] .
8.8
ML Sequence Estimation
The trellis of the vector encoder has the following two properties:
(i) Each input sequence u[0,L−1] = [u0, . . . , uL−1 ] uniquely determines a path of length L in the trellis, and vice versa.
(ii) Each path in the trellis uniquely determines an output sequence x[0,L−1] = [x0, . . . , xL−1 ]
A sequence x[0,L−1] of length L generated by the vector encoder is
called an admissible sequence if the initial and the final state of
the encoder are zero, i.e., if V0 = VL = 0. The set of admissible
sequences is denoted by A.
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Using these definitions, the ML decoding rule is the following:
Search the trellis for an admissible sequence that maximizes the likelihood function.
Or in mathematical terms:
x̂[0,L−1] = argmax p(y [0,L−1] |x[0,L−1] ).
x[0,L−1] ∈A
Using the relation
ky j − xj k2 =
ky j k2 −2hy j , xj i + kxj k2
| {z }
| {z }
= Es
indep. of xj
and the expression for the likelihood function, the rule for ML
sequence estimation (MLSE) can equivalently be formulated
as follows:
x̂[0,L−1]
L−1
1 X
= argmax exp −
ky j − xj k2
N0 j=0
x[0,L−1] ∈A
= argmin
L−1
X
x[0,L−1] ∈A j=0
L−1
X
= argmax
x[0,L−1] ∈A j=0
ky j − xj k2
hy j , xj i.
Hence, the ML estimate of the transmitted sequence is a sequence
P
in A that maximizes the sum L−1
j=0 hy j , xj i.
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Ûj
Uj
Vj+1
Uj Vj
s2
01
s4 11
s3 10
00
s1
Yj,2
Yj,1
Xj,2
Xj,1
q
T
0
T
0
q R
T
2
q R
T
2
2
T
2
T
(.)dτ
(.)dτ
T
T
cos(2πf2τ )
cos(2πf1τ )
cos(2πf2τ )
cos(2πf1τ )
Y (t)
X(t)
W (t)
AWGN Vector Channel
Maximum-Likelihood
Sequence Decoding
T
Vj
Uj
q
8.9
hannel
T
149
Canonical Decomposition of MSK
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8.10
The Viterbi Algorithm
Given Problem:
(i) The admissible sequences
x[0,L−1] = [x0, . . . , xL−1] ∈ A
can be represented in a trellis.
(ii) The value to be maximized (optimization criterion) can be
written as the sum
L−1
X
hy j , xj i.
j=0
(Remember: hy j , xj i = yj,1 · xj,1 + yj,2 · xj,2 .)
The Viterbi Algorithm solves such an optimization problem in
a very effective way.
Branch Metrics and Path Metrics
The branches of the trellis are re-labeled as follows:
g replacements
vj+1
u
xj
/
j
vj
j
,x
yj
vj
j+1
j
h
ji
vj+1
j+1
The new labels are called branch metrics. They may be interpreted as a gain or a profit made when going along this branch.
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When labeling all branches with their branch metrics, we obtain
the following trellis diagram:
π
0
0
π
hy
,s
0
i
3
hy 0 , s1 i
0
hy 1 , s2 i
hy
1,
s4
i
i
3
,s
1
y
h
hy 1 , s1 i
1
π
0
hy 2 , s2 i
hy
2,
s4
i
i
3
,s
2
y
h
hy 2 , s1 i
2
π
0
3
hy 3 , s2 i
hy
3,
s4
i
i
3
,s
3
y
h
hy 3 , s1 i
π
0
4
hy 4 , s2 i
hy
4,
s4
i
i
3
,s
4
y
h
hy 4 , s1 i
π
0
5
The total gain or profit achieved by going along a path in the
trellis is called the path metric. The path corresponding to the
sequence
x[0,L−1] = [x0, . . . , xL−1 ]
has the path metric
L−1
X
j=0
hy j , xj i.
Notice: The branch metric and the path metric are not necessarily
a mathematical metric.
Hence, the ML estimate x̂[0,L−1] of X [0,L−1] is an admissible sequence (∈ A) that maximizes the path metric. The corresponding
path is called an optimum path.
There are 2 · 4L−1 paths of length L. The Viterbi algorithm is
a sequential method that determines an optimum path without
computing the path metrics of all 2 · 4L−1 paths.
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Survivors
The Viterbi algorithm relies on the following property of optimum
paths:
A path of length L is optimum if and only if its subpaths at any depth n = 1, 2, . . . , L − 1 are also optimum.
