The marriage of subband adaptive filtering (SAF) and acoustic echo cancellation (AEC) has been a successful one. Subband adaptive filtering overcomes many of the limitations of traditional time-domain normalized least means squares (NLMS) implementations of echo cancellation. NLMS are commonly implemented on DSPs because of the low memory requirements and computational complexity compared to other adaptive algorithms. The limitation of LMS type algorithms is that convergence speed is inversely proportional to the spectral spread of the input signals. In other words, as the excitation signals moves from a white spectrum to colored spectrum the convergence speed of the system decreases. NLMS in the time domain fails to perform in acoustic echo cancellation because speech is considered a colored signal. SAF provides improved convergence speed and reduced computational complexity.
How does subband adaptive filtering work? First, subband decomposition splits a fullband signal into multiple subband signals that allows for the processing of the information contained in each subband independently. In echo cancellation the excitation and microphone signal are split into several subbands signals. Then adaptive filtering can be applied to each subband signal. The subband decomposition is achieved via a set of bandpass filters called a filter bank. The design of the filter bank plays a crucial role in the performance of a SAF; this will be discussed in more detail later.
Since each subband contains a fraction of the original signal, the subband signal can be downsampled by the number of subbands while still preserving the information of the original samples. For reconstruction of the fullband signals, the downsampled subband signals are upsampled by the same rate as the decimatation, and are combined by a synthesis filter bank.
As mentioned previously, the subband signals can be downsampled by a factor D, where D ≤ N, N being the number of subbands. This reduces the computational complexity of the system as the signals can be processed by lower order subfilters than their fullband counterparts. More explicitly, if we have a fullband filter of length L, it will require L multiplications. For filtering the subband domain with subfilters of length LS = L / D, the number of multiplications for the N subbands is LN / D2. Thus, the amount of the computational complexity savings is dependent the values of the N and D. Besides computational savings, the subband adaptive filters also provide increased convergence speed for LMS type adaptive algorithms over the fullband implementations. The reason for this is because after the subband analysis and downsampling the spectral dynamic range in each subband will be less than the fullband signal.
The design of the filter banks for subband analysis and synthesis plays an important role in the performance limitations of the subband adaptive filter. The limiting factors of the SAFs are due to the fact that a set of ideal bandpass filters cannot be realized. The first limiting factor of a SAFs is the stopband attenuation of the bandpass filter. Using echo cancellation as an example, the SAF will not be able to cancel the echo signal beyond stopband attenuation of the filter bank. The second limiting factor is the overlap of the spectral components of the analysis filters. These overlap components are refered to as aliasing components and act as additional noise components. This additional noise reduces the achievable gain for adaptive algorithms with mean square error cost functions. These limitations do not prevent the practicality of use in real-world application to acoustic echo cancellation. The stopband attenuation can be easily designed to be over 60dB, and the effect of the aliasing components can be minimized by oversampling the analysis filter banks or using adaptive cross filters.