chapter 3 discrete-time signals in the transform-domain

CHAPTER 3 Discrete-Time Signals in the

Transform-Domain

CHAPTER 3 Discrete-Time Signals in the

Transform-Domain

Wang Weilian

[email protected]

School of Information Science and Technology

Yunnan University

云南大学滇池学院课程：数字信号处理 Discrete-Time Signals in the Transform-Domain 2

OutlineOutline

• The Discrete-Time Fourier Transform

• The Discrete Fourier Transform

• Relation between the DTFT and the DFT, and

Their Inverses

• Discrete Fourier Transform Properties

• Computation of the DFT of Real Sequences

• Linear Convolution Using the DFT

• The z-Transform


OutlineOutline

• Region of Convergence of a Rational z-Transform

• Inverse z-Transform

• z-Transform Properties


The Discrete-Time Fourier TransformThe Discrete-Time Fourier Transform

• The discrete-time Fourier transform (DTFT) or, simply, the Fourier transform of a discrete–time sequence x[n] is a representation of the sequence in terms of the complex exponential sequence where is the real frequency variable.

• The discrete-time Fourier transform of a sequence x[n] is defined by

j xe

jX e

[ ]j j n

n

X e x n e



• In general is a complex function of the real variable and can be written in rectangular form as

where and are, respectively, the real and imaginary parts of , and are real functions of .

• Polar form

jX e

j

imX e jreX e

j j jre imX e X e jX e

jX e

where arg

jj j

j

e X e e

X e

X



• Convergence Condition:

If x[n] is an absolutely summable sequence, i.e.,

Thus the equation is a sufficient condition for the existence of the DTFT.

n

j j n

n n

if x n

then X e x n e x n



• Bandlimited Signals:

– A full-band discrete-time signal has a spectrum occupying the whole frequency rang .

– If the spectrum is limited to a portion of the frequency range , it is called a bandlimited signal.

– A lowpass discrete-time signal has a spectrum occupying the frequency range , where is called the bandwidth of the signal.

– A bandpass discrete-time signal has a spectrum occupying the frequency range , where is its bandwidth.

0

0

0 p p

0 L H H L



• Discrete-Time Fourier Transform Properties

There are a number of important properties of the discrete-time Fourier transform which are useful in digital signal processing applications. We list the general properties in Table 3.2, and the symmetry properties in Tables 3.3 and 3.4.



• Energy Density Spectrum

* *

-

2

Parseval's relation:

1 [ ] [ ] ( ) ( )

2

Total energy of a finite-energy sequence [ ] :

If [ ] [ ],

j j

n

gn

g n h n G e H e d

g n

g n

h n g n

2 2

-

2

then from Parseval's relation we observe

1 | [ ] | | ( ) |

2

The quantity:

is called the energy density spectrum of the sequence [ ].

jg

n

j jgg

g n G e d

S e G e

g n


The Discrete Fourier TransformThe Discrete Fourier Transform

• DTFT Computation Using MATLAB

– The Signal Processing Toolbox in MATLAB

– Functions:

• freqz

• abs

• Angle

– The forms of freqz:

• H = freqz(num, den, w)

• [H, w] = freqz(num, den, k, ’whole’)

– Example 3.8: Program 3_1



• Definition

The simplest relation between a finite-length

sequence x[n], defined for , and its

DTFT is obtained by uniformly sampling

on the -axis between at

, .

0 1n N

0 2

2 /k k N 0 1k N

1

2 /

2 /0

From [ ]

[ ] , 0 1

j j n

n

Nj j kn N

k Nn

X e x n e

X k X e x n e k N

jX e

jX e



• The sequence X[k] is called the discrete Fourier transform (DFT) of the sequence x[n].

• Using the commonly used notation

• We can rewrite as

• Inverse discrete Fourier transform (IDFT)

2 /j NNW e

1

0

[ ] [ ] , 0 1N

knN

n

X k x n W k N

1 12 /

0 0

[ ] [ ] [ ] , 0 1N N

j kn N knN

n n

x n X k e X k W n N



• Matrix Relations

The DFT samples defined in can

be expressed in matrix form as

where X is the vector composed of the N DFT samples,

x is the vector of N input samples,

1

0

[ ] [ ]N

knN

n

X k x n W

X xND

X 0 1 1T

X X X N 　

x [0] [1] [ 1]T

x x x N 　



• is the DFT matrix given by

• IDFT relations

ND N N

1 1 1

1

1

1

ND

1 2 N-1N N N

2 4 2(N-1)N N N

N-1 2(N-1) (N-1) (N-1)N N N

　　　 1

　 W　 W　 W

　 W　 W W

W 　 W 　 W

1 *1x X XN ND D

N



• DFT computation Using MATLAB

– MATLAB functions:

fft(x), fft(x,N), ifft(X), ifft(X,N)

– X = fft(x, N)

If N < R=length(x), truncate (截短 ) to the first N samples.

If N > R=length(x), zero-padded (补零 ) at the end.

– Example 3.11, 3.12, 3.13, Program 3_2, 3_3, 3_4.


Relation between the DTFT and the DFT, and their Inverses


• DTFT from DFT by Interpolation

We could express in terms of X[k]:

1 1 1

0 0 0

1 12 /

0 0

12 / 1 / 2

0

1 1[ ] [ ]

1 [ ]

2sin

1 2

2sin

2

N N Nj j n j nkn

Nn n k

N Nj kn N j n

k n

Nj k N N

k

X e x n e eX k WN N

X k e eN

N k

X k eN kNN

jX e




• Sampling the DTFT

– Consider the following question

– We obtain the relation

– Example 3.14

[ ] ( )

2 / , 0 -1

[ ], 0 -1 [ ] ( ), 0

?

