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doi: 10.3934/jimo.2021131
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Two-level optimization approach with accelerated proximal gradient for objective measures in sparse speech reconstruction

Hai Huyen Dam 1, , Siow Yong Low 2, and  Sven Nordholm 1,

1. 

School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University of Technology, Perth, Australia
2. 

Faculty of Engineering and Physical Sciences, University of Southampton Malaysia (UoSM), Iskandar Puteri, Johor, Malaysia

Received  November 2020 Revised  May 2021 Early access August 2021

Compressive speech enhancement makes use of the sparseness of speech and the non-sparseness of noise in time-frequency representation to perform speech enhancement. However, reconstructing the sparsest output may not necessarily translate to a good enhanced speech signal as speech distortion may be at risk. This paper proposes a two level optimization approach to incorporate objective quality measures in compressive speech enhancement. The proposed method combines the accelerated proximal gradient approach and a global one dimensional optimization method to solve the sparse reconstruction. By incorporating objective quality measures in the optimization process, the reconstructed output is not only sparse but also maintains the highest objective quality score possible. In other words, the sparse speech reconstruction process is now quality sparse speech reconstruction. Experimental results in a compressive speech enhancement consistently show score improvement in objectives measures in different noisy environments compared to the non-optimized method. Additionally, the proposed optimization yields a higher convergence rate with a lower computational complexity compared to the existing methods.

Keywords: Accelerate proximal gradient, sparce speech reconstruction, optimization, objective measures.
Mathematics Subject Classification: Primary: 65K10, 90C26, 92C55; Secondary: 94A12.
Citation: Hai Huyen Dam, Siow Yong Low, Sven Nordholm. Two-level optimization approach with accelerated proximal gradient for objective measures in sparse speech reconstruction. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021131
References:
[1]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problem, SIAM Journal on Imaging Sciences, 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[2]

J. Benesty and Y. Huang, A Perspective on Single-Channel Frequency-Domain Speech Enhancement, San Rafael: Morgan and Claypool Publishers, 2010. doi: 10.2200/S00344ED1V01Y201104SAP008.  Google Scholar

[3]

S. F. Boll, Supression of acoustic noise in speech using spectral subtraction, IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP-27 (1979), 113-120.   Google Scholar

[4]

O. Burdakov, Y. Dai and N. Huang, Stabilized Barzilai-Borwein method, J. Comp. Math., 37 (2019), 916-936.  doi: 10.4208/jcm.1911-m2019-0171.  Google Scholar

[5]

E. J. Candés, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509.  doi: 10.1109/TIT.2005.862083.  Google Scholar

[6]

E. J. Candes and T. Tao, Near-optimal signal recovery from random projections: universal encoding strategies, IEEE Transactions on Information Theory, 52 (2006), 5406-5425.  doi: 10.1109/TIT.2006.885507.  Google Scholar

[7]

E. J. Candes and M. B. Wakin, An introduction to compressive sampling, IEEE Signal Processing Magazine, (2008), 21-30. Google Scholar

[8]

H. H. Dam and A. Cantoni, Interior point method for optimum zero-forcing beamforming with per-antenna power constraints and optimal step size, Signal Processing, 106 (2015), 10-14.  doi: 10.1016/j.sigpro.2014.06.028.  Google Scholar

[9]

H. H. Dam and S. Nordholm, Accelerated gradient with optimal step size for second-order blind signal separation, Multidimens. Syst. Signal Process., 29 (2018), 903-919.  doi: 10.1007/s11045-017-0478-8.  Google Scholar

[10]

T. Esch and P. Vary, Efficient musical noise suppression for speech enhancement system, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing, (2009), 4409-4412. doi: 10.1109/ICASSP.2009.4960607.  Google Scholar

[11]

P. K. Ghosh, A. Tsiartas and S. Narayanan, Robust voice activity detection using long-term signal variability, IEEE Transactions on Audio, Speech and Language Processing, 19 (2011), 600-613.  doi: 10.1109/TASL.2010.2052803.  Google Scholar

[12]

S. J. Kim, K. Koh, M. Lustig, S. Boyd and D. Gorinevsky, An interior-point method for large-scale $l_1$-regularized least squares, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 606-617.   Google Scholar

[13]

H. Li, C. Fang and Z. Lin, Accelerated first-order optimization algorithms for machine learning, Proceedings of the IEEE, (2020), 1-16. Google Scholar

[14] P. C. Loizou, Speech Enhancement: Theory and Practice, CRC press, Boca Raton, 2013.  doi: 10.1201/9781420015836.  Google Scholar
[15]

