A generalized maximum correntropy criterion based robust sparse adaptive room equalization

Applied Acoustics 158 (2020) 107036

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A generalized maximum correntropy criterion based robust sparse adaptive room equalization Krishna Kumar, Nithin V. George ⇑ Department of Electrical Engineering, Indian Institute of Technology Gandhinagar, Gujarat 382355, India

a r t i c l e

i n f o

Article history: Received 2 August 2019 Received in revised form 28 August 2019 Accepted 6 September 2019

Keywords: Room equalization Adaptive filter Correntropy criterion Filtered-x least mean square algorithm Acoustic path

a b s t r a c t An adaptive room equalization scheme is usually employed to compensate for the distortion of sound produced by the room impulse response, thereby offering an improved listening experience. In a conventional adaptive room equalizer, an adaptive filter updated using a filtered-x least mean square (FxLMS) algorithm is used to achieve room equalization. Conventional FxLMS algorithm based room equalizers are not robust to strong disturbances picked up by the reference microphone. A robust adaptive room equalization scheme based on a generalized maximum correntropy criteria has been developed in this paper. The performance has been further enhanced by using a proportionate learning strategy to take advantage of the sparse nature of the room impulse response. The proposed algorithm has been shown to provide enhanced room equalization performance over other methods compared, for various types of noise distributions. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction The listening experience in a room is largely dependent on the acoustical properties of the room, which is quantified using the room impulse response (RIR). Adaptive room equalization is a signal processing approach to compensate for this distortion caused by RIR. In an adaptive room equalization scheme, an adaptive finite impulse response (FIR) filter is placed before the loudspeaker, the weights of which are updated in such a way as to minimize the difference between the sound at the receiver (measured using a reference microphone placed near the listener) and the desired sound signal at the receiver. This update is usually carried out using a filtered-x least mean square (FxLMS) algorithm. A simplified schematic diagram of an adaptive room equalization scheme is shown in Fig. 1 [1–7]. As discussed above, in a conventional adaptive room equalization method, an FxLMS algorithm is used for updating the adaptive weights of the room equalizer. It has been reported that FxLMS algorithm is not robust to strong disturbances in the error signal and can diverge in some cases [8]. In a room equalization scenario, this strong disturbances are usually picked up by the reference microphone and is uncorrelated with the input signal applied to the equalizer. A few robust adaptive filtering schemes have been ⇑ Corresponding author. E-mail addresses: [email protected] (K. Kumar), [email protected] (N.V. George). https://doi.org/10.1016/j.apacoust.2019.107036 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.

reported in literature, which uses a robust norm in the design of an adaptive filter. A robust room equalizer which uses a Lorentzian error norm has been reported in [9]. Least mean fourth (LMF) algorithm, which use the fourth power of error as the cost function have also been shown in literature to offer a robust behaviour [10]. Information theoretic adaptive filtering approaches have recently gained significant attention. One of the most popular among them is the class of adaptive filters based on maximum correntropy criterion (MCC) [8,11]. In an MCC based adaptive filter, the weights of the adaptive filter are updated in such a way as to maximize the similarity between the adaptive filter output and the desired signal. MCC based adaptive filters have been shown in literature to provide robust behaviour, when applied for system identification as well a non-linear active noise control [8]. A generalized MCC (GMCC) based adaptive filtering scheme has been recently proposed and has been shown to provide improved performance in comparison with conventional MCC based adaptive filters when applied for de-noising EEG signals [12–14]. A few modified versions of GMCC algorithms have also been proposed in the literature [15,16]. A group constrained GMCC algorithm has been proposed in [17] and has been successfully applied for sparse channel estimation. The computational complexity of a GMCC based robust adaptive filtering scheme has been studied in detail in [12] and it been shown that the computational complexity of GMCC is similar to that of a least mean p-power (LMP) algorithm, which is a generalization of the LMF algorithm.

