FFT-based filtering and the

Using the FFT for DSP Because the FFT provides the means to reduce the computational complexity of the DFT from order (N2) to order (N log2(N)), it is often desirable to do FFT-based processing for DSP systems Even the computational cost of doing both FFT and IFFT may be less than conventional methods

Linear Convolution via FFT

Convolution FIR digital filter computes the linear convolution of the unit sample response with the input signal A filter of length P requires P2 complexity Recall that convolution in time domain is multiplication in frequency domain So, consider multiplying signal and filter transforms in the frequency domain

Circular Convolution The DFT is a sampled version of the Fourier transform, so multiplying DFTs corresponds to circular convolution Circular convolution can be thought of as “time-domain aliasing” If we want linear convolution, we must ensure time-limited input signals to avoid time-domain aliasing (like bandlimiting to avoid frequency-domain aliasing)

Linear convolution with the DFT Consider a unit sample response h[n] with finite length P, and a signal x[n] of length L Linear convolution h*x has length L+P-1 To avoid time-domain aliasing, we zero pad both sequences to at least length L+P-1, do FFT, multiply the transforms, then IFFT to get L+P-1 result

Overlap Processing Now consider filter response of length P, but assume input signal is of arbitrarily long length: need to run filter “on the fly” as blocks of input data become available Plan: break signal into consecutive blocks of length L, pad each with zeros to length L+P-1, and do FFT/multiply/IFFT

Overlap algorithm Note that the last P-1 output samples will overlap the start of the next block, and the overlapping points must be added to get the proper response. This is known as the overlap-add algorithm.

Overlap-Add Process x[n] h[n] y0[n]=h[n]*x0[n] y1[n]=h[n]*x1[n] L 2L 3L 4L x[n] h[n] y0[n]=h[n]*x0[n] y1[n]=h[n]*x1[n] y2[n]=h[n]*x2[n] y3[n]=h[n]*x3[n]

Short-Time Fourier Transform

Short-Time Fourier Transform It is often desirable to have an estimate of the input signal spectrum for a “short” interval, especially for non-stationary signals. Want to see changes in spectrum with time. The Fourier transform gives the frequency response, but it has infinite summation

STFT (cont.) Consider calculating a spectral “snapshot” by calculating Fourier transform of a short interval of the input signal w[n-m] x[n]

STFT (cont.) Express the short-time Fourier transform as a 2-dimensional signal Multiplying signal by a short time function causes smeared spectrum: convolution of the transforms

STFT viewpoints Fourier transform viewpoint: group x[]·w[] Take a sequence of DFTs as the window w[n-m] slides along the signal Filter bank viewpoint: group x[n]e-jwn Transform of w[n] is a low pass function x[n]e-jwn is modulation: shifts spectrum of x[n] by w, where it can be filtered by low pass function

STFT Reconstruction With some restrictions, it is possible to do perfect reconstruction. STFT and inverse STFT are a transform pair. Typically use overlapping window functions during analysis, then overlap-add during synthesis

DSP Analysis/Process/Synthesis Read in then next overlapping block of input samples Apply analysis window Perform FFT Do frequency domain processing Perform IFFT Overlap-add into the output buffer