To evaluate the performance of the developed scale factor ambiguity corrector, a three- channel system of length L = 256 is generated with coefficients drawn from a standard Gaussian distribution. The equivalent subband systems are obtained using (7.18) and scale factors αk are generated randomly with varying variances and applied to each sub- band channel response to simulate the subband channel estimates produced by BSI algo- rithms for the following synthetic experiments.
In the first experiment, it is assumed that perfect estimates of subband systems are obtained, that is, NPMs = −∞ dB such that BSI error would not distort the result. Scale factors with different variances are then introduced to each subband estimate. The es- timation error for the reconstructed full-band channel responses, denoted as NPMf, is measured before and after employing the developed scale factor ambiguity correction algorithm. Figure 7.4 shows the outcome of this experiment, from which it can be ob- served that, as expected, these scale factors cause the reconstructed full-band channels to be inaccurate with NPMf close to 0 dB. Applying the developed correction method re- solves this issue, resulting in channels with NPMfin the vicinity of−80 dB. Additionally, the variance of the scale factors does not correlate with their effect to the full-band chan- nel reconstruction. This means that even little discrepancies in scale factors for different subbands can significantly degrade the full-band channel estimation.
In the second experiment, performance variation of the developed scale factor cor- rector against various levels of subband BSI error in terms of NPMs is demonstrated, where the estimate of the mth channel for the kth subband is simulated by [167]
bhmk = αk(ILsub×Lsub+ Emk)hmk, ∀m, k (7.28)
for Emk = diag{εmk,0 εmk,1 . . . εmk,Lsub−1}, where the variance of εmk = [εmk,0εmk,1 . . . εmk,Lsub−1]is set according to the desired NPMs. These subband channels are then used to investigate the results obtained using the Simplex algorithm in terms of full-band misalignment NPMfand the variance of the corrected scale factors calculated using (7.27). The results, averaged over 100 independent realizations for αk, are shown in Fig. 7.5. Figure 7.5(a) first shows the NPMf against NPMs before and after scale fac- tor correction and for the “ideal” case where αk is known. Figure 7.5(b) then shows the variation of ξ against NPMs. The following can then deduced from these results: (i) the performance of the scale factor correction algorithm degrades with a decreasing NPMs; (ii) scale factor correction has little effect when NPMs > −10 dB, even if the exact val- ues of αk were known; and (iii) the developed method operates with a high accuracy at NPMs ≤ −50 dB.
7.5.3 Application to Speech Dereverberation
The developed scale factor ambiguity corrector is further applied to speech derever- beration over a two-channel recorded acoustic system extracted from the MARDY database [130], which are down-sampled at 8 kHz and truncated to 2400 FIR coefficients. A sample speech signal is obtained by concatenating both male and female utterance ex- tracted from the APLAWD database [68] and resampled at 8 kHz. The time sequence and spectrogram of this speech sample has been shown in Fig. 6.11. Since the objective is to evaluate the effect of scale factor correction to speech dereverberation in the context of subband processing, a noiseless case is considered.
Similar to Section 7.5.2, various levels of subband BSI error NPMs are simulated. Table. 7.1 shows the corresponding full-band BSI error NPMffor cases with and without scale factor correction as well as for the “ideal” case where αkis known. Consistent with the result shown in Fig. 7.5(a), BSI errors on subband systems overwhelm the scale factor
Table 7.1: Effect of scale factor ambiguity correction on BSI error
NPMs(dB) NPMf(dB)
w/o. correction w. correction perfect correction
−80 −0.2 −49.7 −73.6
−30 −0.17 −1.0 −23.9
−20 −0.14 −0.69 −14.7
−10 −0.1 −0.35 −5.14
ambiguity correction. It can also be seen that for the case with known αk, there is still approximately 6 dB performance loss in terms of NPMf during subband to full-band reconstruction.
Figure 7.6 finally shows the spectrograms of the dereverberated speech signal for cases of NPMs = −30 dB and NPMs = −80 dB shown in Table 7.1. The recovered speech signal is obtained by equalizing the reconstructed full-band channel estimates. Alternatively, subband multichannel equalization [126] can be employed. The perfor- mance improvement brought about by scale factor ambiguity correction is clearly seen by comparing Fig. 7.6(b.1) and Fig. 7.6(c.1) with Fig. 7.6(b.2), Fig. 7.6(b.3) and Fig. 7.6(c.2), Fig. 7.6(c.3). For the case where NPMs = −30 dB, however, there still exist spectral errors in Fig. 7.6(b.2) and Fig. 7.6(b.3). This is due to subband BSI inaccuracies which prop- agated into full-band channel estimate during reconstruction. This is not observed in cases where NPMs = −80 dB. The BSD score computed for each subfigure agrees with such observation.
7.6
Summary
In this chapter, the problem of scale factor ambiguity in subband-based BSI has been addressed. These scale factors are the same for all channels in each subband but differ between subbands, hence distorting the reconstruction of the full-band channels. Hav- ing reviewed the complex subband decomposition over oversampled GDFT filter bank, a novel approach was developed to correct such ambiguity by exploiting the relation-
Frequency (Hz)
(a) reverberant speech (BSD: 0.044)
Time (s) 0 1 2 3 4 5 6 0 2000 4000 (c) NPM s= −80 dB (c.1) w/o. correction (BSD: 0.117) 0 1 2 3 4 5 6 0 2000 4000 (c.2) w. correction (BSD: 1.25×10−4 ) 0 1 2 3 4 5 6 0 2000 4000 Time (s) (c.3) perfect correction (BSD: 1.80×10−5 ) 0 1 2 3 4 5 6 0 2000 4000 Frequency (Hz) (b) NPM s= −30 dB (b.1) w/o. correction (BSD: 0.186) 0 1 2 3 4 5 6 0 2000 4000 Frequency (Hz) (b.2) w. correction (BSD: 0.049) 0 1 2 3 4 5 6 0 2000 4000 Time (s) Frequency (Hz) (b.3) perfect correction (BSD: 9.28×10−3 ) 0 1 2 3 4 5 6 0 2000 4000
Figure 7.6: Spectrograms of (a) the reverberant speech, (b) dereverberated speech for the case ofNPMs = −30 dB, and (c) for the cases of NPMs = −80 dB employing the developed scale
factor ambiguity corrector.
ship between subband and full-band CR error. Utilizing the Simplex algorithm, a set of correction terms was found. Simulation results showed that, although the correction performance degrades with an increasing subband BSI error, it solves the scale factor am- biguity problem accurately when the channel identification is good (NPMs ≤ −50 dB). Further results on speech dereverberation show that large subband BSI error can result in poor full-band BSI performance even though an accurate correction performance is achieved. Nevertheless, applying BSI in the context of subbands has been shown to be promising due to the significant reduction of common zeros.