# Old style signal processing?

I am wondering if "an array of overlapping band-pass ﬁlters" it is 'mostly' equivalent to a Short-Time Fourier transform. See this (figure down), where $$s(t)$$ is the signal; LPF a second-order low-pass ﬁlter with a resonance frequency of 4 kHz; BPF an array of band-pass ﬁlters, where frequencies that are spaced equidistantly on a critical band scale; HCM a hair cell model which is assumed identical in all channels, and finally a low-pass ﬁlter at 1250 Hz does an envelope extraction of the patterns in each channel. Processing a signal with this method produces a matrix of intesities that are vertically distributed like in STFT.

What are the differences between them? Could be improved with wavelets as basis?

• if the HCM are identical for all bands, i think you need to bump down the frequency of the outputs of the BPFs down to a common low-frequency baseband. otherwise, if you don't, the HCM for the higher frequency bands need to work with higher frequencies than the HCMs for the lower bands. that might be simply a parameter change for qualitatively identical models, but they can't be exactly identical. – robert bristow-johnson Aug 14 at 2:08

there is a relationship between filterbank analysis and the STFT analysis, but it is not an identity.

perhaps it can be identical, where for filterbank, you are slicing the frequency domain into band slices and for STFT you are slicing the time domain into windowed segments (in audio we might call them "grains" or maybe sometimes "wavelets") in time. but to make the two identical will require some thinking. the BPF can't be simple regular-old BPFs.

• Thank you, as for your experience which will be the best method? I've tried both and stft is much computationally demanding as the channels cannot not be selected [?]. However, I think it can be difficult to evaluate the amount of lost information between them. – fina Aug 14 at 9:05
• i have to confess that, while i have done both STFT and a 60-channel, 5 octave in two different projects and i hadn't had the motivation to compare computational costs or how to meaningfully compare them apples-to-apples. i sorta thought a teeny bit about it, but not seriously. the STFT has the savings from an FFT and the energy in the bins of a 1/12 octave band can be added. but i dunno exactly what the overlap % should be to make it comparable to the filterbank. – robert bristow-johnson Aug 14 at 10:21
• Would you recommend me some R package to use these filters? – fina Aug 14 at 19:51
• i certainly have no R packages. i have never ever coded in R. i wrote some MATLAB and some C code to do this. – robert bristow-johnson Aug 14 at 20:15

Sort of, but not really. In both schemes you are taking a one dimensional signal and decomposing it to a two dimensional signal based on time and frequency. The first disparity would be that using the STFT, the frequency bands are linearly spaced, compared to your logarithmic spacing. Both schemes create a baseband signal from the frequency components. The second disparity is that a STFT modulates the original signal by a complex sinusoid, shifting the desired frequency to the baseband, and lowpass filters it. If one takes the magnitude of the resulting signal, this would be pretty close to what you’ve described. However, it would only be equivalent if the resulting high frequency images from your HCM stages did not alias with the baseband content and are sufficiently rejected by the final LPF stages. This would be the case with sufficiently narrow BPF stages such that no aliasing occurs and sufficiently precise LPFs at the end such that the baseband was not modified but the aliased images were perfectly rejected. Practically speaking, it would be difficult to set up the system such that the result was equivalent to a proper STFT. However, I would expect them to be similar enough for some applications with careful selection of the filter parameters.