Chapter 8 The Discrete Fourier Transform

IntroductionDefinition of the Discrete Fourier TransformProperties of the DFTLinear Convolution Using the DFTDCT Concluding Remarks

1

1. Introduction

Fourier Transform Pair

Required TransformFinite Signal Length Discrete Frequency Samples

DFT ApplicationsSignal AnalysisComputing benefits from the convolution

X e x nj jn

n

( ) ( ) e x pω ω= −

= − ∞

∞

∑

x n X e e dj jn( ) ( )=−∫

1

2πωω ω

π

π

2

2. Discrete Fourier Transform

ConsiderationsThe Discrete-time Fourier transform is defined for sequences with finite or infinite length.The DFT is defined only for sequences with finite length.

Formulation-- Forward TransformConsider the sequence of length Nx[n] for n=0,1, 2, ..., N-1

The discrete-time Fourier transform is

It is periodic with period 2π.

X x n e x n ej n

k

j n

k

N

( ) [ ] [ ]ω ω ω= =−

= − ∞

∞−

=

−

∑ ∑0

1

3

2. Discrete Fourier Transform (c.1)

Formulation-- Forward Transform (c.1)The usually considered frequency interval is (-π , π ]. There are infinitely many points in the interval. If x[n] has N points, we compute N equally spaced w in (-π , π ]. These N points are

So we define

k and n are in the interval from 0 to N-1.

Formulation-- Inverse TransformCompute x[n] from N frequency components

So

2

~( ) ( ) [ ] /X k X k

Nx n e jnk N

n

N

= = −

=

−

∑2 2

0

1π π

x n X e d X eN

jkk

jk n N

k

N

[ ] ( ) ( ) ( )/= ≈ ⋅= =

−

∫ ∑1

2

1

2

2

0

2

2

0

1

πω ω

πω

πω

ω

ππ

x nN

X k e jk n N

m

N

[ ] ( ) /≈=

−

∑1 2

0

1π

ω π= = −k

Nk N

20 1 2 1, , , , . . . ,

4

2. Discrete Fourier Transform (c.2)

The Discrete Fourier Transform

Notes

Periodic property of X[k]– WknN =1; X[k] = X[k+-N]= X[k+-2N] = ...

DFT compute the samples of the discrete-time Fourier transform of x[n].

The Inverse yields a periodic x[n] with period N

The relation with the Discrete Fourier series– X[k] = Nck

1,,2,1,0,;

)(1)(1][

][][)(

/2

1

0

1

0

/2

1

0

1

0

/2

−==

==

==

−

−

=

−−

=

−

=

−

=

−

∑∑

∑∑

NnkeWwhere

WkXN

ekXN

nx

WnxenxkX

Nj

N

k

nkN

n

Njkn

N

n

knN

n

Nknj

Kπ

π

π

X kX k fo r k N

period ic ex ten sion o f X k[ ]

[ ]

[ ]=

≤ ≤ −⎧⎨⎩

0 1

5

2. Properties of the DFT

Let x'[n] is the periodic extension of the x[n]

Linearity PropertiesD[a1x1[n] + a2x2[n]] = a1 D[x1[n]] + a2 D[x2[n]]

Symmetric PropertiesIf x'[n] is real then X[k] is conjugate symmetric, or X[-k] = X*[k].

If x'[n] is real and even, then X[k] is real and even.

If x'[n] is real and odd, then X[k] is imaginary and odd.

Duality PropertiesLet X[k] and x'[n] be the DFT pair. That isX[k] = D[x'[n]] and x'[n] = D-1 [X[k]].

then

x'[n] = {D[X [k]/N]}*

The DFT can be used to compute the inverse DFT.

Time-Shifting & Frequency-Shifting PropertiesD[x'[n-n0]] = e-jkn0 X[k] & D[x'[n]ejnk0 ] = X[k-k0]

6

2. Properties of the DFT-- Periodic Convolution

TopicCompute convolution of two finite sequence using the DFT

Consideration

Convolution– Convolution of two sequences h[n] & u[n] for n=0, 1, 2, ..., N-1.

– The nonzero elements are at most in the interval [0, 2N-2].

Periodic or Cyclic Convolution– If u1[n] and u2[n] are two periodic sequence with period N. The cyclic convolution of the two sequence

is defined as

where n=0, 1, 2, ..., N-1.

Cyclic Convolution & DFT

The DFT of y[n] is equal to the multiplication of the DFT of u1[n] and u2[n].Y[k] = U1 [k]U2 [k]

y n h n i u i h i u n ii

n

i

n

[ ] [ ] [ ] [ ] [ ]= − = −= =

∑ ∑0 0

y k u k i u i u i u k ii

N

i

N

[ ] [ ] [ ] [ ] [ ]= − = −=

−

=

−

∑ ∑1 20

1

1 20

1

7

2. Properties of the DFT-- Periodic Convolution (c.1)

Periodic Convolution

u[0] h[0]

h[1]

h[2]

u[2]

u[1]

u[1] h[0]

h[1]

h[2]

u[2]

u[0]

u[2] h[0]

h[1]

h[2]

u[0]

u[1]y[0]

y[1]

y[2]

8

Periodic Convolution9

2. Properties of the DFT-- Periodic Convolution (c.2)

Definition

1,,2,1,0,;

)(1)(1][

][][)(

/2

1

0

1

0

/2

1

0

1

0

/2

−==

==

==

−

−

=

−−

=

−

=

−

=

−

∑∑

∑∑

NnkeWwhere

WkXN

ekXN

nx

WnxenxkX

Nj

N

k

nkN

n

Njkn

N

n

knN

n

Nknj

Kπ

π

π

10

3. Implementing LTI Systems Using the DFT

Linear ConvolutionFinite length h[n] and indefinite-length x[n]

Method 1Nonoverlapping-and-add

wherex n x n r Lr

r

[ ] [ ]= −=

∞

∑0

x nx n rL n L

otherwiser [ ][ ],

,=

+ ≤ ≤ −⎧⎨⎩

0 1

0

11

3. Implementing LTI Systems Using the DFT (c.1)

Method 1 (c.1)

Method 2Overlapping-and-skipping

The output

where

y n x n h n y n rLr

r

[ ] [ ] * [ ] [ ]= = −=

∞

∑0

y n x n h nr r[ ] [ ] * [ ]=

x n x n r L P P n Lr [ ] [ ( ) ],= + − + − + ≤ ≤ −1 1 0 1

y n y n r L P Pr

r

[ ] [ ( ) ]= − − + + −=

∞

∑ 1 10

y ny n P n L

otherwiserrp[ ][ ],

,=

− ≤ ≤ −⎧⎨⎩

1 1

0

12

The Discrete Cosine Transform (DCT)

General Transform

Four types of DCT is defined for different periodicity

13

Type I Type II

Type III Type IV

The Discrete Cosine Transform (DCT)

Type I DCTExtend x(n) by

We have

14

The Discrete Cosine Transform (DCT)

Type II DCTExtend x(n) by

We have

15

The Discrete Cosine Transform (DCT)

Other Variant with different normalization factor

16

The Discrete Cosine Transform (DCT)

Energy Compaction

17

DCT18

DCT

Error Comparison

19

Homeworks

Deadline= May, 11, Class time8.31, 8.33, 8.40, 8.41, 8.42

20