This document provides an introduction to equalization and summarizes several equalization techniques: 1) Zero forcing equalizers aim to completely eliminate intersymbol interference by inverting the channel response but can amplify noise. 2) The mean square error criterion aims to minimize the error between the received and desired signals when filtered by the equalizer. This can be solved using least squares or adaptive algorithms like LMS. 3) The least mean square algorithm approximates the steepest descent method to iteratively and adaptively update the equalizer filter taps to minimize the mean square error based only on instantaneous measurements. This makes it suitable for time-varying channels.

1. Introduction To Equalization
Presented By :
BHABENDU KUMAR BHAKAT
15304002
Mtech(ECE)

2. TOC
• Communication system model
• Need for equalization
• ZFE
• MSE criterion
• LS
• LMS
• Blind Equalization – Concepts
• Turbo Equalization – Concepts
• MLSE - Concepts

3. InformationInformation
sourcesource
PulsePulse
generatorgenerator
TransTrans
filterfilter
channelchannel
X(t)X(t)
ReceiverReceiver
filterfilterA/DA/D
++
Channel noiseChannel noise
n(t)n(t)
DigitalDigital
ProcessingProcessing
++
HHTT(f)(f)
HHRR(f)(f)
Y(t)Y(t)
HHcc(f)(f)
Basic Digital Communication SystemBasic Digital Communication System

4. TransTrans
filterfilter
channelchannel
ReceiverReceiver
filterfilter
Basic Communication SystemBasic Communication System
HHTT(f)(f) HHRR(f)(f)HHcc(f)(f)
∑ +−−=
k
bdck tnkTtthAtY )()()( 0
The received Signal is the transmitted signal, convolved with the channel
And added with AWGN (Neglecting HTx,HRx)
( ) ( )[ ] ( )tnTAt mb
mK
ckmm
kmhAY 0
+−+= ∑≠
ISI -ISI - IInternter SSymbolymbol
IInterferencenterference
Y(t)
Ak Y(tm)

5. Explanation of ISIExplanation of ISI
tt
ff
FourierFourier
TransformTransform ChannelChannel
ff
FourierFourier
TransformTransform
tt
TTbb 2T2Tbb
3T3Tbb 4T4Tbb
5T5Tbb
tt
6T6Tbb

6. Reasons for ISI
• Channel is band limited in
nature
Physics – e.g. parasitic
capacitance in twisted pairs
– limited frequency response
  unlimited time response
•Tx filter might add ISI when
channel spacing is crucial.
• Channel has multi-path
reflections

7. Channel Model
• Channel is unknown
• Channel is usually modeled as Tap-Delay-
Line (FIR)
D D D
h(0) h(1) h(2) h(N-1) h(N)
y(n)
x(n)
+
+
+
+
+

8. Example for Measured Channels
The Variation of the Amplitude of the Channel Taps is Random
(changing Multipath)and usually modeled as Railegh distribution
in Typical Urban Areas

9. Example for Channel Variation:

10. Equalizer: equalizes the channel – the
received signal would seen like it
passed a delta response.
))(arg())(arg(
)()(1|)(||)(|
|)(|
1
|)(|
fGfG
tthfGfG
fG
fG
CE
totalCE
C
E
−=
=⇒=⋅⇒= δ

11. Need For Equalization
• Need For Equalization:
– Overcome ISI degradation
• Need For Adaptive Equalization:
– Changing Channel in Time
• => Objective:
Find the Inverse of the Channel Response
to reflect a ‘delta channel to the Rx
*Applications (or standards recommend us the channel
types for the receiver to cope with).

12. Zero forcing equalizers
(according to Peak Distortion Criterion)
Tx Ch Eq
qx
∑−= 





±±=
=
=⋅−⋅=
2
2 ...2,1,0
0,1
)()(
n m
m
nmTXCnmTq τ
No ISI
Equalizer taps
is described as matrix)2/( nTmTx −












−−
−−−−
=
)1()5.1()2()5.2()3(
)0()5.0()1()5.1()2(
)1()5.0()0()5.0()1(
)2()5.1()1()5.0()0(
TxTxTxTxTx
xTxTxTxTx
TxTxxTxTx
TxTxTxTxx
X
Example: 5tap-Equalizer, 2/T sample rate:
:Force

13. 















=
−
−
2
1
0
1
2
c
c
c
c
c
C
















=
0
0
1
0
0
q
Equalizer taps as vector
Desired signal as vector
⇒
XC=q Copt=X-1
q
⇒
Disadvantages: Ignores presence of additive noise
(noise enhancement)

14. MSE Criterion
2
1
0
])[][(][ nhnxJ
N
n
θθ −= ∑
−
=
Mean Square Error between the received signal
and the desired signal, filtered by the equalizer filter
LS Algorithm LMS Algorithm
Desired Signal
UnKnown Parameter
)Equalizer filter response(
Received Signal

15. LS
• Least Square Method:
– Unbiased estimator
– Exhibits minimum variance )optimal(
– No probabilistic assumptions )only signal
model(
– Presented by Guass )1795( in studies of
planetary motions(

16. LS - Theory
][][][ mmnhns θ∑ −=
Hns θ=][
2
1
0
])[][(][ nhnxJ
N
n
θθ −= ∑
−
=
∑
∑
−
=
−
=
= 1
0
2
1
0
][
][][
ˆ
N
n
N
n
nh
nhnx
θ
Derivative according to: θ
1.
2.
3.
4.
MSE:

17. The minimum LS error would be obtained by substituting 4 to 3:
][]
])[][(
][
][][ˆ][
])[ˆ][(][ˆ])[ˆ][(][
])[ˆ][])([ˆ][(])[ˆ][(][
2
1
0
2
1
0
1
0
2
min
1
0
1
0
2
)ˆ(0
1
0
1
0
1
0
2
1
0
min
nh
nhnx
nxJ
nhnxnx
nhnxnhnhnxnx
nhnxnhnxnhnxJJ
N
n
N
n
N
n
N
n
N
n
tinhBySubstitu
N
n
N
n
N
n
N
n
∑
∑
∑
∑∑
∑∑
∑∑
−
=
−
=
−
=
−
=
−
=
−
=
−
=
−
=
−
=
−==>
−=
−−−=
−−=−==
θ
θθθ
θθθθ
θ
  
Energy Of
Original Signal
Energy Of
Fitted Signal
][][ nwSignalnx += If Noise Small enough )SNR large enough(: Jmin~0
Back-Up

18. Finding the LS solution
θHns =][
])][(])][(
])[ˆ][])([ˆ][(])[ˆ][(][
1
0
2
1
0
θθ
θθθθ
HnxHnx
nhnxnhnxnhnxJ
T
N
n
N
n
−−=
−−=−= ∑∑
−
=
−
=
θθθ
θθθθθ
HHHxzx
HHxHHxxxJ
TT
scalar
TT
TTTTTT
+−=
+−−=
2
][
)H: observation matrix )Nxp( and
θ
θ
θ
HHxH
J T
scalar
T
22
)(
+−=
∂
∂

xHHH TT 1
)(ˆ −
=θ
T
Nsssns ])1[],...1[],0[(][ −=

19. LS : Pros & Cons
•Advantages:
•Optimal approximation for the Channel- once calculated
it could feed the Equalizer taps.
•Disadvantages:
•heavy Processing )due to matrix inversion which by
It self is a challenge(
•Not adaptive )calculated every once in a while and
is not good for fast varying channels
• Adaptive Equalizer is required when the Channel is time variant
)changes in time( in order to adjust the equalizer filter tap
Weights according to the instantaneous channel properties.

20. Contents:
• Introduction - approximating steepest-descent algorithm
• Steepest descend method
• Least-mean-square algorithm
• LMS algorithm convergence stability
• Numerical example for channel equalization using LMS
• Summary
LEAST-MEAN-SQUARE ALGORITHM

21. SYSTEM BLOCK USING THE LMS
U[n] = Input signal from the channel ; d[n] = Desired Response
H[n] = Some training sequence generator
e[n] = Error feedback between :
A.) desired response.
B.) Equalizer FIR filter output
W = Fir filter using tap weights vector

22. STEEPEST DESCENT METHOD
• Steepest decent algorithm is a gradient based method which
employs recursive solution over problem )cost function(
• The current equalizer taps vector is W)n( and the next
sample equalizer taps vector weight is W)n+1(, We could
estimate the W)n+1( vector by this approximation:
• The gradient is a vector pointing in the direction of the
change in filter coefficients that will cause the greatest
increase in the error signal. Because the goal is to minimize
the error, however, the filter coefficients updated in the
direction opposite the gradient; that is why the gradient term
is negated.
• The constant μ is a step-size. After repeatedly adjusting
each coefficient in the direction opposite to the gradient of
the error, the adaptive filter should converge.
])[(5.0]1[][ nJnWnW −∇++= µ

23. • Given the following function we need to obtain the vector
that would give us the absolute minimum.
• It is obvious that
give us the minimum.
STEEPEST DESCENT EXAMPLE
2
2
2
121 ),( CCccY +=
,021 == CC
1C
2C
y
Now lets find the solution by the steepest descend method

24. • We start by assuming (C1 = 5, C2 = 7)
• We select the constant . If it is too big, we miss the
minimum. If it is too small, it would take us a lot of time to
het the minimum. I would select = 0.1.
• The gradient vector is:
STEEPEST DESCENT EXAMPLE
µ
µ
][2
1
][2
1
][2
1
][2
1
]1[2
1
9.01.02.0
nnnnn
C
C
C
C
C
C
y
C
C
C
C






=





−





=∇∗−





=





+






=












=∇
2
1
2
1
2
2
C
C
dc
dy
dc
dy
y
• So our iterative equation is:

25. STEEPEST DESCENT EXAMPLE






=











=











=





567.0
405.0
:3
3.6
5.4
:2
7
5
:1
2
1
2
1
2
1
C
C
Iteration
C
C
Iteration
C
C
Iteration






=











=





∞→
0
0
lim
013.0
01.0
:60
......
][2
1
2
1
n
n
C
C
C
C
Iteration
As we can see, the vector [c1,c2] convergates to the value
which would yield the function minimum and the speed of
this convergence depends on .µ
1C
2C
y
Initial guess
Minimum

26. MMSE CRITERIA FOR THE LMS
• MMSE – Minimum mean square error
• MSE =
• To obtain the LMS MMSE we should derivative
the MSE and compare it to 0:
•
}])()()({[(})]()({[( 22
∑−=
−−=−
N
Nn
nkunwkdEkykdE
)(
))()()()()(2})({(
)(
)(
2
kdW
mnRmwnwnPnwkdEd
kdW
MSEd
N
Nn
N
Nm
N
Nn
du ∑ ∑∑ −= −=−=
−+−
=
)}()({)(
)}()({)(
)()()()()(2})({}])()()({[( 22
knukmuEmnR
knukdEnP
mnRmwnwnPnwkdEnkunwkdE
uu
du
N
Nn
N
Nm
N
Nn
du
N
Nn
−−=−
−=
−+−=−− ∑ ∑∑∑ −= −=−=−=

27. MMSE CRITERION FOR THE LMS
,...2,1,0),(][2)(2
)(
)(
)( ±±=−+−==∇ ∑−−
kknRnwkP
kdW
MSEd
nJ uu
N
Nn
du
And finally we get:
By comparing the derivative to zero we get the MMSE:
PRwopt •= −1
This calculation is complicated for the DSP (calculating the inverse
matrix ), and can cause the system to not being stable cause if there
are NULLs in the noise, we could get very large values in the inverse
matrix. Also we could not always know the Auto correlation matrix of the
input and the cross-correlation vector, so we would like to make an
approximation of this.

