1、Equalization in a wideband TDMA system,Three basic equalization methodsLinear 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 f
2、requency 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
3、 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
4、tails of previously received symbols.,Decision feedback equalizer structure:,Feed-forward filter (FFF),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 (ML
5、SE-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 trans
6、ition between states,Linear equalization, zero-forcing algorithm,Raised cosine spectrum,Transmitted symbol spectrum,Channel frequency response (incl. T & R filters),Equalizer frequency response,=,Basic idea:,Zero-forcing equalizer,Communication channel,Equalizer,FIR filter contains 2N+1 coefficients
7、,Transmitted impulse sequence,Input to decision circuit,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),FIR filter contains 2M+1 coeffici
8、ents,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 :,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 so
9、urce),Equalizer,Channel,+,Estimate of k:th symbol,Input to decision circuit,Error,MSE vs. equalizer coefficients,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 co
10、efficients 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,R = correlation matrix (M x M) of received (sampled) signal values p = vector (of length M) indicating cross-corre
11、lation 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
12、 in matrix form:,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 Lea
13、st-mean-square (LMS) must be used.,where,M samples,Algorithms,Stochastical 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 an
14、d 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 type,T,T,T,LMS algorithm for adjustment of tap coefficients,Transversal FIR filte
15、r with 2M+1 filter taps,Estimate of k:th symbol after symbol decision,Complex-valued tap coefficients of equalizer filter,+,Widrow,Received complex signal samples,Joint optimization of coefficients and phase,Equalizer filter,Coefficient updating,Phase synchronization,+,Minimize:,Godard,Proakis, Ed.3
16、, 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:,equations,Iteration index,Step size of iteration,LMS algorithm (cont.),Aft
17、er some calculation, the recursion equations are obtained in the form,Effect of iteration step size,Slow acquisition,smaller,larger,Poor tracking performance,Poor stability,Large variation around optimum value,Decision feedback equalizer,T,T,T,LMS algorithm for tap coefficient adjustment,T,T,FFF,FBF
18、,+,+,?,The purpose is again to minimize,Decision 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 commonFeedback filter (FBF) is use
19、d 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 sim
20、ilar 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 11-2,Proakis, Ed.3, Section 10-3-1,Channel estimation circui
21、t,T,T,T,LMS algorithm,Estimated symbols,+,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 estim
22、ated 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),Chann
23、el estimation circuit in receiver,Channel estimationcircuit,Equalizer & decision circuit,Estimated channel coefficients,“Clean” output symbols,Received signal samples,Symbol estimates (with errors),Training symbols (no errors),Mandatory for MLSE-VA, optional for DFE,Theoretical ISI cancellation rece
24、iver,Precursor cancellation of future symbols,Postcursor cancellation of previous symbols,Filter 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),
25、matched filter performance can be achieved.,MLSE-VA receiver structure,Matched filter,MLSE (VA),Channel estimation circuit,NW filter,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 c
26、hannel),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
27、 probability,This is allowed since noise samples are uncorrelated due to NW (= noise whitening) filter,Length 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
28、 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 sta
29、te is defined as a certain combination of K-1 previous symbols causing ISI on the k:th symbol.,K-1,0,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 ex
30、ample (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,1,0,1,0,0,1,1,0,1,Bit detected at time instant,16 states,MLSE-VA receiver structure (cont.),State trellis diagram,k-2,k-1,k,k-3,Number of states,The
31、”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,