Example: Optimality of Subpaths
eplacementsConsider the following trellis. Assume that the thick path (solid
line) is optimum.
π
π
π
π
0
0
0
0
0
0
0
1
2
3
4
5
The dashed path ending at depth 3 cannot have a metric that is
larger than that of the subpath obtained by pruning the optimum
path at depth 3.
3
For each state in each depth n, there are several subpaths ending
in this state. The one with the largest metric (“the best path”) is
called the survivor at this state in this depth.
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From the above considerations, we have the following important
results:
• All subpaths of an optimum path are survivors.
• Therefore, only survivors need to be considered for determining an optimum path.
Accordingly, non-survivor paths can be discarded without loss of
optimality. (Therefore the name “survivor”.)
Viterbi Algorithm
1. Start at the beginning of the trellis (depth 0 and state 0 by
convention).
2. Increase the depth by 1. Extend all previous survivors to the
new depth and determine the new survivor for each state.
3. If the end of the trellis is reached (depth L and state 0 by
convention), the survivor is an optimum path. The associated
sequence is the ML sequence.
Remarks
(i) The complexity of computing the path metrics of all paths is
exponential in the sequence length. (Exhaustive search.)
(ii) The complexity of the Viterbi algorithm is linear in the sequence length.
(iii) The complexity of the Viterbi algorithm is exponential in the
number of shift registers in the vector encoder.
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eplacementsExample: Viterbi Algorithm
Consider the case L = 5. This corresponds to the following trellis.
π
π
π
π
0
0
0
0
0
0
0
1
2
3
4
5
The Viterbi algorithm may be illustrated for a randomly drawn received sequence y [0,L−1] . (The receiver must be capable of handling
every received sequence.)
3
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8.11
Simplified ML Decoder
The MSK trellis has a special structure, and this leads to a special
behavior of the Viterbi algorithm. This can be exploited to derive
a simpler optimal decoder.
Observation
At depth n = 2, the two survivors (solid line and dashed line) will
look as depicted in (a) or (b), but never as in (c) or (d).
PSfrag replacements
0
1
2
0
(a)
0
1
1
2
(c)
2
(b)
0
1
2
(d)
Therefore the final estimate of the state V̂1 is already decided at
time j = 1. This is proved shortly afterwards.
The general result for the given MSK transmitter follows by induction:
The two survivors at any depth n are identical for j < n
and differ only in the current state.
Therefore, final decisions on any state V̂n (value of the vectorencoder state at time n) can already be made at time n.
Land, Fleury: Digital Modulation 1
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placements
156
We proof now that the Viterbi algorithm makes the final decision
about state V̂1 (state at time j = 1) already at time 1.
−hy 0 , s1 i
−hy 0 , s3 i
Proof
The inner product is defined as
hy j , smi = yj,1 · sm,1 + yj,2 · sm,2 .
Due to the signal constellation,
s2 = −s1,
s4 = −s3.
Therefore,
hy j , s2i = hy j , −s1i = −hy j , s1i,
hy j , s4i = hy j , −s3i = −hy j , s3i.
Using these relations, the branches can equivalently be labeled as
π
0
0
π
hy
,s
0
i
3
hy 0 , s1 i
0
1
−hy 1 , s1 i
−h
y
1,
s3
i
i
3
s
,
hy 1
hy 1 , s1 i
π
0
2
Land, Fleury: Digital Modulation 1
−hy 2 , s1 i
−h
y
2,
s3
i
i
3
s
,
hy 2
hy 2 , s1 i
π
0
3
−hy 3 , s1 i
−h
y
3,
s3
i
i
3
s
,
hy 3
hy 3 , s1 i
π
0
4
−hy 4 , s1 i
−h
y
4,
s3
i
i
3
s
,
hy 4
hy 4 , s1 i
π
0
5
NavCom
PSfrag replacements
157
Consider the conditions for selecting the survivors, while making
0 , s1 i
use of these −hy
relations:
−hy
s3i v = 0 (time j = 1):
• Survivor
at0,state
2
−hy 1, s1i
hy 1, s3i
0
PSfrag replacements 0
h
,s
y0
i
3
hy 0, s1i
π
0
1
−h
y
π
1 ,s
3i
hy 1, s1i
hy 0, s3i − hy 1, s3i > hy 0, s1i + hy 1, s1i
hy 0, s3i − hy 1, s3i < hy 0, s1i + hy 1, s1i
−hy 0, s1i
0
2
⇒
⇒
upper path
lower path
−hy
s3i v2 = 1 (“π”) (time j = 1):
• Survivor
at0,state
hy 1, s1i
−hy 1, s1i
π
π
i
i
3
3
s
s
,
,
−hy 1, s3i
hy 1
hy 0
0
0
0
hy 0, s1i
1
2
0
hy 0, s3i − hy 1, s1i > hy 0, s1i + hy 1, s3i
hy 0, s3i − hy 1, s1i < hy 0, s1i + hy 1, s3i
⇒
⇒
upper path
lower path
Obviously, either the two upper paths or the two lower paths are
the survivors. Therefore, the two survivors go through the same
state V̂1.
