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

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).

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-forward filter (FFF)

Feed-forward filter (FFF)

Feed-back filter (FBF)

Feed-back filter (FBF)

Adjustment of filter coefficients

Input Output

++

Symbol decision

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

Sample time instants

State

Allowed transition between states

Linear equalization, zero-forcing algorithm

Raised cosine

spectrum

Raised cosine

spectrum

Transmitted symbol

spectrum

Transmitted symbol

spectrum

Channel frequencyresponse

(incl. T & R filters)

Channel frequencyresponse

(incl. T & R filters)

Equalizer frequency response

Equalizer frequency response

=

Z f B f H f E f

0 ffs = 1/T

B f

H f

E f

Z f

Basic idea:

Zero-forcing equalizer

Communication channel

Equalizer

FIR filter contains 2N+1 coefficients

r k z kTransmitted impulse sequence Input to

decision circuit

N

nn N

h k h k n

Channel impulse response Equalizer impulse response

M

mm M

c k c k m

Coefficients of equivalent FIR filter

( )M

k m k mm M

f c h M k M

(in fact the equivalent FIR filter consists of 2M+1+2N coefficients, but the equalizer can only “handle” 2M+1 equations)

FIR filter contains 2M+1 coefficients

Overall channel

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 :

1, 0

0, 0

M

k m k mm M

kf c h

k

0 1 1 2

1 0 1 2 1

1 1

2 1 2 2 1 1

2 2 1 1 0

... 0

... 0

:

... 1

:

... 0

... 0

M M M M

M M M M

M M M M M M

M M M M M

M M M M M

h c h c h c

h c h c h c

h c h c h c

h c h c h c

h c h c h c

This leads to a set of 2M+1 equations:

(k = –M)

(k = 0)

(k = M)

Minimum Mean Square Error (MMSE)

The aim is to minimize:2

kJ E e

ˆ ˆk k k k ke z b b z (or depending on the source)

EqualizerEqualizerChannelChannelkz kb

ke

r k s k

+

Estimate of k:th symbol

Input to decision circuit

z k b k

Error

MSE vs. equalizer coefficients

2

kJ E e

J

1c

2c

quadratic multi-dimensional function of equalizer coefficient values

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)!

Illustration of case for two real-valued equalizer coefficients (or one complex-valued coefficient)

Wiener solution

opt Rc p

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)

We start with the Wiener-Hopf equations in matrix form:

kb

kr

kr

Correlation matrix R & vector p

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.

*TE k k R r r

*kE k b p r

1 1, ,...,T

k k k Mk r r r rwhere

M samples

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)

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).

opt Rc p 1opt

c R p Inverting a large matrix is difficult!

Conventional linear equalizer of LMS type

k Mr k Mr TT TT TT

LMS algorithm for adjustment of tap coefficients

Transversal FIR filter with 2M+1 filter taps

Estimate of k:th symbol after symbol decision

Complex-valued tap coefficients of equalizer filter

ke

+

kbkz

Mc 1 Mc 1Mc Mc

WidrowReceived complex

signal samples

Joint optimization of coefficients and phase

Equalizer filterEqualizer filter

Coefficient updating

Coefficient updating

Phase synchronization

Phase synchronization

ke

je

+

r kkz

kb

ˆ ˆexpM

k k k m k m km M

e z b c r j b

Minimize:

Godard

Proakis, Ed.3, Section 11-5-2

2

kJ E e

Least-mean-square (LMS) algorithm(derived from “method of steepest descent”) for convergence towards minimum mean square error (MMSE)

2

Re 1 ReRe

kn n

n

ec i c i

c

Real part of n:th coefficient:

2

Im 1 ImIm

kn n

n

ec i c i

c

Imaginary part of n:th coefficient:

2

1 kei i

Phase:

2 2 1 1M equations Iteration index Step size of iteration

2

k k ke e e

LMS algorithm (cont.)

