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This module describes FFT, convolution, filtering, LTI systems, digital filters and circular convolution.

Important application of the fft

How many complex multiplies and adds are required to convolve two N -pt sequences? y n m 0 N 1 x m h n m

There are 2 N 1 non-zero output points and each will be computed using N complex mults and N 1 complex adds. Therefore, Total Cost 2 N 1 N N 1 O N 2

Got questions? Get instant answers now!
  • Zero-pad these two sequences to length 2 N 1 , take DFTs using the FFT algorithm x n X k h n H k The cost is O 2 N 1 2 N 1 O N N
  • Multiply DFTs X k H k The cost is O 2 N 1 O N
  • Inverse DFT using FFT X k H k y n The cost is O 2 N 1 2 N 1 O N N

So the total cost for direct convolution of two N -point sequences is O N 2 . Total cost for convolution using FFT algorithm is O N N . That is a huge savings ( ).

Summary of dft

  • x n is an N -point signal ( ).
  • X k n 0 N 1 x n 2 N k n n 0 N 1 x n W N k n where W N 2 N is a "twiddle factor" and the first part is the basic DFT.

What is the dft

X k X F k N n 0 N 1 x n 2 F n where X F k N is the DTFT of x n and n 0 N 1 x n 2 F n is the DTFT of x n at digital frequency F . This is a sample of the DTFT. We can do frequency domain analysis on a computer!

Inverse dft (idft)

x n 1 N n 0 N 1 X k 2 N k n

  • Build x n using Simple complex sinusoidal building block signals
  • Amplitude of each complex sinusoidal building block in x n is 1 N X k

Circular convolution


x n h n X k H k

Regular convolution from circular convolution

  • Zero pad x n and h n to length length x length h 1
  • Zero padding increases frequency resolution in DFT domain ( )

8-pt DFT of 8-pt signal
16-pt DFT of same signal padded with 8 additional zeros

The fast fourier transform (fft)

  • Efficient computational algorithm for calculating the DFT
  • "Divide and conquer"
  • Break signal into even and odd samples keep taking shorter and shorter DFTs, then build N -pt DFT by cleverly combining shorter DFTs
  • N -pt DFT: O N 2 O N 2 logbase --> N

Fast convolution

  • Use FFT's to compute circular convolution of zero-padded signals
  • Much faster than regular convolution if signal lengths are long
  • O N 2 O N 2 logbase --> N

See .

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
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or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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