Passband Data Transmission
In baseband data transmission, a data stream represented
in the form of a discrete pulse-amplitude modulated (PAM)
signal is transmitted over a low-pass channel.
Example: Nyquist channel
H(f),G(f),C(f)
W
OR
G(f)
H(f)
C(f)
PB.1
Passband Data Transmission
In passband data transmission, the incoming data stream is
modulated onto a carrier with fixed frequency and then
transmitted over a band-pass channel.
Example:
H(f),G(f),C(f)
2W
101
modulator
G(f)
H(f)
C(f)
cos 2πft
PB.2
1
Example
Passband data transmission allows more efficient use of the
allocated RF bandwidth, and flexibility in accommodating
different baseband signal formats.
Example
– Mobile Telephone Systems
• GSM: Gaussian Minimum Shift Keying (GMSK) is
used (a variation of FSK)
• IS-54: π/4-Differential Quaternary Phase Shift Keying
(DQPSK) is used (a variation of PSK)
PB.3
Types
The modulation process making the transmission possible
involves switching (keying) the amplitude, frequency, or
phase of a sinusoidal carrier in accordance with the
incoming data.
There are three basic signaling schemes:
Amplitude-shift keying (ASK)
Frequency-shift keying (FSK)
Phase-shift keying (PSK)
PB.4
2
ASK
Two waveforms
PSK
FSK
PB.5
ASK, PSK and FSK
Unlike ASK signals, both PSK and FSK signals have a
constant envelope.
PSK and FSK are preferred to ASK signals for passband
data transmission over nonlinear channel (amplitude
nonlinearities) such as microwave link and satellite
channels.
PB.6
3
Classification of digital modulation techniques
Coherent and Noncoherent
Digital modulation techniques are classified into coherent
and noncoherent techniques, depending on whether the
receiver is equipped with a phase-recovery circuit or not.
The phase-recovery circuit ensures that the local oscillator
in the receiver is synchronized to the incoming carrier wave
(in both frequency and phase).
PB.7
Phase Recovery (Carrier Synchronization)
Two ways in which a local oscillator can be synchronized with
an incoming carrier wave
Transmit a pilot carrier (similar to the DSB-LC
modulation in analog communication)
Pilot carrier
f
Use a carrier-recovery circuit such as a phase-locked
loop (PPL)
PB.8
4
M-ary signaling
In an M-ary signaling scheme, there are M possible signals
during each signaling interval of duration T.
Usually, M = 2 n and T = nTb where Tb is the bit duration.
T = nTb
10101··01
M bits
modulator
T
PB.9
M-ary signaling
In passband transmission, we have
M-ary ASK
M-ary PSK
M-ary FSK
We can also combine different methods:
M-ary quadrature-amplitude modulation (QAM)
(In baseband data transmission, we have M-ary PAM)
PB.10
5
M-ary signaling
M-ary signaling schemes are preferred over binary signaling
schemes for transmitting digital information over band-pass
channels when the requirement is to conserve bandwidth at
the expense of increased power.
The use of M-ary signaling enables a reduction in
transmission bandwidth by the factor n = log 2 M
over
binary signaling.
Bandwidth ∝ T
T = nTb
10101··01
M bits
modulator
T
PB.11
Coherent PSK
The functional model of passband data transmission system
is
mi
Signal
si
transmission
encoder
Modulator
si (t )
Channel
x(t )
Detector
x
Signal
transmission
m̂
decoder
Carrier signal
• mi is a sequence of symbol emitted from a message
source.
• The channel is linear, with a bandwidth that is wide
enough to transmit the modulated signal and the
channel noise is Gaussian distributed with zero
mean and power spectral density N o / 2 .
PB.12
6
Coherent PSK
The following parameters are considered for a signaling
scheme:
Probability of error
A major goal of passband data transmission systems is the
optimum design of the receiver so as to minimize the
average probability of symbol error in the presence of
additive white Gaussian noise (AWGN)
PB.13
Coherent PSK
Power spectra
Use to determine the signal bandwidth and co-channel
interference in multiplexed systems.
interference
Multiplexer
In practice, the signalings are linear operation, therefore, it is
sufficient to evaluate the baseband power spectral density.
B
2B
PB.14
7
Coherent PSK
Bandwidth Efficiency
– Bandwidth efficiency
ρ=
Rb
bits/s/Hz
B
where Rb is the data rate and B is the used channel
bandwidth.
Example: Nyquist channel for baseband data transmission
Bandwidth B = W = 1/2Tb.
∴ρ =
W
1 / Tb
Rb
=
= 2 bits/s/Hz
B 1 / 2Tb
PB.15
Coherent PSK
In a coherent binary PSK system, the pair of signals s1 (t ) and
s 2 (t ) used to represent binary symbols 1 and 0, respectively,
is defined by
s1 (t ) =
s 2 (t ) =
2 Eb
cos( 2πf c t )
Tb
Tb
2 Eb
2 Eb
cos(2πf c t + π ) = −
cos(2πf c t )
Tb
Tb
where 0 ≤ t ≤ Tb , and Eb is the transmitted signal energy
per bit
PB.16
8
Coherent PSK
Example:
E = ∫ [s1 (t )] dt =
Tb
2
0
2 Eb
Tb
∫
Tb
0
cos 2 (2πf c t )dt =
2 Eb Tb
⋅ = Eb
Tb 2
To ensure that each transmitted bit contains an integral
number of cycles of the carrier wave, the carrier frequency
f c is chosen equal to n / Tb for some fixed integer n.
Tb = 2 / f c
PB.17
Coherent PSK
The transmitted signal can be written as
s1 (t ) = Eb φ (t )
and
s 2 (t ) = − Eb φ (t )
where
2
cos(2πf c t )
Tb
φ (t ) =
0 ≤ t < Tb
2
Note : φ 2 (t ) = ∫
Tb
0
 2

cos(2πf c t )  dt = 1


 Tb
PB.18
9
Generation of coherent binary PSK signals
To generate a binary PSK signal, the first step is
representing the input binary sequence in polar form with
symbols 1 and 0 represented by constant amplitude levels of
and , respectively.
This signal transmission encoder is performed by a polar
nonreturn-to-zero (NRZ) encoder.
 + Eb
si = 
 − Eb
101011
input symbol is 1
input symbol is 0
Signal
si
transmission
encoder
101011
PB.19
Generation of coherent binary PSK signals
The second step is multiplying the carrier encoder output
with the carrier

2 Eb
cos(2πf c t )
if s i = Eb
 s1 (t ) =
Tb

s i (t ) = 
s (t ) = − 2 Eb cos(2πf t ) if s = − E
c
i
b
 2
Tb

si
Product
Modulator
si (t )
f c = n / Tb
φ (t ) =
2
cos(2πf ct )
Tb
PB.20
10
Detection of coherent binary PSK signals
To detect the original binary sequence of 1s and 0s, we
apply the noisy PSK signal to a correlator. The correlator
output is compared with a threshold of zero volts.
x(t )
∫
X
Tb
x1
0
Decision
device
φ (t )
1 if x1 > 0
0 if x1 < 0
0
Correlator
PB.21
Detection of coherent binary PSK signals
Example
If the transmitted symbol is 1, x(t ) =
and the correlator output is
2 Eb
cos(2πf c t )
Tb
Tb
x1 = ∫ x(t )φ (t )dt
0
=∫
Tb
0
2 Eb
2
cos(2πf c t ) ⋅
cos(2πf c t )dt
Tb
Tb
= Eb ⋅
2
Tb
∫
Tb
0
cos 2 (2πf c t )dt
= Eb
Similarly, if the transmitted symbol is 0, x1 = − E b
.PB.22
11
Error probability of binary PSK
We can represent a coherent binary system with a signal
constellation consisting of two message points.
• The coordinates of the message points are all the
possible correlator output under a noiseless
condition.
• The coordinates for BPSK are
Eb and − Eb .
Decision
boundary
φ (t )
− Eb
Eb
PB.23
Error probability of binary PSK
There are two possible kinds of erroneous decision:
– Signal s 2 (t ) is transmitted, but the noise is such that the
received signal point inside region with x1 > 0 and so
the receiver decides in favor of signal s1 (t ) .
– Signal s1 (t ) is transmitted, but the noise is such that the
received signal point inside region with x1 < 0 and so
the receiver decides in favor of signal s 2 (t ) .
si (t ) + w(t )
X
φ (t )
∫
Tb
0
x1
Decision
device
0
1 if x1 > 0
0 if x1 < 0
PB.24
12
Error probability of binary PSK
For the first case, the observable element
the received signal x(t ) by
x1
is related to
Tb
x1 = ∫ x(t )φ (t )dt
0
= ∫ [si (t ) + w(t )]φ (t )dt
Tb
0
Tb
= − Eb + ∫ w(t )φ (t )dt
0
x1 is a Gaussian process with mean :
xi = E[ xi ]
Tb
= E[− Eb + ∫ w(t )φ (t )dt ]
0
= − Eb
PB.25
Variance is
Error probability of binary PSK
σ = E[( xi − xi ) 2 ]
2
2
 Tb


= E  ∫ w(t )φ (t )dt  
 
 0
= E ∫
 0
Tb
=∫
Tb
=∫
Tb
0
0
0
∫
Tb
∫
Tb
0
No
2
N
= o
2
=
∫
Tb
0
∫
E[ w(t ) w(u )]φ (t )φ (u )dtdu
No
δ (t − u )φ (t )φ (u )dtdu
2
Tb
0
w(t ) w(u )φ (t )φ (u )dtdu 

φ 2 (t )dt
PB.26
13
Error probability of binary PSK
Therefore, the conditional probability density function of
x1 , given that symbol 0 was transmitted is
 ( x1 − x1 ) 2 
exp −

2σ 2 
2π σ

1
f ( x1 | 0) =
 ( x1 + Eb ) 2 
exp −
=

N
πN o
o


1
PB.27
Error probability of binary PSK
and the probability of error is
p10 =
∞
∫ f ( x | 0)dx
1
0
1
=
πN o
∫
∞
0
 ( x1 + Eb ) 2 
exp −
 dx1
No


1
( x + Eb ) , we have
No
Putting z =
p10 =
1
1
π∫
∞
Eb / N o 0
[ ]
exp − z 2 dz
 Eb 
1

= erfc

N
2
o


PB.28
14
Error probability of binary PSK
Similarly, the error of the second kind
 Eb 
1
 and hence
p01 = p10 = erfc

N
2
o


 Eb 
1

pe = erfc

N
2
o 

PB.29
Quadriphase-shift keying (QPSK)
QPSK has twice the bandwith efficiency of BPSK, since 2
bits are transmitted in a single modulation symbol. The data
input d k (t ) is divided into an in-phase stream d I (t ) , and a
quadrature stream d Q (t ) .
d k (t )
:1001
d I (t )
:10
d Q (t )
:01
PB.30
15
QPSK
d k (t )
d I (t )
1
0
0
1
1
0
Tb
t
t
d Q (t )
0
1
T = 2Tb
t
PB.31
QPSK
The phase of the carrier takes on one of four equally
spaced values, such as π/4, 3π/4, 5π/4, and 7π/4.
 2E

si (t ) =  T cos[2πf c t + (2i − 1)π / 4] 0 ≤ t ≤ T

0
elsewhere
where i = 1,2,3,4.
E is the transmitted signal energy per symbol;
T is the symbol duration;
fc = n / T ;
(Note : T = 2Tb )
PB.32
16
QPSK
Each possible value of the phase corresponds to a
unique dibit.
For example, 10 for i=1, 00 for i=2, 01 for i=3
and 11 for i=4.
(only a single bit is change from one dibit to the
next)
00
10
11
01
PB.33
QPSK
The transmitted signal can be written as
si (t ) =
=
2E
cos[2πf ct + (2i − 1)π / 4]
T
2E
cos[2πf ct ] cos[(2i − 1)π / 4]
T
2E
sin[ 2πf ct ] sin[(2i − 1)π / 4]
T
= si1φ1 (t ) + si 2φ2 (t )
−
where
φ1 (t ) =
2
cos[2πf c t ];
T
φ 2 (t ) =
2
sin[ 2πf c t ]
T
PB.34
17
QPSK
Input dibit
10
00
01
11
si1
E/2
− E/2
− E/2
E/2
Phase of QPSK
π/4
3π/4
5π/4
7π/4
φ 2 (t )
00
10
E/2
E/2
01
si 2
− E/2
− E/2
E/2
E/2
φ1 (t )
11
PB.35
PB.36
18
Generation of coherent QPSK signals
The incoming binary data sequence is first transformed into
polar form by a nonreturn-to-zero level encoder. The binary
wave is next divided by means of a demultiplexer into two
separate binary sequences.
The result can be regarded as a pair of binary PSK
signals, which may be detected independently due to
the orthogonality of φ1 (t ) and φ 2 (t ) .
PB.37
φ1 (t ) =
X
s1i
10101
Polar NRZ
si
2
cos(2πf ct )
Tb
Demultiplexer
s2i
+
s (t )
X
φ1 (t ) =
2
sin( 2πf ct )
Tb
PB.38
19
Detection of coherent QPSK signals
x(t )
∫
X
φ1 (t )
T
x1
0
1 if x1 > 0
Decision
device
0 if x1 < 0
0
In-phase channel
multiplexer
Quadrature channel
∫
X
T
x2
0
φ2 (t )
Decision
device
0
1 if x2 > 0
0 if x2 < 0
PB.39
Error probability of QPSK
The received signal is
x(t ) = si (t ) + w(t )
and the observation elements are
T
x1 = ∫ x(t )φ1 (t )dt
0
T
= ± E / 2 + ∫ w(t )φ1 (t )dt
0
T
x 2 = ∫ x(t )φ 2 (t )dt
0
T
= ± E / 2 + ∫ w(t )φ 2 (t )dt
0
PB.40
20
As a coherent QPSK is equivalent to two coherent binary
PSK systems working in parallel and using two carriers that
are in phase quadrature.
Hence, the average probability of bit error in each channel
of the coherent QPSK system is
 E/2 1

1
 = erfc E
p = erfc

 2N
2
 No  2

o




PB.41
Error probability of QPSK
As the bit error in the in-phase and quadrature channels of
the coherent QPSK system are statistically independent, the
average probability of a correct decision resulting from the
combined action of the two channels is
pc = (1 − p ) 2
 1
 E
= 1 − erfc
 2No
 2
 E
= 1 − erfc
 2No




2
 1
 E
 + erfc 2 
 4
 2N
o






PB.42
21
The average probability of symbol error for coherent QPSK
is therefore
pe = 1 − pc
 E
= erfc
 2N o
 1

 − erfc 2  E
 4
 2N


o
 E
≈ erfc
 2N o








if E / 2 N o >> 1
PB.43
In a QPSK system, since there are two bits per symbol, the
transmitted signal energy per symbol is twice the signal
energy per bit,
E = 2 Eb
and then
 Eb 

pe ≈ erfc
N
2
o 

d k (t )
1
0
0
1
t
d I (t )
1
0
t
d Q (t )
0
1
t
PB.44
22
With Gray encoding, the bit error rate of QPSK is
 Eb 
1

BER = erfc
2
2
N
o 

Therefore, a coherent QPSK system achieves the same
average probability of bit error as a coherent binary PSK
system for the same bit rate and the same Eb / N o but uses
only half the channel bandwidth.
PB.45
M-ary PSK
During each signaling interval of duration T, one of
the M possible signals
si (t ) =
2E
2π


cos 2πf c t +
(i − 1) 
T
M


i = 1,2,...,
is sent.
PB.46
23
M-ary PSK
The signal constellation of M-ary PSK consists of M
message points which are equally spaced on a circle of
radius E . For example, the constellation of
octaphase-shift keying is
 E
 π 
sin   
Pe ≈ erfc
 M 
 No
M ≥4
PB.47
Power spectra of M-ary PSK signals
The symbol function is
 2E

g (t ) =  T
 0
0≤t ≤T
otherwise
where T = Tb log 2 M and Tb is the bit duration.
As the energy spectral density is the magnitude of the
signal’s Fourier transform, the baseband power
spectral density is
S ( f ) = 2E
sin 2 (πTf )
(πTf ) 2
= 2 Eb log 2 M sinc 2 (Tb f log 2 M )
PB.48
24
(Normalized to fTb )
PB.49
Bandwidth efficiency
The bandwidth required to pass M-ary signal (main
lobe) is given by
2
B=
Q sinc(2) = 0
T
2
=
Tb log 2 M
2 Rb
=
log 2 M
Therefore, the bandwidth efficiency is
R
ρ= b
B
log 2 M
=
2
PB.50
25