# New symmetric decorrelating set-membership NLMS adaptive algorithms for blind speech intelligibility enhancement

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## Abstract

This paper addresses the problem of speech enhancement and acoustic noise reduction by partial and set-membership adaptive algorithms combined with the symmetric decorrelating adaptive (SAD) algorithm structure. In this paper, we propose two new adaptive algorithms based on set-membership principle that improve the original set-membership algorithm behavior in speech enhancement applications. The first proposed algorithm (called Proposed 1), is based on the combination of the SAD algorithm structure with a smart control that uses decorrelating properties between the output and the mixing signal to control and update the SAD adaptive filters. The second proposed algorithm (called Proposed 2) is a modification of Proposed 1 and based on a new regularization relations of the SAD adaptive filters that use a combination between the variance of the mixing and the output signals of the SAD structures. These two proposed algorithms (Proposed 1 and Proposed 2) aim to improve the convergence speed performance and the output signal-to-noise-ratio of the original SAD algorithm when no smart control of the adaptive filters is used. The proposed algorithms have very interesting properties with non-stationary signal like speech when the SAD algorithm is, used alone, fails. The simulation results that are obtained by the comparison between the proposed algorithms (Proposed 1 and Proposed 2) and the original two-channel set-membership NLMS algorithm have shown the best performances of Proposed 1 and Proposed 2 in terms of the following criteria: systems mismatch, segmental SNR and segmental men square error.

## Keywords

Speech enhancement SAD NLMS Set-membership Adaptive algorithm Convergence speed## Abbreviations

- Fs
Sampling frequency

- SNR
Signal to noise ratio

- LMS
Least-mean-squares

- NLMS
Normalized least-mean-squares

- AWGN
Additive white Gaussian noise

- WGN
White Gaussian noise

- USASI
United state of America standard institute

- CSP
Convergence speed performance

- MSE
Mean square error

- BSS
Blind source separation

- CD
Cepstral distance

- FBSS
Forward BSS

- SFT
Short-Fourier-transform

- SegMSE
Segmental mean-square-error

- SegSNR
Segmental signal-to-noise ratio

- SM
System mismatch

- PNLMS
Proportionate NLMS

- SAD
Sysmetric adaptive decorrelating

- SMF
Set-membership filtering

- AP
Affine projection

- SM-AP
Set-membership affine projection

- TC-SM-NLMS
Two-channel set membership NLMS

## List of symbols

- L
Real impulse responses length

*k*Time index

- dB
Decibel

- \(s\left( n \right)\)
Original speech signal

- \(b\left( n \right)\)
Noise

- \(h_{12} \left( n \right)\), \(h_{21} \left( n \right)\)
Cross-coupling impulse responses

- \(p_{1} \left( n \right)\), \(p_{2} \left( n \right)\)
Noisy speech signals

- \(\delta \left( n \right)\)
Dirac impulse

- \(w_{12} \left( n \right)\), \(w_{21} \left( n \right)\)
Adaptive filters

- \(u_{1} \left( n \right)\)
Estimated speech by forward structure

- \(u_{2} \left( n \right)\)
Estimated noise by forward structure

- \(H_{1} \left( k \right)\) and \(H_{2} \left( k \right)\)
Conditions set

- \(\mu_{1}\) and \(\mu_{2}\)
Step-sizes

- \(N\)
Adaptive filter length

- \(\gamma_{1}\) and \(\gamma_{2}\)
Fixed threshold error

- \(SNR1\), \(SNR2\)
Signal to noise ration in the two inputs

- \(w_{12}^{opt} \left( k \right)\), \(w_{21}^{opt} \left( k \right)\)
Optimal filters

- \(\mu_{c1} \left( k \right)\), \(\mu_{c2} \left( k \right)\)
Control step-sizes of Proposed 1

- \(\mu_{v1} \left( k \right)\), \(\mu_{v2} \left( k \right)\)
Control step-sizes of Proposed 2

- \(\beta_{1}\), \(\beta_{2}\)
Smoothing parameters of Proposed 2

## 1 Introduction

Adaptive filtering principle has been largely used in many area of research in the past decades, and several algorithms based on different types of criterion have been proposed in the literature [1, 2]. Several conventional adaptive techniques assume that the problem to be resolved is based on linear form [3, 4], and this assumption, that is not true in practice, reduces the efficiency of the adaptive approach to accomplish and limits the optimal solutions that can be achieved [5, 6].