-1 k

DTFT j

k

DFT j

x n X e

k N k N

y n n N Y k X e k N

[ ] [ ], 0 -1m

y n x n mN n N




• Numerical Computation of the DTFT Using the DFT

– Let be the DTFT of length-N sequence x[n]. We wish to evaluate at a dense grid of frequencies:( )jX e

( )jX e

1 12 /

0 0

12 /

0

2 / , 0 1, where

[ ] [ ]

[ ] 0 1Define a new sequence [ ]

0 -1

then { [ ]} [ ]

k k

k

k

k

N Nj j nj j kn M

n n

e

Mj j kn M

e en

k M k M M N

X e X e x n e x n e

x n n Nx n

N n M

X e DFT x n x n e


Discrete Fourier Transform PropertiesDiscrete Fourier Transform Properties

• Discrete Fourier Transform Properties

Like the DTFT, the DFT also satisfies a number of properties that are useful in signal processing application. A summary of the DFT properties are included in Tables 3.5, 3.6, and 3.7.


Discrete Fourier Transform PropertiesDiscrete Fourier Transform Properties

• Circular Shift of a Sequence

– Time-shifting property of the DTFT

– Circular shifting property of the DFT

0

0 0

0

[ ], 0 -1 [ ], 0 -1

[ ], 0 -1 [ ] [ ], 0 -1

We obtain [ ] [ ] [( )% ]

For >0, [

?

]

DFT

DFT knc c N

c N

c

x n n N X k k N

x n n N X k W X k k N

x n x n n x n n N

n x n

0 0

0 0

[ ] 1

[ ] 0 n

x n n n n N

x n n N n

01 0 1[ ] [ ] ( ) ( )

DTFT j nj jx n x n n X e e X e


Computation of the DFT of Real Sequences


• Computation of the DFT of Real Sequences

Tow N-point DFTs can be computed efficiently using a single N-point DFT X[k] of a complex length-N sequence x[n] defined by

where, and

x n g n jh n

Re{ [ ]}g n x n [ ] Im{ [ ]}h n x n




we arrive at:

Note that

*

*

1{ [ ] },

21

2

N

N

G k X k X k

H k X k X kj

* *

N NX k X N k


Linear Convolution Using the DFTLinear Convolution Using the DFT

• Linear Convolution of Two Finite-Length Sequences

Let g[n] and h[n] be finite-length sequences of lengths N and M, respectively. Denote L=M+N-1. Define two length-L sequences,

, 1

0. 1

, 1

0. 1

e

e

g n n Ng n

n L

h n n Mh n

n L

　 0

　 N

　 0

　 M


Linear Convolution Using the DFTLinear Convolution Using the DFT

obtained by appending g[n] and h[n] with

zero-valued samples. Then

• Linear Convolution of a Finite-Length Sequence with an Infinite-Length Sequence

– Overlap-Add Method

– Overlap-Save Method

L c

e e

y n g n h n y n

g n linear convolution h n


The z-TransformThe z-Transform

• Definition

For a given sequence g[n], its z-transform G(z) is defined as

where is a complex variable.

If we let , then the right-hand side of the above expression reduces to

n

n

G z Z g n g n z

Re Imz z j z jz re

j n j n

n

G re g n r e


The z-TransformThe z-Transform

For a given sequence, the set R of values of z

for which its z-transform converges is called

the region of convergence (ROC).

If

In general, the region of convergence R of a z-transform of a sequence g[n] is an annular region of the z-plane:

n

n

g n r

g gR z R


The z-TransformThe z-Transform• Rational z-Transforms

– An alternate representation as a ration of two polynomials in z:

– An alternate representation in factored form as


Region of Convergence of a Rational z-Transform

Region of Convergence of a Rational z-Transform

• The ROC of a rational z-transform is bounded by the locations of its poles.

– A finite-length sequence ROC:

– A right-sided sequence ROC:

– A left-sided sequence ROC:

– A two-sided sequence ROC:

0 z

gR z

gz R

g gR z R


Inverse z-TransformInverse z-Transform

• General Expression

– By the inverse Fourier transform relation. We have

– By making the change of variable , the above equation can be converted into a contour integral given by

Where is a counterclockwise contour of integration

defined by

1

2n j j ng n r G re e d

jz re

'

11

2n

Cg n G z z dz

j

'C

z r



• Inverse Transform by Partial-Fraction Expansion

can be expressed as

• We can divide P(Z) by D(Z) and re-express G(Z) as

G z

P zG z

D z

0

M N P zG z z

D z



• Simple Poles p168

• Multiple Poles p169


z-Transform Propertiesz-Transform Properties

• P174 Table 3.9


SummarySummary

• Three different frequency-domain representations of an aperiodic discrete-time sequence have been introduced and their properties reviewed .Two of these representations, the discrete-time Fourier transform (DTFT) and the z-transform, are applicable to any arbitrary sequence, whereas the third one , the discrete Fourier transform (DFT), can be applied only to finite-length sequences.

• Relation between these three transforms have been established. The chapter ends with a discussion on the transform-domain representation of a random discrete-time sequence.

• For future convenience we summarize below these three frequency-domain representations.


Assignment and ExperimentAssignment and Experiment

• Assignment

– A03: 3.2, 3.12, 3.20, See p180~182

– A04:

– A05:

• Experiment

– E03: Q3.3 See p32

– E04:

– E05

chapter 3 discrete-time signals in the transform-domain

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