Discrete Fourier Transform

12

Discrete Signal Representation

x(t) x[n] x[n] = x(nTs)

Ts

Small Ts closer samples dense sampling

x(t)

-6T -4T -2T 0 2T 4T 6T 8T 10Tt -6 -4 -2 0 2 4 6 8 10

x[n]

n

3

Relationship

4

Example 1Frequency Domain

Resulted signal

Time Domain

Ideal low pass filter

5

Spectrum of discrete-time signals

x(t): aperiodic continuous-time signal x[n]: samples of x(t) Spectrum of x(t): aperiodic Spectrum of x[n]: periodic

ˆj np

n

ˆX x[n]e

ˆj( 2 k)n

pn

ˆj n j2 nkp

n

ˆX 2 k x[n]e

ˆx[n]e e X ( )

1

6

Spectrum of discrete-time signals

Periodic

0 sˆ T 2-2

p ˆX ( )

7

Relationship

Relationship between the spectrum of x(t) and the spectrum of x[n]?

p s sks

1X ( T ) X( 2 k / T )T

ss

s

ss

1 X( 2 / T )T1 X( )T1 X( 2 / T )T

8

Discrete Fourier transform Spectrum of aperiodic discrete-time

signals: Periodic + continuous Difficult to be handled by computer Periodic spectrum one period is enough Continuous spectrum samples the

spectrum

9

Discrete Fourier Transform

FT of an aperiodic discrete sequence:

Assume x[n] is an aperiodic sequence with N samples x[n] : n=0, 1, …,N-1

ˆj np s

n

ˆ ˆX ( ) x[n]e where T

N 1ˆj n

pn 0

ˆX ( ) x[n]e

10

Discrete Fourier Transform

Interested only in N equally spaced frequencies of 1 period of the FT

2[ ] 0,1, 1pkX k X k NN

11

Discrete Fourier Transform

k2N 1 j nN

n 0N 1 N 1

j2 nk/N nkN

n 0 n 0

X[k] x[n]e

x[n]e x[n]W

nk j2 nk / NNW e

k2ˆN

Discrete Fourier Transform

for k = 0,1,…, N-1

12

Discrete Fourier Transform (DFT)

Discrete Fourier transform= output of the FT of an aperiodic discrete-

time signal at some particular frequenciesN 1

j2 nk/ N

n 0N 1

j2 n(k N)/ N

n 0N 1

j2 nk / N j2 n

n 0

X[k] x[n]e

X[k N] x[n]e

x[n]e e X[k]

13

Inverse DFT

Inverse DFT: Given X[k], Find the time domain signal, x[n]

N 1j2 nk / N

k 0

N 1nk

Nk 0

1x[n] X[k]eN1 X[k]WN

for n = 0,1,…, N-1

Inverse Discrete Fourier Transform

Frequency

DFT of a data sequence x[n] is another data sequence X[k]

What is the frequency represented by each component of X[k]?

2[ ] 0,1, 1pkX k X k NN

0 sˆ T 2-2

k = 0 k = 4

Frequency2[ ] 0,1, 1p

kX k X k NN

15

0 sˆ T 2-2

k = 0 k = 4

s

s s

ˆ 22 fT 2f 1 / T F

s

s

ˆ 4 * 2 / 132 fT 4 * 2 / 13f F * 4 / 13

16

Exercise 1

Given x[n]:

Determine its spectrum using DFT with N=4

Assume sampling frequency is 1000 Hz, what are the frequencies represented by the 4 DFT outputs?

0n

3

x[n]

12 2

0

x[n] = {1, 2, 2, 0}

17

Solution

4 1j2 nk/4

n 0

X[k] x[n]e for k 0,1, ,4 1

18

Frequency resolution of DFT

DFT gives exact frequency response of a signal, sometimes may not give the desired spectrum

0n

9N = 10N = 10

x[n] p ˆX ( )One period of

k

10 X[k] if N = 10So different from

p ˆX ( )

FourierTransform

DFT

19

Frequency resolution of DFT

Improved Resolution! Achieved by padding zeros to the end of

x[n] to make N bigger

0n

50

x[n]

9

x[n] = {1 1 1 1 1 1 1 1 1 1 0 0 0 0 … 0}

Pad 40 zerosPad 40 zeros

49

20

DFT of zero padded sequence

N = 50N = 50

21

DFT of zero padded sequence

N = 100N = 100

22

Properties of DFT

DFT

Linear property: x1[n] X1[k] x2[n] X2[k]

N 1nkN

n 0X[k] x[n]W

1 2 1 2aX [k] bX [k] DFT ax [n] bx [n]

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Properties of DFT

Circulant property (periodicity): x[n] with N samples X[k]

Shift property: x[n] X[k] y[n] = x[n-m]

X[rN k] X[k]

X[ k] X[N k]

0 k

X[k]

-N N

mkNY[k] W X[k]

24

Exercise x(t): contains 256 data, is sampled at the sampling

rate of 25.6 kHz The frequency response of x(t) is computed using a

256-point DFT (i) if it is required to find out the frequency

response of x(t) at 8000 Hz, indicate which output of the DFT should be examined

(ii) If it is required to find out the frequency response of x(t) at 4480 Hz using DFT, discuss how many zeros should be padded to the end of x(t)

25

Summary

DSP system Spectrum of discrete-time signals

Periodic + continuous Shannon sampling theorem DFT:

Discrete spectrum

26

References M.J. Roberts, Fundamentals of

Signals & Systems, McGraw-Hill, 2008. Chapters 11 and 14

James H. McClellan, Ronald W. Schafer and Mark A. Yoder, Signal Processing First, Prentice-Hall, 2003. Chapters 4, 12, 13