28. LMS – APPROXIMATION OF THE
STEEPEST DESCENT METHOD
W(n+1) = W(n) + 2*[P – Rw(n)] <= According the MMSE criterion
We assume the following assumptions:
• Input vectors :u(n), u(n-1),…,u(1) statistically independent vectors.
• Input vector u(n) and desired response d(n), are statistically independent of
d(n), d(n-1),…,d(1)
• Input vector u(n) and desired response d(n) are Gaussian-distributed R.V.
•Environment is wide-sense stationary;
In LMS, the following estimates are used:
Ruu^ = u(n)u
H
(n) – Autocorrelation matrix of input signal
Pud^ = u(n)d*(n) - Cross-correlation vector between U[n] and d[n].
*** Or we could calculate the gradient of |e[n]|2
instead of E{|e[n]|2
}

29. LMS ALGORITHM
≈
[n]}y–[n]u[n]{d{w(n)
[n]w[n]}u[n]u–[n]{u[n]dw(n)
w[n]}R–{PW[n]1]W[n
**
H*
^^
µ
µ
µ
+=
+=
+≅+
We get the final result:
[n]}u[n]e{W[n]1]W[n *
µ+≅+

30. LMS STABILITY
The size of the step size determines the algorithm convergence
rate. Too small step size will make the algorithm take a lot of
iterations. Too big step size will not convergence the weight taps.
Rule Of Thumb:
RPN )12(5
1
+
=µ
Where, N is the equalizer length
Pr, is the received power (signal+noise)
that could be estimated in the receiver.

31. LMS – CONVERGENCE GRAPH
This graph illustrates the LMS algorithm. First we start from guessing
the TAP weights. Then we start going in opposite the gradient vector,
to calculate the next taps, and so on, until we get the MMSE,
meaning the MSE is 0 or a very close value to it.(In practice we can
not get exactly error of 0 because the noise is a random process, we
could only decrease the error below a desired minimum)
Example for the Unknown Channel of 2nd
order:
Desired Combination of tapsDesired Combination of taps

32. LMS Convergence Vs u

33. LMS – EQUALIZER EXAMPLE
Channel equalization
example:
Average Square Error as a
function of iterations number
using different channel
transfer function
(change of W)

35. LMS – Advantage:
• Simplicity of implementation
• Not neglecting the noise like Zero forcing equalizer
• By pass the need for calculating an inverse matrix.
LMS : Pros & Cons
LMS – Disadvantage:
Slow Convergence
Demands using of training sequence as reference
,thus decreasing the communication BW.

36. Non linear equalization
Linear equalization (reminder):
• Tap delayed equalization
• Output is linear combination of the equalizer
input
C
E
G
G
1
=
...
)(
)(
)(
2
3
1
10
1
+++==
−=
−−
−
∏
zazaaC
zX
zY
zaG
E
i
iE  as FIR
...)2()1()()( 210 +−⋅+−⋅+⋅= nxanxanxany

37. Non linear equalization – DFE
(Decision feedback Equalization)
∑ ∑ −⋅−−⋅= )()()( inybinxany ii
Advantages: copes with larger ISI
∏
∏
−
−
−
−
==
i
i
i
i
E
zb
za
G
zX
zY
)(
)(
)(
)(
1
1
 as IIR
A(z)
Receiver
detector
B(z)
+In Output
+
-
The nonlinearity is due the
detector characteristics that
is fed back (MAPPER)
The Decision feedback leads poles in z domain
Disadvantages: instability danger

38. Non linear equalization - DFE

39. Blind Equalization
• ZFE and MSE equalizers assume
option of training sequence for
learning the channel.
• What happens when there is
none?
– Blind Equalization
Adaptive
Equalizer
Decision
+
-
Input Output
Error
Signal
nV
nIˆ
nI
~
nd
neBut Usually employs also :
InterleavingDeInterleaving
Advanced coding
ML criterion
Why? Blind Eq is hard and complicated enough!
So if you are going to implement it, use the best blocks
For decision (detection) and equalizing
With LMS