Land, Fleury: Digital Modulation 1
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ML State Sequence
By induction, we obtain the following optimal decision rule for
state Vj based on y j−1 and y j :
(
0 if hy j−1, s3i − hy j , s3i < hy j−1, s1i + hy j , s1i,
V̂j =
1 if hy j−1, s3i − hy j , s3i > hy j−1, s1i + hy j , s1i,
j = 1, 2, . . . , L − 1. (Remember: VL = 0 by convention.)
This decision rule can be simplified. The modulation symbols are
s1 =
p
(+ Es, 0)T,
s3 = (0, + Es)T.
Thus, the inner products can be written as
hy j , s1i = yj,1 ·
p
Es ,
p
hy j , s3i = yj,2 ·
p
Es .
√
Substituting this in the decision rule and dividing by Es, we obtain the following simplified (and still optimal) decision rule:
(
0 if yj−1,2 − yj,2 < yj−1,1 + yj,1 ,
V̂j =
1 if yj−1,2 − yj,2 > yj−1,1 + yj,1 ,
j = 1, 2, . . . , L − 1.
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159
The following device implements this state decision rule:
Yj,1
Zj,1
Yj−1,1
T
Zj
Yj,2
Yj−1,2
T
zj > 0 → 0
zj < 0 → 1
V̂j
Zj,2
Vector-Encoder Inverse
Given the sequence of state estimates {V̂j }, the sequence of bit
estimates {Ûj } can computed according to
Ûj = V̂j ⊕ V̂j+1,
j = 0, 1, . . . , L − 1. (Compare vector encoder.)
Thereplacements
following device implements this operation:
PSfrag
V̂j−1
V̂j
Ûj−1
T
The time indices are j = 0, 1, . . . , L − 1, and V̂L = 0 as VL = 0 by
convention. Remember that UL−1 does not convey information.
Land, Fleury: Digital Modulation 1
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8.12
Bit-Error Probability
The BER performance of MSK is about 2 dB worse than that
of BPSK and QPSK. This means, when comparing the SNR in
Eb/N0 required to achieve a certain bit-error probability (BEP),
MSK needs about 2 dB more than BPSK and QPSK. This can be
improved by slightly modifying the vector encoder.
The trellis of the vector encoder has the following properties:
(i) Two different paths differ in at least one state.
(ii) Two different paths differ in at least two bits.
(See trellis of vector encoder in Section 8.5.)
The most likely error-event for high SNR is that the transmitted
path and the estimated path differ in one state. This error-event
leads to two bit errors.
Idea for improvement:
• The state sequence should be equal to the bit sequence, such
that the most likely error-event leads only to one bit error.
• This can be achieved by using a pre-coder before the vector
encoder.
The resulting modulation scheme is called precoded MSK
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8.13
Differential Encoding and Differential
Decoding
The two terms “differential encoder” and “differential decoder”
are mainly used due to historical reasons. Both may be used as
encoders.
In the following, bj ∈ {0, 1} denote the bits before encoding, and
cj ∈ {0, 1} denote the bits after encoding.
PSfrag
replacements
Differential
Encoder (DE)
cj
cj+1
bj
T
Operation:
cj+1 = bj ⊕ cj ,
j = 0, 1, 2, . . . . Initialization: c0 = 0.
PSfrag
replacements
Differential
Decoder (DD)
bj
bj−1
cj
T
Operation:
cj = bj ⊕ bj−1,
j = 0, 1, 2, . . . . Initialization: b−1 = 0.
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placements
8.14
Precoded MSK
The DD(!) is used as a pre-coder and is placed before the vector
replacements
encoder. The DE(!) is placed behind the vector-encoder inverse.