S. Y. Low, Compressive speech enhancement in the modulation domain, Speech Communication, 102 (2018), 87-99.  doi: 10.1016/j.specom.2018.08.003.  Google Scholar

[16]

S. Y. Low, D. S. Pham and S. Venkatesh, Compressive speech enhancement, Speech Communication, 55 (2013), 757-768.  doi: 10.1016/j.specom.2013.03.003.  Google Scholar

[17]

R. Martin, Noise power spectral density estimation based on optimal smoothing and minimum statistics, IEEE Transactions on Speech and Audio Processing, 9 (2001), 504-512.  doi: 10.1109/89.928915.  Google Scholar

[18]

R. Miyazaki, H. Saruwatari, T. Inoue, K. Shikano and K. Kondo, Musical-noise-free speech enhancement: Theory and evaluation, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2012), 4565-4568. doi: 10.1109/ICASSP.2012.6288934.  Google Scholar

[19]

M. Nazih, K. Minaoui and P. Comon, Using the proximal gradient and the accelerated proximal gradient as a canonical polyadic tensor decomposition algorithms in difficult situations, Signal Processing, 171 (2020), 107472. doi: 10.1016/j.sigpro.2020.107472.  Google Scholar

[20]

N. Parikh and S. Boyd, Proximal Algorithms, Foundation and Trends in Optimization, 1 (2013), 123-231.   Google Scholar

[21]

A. W. Rix, J. G. Beerends, M. P. Hollier and A. P. Hekstra, Perceptual evaluation of speech quality (PESQ) - a new method for speech quality assessment of telephone networks and codecs, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), 2 (2001), 749-752.  doi: 10.1109/ICASSP.2001.941023.  Google Scholar

[22]

M. Schmidt, Least squares optimization with l1-norm regularization, Technical Report CSP542B, 2005. Google Scholar

[23]

Y. Shi, S. Y. Low and K. F. C. Yiu, Hyper-parameterization of sparse reconstruction for speech enhancement, Applied Acoustics, 138 (2018), 72-79.  doi: 10.1016/j.apacoust.2018.03.020.  Google Scholar

[24]

C. H. Taal, R. C. Hendriks, R. Heusdens and J. Jensen, A short-time objective intelligibility measure for time-frequency weighted noisy speech, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, (2010), 4214-4217. doi: 10.1109/ICASSP.2010.5495701.  Google Scholar

[25]

R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.  Google Scholar

[26]

M. Torcoli, An improved measure of musical noise based on spectral kurtosis, 019 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), (2019), 90-94. doi: 10.1109/WASPAA.2019.8937195.  Google Scholar

[27]

D. Wu, W. Zhu and M. N. S. Swamy, A compressive sensing method for noise reduction of speech and audio signals, 2011 IEEE 54th International Midwest Symposium on Circuits and Systems (MWSCAS), (2011), 1-4. doi: 10.1109/MWSCAS.2011.6026662.  Google Scholar

[28]

Z. Zhang, Y. Xu, J. Yang, X. Li and D. Zhang, A Survey of Sparse Representation: Algorithms and Applications, IEEE Access, 3 (2015), 490-530.   Google Scholar

show all references

References:
[1]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problem, SIAM Journal on Imaging Sciences, 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[2]

J. Benesty and Y. Huang, A Perspective on Single-Channel Frequency-Domain Speech Enhancement, San Rafael: Morgan and Claypool Publishers, 2010. doi: 10.2200/S00344ED1V01Y201104SAP008.  Google Scholar

[3]

S. F. Boll, Supression of acoustic noise in speech using spectral subtraction, IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP-27 (1979), 113-120.   Google Scholar

[4]

O. Burdakov, Y. Dai and N. Huang, Stabilized Barzilai-Borwein method, J. Comp. Math., 37 (2019), 916-936.  doi: 10.4208/jcm.1911-m2019-0171.  Google Scholar

[5]

E. J. Candés, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509.  doi: 10.1109/TIT.2005.862083.  Google Scholar

[6]

E. J. Candes and T. Tao, Near-optimal signal recovery from random projections: universal encoding strategies, IEEE Transactions on Information Theory, 52 (2006), 5406-5425.  doi: 10.1109/TIT.2006.885507.  Google Scholar

[7]

E. J. Candes and M. B. Wakin, An introduction to compressive sampling, IEEE Signal Processing Magazine, (2008), 21-30. Google Scholar

[8]

H. H. Dam and A. Cantoni, Interior point method for optimum zero-forcing beamforming with per-antenna power constraints and optimal step size, Signal Processing, 106 (2015), 10-14.  doi: 10.1016/j.sigpro.2014.06.028.  Google Scholar