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The equalizer is usually a N order adaptive finite impulse response (FIR) filter, with the filter coefficients given by

wðnÞ ¼ ½w1 ðnÞ; w2 ðnÞ; . . . ; wN T ;

ð2Þ

which is updated to minimize the cost function given by nðnÞ ¼ E½e2 ðnÞ  e2 ðnÞ, where E½ is the expectation operator. The output of the adaptive equalizer is given by Fig. 1. Schematic diagram of an adaptive room equalization scheme.

yðnÞ ¼ xT ðnÞwðnÞ;

In most cases, the impulse response of an acoustic path is sparse in nature, with only a very few non-zero coefficients [18]. Same is the case for room impulse responses and the sparsity is partly dependent on the sound absorption properties of the room. The room equalization performance can be further improved, if the adaptive room equalization algorithm exploits the sparse nature of the impulse response. A sparse room equalizer, which uses a proportionate learning strategy has been proposed in [19] and has been shown to offer improved room equalization under sparse impulse response conditions. In an attempt to develop an adaptive room equalizer which offers a robust equalization and takes advantage of the sparse room impulse response, this paper proposes a generalized correntropy criteria based sparse room equalizer. The proposed equalization method is expected to offer improved room equalization performance over other algorithms for different types of noise distributions. The rest of the paper is organized as follows: The proposed adaptive room equalizer is introduced in Section 2 and the update rule is derived. A detailed simulation study has been carried out in Section 3 under different types of measurement noises and a comparison of room equalization performance has been made. The concluding remarks are made in Section 4. 2. Proposed adaptive room equalizer As discussed in the previous section, an adaptive room equalizer tries to compensate for the deterioration in audio quality caused by the acoustic path between loudspeaker and the listener. In its basic form, an adaptive room equalization system consists of an adaptive filter which is added in between the sound source and the loudspeaker. A basic block diagram of a simple room equalizer is shown in Fig. 2. In the figure, xðnÞ is the desired audio signal, yðnÞ is the output of the equalizer, WðzÞ is the transfer function of the equalizer, xðn  DÞ is the desired audio signal delayed by D samples, gðnÞ is the microphone output,

eðnÞ ¼ xðn  DÞ  gðnÞ;

ð1Þ

is the error signal, which is the difference between the delayed version of the desired signal and the microphone output.

ð3Þ

where xðnÞ ¼ ½xðnÞ; xðn  1Þ; . . . ; xðn  N þ 1ÞT is the tap delayed input signal vector. The weight update rule is given by

wðn þ 1Þ ¼ wðnÞ þ 2:l:eðnÞ:x0 ðnÞ;

ð4Þ 0

where l is the step size parameter and x ðnÞ is xðnÞ filtered through a model of the acoustic path. The update rule given in (4) is the filtered-x least mean square (FxLMS) algorithm. The conventional FxLMS algorithm based room equalizer can diverge in the presence of strong error signals. An attempt has been made in this section to design a room equalizer which is robust against strong disturbances sensed by the microphone. The robustness of FxLMS algorithm could be enhanced by using an information theoretic learning approach, which tries to maximize the correntropy between the desired signal and the equalized signal. The cost function considered in this work is the generalized correntropy loss function given by

nðnÞ ¼ c  E½c: expðkjeðnÞja Þ;

ð5Þ

where c is a normalization constant and k ¼

1 ba

is the kernel param-

eter with a is the shape parameter and b representing the scale parameter. Considering an approximation of the cost function given in (5), using a gradient descent approach, we can write the update rule for the proposed GMCC based adaptive room equalizer as

wðn þ 1Þ ¼ wðnÞ  lw

@nðnÞ ; @wðnÞ

ð6Þ

where lw is the step size parameter for the weight update. From (5), we can write

@nðnÞ @½expðkjeðnÞja Þ  @wðnÞ @wðnÞ

ð7Þ

Estimating (7) and substituting in (6), we obtain the update rule given by

wðn þ 1Þ ¼ wðnÞ þ lw ka½expðkjeðnÞja ÞjeðnÞja1 sgn½eðnÞx0 ðnÞ; ð8Þ which is hereafter referred to as the filtered-x GMCC (FxGMCC) algorithm. The performance of the proposed FxGMCC algorithm is dependent on the appropriate selection of the shape parameter a. For a ¼ 2, the algorithm given in (8) becomes the filtered-x maximum correntropy criterion (FxMCC) algorithm. In order to further improve the performance of the proposed approach for sparse room impulse responses, this paper also proposes a sparse learning scheme for FxGMCC algorithm. A proportionate learning strategy [20] has been incorporated into FxGMCC to obtain the following update rule, which is hereafter referred to as the filtered-x improved proportionate GMCC (FxIPGMCC) algorithm. The update rule for Fx-IPGMCC algorithm is given by

