FFT-based filtering and the Short-Time Fourier Transform (STFT) R.C. Maher ECEN4002/5002 DSP Laboratory Spring 2003

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher2 Using the FFT for DSP Because the FFT provides the means to reduce the computational complexity of the DFT from order (N 2 ) to order (N log 2 (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

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher4 Convolution FIR digital filter computes the linear convolution of the unit sample response with the input signal A filter of length P requires P 2 complexity Recall that convolution in time domain is multiplication in frequency domain So, consider multiplying signal and filter transforms in the frequency domain

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher5 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)

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher6 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

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher7 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

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher8 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.

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher9 Overlap-Add Process x[n] 0L2L3L4L y 0 [n]=h[n]*x 0 [n] h[n] y 1 [n]=h[n]*x 1 [n] y 2 [n]=h[n]*x 2 [n] y 3 [n]=h[n]*x 3 [n]

Short-Time Fourier Transform

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher11 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

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher12 STFT (cont.) Consider calculating a spectral “snapshot” by calculating Fourier transform of a short interval of the input signal x[n] w[n-m]

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher13 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

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher14 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 -j  n –Transform of w[n] is a low pass function –x[n]e -j  n is modulation: shifts spectrum of x[n] by , where it can be filtered by low pass function

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher15 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

ECEN4002 Spring 2003Fast Convolution and the STFT R. C. Maher16 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