[9]

H. H. Dam and S. Nordholm, Accelerated gradient with optimal step size for second-order blind signal separation, Multidimens. Syst. Signal Process., 29 (2018), 903-919.  doi: 10.1007/s11045-017-0478-8.  Google Scholar

[10]

T. Esch and P. Vary, Efficient musical noise suppression for speech enhancement system, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing, (2009), 4409-4412. doi: 10.1109/ICASSP.2009.4960607.  Google Scholar

[11]

P. K. Ghosh, A. Tsiartas and S. Narayanan, Robust voice activity detection using long-term signal variability, IEEE Transactions on Audio, Speech and Language Processing, 19 (2011), 600-613.  doi: 10.1109/TASL.2010.2052803.  Google Scholar

[12]

S. J. Kim, K. Koh, M. Lustig, S. Boyd and D. Gorinevsky, An interior-point method for large-scale $l_1$-regularized least squares, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 606-617.   Google Scholar

[13]

H. Li, C. Fang and Z. Lin, Accelerated first-order optimization algorithms for machine learning, Proceedings of the IEEE, (2020), 1-16. Google Scholar

[14] P. C. Loizou, Speech Enhancement: Theory and Practice, CRC press, Boca Raton, 2013.  doi: 10.1201/9781420015836.  Google Scholar
[15]

S. Y. Low, Compressive speech enhancement in the modulation domain, Speech Communication, 102 (2018), 87-99.  doi: 10.1016/j.specom.2018.08.003.  Google Scholar

[16]

S. Y. Low, D. S. Pham and S. Venkatesh, Compressive speech enhancement, Speech Communication, 55 (2013), 757-768.  doi: 10.1016/j.specom.2013.03.003.  Google Scholar

[17]

R. Martin, Noise power spectral density estimation based on optimal smoothing and minimum statistics, IEEE Transactions on Speech and Audio Processing, 9 (2001), 504-512.  doi: 10.1109/89.928915.  Google Scholar

[18]

R. Miyazaki, H. Saruwatari, T. Inoue, K. Shikano and K. Kondo, Musical-noise-free speech enhancement: Theory and evaluation, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (2012), 4565-4568. doi: 10.1109/ICASSP.2012.6288934.  Google Scholar

[19]

M. Nazih, K. Minaoui and P. Comon, Using the proximal gradient and the accelerated proximal gradient as a canonical polyadic tensor decomposition algorithms in difficult situations, Signal Processing, 171 (2020), 107472. doi: 10.1016/j.sigpro.2020.107472.  Google Scholar

[20]

N. Parikh and S. Boyd, Proximal Algorithms, Foundation and Trends in Optimization, 1 (2013), 123-231.   Google Scholar

[21]

A. W. Rix, J. G. Beerends, M. P. Hollier and A. P. Hekstra, Perceptual evaluation of speech quality (PESQ) - a new method for speech quality assessment of telephone networks and codecs, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), 2 (2001), 749-752.  doi: 10.1109/ICASSP.2001.941023.  Google Scholar

[22]

M. Schmidt, Least squares optimization with l1-norm regularization, Technical Report CSP542B, 2005. Google Scholar

[23]

Y. Shi, S. Y. Low and K. F. C. Yiu, Hyper-parameterization of sparse reconstruction for speech enhancement, Applied Acoustics, 138 (2018), 72-79.  doi: 10.1016/j.apacoust.2018.03.020.  Google Scholar

[24]

C. H. Taal, R. C. Hendriks, R. Heusdens and J. Jensen, A short-time objective intelligibility measure for time-frequency weighted noisy speech, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, (2010), 4214-4217. doi: 10.1109/ICASSP.2010.5495701.  Google Scholar

[25]

R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.  Google Scholar

[26]

M. Torcoli, An improved measure of musical noise based on spectral kurtosis, 019 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), (2019), 90-94. doi: 10.1109/WASPAA.2019.8937195.  Google Scholar

[27]

D. Wu, W. Zhu and M. N. S. Swamy, A compressive sensing method for noise reduction of speech and audio signals, 2011 IEEE 54th International Midwest Symposium on Circuits and Systems (MWSCAS), (2011), 1-4. doi: 10.1109/MWSCAS.2011.6026662.  Google Scholar

[28]

Z. Zhang, Y. Xu, J. Yang, X. Li and D. Zhang, A Survey of Sparse Representation: Algorithms and Applications, IEEE Access, 3 (2015), 490-530.   Google Scholar