W (t)
Uj
X(t)
Y (t)
MSK Rx / DE
MSK Tx
DD
Ûj−1
Vector Representation of Precoded MSK
Uj0
s3 10
Uj
Vj = Uj−1
T
01
s1
s2
00
Uj0 Vj
eplacements
Xj,1
Xj,2
s4 11
Initialization: V0 = 0. Notice: Uj0 = Uj ⊕ Vj = Uj ⊕ Uj−1.
Yj,1
T
Yj−1,1
Zj,1
Zj
Yj,2
T
Yj−1,2
zj > 0 → 0
zj < 0 → 1
V̂j = Ûj−1
Zj,2
Notice: The vector-encoder inverse and the DE compensate each
other and can be removed.
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Trellis of Precoded MSK
1
1
s
1/
0
s
1/
3
0/s1
0
1/−s1
0/
−
s3
0
1
s
1/
3
0/s1
1/−s1
0/
−
s3
0
s
1/
3
0/s1
2
1
1
1/−s1
0/
−
s3
0
1
s
1/
3
0/s1
3
1/−s1
0/
−
s3
0
1
3
0/s1
4
0
5
The thick path corresponds to the bit sequence
U [0,4] = [0, 1, 1, 0, 0].
Bit-Error Probability of Precoded MSK
The bit-error probability of precoded MSK is
p
Pb = Q( 2γb),
γb = Eb/N0.
Sketch of the Proof:
The bit-error probability is given as
L−1
1X
Pb = lim
Pr(Ûj 6= Uj ).
L→∞ L
j=0
All terms in this sum can shown to be identical, and thus
Pb = Pr(Û0 6= U0).
Due to the precoding, we have U0 = V1, and therefore
Pb = Pr(V̂1 6= V1).
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This expression can be written using conditional probabilities:
Pb =
1 X
1
X
v=0 u=0
Pr(V̂1 6= V1|V1 = v, U1 = u) · Pr(V1 = v, U1 = u).
(Usual trick.) These conditional probabilities can shown to be all
identical, and thus
Pb = Pr(V̂1 6= V1|V1 = 0, U1 = 0).
As V1 = U0 (property of precoded MSK), we obtain
Pb = Pr(V̂1 = 1|U0 = 0, U1 = 0).
The ML decision rule is
(
0 if
V̂1 =
1 if
y0,2 − y1,2 < y0,1 + y1,1 ,
y0,2 − y1,2 > y0,1 + y1,1 .
Given the hypothesis U0 = U1 = 0, we have
p
X 0 = X 1 = s1 = ( Es, 0)T,
and so
p
Remember:
Y0,1 = Es + W0,1 ,
p
Y1,1 = Es + W1,1 ,
Y 0 = (Y0,1 , Y0,2 )T,
Land, Fleury: Digital Modulation 1
Y0,2 = W0,2 ,
Y1,2 = W1,2 .
Y 1 = (Y1,1 , Y1,2 )T.
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165
Therefore, the bit-error probability can be written as
Pb = Pr(Y0,2 − Y1,2 > Y0,1 + Y1,1 |U0 = 0, U1 = 0)
p
= Pr(W0,2 − W1,2 > 2 Es + W0,1 + W1,1 )
p
= Pr(W0,2 − W1,2 − W0,1 − W1,1 > 2 Es)
|
{z
}
0
W ∼ N (0, 2N0 )
p
p
0
= Pr W / 2N0 > 2Es/N0
p
p = Q 2Es/N0 = Q 2γb .
As (precoded) MSK is a binary modulation scheme, we have Es =
Eb and γs = γb.
Thus, the bit-error probability of precoded MSK is
p
Pb = Q( 2γb),
which is the same as that of BPSK and QPSK.
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8.15
Gaussian MSK
Gaussian MSK is a generalization of MSK, and it is used in the
GSM mobile phone system.
Motivation
The phase function of MSK is continuous, but not its dervative.
These “edges” in the transmit signal lead to some high-frequency
components. By avoiding these “edges”, i.e., by smoothing the
phase function, the spectrum of the transmit signal can be made
more compact.
Concept
The output signal of the transmitter is generated similar to that
in MSK:
L−1
X
√
ul · 2π b(t − lT ) ,
x(t) = 2P cos 2πf1t +
|l=0
{z
φ(t)
}
t ∈ [0, LT ), with the phase pulse


for τ < 0,

0
b(τ ) = monotonically increasing for 0 ≤ τ < JT ,


1/2
for JT ≤ τ ,
for some integer J ≥ 1.
