Dual Views of the Short Time Fourier Transform (STFT)

In the overlap-add formulation of Chapter 8, we used a hopping window to extract time-limited signals to which we applied the DFT. Assuming for the moment that the hop size (the ``sliding DFT''), we have

This is the usual definition of the Short-Time Fourier Transform (STFT) (§6.1). In this chapter, we will look at the STFT from two different points of view: the OverLap-Add (OLA) and Filter-Bank Summation (FBS) points of view. We will show that one is the Fourier dual of the other [10]. Next we will explore some implications of the filter-bank point of view and obtain some useful insights. Finally, some applications are considered.

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Subsections

• Overlap-Add View of the STFT
• Filter Bank View of the STFT
• FBS and Perfect Reconstruction

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Order a Hardcopy of Spectral Audio Signal Processing

Previous: The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
Next: Overlap-Add (OLA) Interpretation of the STFT

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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.