As mentioned, we here explore how signal detection performance depends on whitener parameter setting. In particular, recall that one intriguing phenomenon observed earlier is the worse performance with p = 2 than with p = 1 (both at q = K = 1), although the former is associated with a seemingly more natural-looking band chan-nel matrix and a more expanded MLSE trellis. A comprehensive analysis would require examining the distance property of the received signal after the proposed blockwise whitening. However, a crude understanding can be obtained by looking at the signal-to-interference-plus-noise ratio (SINR) after blockwise whitening.
From (4.1) and (4.4), the pre- and post-whitening SINRs are given by, respec-tively,
SINRpre = E[xHmHHmHmxm]/E[zHmzm], (4.9) SINRpost = E[xHmHHmK−1z Hmxm]/E[zHmK−1z zm]. (4.10) For the power of residual ICI plus noise, we have E[zHmzm] = tr(E[zmzHm]) = tr(Kz) and E[zHmK−1z zm] = tr(E[K−1z zmzHm]) = tr(K−1z Kz) = 2q + 1, where tr(A) de-notes the trace of a matrix A. For the signal power, we have E[xHmHHmHmxm] = tr(E[HHmHmxmxHm]) = Es·tr(E[HHmHm]) = Es·tr(E[HmHHm]) and E[xHmHHmK−1z Hmxm] = tr(E[HHmK−1z HmxmxHm]) = Es· tr(E[HHmK−1z Hm]) = Es· tr(K−1z E[HmHHm]), where Es is as defined previously (the average energy of the transmitted signal samples) and we have assumed that the transmitted signal is independent and identically distributed (i.i.d.).
Note that the factor E[HmHHm] appears in the signal power terms of both SNRs. Employing a procedure similar to that for E[cm,Kc∗m+r,K] in Sec. 3.1, we can derive an expression for E[HmHHm] in terms of the channel parameters as in the case of E[cm,Kc∗m+r,K]. However, although such an expression can provide more pre-cise numerical results, an illuminating insight into the SNR impact of the proposed blockwise whitening technique can already be gathered with a very simple
approx-imation to E[HmHHm], and this insight is sufficient for the purpose of the present work. Specifically, in the limit of little ICI, Hm approaches a diagonal matrix of the channel frequency response. In this case, E[HmHHm] ≈ (PL−1
l=0 σl2)I where I denotes an identity matrix and recall that we have assumed a unity channel power gain, i.e., P
Therefore, the more disparate the eigenvalues of Kz are, the greater gain the pro-posed blockwise whitening can offer. If the eigenvalues are all equal, then no gain is attained.
As examples, we consider the previously considered cases 1) {K = 0, q = 1, p = 1}, 2) {K = 1, q = 1, p = 1}, and 3) {K = 1, q = 1, p = 2}, all at infinite SNR. The corresponding Kz matrices are given in (4.7) and (4.8). For case 1), we obtain the eigenvalues 0.2232σc02, 0.8500σc02 , and 1.9268σc02 ; for case 2), 0.0654σc12, 0.6250σc12, and 3.8796σ2c1; and for case 3), 0.1800σc12 , 0.3550σc12, and 2.4650σc12. The result-ing post- to pre-SINR ratios are 2.0588, 8.7052, and 2.9258, respectively. They do correspond monotonically to the performance gains shown in Figs. 4.2 and 4.3.
However, the mathematical relation between SINR and bit error rate (BER) is not straightforward—a point worth remembering when comparing the SINR perfor-mance of different detection methods and different parameter settings.
With the above caveat, we show some SINR performance results at finite SNR values in Fig. 4.6, both to verify the theory derived in this section and to further illustrate the performance of different detection methods. In the case of the proposed method, the theoretical SINR values shown in the figure have been obtained using (4.11), i.e., SINRpost = Es· tr(K−1z )/(2q + 1), whereas in the case of nonwhitening
10 15 20 25 30 35 5
10 15 20 25 30 35 40
E
b/N
0(dB)
S INR (d B )
Simul., proposed method, K=q=p=1 Theory, proposed method, K=q=p=1 Simul., proposed method, K=q=1, p=2 Theory, proposed method, K=q=1, p=2 Simul., nonwhitening MLSE, K=1 Theory, nonwhitening MLSE, K=1
f
d= 500 Hz
f
d= 3500 Hz
Figure 4.6: SINR performance of different methods in the TU6 channel, with N = 128 and Tsa= 714 ns and assuming perfect CSI.
MLSE, the values of “I” in the theoretical SINR are simply given by σc12 , which are calculated using (3.18) with K = 1. We see that, in the case fd = 500 Hz (normalized peak Doppler frequency ≈ 0.046), the theory and the simulation results agree almost exactly, whereas in the case fd = 3500 Hz (normalized peak Doppler frequency ≈ 0.32), the theory consistently underestimates the SINR performance by a fraction of a dB. The latter phenomenon can be understood by the fact that the σ2c0 as given in (3.17) is a progressively looser upper bound to the actual ICI power as the normalized peak Doppler frequency increases [11]. The figure confirms the earlier observation concerning the superiority of the proposed method with K = q = p = 1, especially in high SNR or high Doppler spread.
4.5 Summary of Results
We considered MLSE-type signal detection in ICI with blockwise whitening of the residual ICI plus noise. Simulations showed that the proposed technique could attain a substantially lower ICI-induced error floor than conventional MLSE.
To capitalize on the above high correlation to improve signal reception over fast varying channels, we consider performing simple blockwise whitening of the residual I+N before signal detection , where the whitener makes use of the ICI characteristics as found. SINR numerical results also show that substantial gains can be achieved with whitening residual ICI plus noise .
Chapter 5
Low Complexity Detection with Whitening of Residual ICI Plus Noise
The ICI is known to be colored [6,24]. Hence it is possible to reduce the error floor by whitening the residual “I+N” (i.e., sum of residual ICI and additive channel noise).
But this would require knowing its autocorrelation function, which remained for a while an unsolved problem. Without knowing the autocorrelation function, one can only resort to less sophisticated techniques, such as simple differencing of the received signals at neighboring subcarriers [8]. Recently, we have obtained a char-acterization of the autocorrelation of the ICI [25]. It is shown that the normalized autocorrelation of the residual ICI is not only high, but also insensitive to a vari-ety of system parameters and channel conditions including the sampling period Tsa, the DFT size N, the signal bandwidth, the average transmitted symbol energy Es, the peak Doppler frequency fd, and the channel power-delay profile (PDP). This is confirmed in [25] by simulation. As a result, the residual I+N can be whitened in a nearly channel-independent manner and, using MLSE as a demonstrator, we have shown that such whitening can facilitate significantly improved signal detection performance.
A main concern with MLSE is its complexity, especially with higher-order mod-ulations. For reduced complexity, in this chapter we consider performing LMMSE and iterative LMMSE detection in association with the above mentioned whitening, together with soft decision feedback. Our LMMSE detector follows a similar princi-ple as that proposed in [22,23], but contains modifications for improved performance.
Simulations show that a good tradeoff between complexity and performance can be achieved.
In what follows, Sec. 5.1 presents the proposed detection method and Sec. 5.2 some simulation results. Finally, Sec. 5.3 gives the summary.