Figure 1.  Convergence for accelerated proximal gradient, proximal gradient methods and interior point methods for babble noise with 0 dB and $ L = 256 $
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Figure 2.  Convergence for the proximal gradient, the accelerated proximal gradient, and the interior point methods for babble noise with 0 dB and $ L = 512 $
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Figure 3.  Convergence for the proximal gradient, the accelerated proximal gradient, and the interior point methods for destroyer noise with 0 dB and $ L = 512 $
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Table 1.  Complexity comparison between the proximal gradient, the accelerated proximal gradient, and the interior point methods for babble noise, destroyer noise and white noise with window length $ L = 256 $
Noise type SNR Accelerated Proximal Gradient Proximal Gradient Interior Point Method
Babble noise 0dB 3.1307s 3.3139s 7.2640s
5dB 2.7209s 2.8722s 7.0281s
10dB 2.3887s 2.4476s 6.8677s
15dB 2.2449s 2.4057s 6.8233s
20dB 2.0481s 2.1050s 6.6695s
Destroyer noise 0dB 2.8322s 2.9187s 6.9363s
5dB 2.4799s 2.5386s 6.8301s
10dB 2.2675s 2.4413s 6.7417s
15dB 2.1390s 2.2119s 6.7070s
20dB 1.8859s 1.9688s 6.4216s
White noise 0dB 3.4491s 3.5234s 6.6548s
5dB 2.8229s 2.9723s 6.9340s
10dB 2.5765s 2.6288s 7.2333s
15dB 2.3393s 2.4726s 7.0217s
20dB 1.9912s 2.0732s 6.5130s
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Noise type SNR Accelerated Proximal Gradient Proximal Gradient Interior Point Method
Babble noise 0dB 3.1307s 3.3139s 7.2640s
5dB 2.7209s 2.8722s 7.0281s
10dB 2.3887s 2.4476s 6.8677s
15dB 2.2449s 2.4057s 6.8233s
20dB 2.0481s 2.1050s 6.6695s
Destroyer noise 0dB 2.8322s 2.9187s 6.9363s
5dB 2.4799s 2.5386s 6.8301s
10dB 2.2675s 2.4413s 6.7417s
15dB 2.1390s 2.2119s 6.7070s
20dB 1.8859s 1.9688s 6.4216s
White noise 0dB 3.4491s 3.5234s 6.6548s
5dB 2.8229s 2.9723s 6.9340s
10dB 2.5765s 2.6288s 7.2333s
15dB 2.3393s 2.4726s 7.0217s
20dB 1.9912s 2.0732s 6.5130s
Table 2.  Complexity comparison between the accelerated proximal gradient, the proximal gradient and the interior point methods for babble noise, destroyer noise and white noise with window length $ L = 512 $
Noise type SNR Accelerated Proximal Gradient Proximal Gradient Interior Point Method
Babble noise 0dB 0.8681s 0.9342s 12.7778s
5dB 0.7779s 0.8346s 12.5931s
10dB 0.7119s 0.7730s 12.2826s
15dB 0.6637s 0.7199s 12.0663s
20dB 0.6138s 0.6703s 11.7910s
Destroyer noise 0dB 0.8096s 0.8760s 12.4143s
5dB 0.7330s 0.7863s 12.4028s
10dB 0.6709s 0.7329s 12.0540s
15dB 0.6263s 0.6908s 11.9282s
20dB 0.5950s 0.6550 11.8206s
White noise 0dB 0.9592s 1.0401s 11.9704s
5dB 0.8137s 0.8761s 12.5119s
10dB 0.7049s 0.7656s 12.8533s
15dB 0.6503s 0.7136s 12.3004s
20dB 0.6193s 0.6818s 11.9545s
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Noise type SNR Accelerated Proximal Gradient Proximal Gradient Interior Point Method
Babble noise 0dB 0.8681s 0.9342s 12.7778s
5dB 0.7779s 0.8346s 12.5931s
10dB 0.7119s 0.7730s 12.2826s
15dB 0.6637s 0.7199s 12.0663s
20dB 0.6138s 0.6703s 11.7910s
Destroyer noise 0dB 0.8096s 0.8760s 12.4143s
5dB 0.7330s 0.7863s 12.4028s
10dB 0.6709s 0.7329s 12.0540s
15dB 0.6263s 0.6908s 11.9282s
20dB 0.5950s 0.6550 11.8206s
White noise 0dB 0.9592s 1.0401s 11.9704s
5dB 0.8137s 0.8761s 12.5119s
10dB 0.7049s 0.7656s 12.8533s
15dB 0.6503s 0.7136s 12.3004s
20dB 0.6193s 0.6818s 11.9545s
Table 3.  PESQ and STOI performance for different SNR with babble noise and $ L = 256 $
SNR Methods PESQ STOI
0 dB Optimized $ \lambda=0.9549 $ 2.0328 0.7147
Fixed value $ \lambda=0.8 $ 2.0073 0.7032
Fixed value $ \lambda=0.9 $ 2.0241 0.7103
Unprocessed $ -- $ 1.8938 0.7145
5 dB Optimized $ \lambda=0.9449 $ 2.4100 0.8200
Fixed value $ \lambda=0.8 $ 2.3896 0.8107
Fixed value $ \lambda=0.9 $ 2.3996 0.8170
Unprocessed $ -- $ 2.2203 0.8130
10 dB Optimized $ \lambda=0.9549 $ 2.7702 0.8999
Fixed value $ \lambda=0.8 $ 2.7522 0.8918
Fixed value $ \lambda=0.9 $ 2.7639 0.8974
Unprocessed $ -- $ 2.5434 0.8899
15dB Optimized $ \lambda=0.9525 $ 3.1247 0.9504
Fixed value $ \lambda=0.8 $ 3.0937 0.9455
Fixed value $ \lambda=0.9 $ 3.1144 0.9489
Unprocessed $ -- $ 2.8556 0.9423
20dB Optimized $ \lambda=0.9549 $ 3.4425 0.9767
Fixed value $ \lambda=0.8 $ 3.3898 0.9731
Fixed value $ \lambda=0.9 $ 3.4317 0.9757
Unprocessed $ -- $ 3.1674 0.9734
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SNR Methods PESQ STOI
0 dB Optimized $ \lambda=0.9549 $ 2.0328 0.7147
Fixed value $ \lambda=0.8 $ 2.0073 0.7032
Fixed value $ \lambda=0.9 $ 2.0241 0.7103
Unprocessed $ -- $ 1.8938 0.7145
5 dB Optimized $ \lambda=0.9449 $ 2.4100 0.8200
Fixed value $ \lambda=0.8 $ 2.3896 0.8107
Fixed value $ \lambda=0.9 $ 2.3996 0.8170
Unprocessed $ -- $ 2.2203 0.8130
10 dB Optimized $ \lambda=0.9549 $ 2.7702 0.8999
Fixed value $ \lambda=0.8 $ 2.7522 0.8918
Fixed value $ \lambda=0.9 $ 2.7639 0.8974
Unprocessed $ -- $ 2.5434 0.8899