wðn þ 1Þ ¼ wðnÞ þ

gKðnÞx0 ðnÞ½expðkjeðnÞja ÞjeðnÞja1 sgn½eðnÞ x0 T ðnÞKðnÞx0 ðnÞ þ dP ð9Þ

Fig. 2. Schematic diagram of a filtered-x adaptive room equalization algorithm.

where

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K. Kumar, N.V. George / Applied Acoustics 158 (2020) 107036

KðnÞ ¼ diag½j0 ðnÞ; j1 ðnÞ; . . . ; jN1 ðnÞ:

ð10Þ

1

with

0.9

ð1  fÞ ð1 þ fÞjwi ðnÞj ji ðnÞ ¼ þ 2N 2jjwðnÞjj1 þ w

ð11Þ

0.8 0.7

dP ¼

ð1  fÞ  dL : 2N

ð12Þ

In the above update rule, dP is a regularization factor with 1 6 f < 1 as a small constant and dL is the regularization factor of a normalized LMS (NLMS) algorithm [20]. In (11), w is a small positive number.

Magnitude

and

0.6 0.5 0.4 0.3 0.2

3. Simulation study

0.1

An attempt has been made in this section to evaluate the performance of the proposed Fx-IPGMCC algorithm. We have compared the robust behaviour of the proposed approach with that of filtered-x NLMS (Fx-NLMS) [19] and filtered-x improved proportionate NLMS (Fx-IPNLMS) algorithms. [3] In addition, we have compared the results with filtered-x improved proportionate normalized least mean fourth (FxIPNLMF) algorithm, which is a modified version of the LMF algorithm, developed in this paper for room equalization under sparse impulse response conditions. The update rule for the FxIPNLMF algorithm is given by

0

500

1000

1500

2000

2500

3000

3500

4000

Samples Fig. 3. Room impulse response considered in all the cases.

40

30

= 0.2 10 -4 )

Fx-IPGMCC ( = 1,

= 0.1,

Fx-IPGMCC ( = 2,

= 0.03,

Fx-IPGMCC ( = 4,

= 0.5,

= 5 10 )

Fx-IPGMCC ( = 6,

= 0.8,

= 8 10 )

-3

= 6 10 ) -2 -2

20

3

gKðnÞx ðnÞ½eðnÞ x0 T ðnÞKðnÞx0 ðnÞ þ dP

ð13Þ

where the variables are similar to that in (9). The ensemble mean square error (EMSE) has been used as the metric for comparison and all simulations were carried out in a MATLAB environment. The input signal used in all the experiments follows uniform distribution, with the magnitude in the range ½0:5; 0:5 and we have used a sampling frequency of 8 kHz. A strong disturbance has been added after 20,000 and 40,000 samples in all the cases to check the robust behaviour. We have considered four different scenarios in this study. In all the cases, the room impulse response was generated using the method given in [21,22], with a room size of 15 m  10 m  5 m, source location of ½7; 8; 1 m and microphone location of ½1; 5; 1:6 m. The number of virtual sources has been taken as 8, the reflection coefficients for all surfaces is 0.3 and the equalizer used is an adaptive FIR filter 6

of length 400. The normalization factor is taken as 10 and delay length of 200. We have also applied a smoothing filter of length 10. Other simulation parameters have been selected in such a way as to offer similar initial convergence for all the algorithms compared. 3.1. Case 1 The room impulse response considered in this case is shown in Fig. 3. A white Gaussian measurement noise with a signal to noise ratio of 40 dB has been considered in this simulation. As discussed earlier, the performance of GMCC based algorithms are largely dependent on the appropriate selection of the shape parameter a. In the first part of this simulation, we analyze the effect of shape parameter on the room equalization performance. Fig. 4 shows the convergence behaviour of the proposed Fx-IPGMCC algorithm for a ¼ 1; 2; 4; 6. As can be seen from the results, for this case a ¼ 2 provides the best performance. It may be noted that all the EMSE curves have been obtained as an average of 100 independent trials. In the second part of this simulation, room equalization performance offered by the proposed GMCC algorithm (for a ¼ 2) has been compared with that provided by other algorithms. Fig. 5 shows the noise as well as the strong disturbance considered in

