Two Dual Interpretations of the STFT

The STFT can be viewed as a function of either frame-time or bin-frequency . We will develop both points of view in this book.

At each frame time , the STFT can be regarded as producing a Fourier transform centered around that time. As advances, a sequence of spectral transforms is obtained. This is depicted graphically in Fig.9.1, and it forms the basis of the overlap-add method for Fourier analysis, modification, and resynthesis [10]. It is also the basis for transform coders [15,262].

In an exact Fourier duality, each bin of the STFT can be regarded as a sample of the complex signal at the output of a lowpass filter whose input is . As discussed in §9.1.2, this signal is obtained from by frequency-shifting it so that frequency is translated down to 0 Hz. For each value of , the time-domain signal , for , is the output of the th ``filter bank channel,'' for . In this ``filter bank'' interpretation, the hop size can be interpreted as the downsampling factor applied to each bin-filter output, and the analysis window is seen as the impulse response of the anti-aliasing filter used prior to downsampling. The window transform is also the frequency response of each channel filter (translated to dc). This point of view is depicted graphically in Fig.9.2 and elaborated further in Chapter 9.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.