15dB Optimized $ \lambda=0.9525 $ 3.1247 0.9504
Fixed value $ \lambda=0.8 $ 3.0937 0.9455
Fixed value $ \lambda=0.9 $ 3.1144 0.9489
Unprocessed $ -- $ 2.8556 0.9423
20dB Optimized $ \lambda=0.9549 $ 3.4425 0.9767
Fixed value $ \lambda=0.8 $ 3.3898 0.9731
Fixed value $ \lambda=0.9 $ 3.4317 0.9757
Unprocessed $ -- $ 3.1674 0.9734
Table 4.  PESQ and STOI performance for different SNR with destroyer noise and $ L = 256 $
SNR Method s PESQ STOI
0 dB Optimized $ \lambda=0.8949 $ 2.1629 0.7532
Fixed value $ \lambda=0.8 $ 2.1543 0.7448
Fixed value $ \lambda=0.9 $ 2.1456 0.7497
Unprocessed $ -- $ 1.9271 0.7524
5 dB Optimized $ \lambda=0.8951 $ 2.5370 0.8337
Fixed value $ \lambda=0.8 $ 2.5186 0.8267
Fixed value $ \lambda=0.9 $ 2.5283 0.8325
Unprocessed $ -- $ 2.2955 0.8281
10 dB Optimized $ \lambda=0.8749 $ 2.8704 0.9001
Fixed value $ \lambda=0.8 $ 2.8543 0.8933
Fixed value $ \lambda=0.9 $ 2.8677 0.8985
Unprocessed $ -- $ 2.6132 0.8902
15dB Optimized $ \lambda=0.8949 $ 3.1914 0.9468
Fixed value $ \lambda=0.8 $ 3.1611 0.9412
Fixed value $ \lambda=0.9 $ 3.1876 0.9455
Unprocessed $ -- $ 2.9256 0.9382
20dB Optimized $ \lambda=0.9451 $ 3.4868 0.9737
Fixed value $ \lambda=0.8 $ 3.4427 0.9696
Fixed value $ \lambda=0.9 $ 3.4722 0.9726
Unprocessed $ -- $ 3.2468 0.9697
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SNR Method s PESQ STOI
0 dB Optimized $ \lambda=0.8949 $ 2.1629 0.7532
Fixed value $ \lambda=0.8 $ 2.1543 0.7448
Fixed value $ \lambda=0.9 $ 2.1456 0.7497
Unprocessed $ -- $ 1.9271 0.7524
5 dB Optimized $ \lambda=0.8951 $ 2.5370 0.8337
Fixed value $ \lambda=0.8 $ 2.5186 0.8267
Fixed value $ \lambda=0.9 $ 2.5283 0.8325
Unprocessed $ -- $ 2.2955 0.8281
10 dB Optimized $ \lambda=0.8749 $ 2.8704 0.9001
Fixed value $ \lambda=0.8 $ 2.8543 0.8933
Fixed value $ \lambda=0.9 $ 2.8677 0.8985
Unprocessed $ -- $ 2.6132 0.8902
15dB Optimized $ \lambda=0.8949 $ 3.1914 0.9468
Fixed value $ \lambda=0.8 $ 3.1611 0.9412
Fixed value $ \lambda=0.9 $ 3.1876 0.9455
Unprocessed $ -- $ 2.9256 0.9382
20dB Optimized $ \lambda=0.9451 $ 3.4868 0.9737
Fixed value $ \lambda=0.8 $ 3.4427 0.9696
Fixed value $ \lambda=0.9 $ 3.4722 0.9726
Unprocessed $ -- $ 3.2468 0.9697
Table 5.  PESQ and STOI performance for different SNR with white noise and $ L = 256 $
SNR Methods PESQ STOI
0 dB Optimized $ \lambda=0.9331 $ 2.0119 0.7661
Fixed value $ \lambda=0.8 $ 1.9895 0.7519
Fixed value $ \lambda=0.9 $ 2.0042 0.7619
Unprocessed $ -- $ 1.6665 0.7377
5 dB Optimized $ \lambda=0.9451 $ 2.3972 0.8615
Fixed value $ \lambda=0.8 $ 2.3716 0.8492
Fixed value $ \lambda=0.9 $ 2.3913 0.8580
Unprocessed $ -- $ 1.9615 0.8387
10 dB Optimized $ \lambda=0.9451 $ 2.8102 0.9275
Fixed value $ \lambda=0.8 $ 2.7735 0.9183
Fixed value $ \lambda=0.9 $ 2.7976 0.9246
Unprocessed $ -- $ 2.2989 0.9146
15dB Optimized $ \lambda=0.9349 $ 3.1973 0.9652
Fixed value $ \lambda=0.8 $ 3.1472 0.9594
Fixed value $ \lambda=0.9 $ 3.1844 0.9636
Unprocessed $ -- $ 2.6442 0.9613
20dB Optimized $ \lambda=0.9501 $ 3.5007 0.9858
Fixed value $ \lambda=0.8 $ 3.4286 0.9797
Fixed value $ \lambda=0.9 $ 3.4796 0.9826
Unprocessed $ -- $ 2.9839 0.9845
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SNR Methods PESQ STOI
0 dB Optimized $ \lambda=0.9331 $ 2.0119 0.7661
Fixed value $ \lambda=0.8 $ 1.9895 0.7519
Fixed value $ \lambda=0.9 $ 2.0042 0.7619
Unprocessed $ -- $ 1.6665 0.7377
5 dB Optimized $ \lambda=0.9451 $ 2.3972 0.8615
Fixed value $ \lambda=0.8 $ 2.3716 0.8492
Fixed value $ \lambda=0.9 $ 2.3913 0.8580
Unprocessed $ -- $ 1.9615 0.8387
10 dB Optimized $ \lambda=0.9451 $ 2.8102 0.9275
Fixed value $ \lambda=0.8 $ 2.7735 0.9183
Fixed value $ \lambda=0.9 $ 2.7976 0.9246
Unprocessed $ -- $ 2.2989 0.9146
15dB Optimized $ \lambda=0.9349 $ 3.1973 0.9652
Fixed value $ \lambda=0.8 $ 3.1472 0.9594
Fixed value $ \lambda=0.9 $ 3.1844 0.9636
Unprocessed $ -- $ 2.6442 0.9613
20dB Optimized $ \lambda=0.9501 $ 3.5007 0.9858
Fixed value $ \lambda=0.8 $ 3.4286 0.9797
Fixed value $ \lambda=0.9 $ 3.4796 0.9826
Unprocessed $ -- $ 2.9839 0.9845
Table 6.  PESQ and STOI performance for different SNR with babble noise and $ L = 512 $
SNR Methods PESQ STOI
0 dB Optimized $ \lambda=0.9601 $ 2.0699 0.7234
Fixed value $ \lambda=0.8 $ 2.0525 0.7129
Fixed value $ \lambda=0.9 $ 2.0634 0.7212
Unprocessed $ -- $ 1.8938 0.7145
5 dB Optimized $ \lambda= 0.9601 $ 2.4185 0.8282
Fixed value $ \lambda=0.8 $ 2.4084 0.8195
Fixed value $ \lambda=0.9 $ 2.4150 0.8258
Unprocessed $ -- $ 2.2203 0.8130
10 dB Optimized $ \lambda=0.9079 $ 2.7672 0.9064
Fixed value $ \lambda=0.8 $ 2.7529 0.8996
Fixed value $ \lambda=0.9 $ 2.7586 0.9045
Unprocessed $ -- $ 2.5434 0.8899
15dB Optimized $ \lambda=0.9077 $ 3.1187 0.9540
Fixed value $ \lambda=0.8 $ 3.0736 0.9507
Fixed value $ \lambda=0.9 $ 3.0790 0.9530
Unprocessed $ -- $ 2.8556 0.9423
20dB Optimized $ \lambda=0.9601 $ 3.3898 0.9785
Fixed value $ \lambda=0.8 $ 3.3703 0.9760
Fixed value $ \lambda=0.9 $ 3.3822 0.9775
Unprocessed $ -- $ 3.1674 0.9734
Table Options