EMSE (dB)

wðn þ 1Þ ¼ wðnÞ þ

0

0

10

0

-10

-20

-30

-40 0

0.5

1

1.5

2

2.5

3

Samples

3.5

4

4.5

5 104

Fig. 4. Case 1: Comparison of convergence characteristics of Fx-IPGMCC for different values of a, for a Gaussian noise distribution.

this case. The improved room equalization performance offered by the proposed method can be seen from the convergence behaviour shown in Fig. 6. The average EMSE, obtained as an average of 2000 samples after the strong disturbances are shown in Tables 1 and 2. A significant improvement in equalization performance can be seen from the results. The other simulation parameters used are:

Fx-NLMS

(g ¼ 0:185),

Fx-IPNLMS

(f ¼ 0:5; w ¼ 4  104 ;

g ¼ 0:0035), Fx-IPNLMF (f ¼ 0:5; w ¼ 4  104 ; g ¼ 1  107 ), and Fx-IPGMCC (f ¼ 0:5; w ¼ 4  104 ; a ¼ 2; k ¼ 0:03; g ¼ 6  103 ). 3.2. Case 2 In this case, we have considered a room impulse response which is same as that used in Case 1. A background noise with a binary probability distribution has been assumed with a lower bound of 0:003 and an upper bound of 0:003. Similar to Case 1, an initial study was carried out to estimate the effect of a on the room equalization performance. As can be seen in Fig. 7, a ¼ 6 offers the best convergence behaviour and thus for the rest of this

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K. Kumar, N.V. George / Applied Acoustics 158 (2020) 107036 120

40 30

100 20

= 0.1,

= 0.2 10 -4 )

Fx-IPGMCC ( = 2,

= 0.1,

= 3.4 10 )

Fx-IPGMCC ( = 4,

= 0.001,

Fx-IPGMCC ( = 6,

= 0.8,

-5

-3

= 0.9 10 ) -2

= 8 10 )

10

MSE (dB)

Magnitude

80

Fx-IPGMCC ( = 1,

60

40

0 -10 -20 -30

20

-40 0 -50 -60

-20 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

5

0.5

1

1.5

2

104

Samples Fig. 5. Strong disturbance signal added in Case 1.

2.5

3

3.5

4

4.5

5 104

Samples

Fig. 7. Case 2: Comparison of convergence characteristics of Fx-IPGMCC for different values of a, for a binary noise distribution.

40 Fx-NLMS ( = 0.185) -3

Fx-IPNLMS ( = 3.5 10 )

30

40

-6

Fx-IPNLMF ( = 0.1 10 ) Fx-IPGMCC ( = 2,

= 0.03,

Fx-NLMS ( = 0.185)

-3

= 6 10 )

-3

Fx-IPNLMS ( = 3.5 10 )

20

30

10

20

0

10

-6

Fx-IPNLMF ( = 0.1 10 )

EMSE (dB)

EMSE (dB)

Fx-IPGMCC ( = 6,

-10

-2

= 8 10 )

0

-20

-10

-30

-20

-40

= 0.8,

-30 0

0.5

1

1.5

2

2.5

3

3.5

Samples

4

4.5

5 10

4

-40

Fig. 6. Case 1: Comparison of convergence characteristics for Gaussian noise distribution.

