The Fast Fourier Transform

Title

Introduction

What is the FFT?

- A collection of “tricks” that exploit the symmetryof the DFT calculation to make its execution muchfaster .

- Speedup increases with DFT size.

Computational Cost of Discrete-Time Filtering

Convolution of an N-point input with an M-point unit sample response .

- Direct convolution:

Number of multiplies ≈ MN

- Using transforms directly:

Computation of N-point DFTs requires N^2 multiplies.

Each convolution requires three DFTs of length N+M-1 plus an additional N+M-1 complex multiplies or

For , for example, the computation is O(N^2)

y[n] x[k]h[n k]k

X[k] x[n]e j2kn / N

n0

N 1

3(N M 1)2 (N M 1)N M

Computational Cost of Discrete-Time Filtering

Convolution of an N-point input with an M-point unit sample response .

- Using overlap-add with sections of length L:

N/L sections, 2 DFTs per section of size L+M-1, plus additional multiplies for the DFT coefficients, plus one more DFT for

- For very large N, still is proportional to

2N

L(L M 1)2

N

L

(L M 1) (L M 1)2

h[n]

2M

The Cooley-Tukey decimation-in-time algorithm

Consider the DFT algorithm for an integer power of 2,

Create separate sums for even and odd values of n:

Letting for n even and for n odd, we obtain

N 2

X[k]n0

N 1 x[n]WN

nk n0

N 1 x[n]e j2nk /N ; WN e j2 /N

X[k] x[n]WNnk

n even x[n]WN

nk

n odd

X[k] x[2r]WN2rk

r0

N / 2 1

x[2r 1]WN2r1 k

r0

N /2 1

12 rnn 2r

The Cooley-Tukey decimation-in-time algorithm

Splitting indices in time , we have obtained

But and

So

X[k] x[2r]WN2rk

r0

N / 2 1

x[2r 1]WN2r1 k

r0

N /2 1

WN2 e j22 / N e j2 /(N / 2) WN / 2

rkN

kN

kN

rkN WWWW 2/2

X[k] n0

(N/ 2) 1

x[2r]WN /2rk WN

k

n0

(N/ 2) 1

x[2r 1]WN / 2rk

N/2-point DFT of x[2r] N/2-point DFT of x[2r+1]

Signal flowgraph representation of 8-point DFT

Recall that the DFT is now of the form

The DFT in (partial) flowgraph notation:

][][][ kHWkGkX kN

Continuing with the decomposition …

So why not break up into additional DFTs? Let’s take the upper 4-point DFT and break it up into two 2-point DFTs:

The complete decomposition into 2-point DFTs

Closer look at the 2-point DFT

The expression for the 2-point DFT is:

Evaluating for we obtain

which in signal flow graph notation looks like …

This topology is referred as the BASIC BUTTERFLY

X[k] n0

1

x[n]W2nk

n0

1

x[n]e j2nk / 2

X[0] x[0] x[1]

X[1] x[0] e j21/ 2x[1] x[0] x[1]

k 0,1

The complete 8-point decimation-in-time FFT

Number of multiplys for N-point FFTs

• Let

(log2(N) columns)(N/2 butterflies/column)(2 mults/butterfly)

or multiplies

N 2 where log2 (N)

N log2(N)

CONCLUSION

Use of the FFT algorithm reduces the number of multiplys required to perform the DFT by a factor of more than 100 for 1024-point DFTs, with the advantage increasing with increasing DFT size.