doi:10.1016/S0165-1684(03)00107-5 
Copyright © 2003 Elsevier B.V. All rights reserved.
QRD-based unconstrained optimal filtering for acoustic noise reduction
KULeuven/ESAT–SCD(SISTA), Kasteelpark Arenberg 10, 3001, Heverlee, Belgium
Received 4 February 2002;
revised 2 December 2002.
Available online 14 May 2003.
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Abstract
We describe a new adaptive filtering algorithm based upon QR-decomposition for optimal multichannel filtering with an “unknown” desired signal, as well as its application to multi-channel acoustic noise reduction. A recursively calculated adaptive filter then optimally estimates the speech component in a noisy signal. The complexity of this algorithm is about 7 times lower than that of existing related algorithms, which are mainly based upon GSVD-decompositions, while performance is kept at the same level. Finally, it is shown how a parameter can be introduced which can be tuned to tradeoff noise reduction for signal distortion.
Author Keywords: Acoustic noise reduction; Adaptive filtering; QR-updating
- M
- number of channels
- N
- number of taps per channel
- x,X
- input vector, Toeplitz input matrix
- Q
- orthogonal matrix in a QR-decomposition
- R
- upper triangular matrix in a QR-decomposition
- W
- matrix of which the columns are filter vectors
- h,H
- room impulse response
- I
- unity matrix
- λ
- forgetting factor
- r
- least-squares residuals
- v,V
- noise data vector, matrix
- d,D
- desired signal vector, matrix
- B
- right-hand side in LS problems
- P
- noise correlation matrix
Fig. 1. Adaptive optimal filtering in the acoustic noise reduction context.
Fig. 2. Givens-rotations to update an
R(
k)-matrix. On top the new input vector is fed in, and for each row of the
R(
k)-matrix a Givens-rotation is executed in order to obtain an upper triangular matrix
R(
k+1).
Fig. 3. Standard QRD-RLS filtering.
Fig. 4. Residual extraction, the filtered signal is calculated without explicitly calculating the filter coefficients.
Fig. 5. Updating scheme for signal+noise mode.
Fig. 6. Signal flow graph for residual extraction.
Fig. 7. Four utterances of the same sentence. The clean speech signal and the noise signal at microphone 1 are plotted (simulation). The GSVD-result is better for this case (λ
v=0.9997 and λ
x=0.9997) than the QRD-result because the negative singular values can be set to zero in the GSVD-method. As shown on the detail below, the distortion is less for the QRD-method though.
Fig. 8. Comparison between GSVD-based and QRD-based unconstrained optimal filtering when the correlation matrices are
not corrected in the GSVD-approach. Again, λ
n=0.9997 and λ
s=0.9997. The QRD-method performs slightly better because of the approximation used in the GSVD-approach.
Fig. 9. QRD-approach versus GSVD-approach with longer estimation window: the difference between both algorithms vanishes, even when the singular values are corrected in the GSVD-approach. (λ
v=0.99995 and λ
x=0.9997)
Fig. 10. When a tradeoff parameter is introduced, more noise reduction is achieved, in exchange for some signal distortion. The upper line is always the algorithm output without the tradeoff parameter, the lower line with a tradeoff parameter μ=2.
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