The most popular adaptive filtering algorithms like the least mean square (LMS) and the normalized LMS (NLMS) algorithms are robust and have a low computational complexity [7, 8]. The adaptive filters updates of LMS and NLMS algorithms are directly controlled by the input vector [9, 10]. This property makes them very limited in terms of performances with a non-stationary signal like speech, but, in the other hand, very convenient for dispersive–impulse response type systems [11, 12, 13]. Several techniques and algorithms based on these principles were proposed to improve the LMS and NLMS algorithms behavior, like convergence speed performance in the transient regime, under similar conditions [14]. In [15, 16], it was proposed the proportionate NLMS (PNLMS) that recovers the convergence speed property of NLMS, nevertheless the PNLMS algorithms are more complexes in comparison with NLMS. The problem’s complexity of PNLMS algorithms were resolved by the set-membership filtering (SMF) family [17, 18]. In set-membership algorithm family, the output filtering control parameter is limited by a predetermined threshold [19]. Available methods based on set-membership (SM) approaches exist for linear and non linear models [20, 21, 22]. In the set-membership context, the estimated unknown system encloses the set of solutions or a union of disconnected sub-sets of solutions which are produced by the uncertainty of experimental data and the predefined system error boundaries [23]. However, the set-membership algorithms allow to get fast convergence speed performance as well as low steady-state error. Several techniques of SMF were proposed with low complexity in the literature like the set-membership NLMS (SMNLMS), [24], and the set-membership affine projection (SM-AP) [25] algorithms.

In this paper, we propose two new adaptive algorithms based on set-membership principle that improve the original set-membership algorithm behavior in speech enhancement applications. The first proposed algorithm (called Proposed 1), is based on the combination of the SAD algorithm structure with a smart control that uses decorrelating properties between the output and the mixing signal to control the SAD filters update. The second proposed algorithm (called Proposed 2) is a modification of Proposed 1 and based on a new regularization relations of the SAD adaptive filters. The two proposed algorithms (Proposed 1 and Proposed 2) aim to improve the convergence speed performance and the output signal-to-noise-ratio of the original SAD algorithm without control system.

This paper is organized as follows: In Sect. 2, the mixing model that we have used in simulation is presented. In Sect. 3, we present the proposed algorithms, i.e. Proposed 1 and Proposed 2. In Sect. 4, the simulation part that contains experiments on three algorithm, i.e. the classical or original TC-SM-NLMS, Proposed 1, and Proposed 2. All the experiments are expressed in terms of three objective criteria as SegSNR, SegMSE, and SM. In Sect. 5, we conclude our work.

## 2 Two channel mixing model

## 3 Proposed TC-SM-NLMS algorithms

- (i)
the two channel correlated set membership NLMS (TC-SM-NLMScor) algorithm that we call, in this paper, “Proposed 1”.

- (ii)
and the two channel correlated set membership regularized NLMS (TC-SM-RNLMScor) that we call, “Proposed 2”.

