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# Overlap-Add (OLA) STFT Processing

This chapter discusses use of the Short-Time Fourier Transform (STFT) to implement linear filtering in the frequency domain. Due to the speed of FFT convolution, the STFT provides the most efficient single-CPU implementation engine for most FIR filters encountered in audio signal processing.

Recall from §7.1 the STFT:     (9.1)

where We noted that if the window has the constant overlap-add property at hop-size , (9.2)

then the sum of the successive DTFTs over time equals the DTFT of the whole signal : (9.3)

Consequently, the inverse-STFT is simply the inverse-DTFT of this sum: We may now introduce spectral modifications by multiplying each spectral frame by some filter frequency response to get (9.4)

Note that can be different for each frame, giving a certain class of time-varying filters. The filtered output signal spectrum is then (9.5)

so that (9.6)

where (9.7)

This chapter discusses practical implementation of the above relations using a Fast Fourier Transform (FFT). In particular, we use an FFT to compute efficiently what may be regarded as a sampled DTFT. We will look at how sampling density must be increased along the unit circle when spectral modifications are to be performed, and we will discuss further the COLA condition on the analysis window and hop-size. In the end, our practical FFT-convolution engine will look as follows: (9.8)

The sum over may be interpreted as adding separately filtered frames . Due to this filtering, the frames must overlap, even when the rectangular window is used. As a result, the overall system is often called an overlap-add FFT processor, or OLA processor'' for short. It is regarded as a sequence of FFTs which may be modified, inverse-transformed, and summed. This hopping transform'' view of the STFT is the Fourier dual of the filter-bank'' interpretation to be discussed in Chapter 9.

Subsections
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