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-back filter (FBF)Adjustment of filter coefficientsInputOutput++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 diagramSample time instantsStateAllowed transition between states

Linear equalization, zero-forcing algorithmRaised cosine spectrumTransmitted symbol spectrumChannel frequencyresponse (incl. T & R filters)Equalizer frequency response=Basic idea:

Zero-forcing equalizerCommunication channelEqualizerFIR filter contains 2N+1 coefficientsTransmitted impulse sequenceInput to decision circuitChannel impulse responseEqualizer impulse responseCoefficients 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) FIR filter contains 2M+1 coefficientsOverall channel

Zero-forcing equalizerWe want overall filter response to be non-zero at decision time k = 0 and zero at all other sampling times k 0 :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:(or depending on the source)EqualizerChannel+Estimate of k:th symbolInput to decision circuitError

MSE vs. equalizer coefficientsquadratic multi-dimensional function of equalizer coefficient valuesMMSE 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 solutionR = correlation matrix (M x M) of received (sampled) signal valuesp = 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:

Correlation matrix R & vector pBefore 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. whereM samples

AlgorithmsStochastical information (R and p) is available:1. Direct solution of the Wiener-Hopf equations:

2. Newtons 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). Inverting a large matrix is difficult!

Conventional linear equalizer of LMS typeTTTLMS algorithm for adjustment of tap coefficientsTransversal FIR filter with 2M+1 filter tapsEstimate of k:th symbol after symbol decisionComplex-valued tap coefficients of equalizer filter+WidrowReceived complex signal samples

Joint optimization of coefficients and phaseEqualizer filterCoefficient updatingPhase synchronization+Minimize:GodardProakis, Ed.3, Section 11-5-2

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:equationsIteration indexStep size of iteration

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

Effect of iteration step sizeSlow acquisitionsmallerlargerPoor tracking performancePoor stabilityLarge variation around optimum value

Decision feedback equalizerTTT LMS algorithm for tap coefficient adjustmentTT FFFFBF++?

The purpose is again to minimizeDecision feedback equalizer (cont.)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 intervalwhere

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.)Proakis, Ed.3, Section 11-2Proakis, Ed.3, Section 10-3-1

Channel estimation circuitTTTLMS algorithmEstimated symbols+Proakis, Ed.3, Section 11-3k:th sample of received signalEstimated channel coefficientsFilter length = CIR length

Channel estimation circuit (cont.)1. Acquisition phase Uses training sequenceSymbols 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)

Channel estimation circuit in receiverChannel estimation circuitEqualizer & decision circuitEstimated channel coefficientsClean output symbolsReceived signal samplesSymbol estimates (with errors)Training symbols (no errors)Mandatory for MLSE-VA, optional for DFE

Theoretical ISI cancellation receiverPrecursor cancellation of future symbolsPostcursor cancellation of previous symbolsFilter matched to sampled channel impulse response+(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 structureMatched filterMLSE(VA)Channel estimation circuitNW filterMLSE-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:Metric to be minimized (select best .. using VA)Objective: find symbol sequence that maximizes this probabilityThis is allowed since noise samples are uncorrelated due to NW (= noise whitening) filterLength of f (k)Since we have AWGN

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-10kNote: 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): 10010100100010001At time k-3At time k-2At time k-1::Bits causing ISI not causing ISI at time instantAt time k101001101Bit detected at time instant16 states

MLSE-VA receiver structure (cont.)State trellis diagramk-2k-1kk-3Number of statesThe best state sequence is estimated by means of Viterbi algorithm (VA)k+1Of 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 sizeProakis, Ed.3, Section 10-1-3