Module dsp::cqt

Constant-Q Transform

Unlike the DFT/FFT (uniform frequency spacing, uniform time resolution across all bins), the CQT uses logarithmically-spaced center frequencies fk = fmin * 2^(k/binsPerOctave) with a constant ratio fk/bandwidth ("Q") at every bin — matching how musical pitch and human hearing are both roughly logarithmic in frequency, which is why the CQT (not the plain DFT) is the standard choice for chromagrams, pitch detection, and other music-analysis tasks. The trade-off is that lower bins need longer analysis windows than higher bins (constant Q means constant relative bandwidth, so absolute bandwidth — and therefore the window length needed to resolve it — scales with 1/fk).

This is the direct-correlation form (Brown 1991): each bin's kernel (a Hann-windowed complex exponential, own length per bin) is correlated directly against the start of 'signal'. It doesn't need an FFT and needs no fast-kernel-matrix precomputation, at the cost of O(numBins * average-kernel-length) per call — fine for analysis/offline use; a production real-time CQT would typically precompute a sparse kernel matrix once and reuse it, which isn't done here.