Digital Signal Processing, MM2.ppt - Aalborg Universitetkom.aau.dk/~zt/courses/Digital_signal_processing/MM2/Digital Signal... · 1 Digital Signal Processing, Fall 2010 Lecture 2:

1

Digital Signal Processing, Fall 2010

Lecture 2: Fourier transform, freq enc response and Z transform

Zheng-Hua Tan

frequency response, and Z-transform

Digital Signal Processing, II, Zheng-Hua Tan1

Department of Electronic SystemsAalborg University, Denmark

[email protected]

Course at a glance

Discrete-time signals and systems

MM1 System

Fourier transform and Z-transform

Filt d i

MM2

Sampling andreconstruction

MM3

Systemanalysis


DFT/FFT

Filter design

MM5

MM4

2

Part I-A: Fourier representation

Fourier transform and frequency response Frequency-domain representation of discrete-time

signals and systemssignals and systems

Fourier transform theorems

Z-transform


Signal representation & system response

Impulse response of LTI systems

][][][][ nhnynnx

A sum of scaled, delayed impulse

k

knkxnx ][][][ ][*][][][][ nhnxknhkxnyk


Convolution sum!

3

System response to complex exponentials

Complex exponentials as input to LTI system h[n]

nj nenx ,][

Define

nj

k

kj

k

knjnj

eekh

ekheTny

)][(

][}{][ )(

k

kjj ekheH ][)(

Frequency and impulse F i


Then njj

k

eeHny )(][ responses are a Fourier

transform pair

signal representation based on complex exponentials, i.e. Fourier representation.

Fourier representation Fourier representation of a sequence: Fourier transform and the inverse

Fourier transform

jjjj

1

FT “is an operation that transforms one function (typically in the time domain) into another (in frequency domain). So the FT is often called the frequency domain representation of the original function & describes which frequencies are present in the original function. The term FT refers both to the frequency domain representation and to the process or f l th t "t f " f ti i t th th ”

deeXnxenxeX njj

n

njj

)(2

1][,][)(

formula that "transforms" one function into the other.”

“Fourier transform defines a relationship between a signal in the time domain and its representation in the frequency domain. … The original signal can be recovered from knowing the Fourier transform, and vice versa.”

6 Digital Signal Processing, II, Zheng-Hua Tan

4

System input and output

)(][ ][ njjnj eeHnyenx

knkxnx ][][][

knhkxny ][][][

)()( :Output )( :Input

)()(2

1][ )(

2

1][

)(][][

jjj

njjjnjj

eHeXeX

deeHeXnydeeXnx

y

][][ ][][ nhnynnx


k

knkxnx ][][][ k

y ][][][

Frequency response

The frequency response of discrete-time LTI systems is always a periodic function of the frequency variable w with period .2frequency variable w with period .

Only specify over the interval

The ‘low frequencies’ are close to 0

)(][][)( 2)2()2( j

n

njnj

n

njj eHeenhenheH

2


The ‘high frequencies’ are close to

5

Frequency response

Frequency response is generally complex

)()()( jI

jR

j ejHeHeH

Frequency response of the ideal delay system

dnj

n

njd

j eenneH

][)(

)(|)(| jeHjj eeH describes changes in magnitude and phase

][][ ][][ dd nnnhnnxny

Calculate Fourier transform


.)(,1|)(|

).sin()(),cos()(

djj

dj

Idj

R

neHeH

neHneH

n

Ideal frequency-selective filters

For which the frequency response is unity over a certain range of frequencies, and is zero at the remaining frequencies.remaining frequencies. Ideal low-pass filter: passes only low and rejects high


6

Sufficient condition for Fourier transform

Condition for the convergence of the infinite sum

|][| |)(| enxeX njj

If x[n] is absolutely summable, its Fourier transform i t ( ffi i t diti )

|][|

|||][|

nx

enx nj


exists (sufficient condition).