*a posterior*errors condition. Further condition is to check if the adaptive filters estimates of \(\varvec{w}_{21} \left( k \right)\) and \(\varvec{w}_{12} \left( k \right)\) are respectively on the outside conditions sets \(\varvec{H}_{1} \left( k \right)\) and \(\varvec{H}_{2} \left( k \right)\). We can write:

### 3.1 Presentation of Proposed 1

From relation (9), the parameter \(\delta\) is a small constant that is introduced to avoid zero division. We can also see from (9) that the step size \(\mu_{c1} \left( k \right)\) is sensible to the correlation vector of \(\varvec{p}_{1} \left( k \right)\) and \(u_{2} \left( k \right)\), and from relation (10) that the step size \(\mu_{c2} \left( k \right)\) is sensible to the vector correlation of \(\varvec{ p}_{2} \left( k \right)\) and \(u_{1} \left( k \right)\). These two relations means that when the two output signals \(u_{1} \left( k \right)\) and \(u_{2} \left( k \right)\) converge to their optimal solution, \({\text{i}} . {\text{e}} .\, w_{12}^{opt} \left( k \right) = h_{12} \left( k \right)\) and \(w_{21}^{opt} \left( k \right) = h_{21} \left( k \right)\), under the condition that the step-size \(\mu_{c1} \left( k \right)\) takes important values (close to 1) when speech is absent and small values (close to 0) in the other case (speech presence), in such condition, the FBSS structure combined with our proposed algorithm (Proposed 1) provides a recovered speech signal without noise at the output \(u_{1} \left( k \right)\). The same conclusions can be generated for the step size \(\mu_{c2} \left( k \right)\), but as we are only interested in the output \(u_{1} \left( k \right)\), we will focus only on this last output.

Proposed 1

### 3.2 Presentation of Proposed 2

Proposed 2

## 4 Simulation results of the proposed algorithms (Proposed 1 and 2)

Simulation parameters of TC-SM-NLMS and Proposed algorithms (i.e. Proposed 1 and 2)

TC-SM-NLMS [2] | Proposed 1 (in this paper) | Proposed 2 (in this paper) | |
---|---|---|---|

Simulation parameters |
\(\mu_{1} = \mu_{2} = 0.2 , 0.4\,and\,0.6\)
\(N = 64 , 128\,and\,256\)
\(\gamma_{1} = \gamma_{2} = 20\) and 80 Input \(SNR1 = 3\) dB Input \(SNR2 = 3\) dB |
\(\mu_{1} = \mu_{2} = 0.2 , 0.4\,and\,0.6\)
\(N = 64 , 128\,and\,256\)
\(\gamma_{1} = \gamma_{2} = 20\) and 80 Input \(SNR1 = 3\,{\text{dB}}\) Input \(SNR2 = 3\) dB |
\(\mu_{1} = \mu_{2} = 0.2 , 0.4\,and\,0.6\)
\(N = 64 ,128\,and\,256\)
\(\gamma_{1} = \gamma_{2} = 20\) and 80 Input \(SNR1 = 3\) dB Input \(SNR2 = 3\) dB \(\beta_{1} = \beta_{2} = 0.5\) |

In order to evaluate the convergence speed performance of the proposed algorithms (Proposed 1 and Proposed 2), we have used 3 objective criteria [30].

*P*is the number of only-speech periods.

### 4.1 System mismatch (SM) criterion evaluation

*N*= 64, 128 and 256. In Fig. 6, the original speech and the white Gaussian noise (WGN) are used as the input of the system. In Fig. 7, we use the same speech signal with USASI noise as a second source. According to Figs. 6 and 7, we observe that the three algorithms converge toward the optimal solution with different noise types and adaptive filter lengths. We clearly show also the fast convergence of the two proposed algorithms in comparison with the classical TC-SM-NLMS. These experiments prove that the Proposed 1 is more efficient than the classical TC-SM-NLMS algorithms in terms of convergence speed to the optimal solution, and the Proposed 2 behaves more faster than the two other ones. This means that the proposed modification that incorporate the correlation parameters in the variance normalization has improved the behavior of Proposed 2.

This good performance still the same even with important adaptive filter lengths. The good behavior of Proposed 2 is managed by the new formulas of the step-size that accelerates the algorithm when the input signal variance is low (in the speech absence periods) and to slow down it when the variance is high (in the presence of speech plus noise), this smart automatic mechanism lead to an ANC system with only noise at the reference signal and this is exactly why Proposed 2 has the best performance.

