This chapter explores in detail the topic of spectral analysis and synthesis. It begins by looking at the concept of the Fourier transform and series and examines how we can derive the spectrum of a signal from its time-domain waveform. We then explore the discrete-time discrete-frequency version of the Fourier transform, the DFT, its formula and its implementation. This allows us to look in more detail into the characteristics of the spectral representation. Following this, we study the radix-2 fast Fourier transform algorithm, which is the most commonly used implementation of the DFT. The application of spectral processing to convolution and reverb completes the section. The second half of the chapter is dedicated to timevarying spectral analysis, where we introduce the short-time Fourier transform and its applications in spectral processing. We look at instantaneous frequency analysis and the phase vocoder, completing the study with a number of application examples. The chapter concludes with some notes on real-time spectral processing.
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