Simulation Results

In [8] simulation results are presented that illustrate that the MAP sequence estimator described above, and its suboptimal PSP implementation, performs better than the RLS and LMS based PSP algorithms proposed in [10]. It is also shown that near optimal performance is realized using the list Viterbi PSP algorithm mentioned above, with as low as L = 3 paths kept per trellis node. Here we concentrate on two issues: the accuracy to which symbols can be estimate as influenced by the channel fading rate; and the sensitivity of the Gauss-Markov channel model based MAP estimator on inaccuracies in assumed model parameters.

Here we model the channel as described in Section III and illustrated in Figure 1. A M = 3 tap channel is considered, with zero-mean coefficients (i.e. ${\bf d} = {\bf0}$) which are uncorrelated across delay and have variances $\sigma_h^2 = 0.01$. For each trial, a n=100 symbol transmission is simulated, with ${\bf h}_0$generated using the assumed ${\bf h}$ statistics noted above. Figure 2 shows achieved bit-error-rates (BER's) for three values of $\alpha$: 0.98, 0.99, and 0.995. Performance is probably unexceptable for $\alpha < 0.97$, and as expected performance improves for increasing $\alpha$ (i.e. reduced fading rate). Eventually, as SNR increases, performance improvement becomes negligible, corresponding to a channel-uncertainty limited mode of operation. Even for very fast fading channels, very good performance can be achieved if the channel is known. However, when the channel is unknown, temporal correlation is an important performance factor. The larger the temporal correlation (i.e. the closer $\vert \alpha \vert$ is to one for a Gauss-Markov channel), the better the sequence of symbols can be estimated.

  

Figure 2: Bit-Error-Rate (BER) vs. SNR for several values of the Gauss-Markov memory factor $\alpha$.

Next we consider the sensitivity of the Gauss-Markov channel model based MAP sequence estimator to assumed model parameters. For Figures 3-5, n=10 symbols were used, and the optimum Gauss-Markov model results were obtained by exhaustive search. This allows us to accurately measure the model parameter sensitivity without affects which may be caused by the sub-optimum PSP approach.

Figures 3 and 4 illustrate the sensitivity to the assumed value of $\alpha$ for two different values of true $\alpha$. For model $\alpha$ = 0, the Gauss-Markov channel model degenerates to a temporally-independent Gaussian model (see [8]). Although there is significant performance variation over the range $0 \leq \alpha \leq 1$, over the range of useful $\alpha$(say $0.975 \leq \vert \alpha \vert < 1$), performance does not vary significantly.

  

Figure 3: BER vs. model $\alpha$. True $\alpha$ = 0.975

  

Figure 4: BER vs. model $\alpha$. True $\alpha$ = 0.99

For Figure 5, the true value of $\alpha$ was used in the Gauss-Markov model, but the assumed value of ${\bf h}_0$ was randomly perturbed by adding a zero-mean Gaussian component. For variance of the perturbation less than .001, performance is unaffected. As the variance increases, performance degrades and approaches the Gaussian model. This illustrates that the algorithm is not overly sensitive to errors in the assumed value of ${\bf h}_0$. Thus, using an ${\bf h}_0$ estimated from a previous data block is possible.

  

Figure 5: BER vs. model h₀ variance. $\alpha$ = 0.975