UNIT IV
SIGNAL PROCESSING IN WIRELESS COMMUNICATIONS
Ref.: 1.“Wireless Communications”, Molisch
2. "Wireless Communications", Rappaport
3. "Wireless Communications", Andrea Goldsmith
Diversity
- Principle of Diversity
- Macrodiversity
- Microdiversity
- Signal Combining Techniques
- Transmit Diversity
Channel Impairments
1) ACI/CCI → system generated interference
[ACI - Adjacent Channel Inteference,
CCI - Co-channel Interference]
2) Shadowing → large-scale path loss from LOS obstructions
3) Multipath Fading → rapid small-scale signal variations
4) Doppler Spread → due to motion of mobile unit
Note:All can lead to significant distortion or attenuation of Rx
signal
Degrade Bit Error Rate (BER) of digitally modulated signal
Techniques used to improve Rx signal quality
• Three techniques are used to improve Rx signal quality and
lower BER:
1) Equalization
2) Diversity
3) Channel Coding
- They can be used independently or combined
Diversity Techniques
Principle of Diversity
- Primary goal is to reduce depth & duration of small-scale fades
- To ensure that the same information reaches the receiver on
statistically indeendent channels.
Types of Diversity
Spatial or antenna diversity → most common
•Use multiple Rx antennas in mobile or base station
•Even small antenna separation (∝ λ ) changes phase of signal
→ constructive /destructive nature is changed
Other diversity types → polarization, frequency, & time diversity
Diversity arrangements
Let’s have a look at fading again
Received power [log scale]
Illustration of interference pattern from above
Movement
A
Position
Transmitter
A
B
Reflector
B
Diversity arrangements
The diversity principle
The principle of diversity is to transmit the same information on
M statistically independent channels.
By doing this, we increase the chance that the information will
be received properly.
The example given on the previous slide is one such arrangement:
antenna diversity.
Diversity arrangements
General improvement trend
Bit error rate (4PSK)
100
Rayleigh fading
No diversity
10 dB
10-1
10 x
10-2
10 dB
Rayleigh fading
Mth order diversity
10-3
10-4
10M x
No fading
10-5
10-6
0
2
4
6
8
10
12
Eb/N0 [dB]
14
16
18
20
Microscopic diversity
- Most widely used
- Combat small-scale fading (fading created by interference
effects)
- Use multiple antennas separated in space
- At a mobile, signals are independent if separation > λ / 2
- But it is not practical to have a mobile with multiple antennas
separated by λ / 2 (7.5cm apart at 2 GHz)
- Can have multiple receiving antennas at base stations, but
must be separated on the order of ten wavelengths (1 to 5
meters).
Microscopic diversity
- Since reflections occur near receiver, independent signals
spread out a lot before they reach the base station.
- a typical antenna configuration for 120 degree sectoring.
- For each sector, a transmit antenna is in the center, with two
diversity receiving antennas on each side.
- If one radio path undergoes a deep fade, another independent
path may have a strong signal.
- By having more than one path selection can be made,
instantaneous and average SNRs at the receiver may be
improved
Microscopic diversity Techniques
- Spatial Diversity (several antenna elemenst separated in space)
- Temporal Diversity (repetition of the transmit signal at different
times)
- Frequency Diversity (transmission of the signal on different
frequencies)
- Angular Diversity (multiple antennas with different antenna
patterns)
- Polarization Diversity (multiple antennas receiving different
polarizations)
Diversity arrangements
Some techniques
Spatial (antenna) diversity
We will focus on this
one today!
TX
Signal combiner
Frequency diversity
D
TX
D
D
Signal combiner
Temporal diversity
Coding
Interleaving
De-interleaving
De-coding
(We also have angular and polarization diversity)
Spatial (antenna) diversity
Fading correlation on antennas
Isotropic
uncorrelated
scattering.
Macroscopic diversity
- Combat large-scale fading (fading created by shadowing effects)
- Frequency diversity/Polarization Diversity/Spatial Diversity/
Temporal Diversity are not suitable here.