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Notice that there are two new degrees of freedom:
(i) The phase pulse may stretch over more than one symbol
duration T .
(ii) The phase pulse may be be selected such that its derivative
is zero for t = and t = JT .
Phase Pulse and Frequency Pulse
In GMSK, the phase pulse b(τ ) is constructed via its derivative c(τ ), called the frequency pulse:
(i) The frequency pulse c(τ ) is the convolution of a rectangular pulse (as in MSK!) with a Gaussian pulse:
c(τ ) = rectangular pulse ∗ Gaussian pulse,
R
where c(τ ) = 0 for τ ∈
/ [0, JT ), and c(τ )dτ = 1/2. The
resulting pulse is
2πB
2πB
τ − T /2 − Q √
τ + T /2 .
c(τ ) = Q √
ln 2
ln 2
The value B corresponds to a bandwidth and is a design
parameter.
(ii) The phase pulse is the integral over the frequency pulse:
b(τ ) =
Zτ
c(τ 0)dτ 0.
0
Due to the definition of the frequency pulse, we have b(τ ) =
1/2 for τ ≥ JT .
The resulting phase pulse fulfills the conditions given above.
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In GSM, the parameter B is chosen such that BT ≈ 0.3. The
length of the phase pulse is then L ≈ 4.
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9
Quadrature Amplitude Modulation
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10
BPAM with Bandlimited Waveforms
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Part III
Appendix
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172
A
A.1
Geometrical
Signals
Representation
of
Sets of Orthonormal Functions
The D real-valued functions
ψ1(t), ψ2 (t), . . . , ψD (t)
form an orthonormal set if
(
Z+∞
0 for k =
6 l (orthogonality),
ψk (t) ψl (t) dt =
1 for k = l (normalization).
−∞
Example: D = 3
ψ1(t)
√
1/ T
√
−1/ T
T
t
T
t
T
t
ψ2(t)
√
1/ T
PSfrag replacements
√
−1/ T
ψ3(t)
√
1/ T
√
−1/ T
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3
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A.2
Signal Space Spanned by an Orthonormal Set of Functions
Let ψ1(t), ψ2 (t), . . . , ψD (t) be a set of orthonormal functions.
We denote by Sψ the set containing all linear combinations of these
functions:
n
Sψ := s(t) : s(t) =
D
X
i=1
o
siψi(t); s1, s2, . . . , sD ∈ R .
Example: (continued)
The following functions are elements of Sψ :
replacements s1(t) = ψ1(t) + ψ2(t),
s2(t) = 12 ψ1(t) − ψ2(t) + 21 ψ3(t)
s1 (t)
√
2/ T
√
−2/ T
s2 (t)
√
2/ T
T
t
√
−2/ T
T
t
3
The set Sψ is a vector space with respect to the following operations:
• Addition
+ : s1(t), s2 (t) →
7 s1(t) + s2(t)
• Multiplication with a scalar
· : a, s(t) 7→ a · s(t)
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A.3
Vector Representation of Elements in
the Vector Space
With each element s(t) ∈ Sψ ,
s(t) =
D
X
siψi(t),
i=1
we associate the D-dimensional real vector
s = [s1, s2, . . . , sD ]T ∈ RD .
Example: (continued)
s1(t) = ψ1(t) + ψ2(t)
s2(t) = 12 ψ1(t) − ψ2(t) + 12 ψ3(t)
7→
7→
s1 = [1, 1, 0]T ,
s2 = [ 12 , −1, 12 ]T .
3
Vectors in R, R2, R3, can be represented geometrically by points
on the line, in the plane, an in the 3-dimensional Euclidean space,
respectively.
Example: (continued)
ψ2
PSfrag replacements
s1
1
0.5
1
0.5
ψ3
1
ψ1
s2
3
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A.4
Vector Isomorphism
The mapping Sψ → RD ,
s(t) =
D
X
siψi(t)
i=1
7→
s = [s1, s2, . . . , sD ]T,
is a (vector) isomorphism, i.e.:
(i) The mapping is bijective.
(ii) The mapping preserves the addition and the multiplication
operators:
s1(t) + s2(t)
as(t)
7
→
7→
s 1 + s2 ,
as.
Since the mapping Sψ → RD is an isomorphism, Sψ and RD are
“the same” or “equivalent” from an algebraic point of view.