After some calculation, the recursion equations are obtained in the form

ˆRe 1 Re 2 ReM

j jn n m k m k k n

m M

c i c i e c r b r e

ˆIm 1 Im 2 ImM

j jn n m k m k k n

m M

c i c i e c r b r e

ˆ1 2 ImM

jk m k m

m M

i i b e c r

ke

Effect of iteration step size

Slow acquisitionSlow acquisition

smaller larger

Poor tracking performance

Poor tracking performance

Poor stabilityPoor stability

Large variation around optimum

value

Large variation around optimum

value

Decision feedback equalizer

TT TT TT

LMS algorithm

for tap coefficient adjustment

TT TT

FFFFFF

FBFFBF +

+ ke

kb

kz

k Qb 1kb

k Mr k Mr

Mc 1 Mc 1Mc Mc

1Qq Qq1q?

The purpose is again to minimize

Decision feedback equalizer (cont.)

1

ˆ ˆ ˆQM

k k k m k m n k n km M n

e z b c r q b b

2 2

k kJ E e e

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

where

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):

Decision feedback equalizer (cont.)

1, 2, ,M

n m n mm M

q c h n Q

Proakis, Ed.3, Section 11-2

Proakis, Ed.3, Section 10-3-1

Channel estimation circuit

TT TT TT

LMS algorithm

Estimated symbols

+

Mc1Mc 1c0c

k Mb kb

krkr

ˆm mh c

Proakis, Ed.3, Section 11-3

k:th sample of received signal Estimated channel coefficients

Filter length = CIR length

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.

k kb b

Alternatively: blind estimation (no training sequence)

Channel estimation circuit in receiver

Channel estimation

circuit

Channel estimation

circuit

Equalizer & decision

circuit

Equalizer & decision

circuit

h m b k

b k

r k

Estimated channel coefficients

“Clean” output symbolsReceived signal samples

Symbol estimates (with errors)

Training symbols

(no errors)

Mandatory for MLSE-VA, optional for DFE

Theoretical ISI cancellation receiver

Precursor cancellation of future symbols

Precursor cancellation of future symbols

Postcursor cancellation of previous symbols

Postcursor cancellation of previous symbols

Filter matched to sampled channel impulse response

Filter matched to sampled channel impulse response

1ˆ ˆk k Qb b 1

ˆ ˆk P kb b

k Pr

+

kb

(extension of DFE, for simulation of matched filter bound)

If previous and future symbols can be estimated without error (impossible in a practical system), matched filter performance can be achieved.

MLSE-VA receiver structure

Matched filter

Matched filter

MLSE(VA)

MLSE(VA)

Channel estimation circuit

Channel estimation circuit

NW filter

NW filter

r t

f k

y k b k

f k

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)

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:

21

2 21 01

1 1 ˆ ˆˆ ˆˆ ˆ, , exp22

N N K

k k n k nN Nk nk

p p y y f b

y b f b f

Metric to be minimized

(select best .. using VA)

Objective: find symbol

sequence that maximizes this

probability

This is allowed since noise samples are

uncorrelated due to NW (= noise whitening) filter

Length of f (k)Since we have AWGN

b

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.

K-10 k

f k Note: this is overall CIR, including response of matched

filter and NW filter

MLSE-VA receiver structure (cont.)

At adjacent time instants, the symbol sequences causing ISI are correlated. As an example (m=2, K=5):

1 0 0 1 0

1 0 0 1 0

0 0 1 0 0

0

1

At time k-3

At time k-2

At time k-1

:

:

Bits causing ISI not causing ISI at time instant

At time k

1

0 1 0 0 11 0 1

Bit detected at time instant16 states

MLSE-VA receiver structure (cont.)

State trellis diagram

k-2 k-1 kk-3

1Km

Number of states The ”best” state

sequence is estimated by

means of Viterbi algorithm (VA)

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.

Alphabet size

Proakis, Ed.3, Section 10-1-3