- If there is a hill in between Tx and Rx antennas on either the BS or
MS does not help.
Macroscopic diversity (contd.)
- To solve the problem use a separate BS
- Large distance between BS1 and BS2 gives rise to macrodiversity.
- Use on-frequency repeaters (receive the signal and retransmit the
amplified version). It is simpler as synchronization is not necessary but
delay and dispersion are larger.
- Simulcast (same signal transmitted simultaneously from different
BSs.)
- Simulcast widely used for broadcast applications like digital TV.
- Disadvantage of simulcast is the large amount of signaling
information that has to be carried on landlines, synchronization
information and transmit data have to be transported on landlines to
BSs.
Signal Combining
- Select
path with best SNR or combine multiple paths
→ improve overall SNR performance
Selection diversity - 'Best' signal copy is selected and
processed (demodulated and decoded) and all other
copies are discarded
Combining Diversity - All signal copies are combined and
combined signal decoded
Note: Combining diversity leads to better performance but
Rx complexity higher than Selection Diversity.
Gain of Multiple Antennas - Diversity Gain and
Beamforming Gain.
Spatial (antenna) diversity
Selection diversity
RSSI = received
signal strength
indicator
Spatial (antenna) diversity
Selection diversity, cont.
Disadvantages of Selection Diversity
- Selection criteria (Power/BER) of all diversity branches
monitored to select the best.
- Alternately to reduce hardware cost and spectral inefficiency,
switched diversity done.
Switched Diversity - Active branch monitored if signal strength
falls below threshold Rx switches to a different antenna
Demerits
- Works well if sufficient signal quality in one of the branches
- If all branches signal strength < threshold then repeated switching
Free Parameters of Switched Diversity - switching threshold
(neither too low nor too high), hysteresis time (not too long or short)
Disadvantages of Selection Diversity
- Selection Diversity wastes signal energy by discarding M-1 copies
of Rxd signal
- Combining Diversity - All branches are considered
- Combining Diversity Types - Maximal Ratio Combining, Equal
Gain Combining
Spatial (antenna) diversity
Maximum ratio combining
Spatial (antenna) diversity
Spatial (antenna) diversity
Performance comparison
Cumulative distribution of SNR
MRC
Comparison of
SNR distribution
RSSI selection
for different number
of antennas M and
two different diversity
techniques.
[Fig. 13.9]
Copyright: Prentice-Hall
Spatial (Antenna) Diversity
• Spatial
or Antenna Diversity
– M independent branches
– Variable gain & phase at each branch → G∠ θ
– Each branch has same average SNR:
Eb
SNR   
N0
– Instantaneous
i
SNR   i
– the pdf of
i




1 i
Pr  i      p( i )d i   e d i  1  e 

0
0
Spatial (Antenna) Diversity
–- The probability that all M independent diversity branches
Rx signal which are simultaneously less than some
specific SNR threshold γ
Pr  1 ,... M     (1  e  /  ) M  PM ( )
Pr  i     1  PM ( )  1  (1  e /  ) M
d
M
pM ( ) 
PM ( )  1  e
d


 M 1
e 

– The pdf of  :
– Average SNR improvement offered by selection diversity


0
0
    pM ( )d     Mx 1  e

M
1

 k 1 k

 x M 1
e  x dx, x   
Spatial (antenna) diversity
Performance comparison
Cumulative distribution of SNR
MRC
Comparison of
SNR distribution
for different
fnumber
of antennas M
RSSI selection
[Fig. 13.9]
Copyright: Prentice-Hall
Space diversity types/methods:
1) Selection diversity
2) Feedback diversity
3) Maximal radio combining
4) Equal gain diversity
Selection diversity Technique
Selection Diversity → simple & cheap
– Rx selects branch with highest instantaneous SNR
• new selection made at a time that is the reciprocal of the
fading rate
• this will cause the system to stay with the current signal
until it is likely the signal has faded
– SNR improvement :
•  is new avg. SNR
• Γ : avg. SNR in each branch
Selection Diversity Technique (Contd.):
Selection Diversity Technique:
Ref: Rappaport (Wireless Communications)
Scanning/Feedback Diversity
Scanning/Feedback Diversity
– scan each antenna until a signal is found that is above
predetermined threshold
– if signal drops below threshold → rescan
– only one Rx is required (since only receiving one signal
at a time), so less costly → still need multiple antennas
Maximal Ratio Combiner Diversity
–
–
–
–
signal amplitudes are weighted according to each SNR
summed in-phase
most complex of all types
a complicated mechanism, but modern DSP makes this
more practical → especially in the base station Rx where
battery power to perform computations is not an issue
Maximal Ratio Combiner Diversity
The resulting signal envelop applied to detector:
M
rM   Gi ri
i 1
Total noise power:
M
NT  N  Gi2
i 1
SNR applied to detector:
M
rM2

2 NT
Maximal Ratio Combiner Diversity
The voltage signals from each of the M diversity branches are co-phased to provide
coherent voltage addition and are individually weighted to provide optimal SNR
( rMis maximized whenGi
 ri / N )
The SNR out of the diversity combiner is the sum of the SNRs in each branch.
