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CE 40763 Digital Signal Processing Fall 1992 Fast Fourier transform (FFT)

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CE 40763Digital Signal ProcessingFall 1992Fast Fourier transform (FFT)

HosseinSameti

Department of Computer Engineering

Sharif University of Technology

- Many real-life systems can be modeled by LTI systems use convolution for computing the output use DFT to compute convolution
- Fast Fourier Transform (FFT) is a method for calculating Discrete Fourier Transform (DFT) Only faster!
- Definition of DFT:
- How many computations?

N pt. DFT of x(n)

Q: For each k:

How many adds and how many mults?

A: (N-1) complex adds and N complex mults.

How many k values do we have?

N

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Direct computation:

Ideal case:

FFT:

Example:

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

FFT

Decimation in time

Decimation in frequency

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

- The main idea: use the divide and conquer method

- It works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly.

- The solutions to the sub-problems are then combined to give a solution to the original problem.

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

N: power of 2

n: even

n: odd

n: even

n=2r

n:0N-2

r:0N/2-1

n: odd

n=2r+1

n:1N-1

r:0N/2-1

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Suppose:

- What are G(k) and H(k)?

- In G(k) and H(k), k varies between 0 and N/2-1.

- However, in X(k) , k varies between 0 and N-1.

Solution: use the relationship between DFS and DFT.

We thus need to replicate G(k) and H(k) “once”, to get X(k).

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

pt. DFT

g(r)

After replication

+

pt. DFT

h(r)

(twiddle factor)

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

pt. DFT

g(r)

pt. DFT

h(r)

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

N/2 pt. DFT block

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

r(0)

r(1)

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

r(0)

r(1)

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Flow graph of a the 2-pt. DFT

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

How many stages do we have?

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

2 mults+ 2 adds

1 mult+ 2 adds

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

In-place computation (only N storage locations are needed)

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

- How many stages do we have

- Each stage has N inputs and N outputs.

- Each butterfly has 2 inputs and 2 outputs.

- Each stage has butterflies.

- Each butterfly needs 1 mult and 2 adds.

Total number of operations:

adds

mults

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Output indexing is in order.

input indexing is shuffled.

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

- The main idea: use the divide and conquer method (this time in the frequency domain)

- Divide the computation into two parts: even indices of k and odd indices of k.

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

1

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

N/2 pt. DFT of g(n)

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

-1

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

N/2 pt. DFT of h(n)

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

- Change x with X (i.e., input nodes with output nodes)
- Change X with x (i.e., output nodes with input nodes)
- Reverse the order of the flow graphs.
- The same system function is achieved.

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

- How can we deal with twiddle factors?
- Should we store them in a table (i.e, use a lookup table) or should we calculate them?
- What happens if N is not a factor of 2?
- It can be shown that if N=RQ, then an N pt. DFT can be expressed in terms of R Q-pt. DFT or Q R pt. DFTs (Cooley-Tukey algorithm).

Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology