1. FFT-based filtering and the

2. 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

3. Linear Convolution via FFT

4. 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

5. 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)

6. 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

7. 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

8. 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.

9. 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]

10. Short-Time Fourier Transform

11. 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

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

13. 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

14. 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

15. 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

16. 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