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A.5
Scalar Product and Isometry
Scalar product in Sψ :
Z+∞
s1(t)s2 (t) dt
hs1(t), s2 (t)i :=
−∞
for s1(t), s2 (t) ∈ Sψ .
Scalar product in RD :
hs1, s2i :=
for s1, s2 ∈ RD .
D
X
s1,i s2,i
i=1
Relation of scalar products:
Each element in Sψ
ψ1(t), ψ2 (t), . . . , ψD (t):
s1(t) =
D
X
s1,k ψk (t),
is
a
linear
s2(t) =
combination
D
X
of
s2,l ψl (t).
l=1
k=1
Inserting the two sums into the definition of the scalar product, we
obtain
Z X
D
D
X
hs1(t), s2 (t)i =
s1,k ψk (t)
s2,l ψl (t) dt =
k=1
=
D X
D
X
k=1 l=1
s1,k s2,l
Z
l=1
ψk (t)ψl (t)dt =
{z
}
|
“orthonormality”
D
X
k=1
s1,k s2,k = hs1, s2i
Hence the mapping Sψ → RD preserves the scalar product, i.e., it
is an isometry.
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Norms in Sψ
ks(t)k2 := hs(t), s(t)i =
Z
s2(t) dt = Es
The value Es is the energy of s(t).
Norms in RD :
2
ksk := hs, si =
D
X
s2i
i=1
From the isometry and the definitions of the norms, we obtain the
Parseval relation
ks(t)k2 = ksk2.
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A.6
Waveform Synthesis
Let ψ1(t), ψ2(t), . . . , ψD (t) be a set of orthonormal functions
spanning the vector space Sψ . Every element s(t) ∈ Sψ can be
synthesized from its vector representation s = [s1, s2, . . . , sD ]T by
one of
the following devices.
PSfrag
replacements
s
Multiplicative-based
Synthesizer
ψ1(t)
s1
ψ2(t)
s(t) =
s2
D
X
sk ψk (t)
k=1
δ(t)
ψD (t)
sD
Filter-bank based Synthesizer
Sfrag replacements
δ(t)
s
Filter
s1
ψ1(t)
Filter
s2
ψ2(t)
Filter
sD
s(t) =
s(t)
ψD (t)
D
X
k=1
sk δ(t) ∗ ψk (t) with Dirac function δ(t)
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A.7
Implementation of the Scalar Product
Let s1(t) and s2(t) be time-limited to the interval [0, T ]. Then
ZT
hs1(t), s2 (t)i :=
s1(t)s2 (t) dt.
0
The scalar product can be computed by the following two devices.
ag replacements
Correlator-based Implementation
RT
s1(t)
0
s2(t)
(.)dt
hs1(t), s2 (t)i
Matched-filter based Implementation
ag replacements
We seek a linear filter that outputs the value hs1(t), s2 (t)i at a
certain time, say t0.
sample at
time t0
linear filter
s1(t)
hs1(t), s2 (t)i
g(t)
We want
hs1(t), s2 (t)i =
ZT
0
Z+∞
!
0
s1(τ )s2 (τ ) dτ =
s1(τ )g(t − τ ) dτ 0
t =t0
−∞
The second equality holds for
s2(τ ) = g(t0 − τ )
Land, Fleury: Digital Modulation 1
⇔
g(t) = s2(t0 − t)
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180
Example:
g replacements
s2(t0 − t)
s2(t)
t0
T t
T t
3
The filter is causal and thus realizable for a function time-limited
to the interval [0, T ] if
t0 ≥ T.
The value t0 is the time at which the value hs1(t), s2 (t)i appears
replacements
at the filter output. Choosing the minimum value, i.e., t 0 = T , we
obtain the following (realizable) implementation.
sample at
time T
s1 (t)
matched filter
s2(T − t)
hs1(t), s2 (t)i
The filter is matched to s2(t).
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A.8
Computation of the Vector Representation
Let ψ1(t), ψ2(t), . . . , ψD (t) be a set of orthonormal functions
time-limited to [0, T ] and spanning the vector space Sψ .
We seek a device that computes the vector representation s ∈ RD
for any function s(t) ∈ Sψ , i.e.,
s = [s1, s2, . . . , sD ]T
s(t) =
such that
D
X
sk ψk (t).
k=1
The components of s can be computed as
sk = hs(t), ψk (t)i,
k = 1, 2, . . . , D.