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SNR Methods PESQ STOI
0 dB Optimized $ \lambda=0.9601 $ 2.0699 0.7234
Fixed value $ \lambda=0.8 $ 2.0525 0.7129
Fixed value $ \lambda=0.9 $ 2.0634 0.7212
Unprocessed $ -- $ 1.8938 0.7145
5 dB Optimized $ \lambda= 0.9601 $ 2.4185 0.8282
Fixed value $ \lambda=0.8 $ 2.4084 0.8195
Fixed value $ \lambda=0.9 $ 2.4150 0.8258
Unprocessed $ -- $ 2.2203 0.8130
10 dB Optimized $ \lambda=0.9079 $ 2.7672 0.9064
Fixed value $ \lambda=0.8 $ 2.7529 0.8996
Fixed value $ \lambda=0.9 $ 2.7586 0.9045
Unprocessed $ -- $ 2.5434 0.8899
15dB Optimized $ \lambda=0.9077 $ 3.1187 0.9540
Fixed value $ \lambda=0.8 $ 3.0736 0.9507
Fixed value $ \lambda=0.9 $ 3.0790 0.9530
Unprocessed $ -- $ 2.8556 0.9423
20dB Optimized $ \lambda=0.9601 $ 3.3898 0.9785
Fixed value $ \lambda=0.8 $ 3.3703 0.9760
Fixed value $ \lambda=0.9 $ 3.3822 0.9775
Unprocessed $ -- $ 3.1674 0.9734
Table 7.  PESQ and STOI performance for different SNR with destroyer noise and 512 subbands
SNR Method s PESQ STOI
0 dB Optimized $ \lambda=0.7601 $ 2.2328 0.7629
Fixed value $ \lambda=0.8 $ 2.2256 0.7602
Fixed value $ \lambda=0.9 $ 2.2078 0.7622
Unprocessed $ -- $ 1.9271 0.7524
5 dB Optimized $ \lambda=0.7700 $ 2.5651 0.8441
Fixed value $ \lambda=0.8 $ 2.5589 0.8414
Fixed value $ \lambda=0.9 $ 2.5569 0.8436
Unprocessed $ -- $ 2.2955 0.8281
10 dB Optimized $ \lambda=0.7700 $ 2.8773 0.9084
Fixed value $ \lambda=0.8 $ 2.8699 0.9056
Fixed value $ \lambda=0.9 $ 2.8742 0.9081
Unprocessed $ -- $ 2.6132 0.8902
15dB Optimized $ \lambda=0.8301 $ 3.1775 0.9530
Fixed value $ \lambda=0.8 $ 3.1710 0.9509
Fixed value $ \lambda=0.9 $ 3.1732 0.9529
Unprocessed $ -- $ 2.9256 0.9382
20dB Optimized $ \lambda=0.9270 $ 3.4819 0.9768
Fixed value $ \lambda=0.8 $ 3.4375 0.9750
Fixed value $ \lambda=0.9 $ 3.4469 0.9766
Unprocessed $ -- $ 3.2468 0.9697
Table Options