Table 1 Comparison of EMSE (in dB) for different noise distributions, obtained as an average of EMSE from 20,001 to 22,000 samples. Noise Distribution

Gaussian Binary Laplacian Uniform

Algorithm Fx-NLMS

Fx-IPNLMS

Fx-IPNLMF

Fx-IPGMCC

4.5217 4.5178 4.0227 4.5321

18.4472 18.4838 17.6639 18.4345

10.6143 10.6195 10.5919 10.6054

21.7132 19.8024 21.3404 21.5686

0

0.5

1

1.5

2

2.5

Samples

3

3.5

4

4.5

5 104

Fig. 8. Case 2: Comparison of convergence characteristics for binary noise distribution.

case, we have considered Fx-IPGMCC algorithm with a ¼ 6. The comparison of convergence characteristics across the different algorithms is shown in Fig. 8. The average EMSE obtained after strong disturbances are indicated in Table 1 and Table 2. FxIPGMCC algorithm can be seen to provide best room equalization performance in comparison with other methods. The other simulation parameters used are: Fx-NLMS (g ¼ 0:185), Fx-IPNLMS (f ¼ 0:5; w ¼ 4  104 ; g ¼ 0:0035), Fx-IPNLMF (f ¼ 0:5;w ¼ 4  104 ;

Table 2 Comparison of EMSE (in dB) for different noise distributions, obtained as an average of EMSE from 40,001 to 42,000 samples. Noise Distribution

Gaussian Binary Laplacian Uniform

Algorithm

g ¼ 1  107 ), and k ¼ 0:8; g ¼ 0:08).

Fx-IPGMCC

(f ¼ 0:5; w ¼ 4  104 ; a ¼ 6;

3.3. Case 3 and 4

Fx-NLMS

Fx-IPNLMS

Fx-IPNLMF

Fx-IPGMCC

4.0830 4.5767 4.5997 4.6195

19.1099 19.2279 18.5781 19.2204

10.6093 10.6053 10.5799 10.5986

23.0204 20.3777 22.7769 22.8943

Two other room equalization scenarios are considered in this portion of the simulation study. In the first case (Case 3), we consider a background noise with a Laplacian distribution and in the second case (Case 4), the distribution of the background noise is

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K. Kumar, N.V. George / Applied Acoustics 158 (2020) 107036 40

40 Fx-IPGMCC ( = 1,

30

Fx-NLMS ( = 0.185)

= 9 10 )

Fx-IPNLMS ( = 5.5 10 -3 )

-3

Fx-IPGMCC ( = 2,

= 0.1,

Fx-IPGMCC ( = 4,

= 0.03,

Fx-IPGMCC ( = 6,

= 6 10 )

= 0.01,

30

-3

= 2.3 10 )

Fx-IPGMCC ( = 1,

= 9 10 )

20

20

10

10

0

-10

-20

-20

-30

-30

-40 0.5

1

1.5

2

2.5

3

3.5

4

4.5

Fig. 9. Case 3: Comparison of convergence characteristics of Fx-IPGMCC for different values of a, for a Laplacian noise distribution.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 104

Samples

Fig. 11. Case 3: Comparison of convergence characteristics for Laplacian noise distribution.

40

40

30

Fx-IPGMCC ( = 1,

= 0.1,

= 0.2 10 -3 )

Fx-IPGMCC ( = 2,

= 0.1,

= 6 10 )

Fx-IPGMCC ( = 4, Fx-IPGMCC ( = 6,

= 0.03, = 2.3 10 ) = 0.6, = 0.1)

Fx-NLMS ( = 0.185) -3

-3

Fx-IPNLMS ( = 3.5 10 )

30

-3

-6

Fx-IPNLMF ( = 0.1 10 ) Fx-IPGMCC ( = 2,

20

20

10

10

EMSE (dB)

EMSE (dB)

-4

= 9 10 )

-40

5 104

Samples

= 0.1,

0

-10

0

-6

Fx-IPNLMF ( = 0.1 10 )

-4

EMSE (dB)

EMSE (dB)

-4

= 0.1,

0

= 0.1,

-3

= 6 10 )

0

-10

-10

-20

-20

-30

-30

-40

-40 0

0.5

1

1.5

2

2.5

3

Samples

3.5

4

4.5

5

0

0.5

1

1.5

104

2

2.5

3

Samples

3.5

4

4.5

5 104

Fig. 10. Case 4: Comparison of convergence characteristics of Fx-IPGMCC for different values of a, for an uniform noise distribution.

