In the application of acoustic echo cancellation, the correlation of the far-end input speech signal is high . However, the traditional method is based on the assumption of "signal independence". Traditional full-band LMS and NLMS and other stochastic gradient algorithms with low computational complexity It is difficult to meet the requirements of the system on the convergence speed.
Far-end voice signal correlation has two meanings:
- In the time domain: it characterizes the diffusion of the eigenvalues of the correlation matrix of the speech signal
- In the frequency domain: it characterizes the spectral dynamic range of the far-end voice signal
In general, the speech signal has a significantly larger spectral dynamic range than the white signal, that is, a greater signal correlation. Therefore, the convergence speed of the algorithm can be accelerated by reducing the correlation of the input signal . An effective method is the sub-band adaptive filtering algorithm. The sub-band structure is a processing of the signal based on the frequency domain (saving calculation , improve the convergence speed).
Subband adaptive filter (subband adaptive filter, SAF): The relevant signal is divided into approximately irrelevant subband independent signals through the filter bank (subband segmentation). Then the multi-rate extraction is performed on the sub-band signal to obtain the sampling signal, and then the signal is adaptively processed. To study subband adaptive filters, it is first necessary to understand multirate signal decimation systems and filter banks .
multi-rate system
There are two types of multi-rate decimation systems for sub-band adaptive filters: down-sampling and up-sampling , and the system can obtain different sampling rates mainly through decimation and interpolation methods. After the input signal is divided by N filters, the total number of sampling points is N times that of the original signal, and the greatly increased number of samples increases the amount of calculation. Assuming that the sampling factor is K, the sampling rate can be reduced from f to f/K by retaining K times the sampling points of the signal, thus reducing the calculation amount of the adaptive algorithm. Conversely, upsampling increases the original sampling rate from f to Kf by inserting K-1 zeros between adjacent points of the signal.
Downsampling is achieved through the decimator and delay shown below.
The time domain expression of downsampling is xD(n)=x(mK), n is the block sequence number, and its frequency domain expression is:
This process can be understood as expanding the frequency of x(n) by K times and extending the period by 2π , that is, generating the spectrum of xD(n).
Upsampling can be obtained by adding K−1 zero points between adjacent sampling points, and the expression of the upsampling signal xI(n) is:
The frequency domain expression is to compress the frequency of x(n) by K times, and then extend it with a period of 2π to obtain :
filter bank
Signal sub-band segmentation is achieved by filter banks. The filter bank is composed of a series of bandpass filters , mainly including the following links: 1) analysis filter bank, 2) decimation, 3) interpolation, 4) synthesis filter bank
The analysis filter bank divides the digital signal and extracts it into multiple sub-band signals . After signal processing, the synthesis filter bank interpolates and filters the sub-band signals to restore the original signal.
After the signal passes through the analysis filter bank, the subband signal expression is:
The sampled subband signal can be written as:
It can be seen that it is a common part of the output signal of the integrated filter bank .
By combining the common terms, the output of the N-channel QMF filter bank is
In practical applications, it can be realized using a cosine modulated filter bank. Scholars have conducted in-depth research on the theory and design of cosine modulated filter banks. Both the analysis filter bank and the synthesis filter bank of the cosine modulation filter bank are N-channel critical sampling filter banks, and they can be expressed as E(z) and R(z) respectively by unitary matrix. A fully reconstructed system of the analytical filter bank E(z) can be realized by the pseudo-unitary property:
where K-1 is the order of the polyphase component matrix. The pseudo-unitary condition means . The polyphase component in the synthesis filter bank can be realized by the above formula, ie.
Both the analysis filter bank and the synthesis filter bank of the N-channel cosine-modulated filter bank are based on low-pass filters with a cutoff frequency of . The analysis filter for cosine modulation is expressed as
By optimizing the design of the prototype filter to satisfy a set of predetermined constraints, the signal can be fully reconstructed. Different constraints and optimization methods lead to two different classes of cosine-modulated filterbanks, namely pseudo-QMF cosine-modulated filterbanks and pseudo-unitary cosine-modulated filterbanks.
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