# Summary and Conclusions

This book presented some basic methods for audio signal processing using frequency-domain methods, and a first look at some applications. At the outset, we studied a number of Fourier theorems in order to build up our intuition regarding signals in the time domain and their counterparts in the frequency domain. Secondly, we looked at quite a few FFT windows, and the Fourier theorems provided effective tools of comparison. Next, we utilized both Fourier theorems and FFT-window properties in the context of FIR filter design by the window method and others. We then looked at elementary methods for spectrum analysis of tonal signals (sinusoids) and noise-like signals (specifically any filtered white noise). The subject of audio time-frequency displays was addressed, with the goal of progressing from the classic audio spectrogram to a more accurate ``what you see is what you hear'' spectrogram. Next we considered the topic of modifications in the frequency domain using the Short Time Fourier Transform (STFT). We saw that the STFT can be regarded as simply a sequence of FFTs (the overlap-add point of view), or as a downsampled uniform filter bank output (the filter-bank summation point of view). We concluded with a discussion of applications in computer music and digital audio, followed by an introduction to the relatively advanced topics of multirate polyphase and wavelet filter banks.

The following appendices provide elementary background material (such as a summary of notation, and a first introduction to statistical signal processing), coverage of supporting topics (such as matlab examples), and further extensions of certain chapters (such as continuous-time Fourier theorems). Finally, a history appendix is provided which tours many of the topics addressed in this book in historical order.

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Multirate Filter Banks