lec dft fft

Dr. Shoab Khan

Digital Signal Processing

Lecture

Discrete-Time Fourier Series

The conventional (continuous-time) FS represent a periodic signal using an infinite number of complex exponentials, whereas the DFS represent such a signal using a finite number of complex exponentials

Example 2 DFS of an periodic rectangular pulse train

The DFS coefficients

10/ksin

2/ksine

e1

e1ekX

~ 10/k4j

k10/2j

5k10/2j4

0n

kn10/2j

Properties of DFS Linearity

Shift of a Sequence

Duality

kX~

bkX~

anx~bnx~a

kX~

nx~kX

~nx~

21DFS

21

2DFS

2

1DFS

1

mkX~

nx~e

kX~

emnx~kX

~nx~

DFSN/nm2j

N/km2jDFS

DFS

kx~NnX

~kX

~nx~

DFS

DFS

Periodic Convolution Take two periodic sequences

Let’s form the product

The periodic sequence with given DFS can be written as

Periodic convolution is commutative

kX

~nx~

kX~

nx~

2DFS

2

1DFS

1

kX~

kX~

kX~

213

1N

0m213 mnx~mx~nx~

1N

0m123 mnx~mx~nx~

Graphical Periodic Convolution

DTFT to DFT

Sampling the Fourier Transform Consider an aperiodic sequence with a Fourier transform

Assume that a sequence is obtained by sampling the DTFT

Since the DTFT is periodic resulting sequence is also periodic

We can also write it in terms of the z-transform

The sampling points are shown in figure

could be the DFS of a sequence

Write the corresponding sequence

kN/2j

kN/2

j eXeXkX~

jDTFT eX]n[x

kN/2j

ezeXzXkX

~kN/2

kX~

1N

0k

knN/2jekX~

N

1]n[x~

DFT Analysis and Synthesis

DFT is Periodic with period N

Properties of DFT Linearity

Duality

Circular Shift of a Sequence

kbXkaXnbxnax

kXnx

kXnx

21DFT

21

2DFT

2

1DFT

1

mN/k2jDFT

N

DFT

ekX1-Nn0 mnx

kXnx

N

DFT

DFT

kNxnX

kXnx

DFT Properties

Example: Circular Shift

Example: Circular Shift

Example: Circular Shift

Duality

Circular Flip

Properties: Circular Convolution

Example: Circular Convolution

Example: Circular Convolution

illustration of the circular convolution process

Example (continued)

Illustration of circular convolution for N = 8:

03/17/2011 CS267 Lecture 18

Using the 2D FFT for image compression

° Image = 200x320 matrix of values

° Compress by keeping largest 2.5% of FFT components

° Similar idea used by jpeg

void dft(complex_float* in, complex_float* out, int N, int inv)

{

int i, j;

float a, f;

complex_float s, w;

f = inv ? 1.0/N : 1.0;

for (i = 0; i < N; i++) {

s.re = 0;

s.im = 0;

for (j = 0; j < N; j++) {

a = -2*PI*i*j/N;

if (inv) a = -a;

w.re = cos(a);

w.im = sin(a);

s.re += in[j].re * w.re - in[j].im * w.im;

s.im += in[j].im * w.re + in[j].re * w.im;

}

out[i].re = s.re*f;

out[i].im = s.im*f;

}

DFT & IDFT

DFT Properties

Circular Convolution

Filtering of Long Data Sequences

The input signal x(n) is often very long especially in real-time signal processing applications.

For linear filtering via the DFT, for example, the signal must be limited size due to memory requirements

Filtering of Long Data Sequences

Segment the input signal into fixed-size blocks prior to processing.

Compute DFT-based linear filtering of each block separately via the FFT.

Fit the output blocks together in such a way that the overall output is equivalent to the linear filtering of x(n) directly.

Main advantage: samples of the output y(n) = h(n) ⊗ x(n) will be available real-time on a block-by-block basis.

Filtering of Long Data Sequences

Goal: FIR filtering: y(n) = x(n) * h(n)

Two approaches to real-time linear filtering of long inputs:

Overlap-Add Method

Overlap-Save Method

Assumptions:

FIR filter h(n) length = P

Block length = L ≫ P

For example, consider two sequences h[n] and x[n] as follows.

Segmenting x[n]

into L-length

sequences. Each

segment is padded

by P1 zero values.

Fir filtering by

using the

overlapping-add

method.

Overlap-Add Method

1. Break the input signal x(n) into non-overlapping blocks xm(n) of length L.

2. Zero pad h(n) to be of length N = L + M -1

3. Take N-DFT of h(n) to give H(k), k = 0; 1, …,N-1

4. For each block m:

1. Zero pad xm(n) to be of length N = L + M-1

2. Take N-DFT of xm(n) to give Xm(k), k = 0,1,…,N-1

3. Multiply: Ym(k) = Xm(k)H(k), k = 0,1,…,N -1

4. Take N-IDFT of Ym(k) to give ym(n), n = 0,1,…,N-1

5. Form y(n) by overlapping the last M-1 samples of ym(n) with the first M -1 samples of ym+1(n) and adding the result

Example of overlapping-save method

Decompose x[n] into overlapping sections

of length L

Example of overlapping-save method (continue)

Result of circularly convolving each section with h[n]. The portions

of each filter section to be discarded in forming the linear convolution

are indicated

Overlap-Save Method

Overlap-Save Method

Using DFT for Circular Convolution

N = L + M -1

Let xm(n) have support n = 01,2,…,N -1

Let h(n) have support n = 01,2,…,M-1

We zero pad h(n) to have support n = 01,2,…,N-1

1. Take N-DFT of xm(n) to give Xm(k), k = 0,1,2,….,N-1

2. Take N-DFT of h(n) to give H(k), k = 0,1,2,….,N-1

3. Multiply: Ym(k) = Xm(k) H(k), k = 0,1,2,….,N-1

4. Take N-IDFT of Ym(k) to give yC,m(n), n = 01,2,…,N-1