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Inverse DFT

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Inverse DFT

- 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:

- 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

- To speed up the manual analysis, remember:

- Relate this to the argand diagram…

- Similarly

- So the vector rotates clockwise

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

- 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

- Processing cost for DFT is:

- Processing cost for FFT is:

- 1024 point:
DFT: 1048576x and 1047552+

FFT: 5120x and 10240+