40. Turbo Equalization
MAP
Decoder
+ Π
Π
−1
MAP
Equalizer
+
Channel
Estimator
r
L
D
e(c’) L
D
e(c) L
D
(c)
L
D
(d)
L
E
e(c)L
E
e(c’)L
E
(c’)
Iterative :
Estimate
Equalize
Decode
ReEncode
Usually employs also :
InterleavingDeInterleaving
TurboCoding (Advanced iterative code)
MAP (based on ML criterion)
Why? It is complicated enough!
So if you are going to implement it, use the best blocks
Next iteration would rely on better estimation
therefore would lead more precise equalization

41. Performance of Turbo Eq Vs
Iterations

42. ML criterion
• MSE optimizes detection up to 1st
/2nd
order
statistics.
• In Uri’s Class:
– Optimum Detection:
• Strongest Survivor
• Correlation (MF)
(allow optimal performance for Delta ch and Additive noise.
 Optimized Detection maximizes prob of detection
(minimizes error or Euclidean distance in Signal Space)
• Lets find the Optimal Detection Criterion while in
presence of memory channel (ISI)

43. ML criterion –Cont.
• Maximum Likelihood :
Maximizes decision probability for the received trellis
Example BPSK (NRZI)
bESS =−= 01
2possible transmitted signals
Energy Per Bit
kbk nEr +±=
Received Signal occupies AWGN







 −
−= 2
2
1
2
)(
exp
2
1
)|(
n
bk
n
k
Er
srp
σπσ







 +
−= 2
2
0
2
)(
exp
2
1
)|(
n
bk
n
k
Er
srp
σπσ
Conditional PDF (prob of correct decision on r1 pending s1 was transmitted…)
N0/2
Prob of correct decision on a sequence of symbols
∏=
=
K
k
m
kk
m
k srpsrrrp
1
)()(
21 )|()|,...,,(
Transmitted sequence
optimal

44. ML – Cont.
With logarithm operation, it could be shown that this is equivalent to:
Minimizing the Euclidean distance metric of the sequence:
∑=
−=
K
k
m
kk
m
srsrD
1
2)()(
)(),(
How could this be used?
Looks Similar?
while MSE minimizes Error (maximizes Prob) for decision on certain Sym,
MLSE minimizes Error (maximizes Prob) for decision on certain Trellis ofSym,
(Called Metric)

45. Viterbi Equalizer
(On the tip of the tongue)
bE−bE
bE/1
Example for NRZI:
Trasmit Symbols:
(0=No change in transmitted Symbol
(1=Alter Symbol)
Tt = Tt 2= Tt 3= Tt 4=
S0
S1
bE−/0 bE−/0 bE−/0 bE−/0
bE/1 bE/1 bE/1
bE−/1 bE−/1 bE−/1
2
2
2
10 )()()0,0( bb ErErD +++=
2
2
2
10 )()()1,1( bb ErErD ++−=
2
2
2
10 )()()1,0( bb ErErD −++=
2
2
2
10 )()()0,1( bb ErErD −+−=
bE/0 bE/0 bE/0
Metric
(Sum
of
Euclidean
Distance)
2
300 )()0,0()0,0,0( bErDD ++=
2
300 )()1,0()1,1,0( bErDD ++=
2
300 )()0,0()1,0,0( bErDD −+=
2
300 )()1,0()0,1,0( bErDD −+=

46. We Always disqualify one metric for possible S0 and possible S1.
Finally we are left with 2 options for possible Trellis.
Finally are decide on the correct Trellis with the Euclidean
Metric of each or with Apost DATA

48. References
• John G.Proakis – Digital Communications.
• John G.Proakis –Communication Systems Eng.
• Simon Haykin - Adaptive Filter Theory
• K Hooli – Adaptive filters and LMS
• S.Kay – Statistical Signal Processing – Estimation Theory