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
DAG: Ville Kuvaja
Koki Sugawara

2. What is ment by equalization in a TDMA system?
To remove ISI due to multipaths
Short training sequence  quick adjustment

3. 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).

4. 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

5. Zero-Forcing Equalization
Zero ISI at the receiver output
B(f)H(f)E(f)=Z(f)
Z(f): Nyquist spectrum e.g. raised cosine T 
Estimate of symbol
tap coefficients for equalizer E
Received signal
Nyquist spectrum:

6. Least-mean-square (LMS) algorithm
(simplification of “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

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

8. LMS and RLS Algorithm Minimize square error
RLS uses accumulated square error of the past
New coefficients at (k+1) step
LMS:
RLS:
Received signal:
equalizer coefficients:
error
LMS algorithm
(stepsize 0<Δ<1)

9. Effect of iteration step size
smaller
larger 
Slow acquisition
Poor stability
Poor tracking performance
Large variation around optimum value
Above characteristics holds both for coefficients and phase.
Convergence condition
0<<2/max
 : eigenvalue of autocorrelation matrix of r

10. 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 + 
The shed part is a transversal filter as a ZF equalizer.
An error calculation module and a tap weight control part should be added.
Complex-valued tap coefficients of equalizer filter
Estimate of k:th symbol after symbol decision

11. 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

12. Decision feedback equalizerT T
FBF +  + T T T
LMS algorithm
for tap coefficient adjustment
The shed part is same as an LMS equalizer.
The estimated symbol is fed back to the FBF to cancel ISI due to previous symbols.
The equalizing algorithm is same when considered
FFF 

13. Channel estimation circuit
Proakis, Ed.3, Section 11-3 T 
LMS algorithm
Estimated symbols +
k:th sample of received signal
Estimated channel coefficients
Filter length = CIR length
Again a transversal equalizer
Replicates received signal from estimated symbols
Now input signals and estimated signals are reversely applied to a transversal filter.
The optimized coefficients represent the channel response.

14. Training Sequence & Decision Directed Mode
training (known) sequence
weight setup
decision directed mode
adaptation + T  bk
Weight
update

15. How should the length of training sequence be?
At least the number of taps in the transversal filter (equalizer)
all the registers are filled with data sequence
the effect of each delayed component is considered at least once

16. How does a system adapt? T  +

17. 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 L states, where m is the symbol alphabet size and L is the length of the overall channel impulse response (in symbol intervals).
State trellis diagram
Allowed transition between states
State
Sample time instants

18. MLSE-VA Consider the ML of estimated symbol sequence
The probability that the received signal is yk
under certain symbol estimates in the past
under certain channel estimate
Extension to N signal sequence
The probability p shows a Gaussian distribution.

19. Channel estimation circuit
MLSE-VA (cont.)
p goes to maximum when
is minimized.
Transversal filter (again!)
Matched filter
MLSE
(VA)
Channel estimation circuit
NW filter