Maximal Ratio Combiner Diversity
The probability that less than some specific SNR threshold γ
gives optimal SNR improvement :
Γi: avg. SNR of each individual branch
Γi = Γ if the avg. SNR is the same for each branch
M
M
i 1
i 1
 M    i  i M 
Maximal Ratio Combiner Diversity
Equal Gain Combining Diversity
• Combine multiple signals into one
• G = 1, but the phase is adjusted for each received signal.
• The signal from each branch are co-phased vectors add
in-phase.
• Better performance than selection diversity
Transmit Diversity
• Multiple antennas installed at just one link (usually at BS)
•Uplink transmission from MS to BS - multiple antennas act as
Rx diversity branches
•For downlink diversity branches originate at Txr.
- Transmit Diversity with channel-state information
- Transmit Diversity without channel-state information
Time Diversity
• Time Diversity → transmit repeatedly the information at
different time spacings
• Time spacing > coherence time (coherence time is the time
over which a fading signal can be considered to have similar
characteristics)
• So signals can be considered independent
• Main disadvantage is that BW efficiency is significantly
worsened – signal is transmitted more than once BW must ↑ to
obtain the same Rd (data rate)
Note: If data stream repeated twice then either BW doubles for the
same Rd or Rd is reduced by ½ for the same BW
Time Diversity - RAKE Receiver
• Powerful form of time diversity available in spread spectrum (DS)
systems → CDMA
• Signal is transmitted only once
• Propagation delays in the MRC provide multiple copies of Tx signals
delayed in time
• If time delay between multiple signals > chip period of spreading
sequence (Tc) → multipath signals can be considered uncorrelated
(independent)
• In a basic system, these delayed signals only appear as noise,
since they are delayed by more than a chip duration and ignored.
• Multiplying by the chip code results in noise because of the time
shift.
• But this can be used to our advantage by shifting the chip sequence to
receive that delayed signal separately from the other signals.
Time Diversity - RAKE Receiver
• attempts to collect the time-shifted versions of the original signal
by providing a separate correlation receiver for each of the
multipath signals.
• Each correlation receiver may be adjusted in time delay, so that
a microprocessor controller can cause different correlation
receivers to search in different time windows for significant
multipath.
• The range of time delays that a particular correlator can search
is called a search window.
Time Diversity - RAKE Receiver
The RAKE Rx is a time diversity Rx that collects time-shifted versions of the
original Tx signal
Time Diversity - RAKE Receiver
The RAKE Rx is a time diversity Rx that collects time-shifted
versions of the original Tx signal
M branches or “fingers” of correlation Rx’s
Separately detect the M strongest signals
Weighted sum computed from M branches
faded signal → low weight
strong signal → high weight
overcomes fading of a signal in a single branch
SNR statistics for diversity receivers
Nr.
1
1
BER of diversity receivers
1
Computation via moment-generating function
Spatial (antenna) diversity
Performance comparison, cont.