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SNR Method s PESQ STOI
0 dB Optimized $ \lambda=0.7601 $ 2.2328 0.7629
Fixed value $ \lambda=0.8 $ 2.2256 0.7602
Fixed value $ \lambda=0.9 $ 2.2078 0.7622
Unprocessed $ -- $ 1.9271 0.7524
5 dB Optimized $ \lambda=0.7700 $ 2.5651 0.8441
Fixed value $ \lambda=0.8 $ 2.5589 0.8414
Fixed value $ \lambda=0.9 $ 2.5569 0.8436
Unprocessed $ -- $ 2.2955 0.8281
10 dB Optimized $ \lambda=0.7700 $ 2.8773 0.9084
Fixed value $ \lambda=0.8 $ 2.8699 0.9056
Fixed value $ \lambda=0.9 $ 2.8742 0.9081
Unprocessed $ -- $ 2.6132 0.8902
15dB Optimized $ \lambda=0.8301 $ 3.1775 0.9530
Fixed value $ \lambda=0.8 $ 3.1710 0.9509
Fixed value $ \lambda=0.9 $ 3.1732 0.9529
Unprocessed $ -- $ 2.9256 0.9382
20dB Optimized $ \lambda=0.9270 $ 3.4819 0.9768
Fixed value $ \lambda=0.8 $ 3.4375 0.9750
Fixed value $ \lambda=0.9 $ 3.4469 0.9766
Unprocessed $ -- $ 3.2468 0.9697
Table 8.  PESQ and STOI performance for different SNR with white noise and 512 subbands
SNR Methods PESQ STOI
0 dB Optimized $ \lambda=0.8599 $ 2.0454 0.7829
Fixed value $ \lambda=0.8 $ 2.0395 0.7721
Fixed value $ \lambda=0.9 $ 2.0403 0.7775
Unprocessed $ -- $ 1.6665 0.7377
5 dB Optimized $ \lambda=0.8801 $ 2.4148 0.8735
Fixed value $ \lambda=0.8 $ 2.4129 0.8638
Fixed value $ \lambda=0.9 $ 2.4101 0.8695
Unprocessed $ -- $ 1.9615 0.8387
10 dB Optimized $ \lambda=0.9496 $ 2.8009 0.9338
Fixed value $ \lambda=0.8 $ 2.7937 0.9261
Fixed value $ \lambda=0.9 $ 2.7948 0.9308
Unprocessed $ -- $ 2.2989 0.9146
15dB Optimized $ \lambda=0.9550 $ 3.1967 0.9673
Fixed value $ \lambda=0.8 $ 3.1421 0.9623
Fixed value $ \lambda=0.9 $ 3.1514 0.9652
Unprocessed $ -- $ 2.6442 0.9613
20dB Optimized $ \lambda=0.9601 $ 3.4393 0.9864
Fixed value $ \lambda=0.8 $ 3.3930 0.9807
Fixed value $ \lambda=0.9 $ 3.4175 0.9823
Unprocessed $ -- $ 2.9839 0.9845
Table Options