Example 1: Exponential sequence

Let nuanx n][

The Fourier transform of it

1|| if 1

1

)()( 00

aae

eaeaeX

j

nj

n

nj

n

nj


1 ae

7

Example 2: ideal lowpass filter

Frequency response

|| ,0

|| ,1)(

c

cjlp eH

nsin1

Noncausal.

Not absolutely summable.

nn

ndenh cnj

lpc

c,

sin

2

1][

njcjlp e

neH sin

)(


n

p n

M

Mn

njcjM e

n

neH

sin

)(

Example 3: a constant sequence

Constant sequence

1][ nxnjj

njj

enxeX

deeXnx

][)(where

)(2

1][

Its Fourier transform is defined as the periodic impulse train

)2(2)( reXr

j

enxeX ][)( where


8

Part I-B: Fourier transform theorems




Z-transform


Linearity of the Fourier Transform

deeXnxenxeX njj

n

njj

)(2

1][,][)(

)()(][][

)(][

)(][

22

11

jjF

jF

jF

bXXb

eXnx

eXnx

n


)()(][][ 2121 jj ebXeaXnbxnax

9

Time shifting and frequency shifting

deeXnxenxeX njj

n

njj

)(2

1][,][)(

)(][

)(][

)(][

)( 00

jF

nj

jnjF

d

jF

eXnxe

eXennx

eXnx

d

n


)(][ )( 00 jj eXnxe

Time reversal

deeXnxenxeX njj

n

njj

)(2

1][,][)(

)(][

)(][

jF

jF

eXnx

eXnx

n


10

The convolution theorem

deeXnxenxeX njj

n

njj

)(2

1][,][)(

][*][][][][

)(][

)(][

jF

jF

nhnxknhkxny

eHnh

eXnx

n


)()()(

][][][][][

jjj

k

eXeHeY

y

The modulation or Windowing theorem

deeXnxenxeX njj

n

njj

)(2

1][,][)(

deWeXeY

nwnxny

eWnw

eXnx

jjj

jF

jF

)()(2

1)(

][][][

)(][

)(][

)(

n


)()(

2)(

11


Table 2.2


Fourier transform pairs


12

Example

Determining a Fourier transform using Tables 2.2 and 2.3

]5[][ nuanx n

Solution

jjjj

n

jj

Fn

eeXeeX

nuanxnx

aeeXnuanx

)()(

])5[(i.e. ]5[][

1

1)(][][

55

512

11

]5[][ nuanx


j

jj

n

jjjj

ae

eaeX

nuanxanx

aeeXeeX

1)(

])5[(i.e. ][][

1)()(

55

25

12

Part II-A: Z-transform

Fourier transform and frequency response

z-transform Z-transform

Properties of the ROC

Inverse z-transform

Properties of z-transform


13

Limitation of Fourier transform

Fourier transform

deeXnxenxeX njj

n

njj

)(2

1][,][)(

Condition for the convergence of the infinite sum

If x[n] is absolutely summable, its Fourier transform exists ( ffi i t diti )

n

nj

n

nj

n

j nxenxenxeX |][| |||][| |][| |)(|

n


(sufficient condition).

Example

No :1||

)2(1

1 )( :1

1

1 )( :1|| ][][

a

ke

eXa

aeeXanuanx

kj

j

jjn

z-transform

Fourier transform

njj enxeX ][)(

z-transform

The complex variable z in polar form jωrez

n

)(][ ][)( zXnxznxzXZ

n

n


)()( 1 jωeXzX,r|z|

njnnjj ernxrenxreXzX

)][()(][)()(

14

z-plane

z-transform is a function of a complex variable using the complex z-plane

Z-transform on unit circle<-> Fourier transform

Linear frequency axis in Fourier transform Unit circle in z-transform(periodicity in freq. of


(periodicity in freq. of Fourier transform)

Region of convergence – ROC

Fourier transform does not converge for all sequences

njj enxeX

][)(

z-transform does not converge for all sequences or for all values of z.