MRC
Comparison of
2ASK/2PSK BER
for different number
of antennas M and
two different diversity
techniques.
RSSI selection
Copyright: Prentice-Hall
Spatial (antenna) diversity
Errors due to signal distortion
Comparison of
2ASK/2PSK BER
for different number
of antennas M and
two different diversity
techniques.
Copyright: Prentice-Hall
Optimum combining in flat-fading
channel
• Most systems interference limited
• OC reduces not only fading but also interference
• Each antenna can eliminate one interferer or give one
diversity degree for fading reduction:
(“zero-forcing”).
• MMSE or decision-feedback gives even better results
• Computation of weights for combining
K
wopt
R
1h
d
R
2I
E r rT
k
1
Performance of Optimum Combining
• Define channel matrix H:
Hkm is transfer function for
k-th user to m-th diversity
antenna
• Error of BPSK, QPSK for one
channel constellation bounded
as
BER
static
≤ exp [−hd Rni hd ]
−1
H
2 interferers, optimum combining
5x10
-1
10 -1
M=1
10 -2
M=2
10 -3
M=5
• average behavior:
BER≤[1+SNR
10 -4
-10
]
−(M−K)
-5
M=3
0
5
Γ / dΒ
From Winters 1984,
10
15
20
Review of Channel coding &
Speech Coding Techniques
Contents
• Overview
• Block codes
• Convolution codes
• Trellis-coded modulation
• Turbo codes and LDPC codes
• Fading channel and interleaving
OVERVIEW
Basic types of codes
Channel codes are used to add protection against errors in the channel.
It can be seen as a way of increasing the distance between transmitted
alternatives, so that a receiver has a better chance of detecting the
correct one in a noisy channel.
We can classify channel codes in two principal groups:
BLOCK CODES
CONVOLUTION CODES
Encodes data in
blocks of k, using
code words of
length n.
Encodes data in a stream,
without breaking it into
blocks, creating code
sequences.
Information and redundancy (1)
EXAMPLE
Is the English language protected by a code, allowing us to correct
transmission errors?
When receiving the following sentence with errors marked by ´-´:
“D- n-t w-rr- c-n -ss-r-
-b--t ---r d-ff-cult--s -n M-th-m-t-cs.
--- m-n- -r- st-ll gr--t-r.”
it can still be “decoded” properly.
What does it say, and who is quoted?
There is something more than information in the original sentence
that allows us to decode it properly, redundancy.
Redundancy is available in almost all “natural” data, such as text, music,
images, etc.
Information and redundancy (2)
Electronic circuits do not have the power of the human brain and
needs more structured redundancy to be able to decode “noisy”
messages.
”Pure information”
without
redundancy
Original source data
with redundancy
E.g. a speech
coder
Source
coding
Channel
coding
”Pure information” with
structured redundancy.
The structured redundancy added
in the channel coding is often called
parity or check sum.
Illustration of code words
Assume that we have a block code, which consists of k information
bits per n bit code word (n > k).
Since there are only 2k different information sequences, there can be
only 2k different code words.
2n different
binary sequences
of length n.
Only 2k are valid
code words in
our code.
Illustration of decoding
Received word
Distances
Two common ones:
Hamming distance Measures the number of bits
being different between two
binary words.
Used for binary
channels with
random bit errors.
Euclidean distance Same measure we have used
for signal constellations.
Used for AWGN
channels.
Coding gain
When applying channel codes we decrease the Eb/N0 required to
obtain some specified performance (BER).
BER
BERspec
Gcode
Eb/N0 [dB]
BLOCK CODES
Channel coding
Linear block codes
Channel coding
Some definitions
min
i≠ j
0
G G
= 1
x+x
ij
0
+
1
0
=
0
1
0
G
w( x)
d(
i
j
1
G
i
j
i
+x
j
)
Channel coding
Encoding example
For a specific (n,k) = (7,4) code we encode the information
sequence 1 0 1 1 as
1
0
0
0
1
0
1
0
0
0
0
0
1
0
1
0
0
1
0
1
0
0
1
1
1
Systematic bits
1
=
1
0
1
1
0
1
1
1 parity bits.