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SNR Methods PESQ STOI
0 dB Optimized $ \lambda=0.8599 $ 2.0454 0.7829
Fixed value $ \lambda=0.8 $ 2.0395 0.7721
Fixed value $ \lambda=0.9 $ 2.0403 0.7775
Unprocessed $ -- $ 1.6665 0.7377
5 dB Optimized $ \lambda=0.8801 $ 2.4148 0.8735
Fixed value $ \lambda=0.8 $ 2.4129 0.8638
Fixed value $ \lambda=0.9 $ 2.4101 0.8695
Unprocessed $ -- $ 1.9615 0.8387
10 dB Optimized $ \lambda=0.9496 $ 2.8009 0.9338
Fixed value $ \lambda=0.8 $ 2.7937 0.9261
Fixed value $ \lambda=0.9 $ 2.7948 0.9308
Unprocessed $ -- $ 2.2989 0.9146
15dB Optimized $ \lambda=0.9550 $ 3.1967 0.9673
Fixed value $ \lambda=0.8 $ 3.1421 0.9623
Fixed value $ \lambda=0.9 $ 3.1514 0.9652
Unprocessed $ -- $ 2.6442 0.9613
20dB Optimized $ \lambda=0.9601 $ 3.4393 0.9864
Fixed value $ \lambda=0.8 $ 3.3930 0.9807
Fixed value $ \lambda=0.9 $ 3.4175 0.9823
Unprocessed $ -- $ 2.9839 0.9845
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