ROC for any given seq the set of values of z for

n

nj

n

nn

n

j ernxznxreX

)][( ][)(X(z)


ROC – for any given seq., the set of values of z for which the z-transform converges

|| |][| n

nznx

n

nrnx |][| ROC is ring!

15

ROC

Outer boundary is a circle (may extend to infinity)

Inner boundary is a circle (may extend to incl. origin)

If ROC incl des nit circle Fo rier transform If ROC includes unit circle, Fourier transform converges


Zeros and poles

The most important and useful z-transforms –rational function:

zP )(

Zeros: values of z for which X(z)=0.

Poles : values of z for which X(z) is infinite.

Cl l ti b t l d ROC

zzQzP

zQ

zPzX

in spolynomial are )( and )(

)(

)()(


Close relation between poles and ROC

16

Right-sided exponential sequence

1)(][)(

][][

nnn

n

azznuazX

nuanx

ROC

z-transform

0nn

0

1 ||n

naz


1||,1

1][

1

z

znu

Z

||||,1

1)()(

10

1 azaz

z

azazzX

n

n

Left-sided exponential sequence

]1[)(

]1[][

n

nn

n

znuazX

nuanx

ROC

0

11

)(1 n

n

n

nn

n

zaza

0

1 ||n

nza


z-transform

||||,1

1)(

1az

az

z

azzX

17

Sum of two exponential sequence

)3

1()

2

1()(

][)3

1(][)

2

1(][

11

zzzX

nununx

nn

nn

ROC

)31

)(21

(

)121

(2

31

1

1

21

1

1

)3

()2

()(

11

00

zz

zz

zz

nn


2

1||

3

1|| and

2

1||

z

zz

Sum of two exponential sequence

Another way to calculate:

][)1

(][)1

(][ nununx nn

3

1|| ,

11

1][)

3

1(

2

1|| ,

21

1

1][)

2

1(

][)3

(][)2

(][

1

1

znu

zz

nu

nununx

Zn

Zn


2

1||

31

1

1

21

1

1][)

3

1(][)

2

1(

33

13

11

1

z

zznunu

z

Znn

18

Two-sided exponential sequence

1||

1][)

1( znu

Zn

]1[)2

1(][)

3

1(][ nununx nn

2

1|| ,

3

1|| ,

11

11

)(

2

1|| ,

21

1

1]1[)

2

1(

3|| ,

31

1][)

3(

11

1

1

zzzX

zz

nu

zz

nu

Zn


2321

131

1 11 zz

)31

)(21

(

)121

(2

zz

zz

Some common z-transform pairs


19

Part II-B: Properties of the ROC




Inverse z-transform





20


The algebraic expression or pole-zero pattern does not completely specify the z-completely specify the ztransform of a sequence the ROC must be specified!

Stability, causality and the ROC


Part II-C: Inverse Z-transform




Inverse z-transform



21

Inverse z-transform

Needed for system analysis: 1) z-transform, 2) manipulation, 3) inverse z-transform.

Approaches: Inspection method

Partial fraction expansion


Inspection method

By inspection, e.g.

2

1|| ),

11

1()(

1

zzX

Make use of

22

1 1 z

|||| ),1

1()(

1az

aznua

Zn


?2

1|| if z

][)2

1(][ nunx n

]1[)2

1(][ course, of nunx n

22

Partial fraction expansion

For rational function, get the format of a sum of simpler terms, and then use the inspection methodmethod.


Second-order z-transform

111

)( zX

Nk

M

k

kk zb

zX 0)(

2

1|| ,

8

1

4

31

1)(

21

zzz

zX

)21

1)(41

1( 11 zz

2|)()1

1(

1|)()4

11(

)21

1()41

1()(

1

4/11

1

1

2

1

1

zXzA

zXzA

z

A

z

AzX

z

k

kk za

0

N

kAX )(

N

kk

M

kk

zd

zc

a

bzX

1

1

1

1

0

0

)1(

)1()(


][)4

1(][)

2

1(2][ nununx nn

)21

1(

2

)41

1(

1)(

2|)()2

1(

11

2/12

zzzX

zXzA z

k k

k

zdzX

111

)(

kdzkk zdzXA |)1)(( 1

23

What about M>=N?