Fig. 12. Case 4: Comparison of convergence characteristics for uniform noise distribution.

uniform in nature. The noise signal with a Laplacian distribution is given by

compared. This is also evident from the comparison of EMSE shown in Tables 1 and 2. The other simulation parameters used are: Fx-

/ ¼ BL þ n:ðBU  BL Þ

ð14Þ

NLMS (g ¼ 0:185), Fx-IPNLMS (f ¼ 0:5; w ¼ 4  104 ; g ¼ 0:0055), Fx-IPNLMF (f ¼ 0:5; w ¼ 4  104 ; g ¼ 1  107 ), and Fx-IPGMCC

where BL is the lower bound, BU is the upper bound and (14) with

(f ¼ 0:5; w ¼ 4  104 ; a ¼ 1; k ¼ 0:1; g ¼ 0:9  103 ) for Case 3 and

1 n ¼ pffiffiffi :sgnð0:5  bÞ: log½2: minðb; 1  bÞ; 2

Fx-NLMS

ð15Þ

with b denoting a random number with magnitude uniformly distributed in the range ½0; 1. The lower and upper bound used are same as that in Case 1. Similar to the previous cases, we can see from the comparison of convergence behaviour for various values of a shown in Fig. 9 and Fig. 10, that the best convergence characteristics is offered by a ¼ 1 for Case 3 and a ¼ 6 for Case 4. Thus, for the rest of the simulations in this case, we have used a ¼ 1 (Case 3) and a ¼ 2 (Case 4). From the comparison of room equalization performance shown in Figs. 11 and 12, we can observe that the proposed Fx-IPGMCC algorithm performs better than other algorithms

(g ¼ 0:185),

(f ¼ 0:5; w ¼ 4  104 ; g ¼

Fx-IPNLMS 4

0:0035),Fx-IPNLMF (f ¼ 0:5; w ¼ 4  10 ; g ¼ 0:1  106 ), and FxIPGMCC (f ¼ 0:5; w ¼ 4  104 ; a ¼ 2; k ¼ 0:1; g ¼ 6  103 ) for Case 4 respectively.

4. Conclusions A robust room equalization method based on an information theoretic learning approach has been developed in this paper. The proposed scheme also takes advantage of the sparse nature of the room impulse response to further enhance the room equal-

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K. Kumar, N.V. George / Applied Acoustics 158 (2020) 107036

ization performance. A generalized correntropy based learning approach, which also incorporates proportionate weight update method has been developed. The improved room equalization offered by the proposed algorithm under different types of noise distributions has been shown through a simulation study. Role of the funding source This work is supported by the Department of Science and Technology, Government of India under the Core Grant Scheme (CRG/2018/002919) and TEOCO Chair of Indian Institute of Technology Gandhinagar. References [1] Fielder LD. Analysis of traditional and reverberation-reducing methods of room equalization. J Audio Eng Soc 2003;51(1/2):3–26. [2] Schneider M, Kellermann W. Adaptive listening room equalization using a scalable filtering structure in the wave domain. In: 2012 IEEE international conference on acoustics, speech and signal processing (ICASSP). IEEE; 2012. p. 13–6. [3] Fuster L, de Diego M, Ferrer M, Gonzalez A, Pinero G. A biased multichannel adaptive algorithm for room equalization. In: 2012 proceedings of the 20th european signal processing conference (EUSIPCO). IEEE; 2012. p. 1344–8. [4] Cecchi S, Romoli L, Carini A, Piazza F. A multichannel and multiple position adaptive room response equalizer in warped domain: real-time implementation and performance evaluation. Appl Acoust 2014;82:28–37. [5] Fuster L, de Diego M, Azpicueta-Ruiz LA, Ferrer M. Adaptive filtered-x algorithms for room equalization based on block-based combination schemes. IEEE/ACM Trans Audio Speech Language Process 2016;24 (10):1732–45. [6] Cecchi S, Carini A, Spors S. Room response equalization a review. Appl Sci 2018;8(1):16. [7] Bergner J, Preihs S, Hupke R, Peissig J. A system for room response equalization of listening areas using parametric peak filters. In: Audio engineering society

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