0
1
1
1
0
Generator matrix
Channel coding
Encoding example, cont.
Encoding all possible 4 bit information sequences gives:
Information
Code word
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Hamming
weight
0
1
0
1
1
0
1
0
1
0
1
0
0
1
0
1
0
1
1
0
0
1
1
0
1
0
0
1
1
0
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
4
3
3
3
3
4
4
3
3
4
4
4
4
3
7
This is a (7,4) Hamming code, capable of correcting one bit error.
Channel coding
Error correction capability
t
=
dmin −1
2
t
t
dmin
From Ericsson radio school
Channel coding
Performance and code length
Eb/N0
CONVOLUTION CODES
Channel coding
Encoder structure
L=3
Copyright: Ericsson
Channel coding
Encoding example
Input State
0
1
0
1
0
1
0
1
00
00
01
01
10
10
11
11
Output Next state
000
111
001
110
011
100
010
101
We usually start the encoder in the all-zero state!
Copyright: Ericsson
00
10
00
10
01
11
01
11
Channel coding
Encoding example, cont.
We can view the encoding process in a trellis created from the table on
the previous slide.
Copyright: Ericsson
Channel coding
Termination
Copyright: Ericsson
Channel coding
A Viterbi decoding example
Received sequence:
010
000
000
1
000
2
100
1
000
001
2
4
3
0
0
5
000
5
4
8
6
7
5
4
6
4
2
7
101 5
101 7
0
0
6
6
8
Tail bits
0
5
000
5
5
5
001
000
2
4
4
1
110
4
101 4
Decoded data:
3
000
6
3
4
011
0
Channel coding
Surviving paths
Copyright: B. Mayr
TRELLIS-CODED MODULATION
Principle of TCM
• Goal: improve BER performance while leaving the
bandwidth requirement unchanged
• “Conventional” coding introduces redundancy, and
therefore increases the requirement for bandwidth
• Therefore, TCM increases the constellation size of the
modulation, while at the same time using a convolutional
code
Trellis-coded modulation (1)
• Simple example: TCM with 8-PSK and rate 2/3 coding
Copyright: B. Mayr
Trellis-coded modulation (2)
Signal-space diagram
Admissible transitions
Copyright: B. Mayr
TCM: BER computation (1)
d2
Copyright: B. Mayr
8EB
TCM: BER computation (2)
• Asymptotic coding gain of 3 dB
- Euclidean distance is 8E, compared to 4E for QPSK
Copyright: B. Mayr
Set partitioning
Copyright: B. Mayr
TURBO CODES AND LDPC CODES
Turbocoders
• Generates long codewords by
- encoding data with two different convolutional encoders
- for each of the encoders, data are interleaved with
different interleavers
Copyright: M. Valenti
Decoding of turbocodes
• Iterative decoding
• Two separate decoders (corresponding to the two
convolutional encoders) that exchange information
• Quantity of interest is the log-likelihood ratio
log
Pr
bi
1|x
Pr
bi
1|x
Block diagram of turbo decoder
Copyright: IEEE
Performance of turbo codes
#18
#6
#2
#2
Copyright: IEEE
Principle of LDPC codes
• LDPC: low density parity check codes
• Block codes with large block length
• Defined by the parity-check matrix H, not the generator
matrix
Construction of parity-check matrix
1. Divide matrix horizontally into p submatrices
2. Put a “1” into each column of the submatrix. Make sure that there are
q “1”s per row
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
3. Let other submatrices be column permutations of first submatrix
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0
0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0
H
0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0
0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0
0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0
0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
Encoding of bits
• Generator matrix has to be computed
• First step:
H
PT I
• Second step: generator matrix is
G
IP
Decoding: Tanner graph
• Method for iterative decoding
• Represent code in a Tanner graph (bipartite graph)
Check nodes
Variable nodes
Tanner graph for parity check matrix
H=
1 0 1 1
0 1 1 1
Decoding: step-by-step procedure
1. Variable nodes decide what they think they are, given external
evidence only
0, for all i
,0
2/
,0
2
rj , for all j
2. Constraint nodes compute what they think variable nodes have to be
l 1
l
2tanh
tanh
1
k A i
A i
j
i,k
2
j is "all the members of ensemble A i
with the exception of j"
3. Update opinion of what variable nodes have to be
l
i,j
B
j
2/ 2n
rj
k
l
i
k,j
B j
i is "all variable nodes that connect to the j
th constraint node, with the exception of i. "
4. compute the pseudoposterior probabilities that a bit is 1 or 0
Lj
2/
2
rj
i
l
,
5. If codeword has syndrome 0, stop iteration; otherwise goto 2
FADING CHANNELS
AND INTERLEAVING
Channel coding
Distribution of low-quality bits
Without interleaving
With interleaving
Eb/N0
Eb/N0
Fading dip gives many
low-quality bits in
the same code word
Code words
bit
bit
With interleaving the fading dip
spreads more evenly across
code words
Code words
Channel coding
Block interleaver
Channel coding
Interleaving - BER example
BER of a R=1/3 repetition code over a Rayleigh-fading channel,
with and without interleaving. Decoding strategy: majority selection.
10 dB
Div. order 2
Div. order 1
10 dB
100x
10x
Summary
• Channel coding is used to improve error performance
• For a fixed requirement, we get a coding gain that
translates to a lower received power requirement.
• The two main types of codes are block codes and
convolution codes
• Depending on the channel, we use different metrics to
measure the distances
• Decoding of convolution codes is efficiently done with the
Viterbi algorithm
• In fading channels we need interleaving in order to break
up fading dips (but causes delay)
Equalization
30
Contents
• Inter-symbol interference
• Linear equalizers
• Decision-feedback equalizers
• Maximum-likelihood sequence estimation
INTER-SYMBOL INTERFERENCE
Inter-symbol interference - Background
Transmitted symbols
Received symbols
Channel with
delay spread
Modeling of channel impulse response
What we have used so far (PAM and optimal receiver):
n( t)
c δ (t −kT )
k
kT
g (t)
g (T−t)
PAM
Matched filter
Including a channel impulse response h(t):
cδ
(t −kT)
k
n (t)
g (t)
h(t)
PAM
Can be seen as a “new”
basis pulse
ISI-free and
white noise
with proper
pulses g(t)
kT
(g∗h)*(T−t)
Matched filter
ϕk
ϕk
This one is no
longer ISI-free and
noise is not white
Modeling of channel impulse response
We can create a discrete time equivalent of the “new” system:
nk
ck
F ( z)
F
(z )
1
−
ϕk
where we can say that F(z) represent the basis pulse and channel, while
F*(z-1) represent the matched filter. (This is an abuse of signal theory!)
We can now achieve white noise quite easily, if (the not unique) F(z) is
chosen wisely (F*(z-1) has a stable inverse) :
nk
ck
F ( z)
F * (z
−1
)
ϕk
1/ F *( z
u
−1
Noise
whitening
filter
)
k
NOTE:
F*(z )/F *(z -1)=1
-1
The discrete-time channel model
With the application of a noise-whitening filter, we arrive at a discrete-time
model
ck
nk
F(z)
uk
where we have ISI and white additive noise, in the form
u = ∑L
k
j=0
f j ck− j +n k
This is the
model we are
going to use
when
designing
equalizers.
The coefficients f j represent the causal impulse response of the
discrete-time equivalent of the channel F(z), with an ISI that extends
over L symbols.
Channel estimation
LINEAR EQUALIZER
Principle
The principle of a linear equalizer is very simple: Apply a filter E(z) at the
receiver, mitigating the effect of ISI:
ck
F(z)
nk
uk
E(z)
ck
Linear
equalizer
Now we have two different strategies:
1) Design E(z) so that the ISI is totally removed
Zero-forcing
2) Design E(z) so that we minimize the mean
squared-error ofε=c−c
kk
k
MSE
Zero-forcing equalizer
c
nk
uk
F(z)
k
ck
1/ F(z)
ZF
equalizer
Information
Channel
f
Noise
f
Information
and noise
Equalizer
f
Noise enhancement!
f
f
MSE equalizer
The MSE equalizer is designed to minimize the error variance
ck
nk
uk
F(z)
σ s2F
−1
(z )
σ s F(z)2+N
ck
2
0
MSE
equalizer
Information
Noise
Channel
f
f
Information
and noise
Equalizer
f
f
Less noise enhancement than Z-F!
f
DECISION-FEEDBACK
EQUALIZER
DFE - Principle
Decision
device
nk
ck
F(z)
E(z)
ck
+
-
Forward
filter
D(z)
Feedback
filter
This part shapes
the signal to work
well with the
decision
feedback.
This part removes ISI on
“future” symbols from the
currently detected symbol.
If we make
a wrong
decision
here, we
may
increase the
ISI instead
of remove it.
Zero-forcing DFE
In the design of a ZF-DFE, we want to completely remove all ISI before
the detection.
ISI-free
nk
ck
F(z)
E(z)
ck
+
D(z)
This enforces a relation between the E(z) and D(z), which is (we assume
that we make correct decisions!)
F ( z ) E ( z) −D ( z ) = 1
MSE-DFE
minimal MSE
nk
ck
F(z)
E(z)
ck
+
D(z)
MAXIMUM-LIKELIHOOD
SEQUENCE ESTIMATION
Principle
“noise free signal alternative”
um
NF
L
=∑ fcj m− j
j=0
The squared Euclidean distance (optimal for white Gaussian noise) to
the received sequence {um} is
d ({um},{ um
2
NF
}) = ∑ um −u m
NF 2
m
=
∑ u − ∑ fc
L
m
m
2
j m− j
j=0
The MLSE decision is then the sequence of symbols {cm} minimizing this
distance
cm =arg min
{cm}
∑ u − ∑ fc
L
m
m
j m− j
j=0
2
The Viterbi-equalizer
Let’s use an example to describe the Viterbi-equalizer.
Discrete-time channel:
ck
F(z)2
1
−
This would case
serious noise
enhancement in
linear equalizers.
z
-0.9
f
Further, assume that our symbol alphabet is -1 and +1 (representing
the bits 0 and 1, respectively).
The fundamental
trellis stage:
State
-1
-0.1
1
-1.9
1.9
0.1
Input cm
-1
+1
The Viterbi-equalizer (2)
Transmitted:
1 1 -1 1
Noise free sequence:
1.9 0.1 -1.9 1.9 -1.9
-1
Noise
The filter starts
in state -1.
1− 0.9z−1
Received noisy sequence:
0.72
0.19
-1.70
1.09
VITERBI
DETECTOR
State
-1
-0.1
0.68
1.9
-0.1 0.76
1.9
5.75
3.60
1.39
1
-0.1 3.32
1.9
-1.9
0.1 1.40
-1.9
0.1 4.64
Detected sequence:
1
1
-1
-1.06
-0.1 2.86
1.9
1.44
13.72
At this stage,
the path ending
here has the best
metric!
-0.1 3.78
1.9
13.58
2.09
-1.9
2.79
11.62
-1.9
0.1 5.62
0.1 3.43
1
-1
Correct!
Summary
• Linear equalizers suffer from noise enhancement.
• Decision-feedback equalizers (DFEs) use decisions on data
to remove parts of the ISI, allowing the linear equalizer part
to be less ”powerful” and thereby suffer less from noise
enhancement.
• Incorrect decisions can cause error-propagation in DFEs,
since an incorrect decision may add ISI instead of removing
it.
• Maximum-likelihood sequence estimation (MLSE) is optimal
in the sense of having the lowest probability of detecting the
wrong sequence.
• Brut-force MLSE is prohibitively complex.
• The Viterbi-equalizer (detector) implements the MLSE with
considerably lower complexity.