)1)(21

1(

)1()(

11

21

zz

zzX1|| ,

21

23

1

21)(

21

21

zzz

zzzX

9|)()2

11(

division. longby Found 2

)1()21

1()(

2/11

1

0

12

1

10

zXzA

B

z

A

z

ABzX

z

15

23

2121

2

3

2

1

1

12

1212

z

zz

zzzz


][8][)2

1(9][2][ nununnx n

)1(

8

)21

1(

92)(

8|)()1(

11

11

2

zzzX

zXzA z

Finite-length sequence

12

1112

11

1)(

)1)(1)(2

11()(

zzzzX

zzzzzX

21

2)( zzzzX

]1[2

1][]1[

2

1]2[][ nnnnnx


24

Part II-D: Properties of Z-transform




Inverse z-transform



Linearity

1

ROC ),(][

ROC ),(][

22

11

x

Z

x

Z

RzXnx

RzXnx

][)( n

n

znxzX

Linearity

21 contains ROC ),()(][][ 2121 xx

Z

RRzbXzaXnbxnax

znuZ

1|| ,1

1][

1

2OC),(][ 22 xnx

at least


zz

znnunu

zz

znu

z

Z

Z

All ,11

1][]1[][

1|| ,1

]1[

||,1

][

1

1

1

1

1

25

Time shifting

)or 0for (except ROC

),(][

1

00

x

nZ

R

zXznnx ][)( n

n

znxzX

Example

)

4

11

1(

4

11

)(

4

1|| ,

41

1)(

1

1

1

1

zz

z

zzX

zz

zX


)

4

11

1(]1[)

4

1(

41

1

1][)

4

1(

1

11

1

zznu

znu

Zn

Zn

Multiplication by exponential sequence

)(][

ROC ),/(][

)(

000

00

jF

nj

x

Zn

eXnxe

|R|zzzXnxz ][)( n

n

znxzX

Examples

azazaz

nua

zz

nu

Zn

Z

|| ,1

1

)/(1

1][

1|| ,1

1][

11

1

)(][


1|| ,cos21

)cos1(

121

121

)(

][)(2

1][)(

2

1][)cos(][

210

10

11

0

00

00

zzz

z

zezezX

nuenuenunnx

jj

njnj

26

Time reversal

x

Z

RzXnx /1ROC ),/1(][ ][)( n

n

znxzX

Example

|||| ,1

1)(

][][

1

azzX

nuanx n

|||| ,1

1)(

][][

1azzX

nuanx n


||||1

)( az 1 az

Convolution of sequences

21 contains ROC

),()(][*][ 2121

xx

Z

RR

zXzXnxnx

][)( n

n

znxzX

Example

?][

][][

][)2

1(][

ny

nunh

nunx n

2

1|| ,

11

)(

1|| ,1

1)(

2

1|| ,

21

1

1)(

1

1

zzY

zz

zH

zz

zX


?][ny

))

21

1(

21

)1(

1(

21

1

1

2||

)1)(21

1()(

11

11

zz

zz

])[)2

1(][(

21

1

1][ 1 nununy n

27



Summary







Inverse z-transform


28

Course at a glance

Discrete-time signals and systems

MM1 System

Fourier transform and Z-transform

Filt d i

MM2

Sampling andreconstruction

MM3

Systemanalysis


DFT/FFT

Filter design

MM5

MM4

Documents

Digital Signal Processing, MM2.ppt - Aalborg Universitetkom.aau.dk/~zt/courses/Digital_signal_processing/MM2/Digital Signal... · 1 Digital Signal Processing, Fall 2010 Lecture 2: