5 l lEC533: Digital Signal Processing

Lecture 6The Z-Transform and its application in

signal processingsignal processing

6.1 - Z-Transform and LTI System S t F ti f LTI S t ( ) ( )∞• System Function of LTI Systems:

( )nx ( )nh( )zH( )zX

( )ny

( )zY ( ) ( )( )zYzH =

( ) ( )∑=−∞=

−

n

nznhnh

• As the LTI system can be characterized by the difference equation, written as

( )zH( )zX ( )zY ( ) ( )zXzH

• The difference equation specifies the actual operation that must be performed by he diffe ence equation specifies the actual ope ation that must be pe fo med bythe discrete-time system on the input data, in the time domain, in order to generate the desired output.

In Z-domainIn Z domain

If the O/p of the system depends only on the present & past I/p samples but not on previous outputs, i.e., bk=0’s FIR system, Else, Infinite Impulse Responseon previous outputs, i.e., bk 0 s FIR system, Else, Infinite Impulse Response (IIR)

6.1.1 - LTI System Transfer Function

If the O/p of the system depends only on the present & past i/p samples but not on previous outputs i e bk=0’s Finite Impulse Response (FIR) systemon previous outputs, i.e., bk 0 s Finite Impulse Response (FIR) system ,

h(n) = 0, n < 0, h(n) = 0, n >N,( ) ∑==

−N

k

kk zazH

0

FIR system is an all zero system and are always stable

If bk≠0’s, the system is called Infinite Impulse Response (IIR) system

≤≤h )(IIR filter has poles

∞≤≤∞ n- nh ),(

6.1.2 - Properties of LTI Systems Using the Z-Transform

ImCausal Systems : ROC extends outward from the outermost pole.

ReR

Stable Systems (H(z) is BIBO): ROC includes the unit circle.Im

Re

1A stable system requires that its Fourier transform is uniformly convergent.

Re

6.2 - Z-transform and Frequency Response Estimation

• The frequency response of a system (as digital filter spectrum) can be readily obtained from its z-transform.

H( )H(z)as,

where σ is a transient term & it tends to zero as f

at steady state, Steady‐state frequency response of a system (DTFT).

where, A(ω)≡ Amplitude (Magnitude) Response , B(ω) ≡ Angle (Phase) Response

Steady‐state

6.2 - Frequency Response Estimation – cont.

• Phase DelayThe amount of time delay each frequency component of the signal suffers inThe amount of time delay, each frequency component of the signal suffers in going through the system.

• Group DelayGroup DelayThe average time delay the composite signal suffers at each frequency.

6.3 - Inverse Z-Transform

wherewhereDTFT

IDTFT

(A contour integral)

where, for a fixed r,

6.3 - The Inverse Z-Transform – cont.

• There is an inversion integral for the z transform,

x n[ ]= 1j2π

X z( )zn−1dzC∫

but doing it requires integration in the complex plane andit is rarely used in engineering practice.

• There are two other common methods,

Power series method (long division method)Partial-Fraction Expansionp

6.3.1 - Power Series (Long Division) Method

Suppose it is desired to find the inverse z transform of

3 z2

H z( )=z3 −

z2

z3 −1512

z2 +1736

z − 118

Synthetically dividing the numerator by the d i t i ld th i fi it i

12 36 18

denominator yields the infinite series

1+34

z−1 +67

144z−2 +L

This will always work but the answer is not in closed form (Disadvantage).

4 144

in closed form (Disadvantage).

6.3.2 - Partial-Fraction Expansion

•Put H(z-1) in a fractional form with the degree of numerator less than degree of denominator.

•Put the denominator in the form of simple poles.

A l i l f i i• Apply partial fraction expansion.

• Apply inverse z transform for those simple fractions.

NoteIf N: order of numerator, & M: order of denominator.then,

If N=M  Divide

If 1N<M Make direct PFIf 1N<M  Make direct P.F.

If N>M  Make long division then P.F.

Example 1:

find, a) Transfer Function b) Impulse Response

-1 -1/3 7/3

*33

3 3 3

Example 2:

) D h l &

Consider the discrete system,

a) Determine the poles & zeros.b) Plot (locate) them on the z-plane.c) Discuss the stability.d) Find the impulse response.e) Find the first 4 samples of h(n)

Solution:

a) zeros when N(z)=0z1=2 z2= - 0.5

b)1 2

poles when D(z)=0p1=0.3 p2= - 0.6

xxo o20.3

‐ 0.5‐ 0.6

c) Since all poles lies inside the unit circle, therefore the system is stable.

Example 2 – cont.d) As discussed before, use partial fraction expansion and the table of transformation to

get the inverse z-transformc) Divide N(z) by D(z) using long division,) ( ) y ( ) g g ,

1

1 2 3

‐ 1.8

‐ 0.28‐ 0.24

Example 3:

Consider the system described by the following difference equation,

Find,a) The transfer function.b) Th t d t t fb) The steady state frequency response.c) The O/p of the system when a sine wave of frequency 50 Hz & amplitude of 10 is

applied at its input, the sampling frequency is 1 KHz.

Solution:a)

Example 3– cont.

b)

Example 3– cont.

)sin()( 0 θω += nTAnx)1.0sin(10))

10001..50.2sin((10 nn πθπ =+=

c)

1000Since the I/p is sinusoidal, the O/p should be sinusoidal,

)1.0sin()( yy nAny θπ +=)(AAAwhere,

)(

)(

0

0

ωθθθ

ω

+=

=

xy

xy AAA

10xy

20

20 ))sin(5.0())cos(5.0(1

110TT

Ayωω +−

×=

10 3.18))1.0sin(5.0())1.0cos(5.0(1

1022=

+−=

ππ⎞⎛

42178)1.0cos(5.01

)1.0sin(5.0tan 01 ′=⎟⎟⎠

⎞⎜⎜⎝

⎛−

−= −

ππθθ xy

6.4 Relationships between System Representationsp

H(z) Take inverse z-transform

Express H(z) in1/z cross

multiply and

h(n)Difference

Takez-transform

take inverse Take z-transformsolve for Y/X

h(n)Equation

Take inverseDTFT

Substitutez=ejwT

H(ejwT)Take Fourier

transform

Take DTFT solvefor Y/X