Inverse DFT

1. Inverse DFT

2. 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

3. 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

4. To speed up the manual analysis, remember: • Relate this to the argand diagram…

5. Similarly • So the vector rotates clockwise

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

7. 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

8. Fast Fourier Transform • Processing cost for DFT is: • Processing cost for FFT is: • 1024 point: DFT: 1048576x and 1047552+ FFT: 5120x and 10240+