Correlator-based Implementation
frag replacements
RT
0
s(t)
(.)dt
s1
(.)dt
sD
ψ1(t)
RT
0
ψD (t)
frag replacements
Matched-filter based Implementation
s(t)
Land, Fleury: Digital Modulation 1
MF
ψ1(T − t)
s1
MF
ψD (T − t)
sD
sample at
time T
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A.9
Gram-Schmidt Orthonormalization
Assume a set of M signal waveforms
n
o
s1(t), s2 (t), . . . , sM (t)
spanning a D-dimensional vector space. From these waveforms,
we can construct a set of D ≤ M orthonormal waveforms
n
o
ψ1(t), ψ2 (t), . . . , ψD (t)
by applying the following procedure.
Remember:
hs1(t), s2 (t)i =
Z
s1(t)s2 (t) dt
ks(t)k2 := hs(t), s(t)i = Es
Gram-Schmidt orthonormalization procedure
1. Take the first waveform s1(t) and normalize it to unit energy:
s1(t)
ψ1(t) := p
.
2
ks1 (t)k
2. Take the second waveform s2(t). Determine the projection
of s2(t) onto ψ1(t), and subtract this projection from s2(t):
c2,1 = hs2(t), ψ1(t)i,
s02(t) = s2(t) − c2,1 ψ1(t).
Normalize s02(t) to unit energy:
s02(t)
.
ψ2(t) := p 0
ks2 (t)k2
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3. Take the third waveform s3(t). Determine the projections
of s3(t) onto ψ1(t) and onto ψ2(t), and subtract these projections from s3(t):
c3,1 = hs3(t), ψ1(t)i,
c3,2 = hs3(t), ψ2(t)i,
s03(t) = s3(t) − c3,1 ψ1(t) − c3,2 ψ2(t).
Normalize s03(t) to unit energy:
s03(t)
.
ψ3(t) := p 0
ks3 (t)k2
4. Proceed in this way for all other waveforms sk (t) until k = D.
Notice that s0k (t) = 0 for k > D. (Why?)
The general rule is the following. For k = 1, 2, . . . , D,
ck,i = hsk (t), ψi(t)i,
k−1
X
0
ck,i ψi(t),
sk (t) = sk (t) −
ψk (t) :=
i = 1, 2, . . . , k − 1
i=1
0
s (t)
p k0
.
ksk (t)k2
Due to this procedure, the resulting waveforms ψk (t) are orthogonal and normalized, as desired, and they form an orthonormal
basis of the D-dimensional vector space of the waveforms si(t).
Remark: The set of orthonormal waveforms is not unique.
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Example: Orthonormalization
replacements
Original signal waveforms:
s1 (t)
1
s3 (t)
1
0
1
−1
2
3 t
0
−1
s2 (t)
1
s4 (t)
1
√ 0
1/√2
−1/ 2 −1
0
1
2
3 t
Set 1 of orthonormal waveforms:
ψ1(t)
√
1/ 2
0
√
eplacements
−1/ 2
ψ2(t)
√
1/ 2
0
√
−1/ 2
1
1
−1
3 t
2
3 t
Set 2 of orthonormal waveforms:
ψ1(t)
1
1 PSfrag
2 replacements
3 t
0
1
−1
2
3 t
ψ2(t)
1
1
2
3 t
0
−1
ψ3(t)
1
ψ3(t)
1
0
√
0
3 t 1/√2
−1/ 2 −1
−1
2
1
2
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2
3 t
1
2
3 t
3
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B
Orthogonal Transformation of a
White Gaussian Noise Vector
The statistical properties of a white Gaussian noise vector
(WGNV) are invariant with respect to orthonormal transformations. This is first shown for a vector with two dimensions, and
then for a vector with an arbitrary number of dimensions.
B.1
The Two-Dimensional Case
Let W denote a 2-dimensional WGNV, i.e.,
2
!
0
σ 0
W1
,
W =
∼N
.
0
0 σ2
W2
Consider
the original coordinate system (ξ1, ξ2) and the rotated
PSfrag
replacements
coordinate system (ξ10 , ξ20 ).
ξ2
ξ20
w
w2
ξ10
w10
α = π/4
w20
w1
ξ1
Let [W10 , W20 ]T denote the components of W expressed with respect
to the rotated coordinate system (ξ10 , ξ20 ):
W10 = +W1 cos α + W2 sin α
W20 = −W1 sin α + W2 cos α.