### 4.2 Segmental Mean Square Error (SegMSE) criterion evaluation

*N*= 64, 128 and 256. We have done two experiments according to the source signals. First, we use the speech signal shown in Fig. 5 and WGN as the source signal and noise, respectively, in the mixing model. The obtained results for different adaptive filters length are presented in Fig. 8. In the second experiment, the noise source is USASI, and the results are reported on Fig. 9. We confirm again that the output speech signal obtained by Proposed 2 is more intelligible than the other results.

Basing on Figs. 8 and 9, we confirm that the three algorithms converge to the optimal solution. However, we see clearly the better performance of Proposed 1 and 2 in comparison with the TC-SM-NLMS algorithms in term of convergence speed. Also, we confirm that the introduced modification on Proposed 2 algorithm has improved the MSE values of Proposed 1, which means that Proposed 2 algorithm is more efficient than the other algorithms and allows to quickly converge to the smallest MSE values in the permanent regime. This allows low distortion of the output speech signal (more Intelligible speech signal) and more noise reduction at the output result.

### 4.3 Segmental signal to noise ratio (SegSNR) criterion evaluation

From the obtained results, we clearly show the supremacy of both proposed algorithms (Proposed 1 and 2) in the transient phase in comparison with the classical partial TC-SM-NLMS algorithm. Also, we have observed the best performance of the second proposed (Proposed 2) algorithm in all cases (N = 64, 128 and 256 and with both types noises, i.e. white Gaussian in Fig. 10 or USASI in Fig. 11). Also, we confirm that the processing speech signal obtained by Proposed 2 is more intelligible than the other results obtained with other algorithms.

### 4.4 Performance quantification of the proposed algorithms

According to the obtained results of Figs. 12 and 13, we can easily see the best performance of Proposed 2 in comparison with the other algorithms. The same remark is noted when we use a speech signal or USASI noise as an input. We have also noted that the threshold parameters must be experimentally selected and when this items is selected around 50, the convergence speed performance of Proposed 2 get its good values in both experiments.

## 5 Conclusion

In this paper, two new algorithms based set-membership principle are proposed. Both of them combine the advantages of the SAD algorithm and regularization techniques based on the cross-correlation between the output signals of the SAD structure and the mixing ones. Three criteria were used to evaluate the performances of the proposed algorithms in comparison with the original TC-SM-SAD algorithm, i.e. The system mismatch (SM), the output SegSNR, and SegMSE. All the criteria have shown the good behavior of the proposed algorithms (Proposed 1 and Proposed 2) even when non-stationary signals like speech are used in the input. Intensives experiments were done on both algorithms and show, in terms of these criteria, the efficiency of Proposed 2 in comparison with Proposed 1 and the original TC-SM-SAD algorithms. The proposed combination between the variance of the output signals and the mixing signal and its used as a regularization factor have shown efficiency in improving the behavior of Proposed 2 algorithm. In conclusion, we can say that we get a less distortion at the output speech signal when small values of the TC-SM-SAD algorithm step-sizes are chosen, nevertheless, the convergence speed will be degraded. The new Proposed 1 and Proposed 2 algorithms allow to improve in the same time the convergence speed performance and the distortion with low step-sizes values. We have also noted that the intelligibility property of the processed speech signal has been improved in all the cases especially with the Proposed 2 algorithm. This is why we recommend the use of these two proposed algorithms in speech enhancement and acoustic noise reduction applications.

## Notes

### Acknowledgements

This study was carried out without funding. Authors are grateful to the Blida University, Algeria, and Professor Abderreak Guessoum, Professor at Blida University, Algeria, and Director of Signal Processing and imaging Laboratory (LATSI) for providing the infrastructure to achieve this work.

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that they have no conflict of interest.

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