1. Equalization in a wideband TDMA system
Three basic equalization methods
Linear equalization (LE)
Decision feedback equalization (DFE)
Sequence estimation (MLSE-VA)
Example of channel estimation circuit

2. Three basic equalization methods (1)
Linear equalization (LE):
Performance is not very good when the frequency response of the frequency selective channel contains deep fades.
Zero-forcing algorithm aims to eliminate the intersymbol interference (ISI) at decision time instants (i.e. at the center of the bit/symbol interval).
Least-mean-square (LMS) algorithm will be investigated in greater detail in this presentation.
Recursive least-squares (RLS) algorithm offers faster convergence, but is computationally more complex than LMS (since matrix inversion is required).

3. Three basic equalization methods (2)
Decision feedback equalization (DFE):
Performance better than LE, due to ISI cancellation of tails of previously received symbols.
Decision feedback equalizer structure:
Feed-back filter (FBF)
Input
Output
Feed-forward filter (FFF) + +
Symbol decision
Adjustment of filter coefficients

4. Three basic equalization methods (3)
Maximum Likelihood Sequence Estimation using the Viterbi Algorithm (MLSE-VA):
Best performance. Operation of the Viterbi algorithm can be visualized by means of a trellis diagram with m K-1 states, where m is the symbol alphabet size and K is the length of the overall channel impulse response (in samples).
State trellis diagram
Allowed transition between states
State
Sample time instants

5. Linear equalization, zero-forcing algorithm
Basic idea:
Raised cosine spectrum
Transmitted symbol spectrum
Channel frequency
response
(incl. T & R filters)
Equalizer frequency response = f
fs = 1/T

6. Zero-forcing equalizer
Transmitted impulse sequence
Communication channel
Equalizer
Input to decision circuit
FIR filter contains 2N+1 coefficients
FIR filter contains 2M+1 coefficients
Overall channel
Channel impulse response
Equalizer impulse response
Coefficients of equivalent FIR filter
(in fact the equivalent FIR filter consists of 2M+1+2N coefficients, but the equalizer can only “handle” 2M+1 equations)

7. Zero-forcing equalizer
We want overall filter response to be non-zero at decision time k = 0 and zero at all other sampling times k  0 :
(k = –M)
This leads to a set of 2M+1 equations:
(k = 0)
(k = M)

8. Minimum Mean Square Error (MMSE)
The aim is to minimize:
(or
depending on the source)
Input to decision circuit
Estimate of k:th symbol
Error +
Channel
Equalizer

9. MSE vs. equalizer coefficients
quadratic multi-dimensional function of equalizer coefficient values
Illustration of case for two real-valued equalizer coefficients (or one complex-valued coefficient)
MMSE aim: find minimum value directly (Wiener solution), or use an algorithm that recursively changes the equalizer coefficients in the correct direction (towards the minimum value of J)!

10. Wiener solution
We start with the Wiener-Hopf equations in matrix form:
R = correlation matrix (M x M) of received (sampled) signal values
p = vector (of length M) indicating cross-correlation between received signal values and estimate of received symbol
copt = vector (of length M) consisting of the optimal equalizer coefficient values
(We assume here that the equalizer contains M taps, not 2M+1 taps like in other parts of this presentation)

11. Correlation matrix R & vector p
where
M samples
Before we can perform the stochastical expectation operation, we must know the stochastical properties of the transmitted signal (and of the channel if it is changing). Usually we do not have this information => some non-stochastical algorithm like Least-mean-square (LMS) must be used.

12. Inverting a large matrix is difficult!
Algorithms
Stochastical information (R and p) is available:
1. Direct solution of the Wiener-Hopf equations:
2. Newton’s algorithm (fast iterative algorithm)
3. Method of steepest descent (this iterative algorithm is slow but easier to implement)
Inverting a large matrix is difficult!
R and p are not available:
Use an algorithm that is based on the received signal sequence directly. One such algorithm is Least-Mean-Square (LMS).