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Writing this in matrix notation as
0 W1
cos α sin α W1
=
,
W20
− sin α cos α W2
the vector W 0 can seen to be the orthonormal transformation of
the vector W .
The resulting vector W 0 = [W10 , W20 ]T is also a two-dimensional
WGNV with the same covariance matrix as W = [W1, W2]T:
0
2
!
W1
0
σ 0
∼
N
.
W0 =
,
W20
0
0 σ2
Proof: See problem.
B.2
The General Case
Let W = [W1, . . . , WD ]T denote a D-dimensional WGNV with
zero mean and covariance matrix σ 2I, i.e.,
2
W ∼ N 0, σ I .
(0 is a column vector of length D, and I is the D × D identity
matrix.) Let further V denote an orthonormal transformation
RD → RD , i.e., a linear transformation with the property
V V T = V TV = I.
Then the transformed vector
W0 = V W
is also a D-dimensional WGNV with the same covariance matrix
as W :
0
2
W ∼ N 0, σ I .
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Proof
(i) A linear transformation of a Gaussian random vector is again
a Gaussian random vector; thus W 0 is Gaussian.
(ii) The mean value is
E W 0 = E V W = V E W = 0.
(iii) The covariance matrix is
E W 0W 0T = E V W W TV T
T
T
= V E W W V = σ2 V
V T} = σ 2I.
|
{z
| {z }
I
σ2I
Comment
In the case D = 2, V takes the form
cos α sin α
V =
− sin α cos α
for some α ∈ [0, 2π).
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C
C.1
The Shannon Limit
Capacity of the Band-limited AWGN
Channel
Consider an AWGN channel with bandwidth W [1/sec]. The capacity of this channel is given by
CW
S = W log2 1 +
[bit/sec].
W N0
Consider further a digital transmission scheme operating at transmission rate R [bit/sec] with limited transmit power S.
Channel Coding Theorem (C.E. Shannon, 1948):
Consider a binary random sequence generated by a BSS
that is transmitted over an AWGN channel with bandwidth W using some transmission scheme.
Transmission with an arbitrarily small probability of error (using an appropriate coding/modulation scheme)
is possible provided that the transmission rate is less
than the channel capacity, i.e.,
Pb → 0
is possible if
R < CW .
Conversely, error-free transmission is impossible if the
rate is larger than the channel capacity, i.e.,
Pb → 0
is impossible if
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A coding/modulation scheme can be characterized by its spectral
efficiency and its power efficiency. The spectral efficiency is
the ratio R/W , and it has the unit secbitHz . The power efficiency
is the SNR γb = Eb/N0 necessary to achieve a certain error probability (e.g., 10−5).
Example: MPAM
The rate is given by R = T1 log2 M . The bandwidth required to
transmit pulses of duration T is about W = 1/T . The spectral
efficiency is thus
bit
R
= log2 M
.
W
sec Hz
Error probabilities can be found in [1, p. 412, 435].
3
Example: MPPM
The rate is given by R = T1 log2 M . The bandwidth required
to transmit pulses of duration T /M is about W = M/T . The
spectral efficiency is thus
R
log2 M bit
=
.
W
M sec Hz
Error probabilities can be found in [1, p. 426, 435].
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A theoretical limit for the relation between the spectral efficiency
and the power efficiency (for error-free transmission) can be established using the channel coding theorem.
The transmit power is given by S = REb. Combining this relation,
the channel capacity, and the condition R < CW , we obtain
R
REb R < log2 1 +
= log2 1 + γb .
W
W N0
W
This can be rewritten as
2R/W − 1
.
γb >
R/W
Using the maximal rate for error-free transmission, R = C, we
obtain the theoretical bound
2C/W − 1
γb =
.
C/W
The plot can be found in [1, pp. 587].
The lower limit of γb for R/W → 0 is called the Shannon limit
for the AWGN channel:
2R/W − 1
γb > lim
= ln 2.
R/W →0 R/W
Notice that 10 log10(ln 2) ≈ −1.6 dB. Thus, error-free transmission over the AWGN channel is impossible if the SNR per bit is
lower than −1.6 dB.
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C.2
Capacity of the
AWGN Channel
Band-Unlimited
For the AWGN channel with unlimited bandwidth (but still
limited transmit power), the capacity is given by
S 1 S
C∞ = lim W log2 1 +
=
.
W →∞
W N0
ln 2 N0
The condition R < C∞ becomes then equivalent with the condition
γb > ln 2. (Remember: S = REb.)
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