Inverse dft
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Inverse DFT. Frequency to time domain. Sometimes calculations are easier in the frequency domain then later convert the results back to the time domain Convert Time -> Frequency with DFT Convert Frequency -> Time with the Inverse Discrete Fourier Transform. From Last week, the DFT is:.

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Inverse DFT

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Inverse dft

Inverse DFT


Frequency to time domain

Frequency to time domain

  • Sometimes calculations are easier in the frequency domain then later convert the results back to the time domain

  • Convert Time -> Frequency with DFT

  • Convert Frequency -> Time with the Inverse Discrete Fourier Transform


Inverse dft

  • From Last week, the DFT is:

  • The IDFT is:

Where x is effectively a row matrix of size N

h is the required harmonic

N is number of Fourier coefficients

F(h) is the complex DFT value


Inverse dft

  • To speed up the manual analysis, remember:

  • Relate this to the argand diagram…


Inverse dft

  • Similarly

  • So the vector rotates clockwise


Example

Example

  • Consider the 4 DFT values generated from last week’s example: {2,1+j,0,1-j}


Dft processing cost

DFT processing cost

  • DFT processing cost is expensive

    • Each term is a product of a complex number

    • Each term is added so for an 8 point DFT need 8 multiplies and 7 adds (N and N-1)

    • There are 8 harmonic components to be evaluated (h=0 to 7)

    • So an 8 point DFT requires 8x8 complex multiplications and 8x7 complex additions

    • An N point transform needs N2 Complex multiplications and N(N-1) complex adds


Fast fourier transform

Fast Fourier Transform

  • Processing cost for DFT is:

  • Processing cost for FFT is:

  • 1024 point:

    DFT: 1048576x and 1047552+

    FFT: 5120x and 10240+


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