13. Conventional linear equalizer of LMS type
Widrow
Received complex signal samples
Transversal FIR filter with 2M+1 filter taps
LMS algorithm for
adjustment of tap coefficients T T T + 
Complex-valued tap coefficients of equalizer filter
Estimate of k:th symbol after symbol decision

14. Joint optimization of coefficients and phase
Equalizer filter
Coefficient updating
Phase synchronization +
Godard
Proakis, Ed.3, Section
Minimize:

15. Least-mean-square (LMS) algorithm
(derived from “method of steepest descent”)
for convergence towards minimum mean square error (MMSE)
Real part of n:th coefficient:
Imaginary part of n:th coefficient:
Phase:
Iteration index
Step size of iteration
equations

16. LMS algorithm (cont.)
After some calculation, the recursion equations are obtained in the form

17. Effect of iteration step size
smaller
larger 
Slow acquisition
Poor stability
Poor tracking performance
Large variation around optimum value

18. Decision feedback equalizerT T ?
FBF +  + T T T
LMS algorithm
for tap coefficient adjustment
FFF 

19. Decision feedback equalizer (cont.)
The purpose is again to minimize
where
Feedforward filter (FFF) is similar to filter in linear equalizer
tap spacing smaller than symbol interval is allowed => fractionally spaced equalizer
=> oversampling by a factor of 2 or 4 is common
Feedback filter (FBF) is used for either reducing or canceling (difference: see next slide) samples of previous symbols at decision time instants
tap spacing must be equal to symbol interval

20. Decision feedback equalizer (cont.)
The coefficients of the feedback filter (FBF) can be obtained in either of two ways:
Recursively (using the LMS algorithm) in a similar fashion as FFF coefficients
By calculation from FFF coefficients and channel coefficients (we achieve exact ISI cancellation in this way, but channel estimation is necessary):
Proakis, Ed.3, Section 11-2
Proakis, Ed.3, Section

21. Channel estimation circuit
Proakis, Ed.3, Section 11-3
Estimated symbols
Filter length = CIR length T T T
LMS algorithm  +
k:th sample of received signal
Estimated channel coefficients

22. Channel estimation circuit (cont.)
1. Acquisition phase
Uses “training sequence”
Symbols are known at receiver,
2. Tracking phase
Uses estimated symbols (decision directed mode)
Symbol estimates are obtained from the decision circuit (note the delay in the feedback loop!)
Since the estimation circuit is adaptive, time-varying channel coefficients can be tracked to some extent.
Alternatively: blind estimation (no training sequence)

23. Channel estimation circuit in receiver
Mandatory for MLSE-VA, optional for DFE
Symbol estimates (with errors)
Training symbols (no errors)
Estimated channel coefficients
Channel
estimation
circuit
Equalizer
& decision
circuit
Received signal samples
“Clean” output symbols

24. Theoretical ISI cancellation receiver
(extension of DFE, for simulation of matched filter bound)
Precursor cancellation of future symbols
Postcursor cancellation of previous symbols
Filter matched to
sampled channel impulse response +
If previous and future symbols can be estimated without error (impossible in a practical system), matched filter performance can be achieved.

25. MLSE-VA receiver structure
Matched filter
NW filter
MLSE
(VA)
Channel estimation circuit
MLSE-VA circuit causes delay of estimated symbol sequence before it is available for channel estimation
=> channel estimates may be out-of-date
(in a fast time-varying channel)

26. MLSE-VA receiver structure (cont.)
The probability of receiving sample sequence y (note: vector form) of length N, conditioned on a certain symbol sequence estimate and overall channel estimate:
Since we have AWGN
Length of f (k)
Objective: find symbol sequence that maximizes this probability
This is allowed since noise samples are uncorrelated due to NW (= noise whitening) filter
Metric to be minimized (select best .. using VA)

27. MLSE-VA receiver structure (cont.)
We want to choose that symbol sequence estimate and overall channel estimate which maximizes the conditional probability.
Since product of exponentials <=> sum of exponents, the metric to be minimized is a sum expression.
If the length of the overall channel impulse response in samples (or channel coefficients) is K, in other words the time span of the channel is (K-1)T, the next step is to construct a state trellis where a state is defined as a certain combination of K-1 previous symbols causing ISI on the k:th symbol.
Note: this is overall CIR, including response of matched filter and NW filter
K-1 k

28. MLSE-VA receiver structure (cont.)
At adjacent time instants, the symbol sequences causing ISI are correlated. As an example (m=2, K=5): :
At time k-3 1 1
At time k-2 1 1
At time k-1 1 1 1
At time k 1 1 1 1 :
16 states
Bit detected at time instant
Bits causing ISI not causing ISI at time instant

29. MLSE-VA receiver structure (cont.)
State trellis diagram
Number of states
The ”best” state sequence is estimated by means of Viterbi algorithm (VA)
Alphabet size
k-3
k-2
k-1 k
k+1
Of the transitions terminating in a certain state at a certain time instant, the VA selects the transition associated with highest accumulated probability (up to that time instant) for further processing.
Proakis, Ed.3, Section