Some Data about FFT

Periodicity & Symmetry of DFT

Decimation抽取 in Time (DIT)

Decimation in Frequency (DIF)

Computation Costs

DIT Radix基-4 FFT Structure

Homework

           p486: 11.27,  p486: 11.28

Some Data about FFT

Periodicity & Symmetry of DFT

                          X [k+rN] = X [k]

Decimation in Time FFT

      Break x[n] into Even & Odd indexed sequence:

               y [n] = x[2n]            z [n] = x [2n+1]

      where  n = 0, 1, 2, ..., N/2-1.

      Then for any fixed  k ≤ N/2-1,

         so

  2-point Butterfly or Twiddle:

   DIT N-Point FFT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Decimation in Frequency  FFT

 

      Break down x[n] into 2 halves:

             

 

 

      For Even & Odd k:

           

 

      where  r = 0, 1, 2, ..., N/2-1.  Now Set

 

      Then

            

 

 

 

   2-point Butterfly or Twiddle:

 

   DIF N-Point FFT

Computation Costs

          For all the N samples of X [k],

DIT Radix-4 FFT Structure