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# CE 40763 Digital Signal Processing Fall 1992 Fast Fourier transform (FFT) - PowerPoint PPT Presentation

CE 40763 Digital Signal Processing Fall 1992 Fast Fourier transform (FFT). Hossein Sameti Department of Computer Engineering Sharif University of Technology. Motivation.

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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

### Motivation

• 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

### Motivation

Direct computation:

Ideal case:

FFT:

Example:

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

### Algorithms for calculating FFT

FFT

Decimation in time

Decimation in frequency

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

### Decimation in time

• 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

### Decimation in time

N: power of 2

n: even

n: odd

n: even

n=2r

n:0N-2

r:0N/2-1

n: odd

n=2r+1

n:1N-1

r:0N/2-1

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

### Decimation in time

Suppose:

• What are G(k) and H(k)?

### Decimation in time

• 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

### Decimation in time

pt. DFT

g(r)

After replication

+

pt. DFT

h(r)

(twiddle factor)

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

### Decimation in time

pt. DFT

g(r)

pt. DFT

h(r)

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

### Example of Decimation in time (N=8)

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

### Example of Decimation in time (N=8)

N/2 pt. DFT block

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

### Example of Decimation in time (N=8)

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

### Example of Decimation in time (N=8)

r(0)

r(1)

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

### Example of Decimation in time (N=8)

r(0)

r(1)

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

### Example of Decimation in time (N=8)

Flow graph of a the 2-pt. DFT

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

### Example of Decimation in time (N=8)

How many stages do we have?

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

### General form of a butterfly

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

### Revised form of a butterfly

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

### Revised form of a butterfly

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

### Final figure for 8-pt DFT

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

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

### Computational complexity

• 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:

mults

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

### Indexing of the inputs and outputs

Output indexing is in order.

input indexing is shuffled.

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

### Bit reversing

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

### Re-arranging the input order

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

### Re-arranging the input order

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

### Decimation in frequency

• 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

### Decimation in frequency

1

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

### Decimation in frequency

N/2 pt. DFT of g(n)

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

### Decimation in frequency

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

### Decimation in frequency

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

### Decimation in frequency

-1

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

### Decimation in frequency

N/2 pt. DFT of h(n)

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

### Decimation in frequency

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

### Format of the Last stage Butterfly in Decimation in frequency

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

### Decimation in frequency

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

### Decimation in frequency (re-order the output)

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

### Decimation in frequency (ordered input and output)

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

### Transposition theorem

• 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

### Decimation in frequency

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

### Transposed version of the previous figure (Decimation in Time)

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

### Practical issues

• 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