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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

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  • 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

Dft

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

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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
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Graphene has a hexagonal structure
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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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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