Topic 10: The Z Transform o Introduction to Z Transform o - - PowerPoint PPT Presentation

1

• Introduction to Z Transform
• Relationship to the Fourier transform
• Z Transform and Examples
• Region of Convergence of the Z Transform
• Inverse Z Transform and Examples
• Properties of Z Transform and Examples
• Analysis and characterization of LTI systems using z-transforms
• Geometric evaluation of the Fourier transform from the pole-zero plot
• Summary
• Dr. Aishy Amer

Concordia University Electrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

• A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997
• M.J. Roberts, Signals and Systems, McGraw Hill, 2004
• J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

ELEC361: Signals And Systems

Topic 10: The Z Transform

2

The Z Transform

The Laplace transform: an extension of the continuous-time Fourier

transform

Advantage: to perform analysis of continuous-time signals &

systems whose Fourier transform does not exist

The z−transform: an extension of the discrete-time Fourier

transform

Let h[n] be the impulse response of a LTI system The response of this system to a complex exponential input of the

form zn is

The expression H(z) is referred to as the z−transform of h[n] where

z is a complex variable, z = rejω

• Since z is a complex quantity, H(z) is a complex function
• (Why do we deal with complex signals? They are often analytically simpler

to deal with than real signals.)

; ] [

n

z n x =

3

The Z Transform

• The Fourier transform of h[n] can be obtained by evaluating the z−transform at z =

ejω with ω real

• ZT:
• z = rejω

ω real H(z) is defined for a region in z – called the ROC- for which the sum exists

• Since the Z-Transform is a power series, it converge when h[n]z-n is absolutely

summable, i.e.,

• DT-FT

z = ejω

(r=1)

ω real

• Recall:
• h[n] is the impulse response of an LTI system
• H(ei ω) is the frequency response
• H(z) is the transfer function

∞ <

∑

∞ −∞ = − n n

z n h | ] [ |

4

The Z Transform:

Rational function/ Poles and Zeros

The Z-transform will have the below structure, based on rational Functions:

For any two polynomials A and B, their ratio is called a rational function The numerator and denominator can be polynomials of any order The rational function is undefined when the denominator equals zero,

i.e., we have a discontinuity in the function

The z−transform is characterized by its zeros and poles Zeros: The value(s) for z where P (z) = 0, i.e., the complex frequencies

that make the transfer function zero

Poles: The value(s) for z where Q(z) = 0, i.e., the complex frequencies

that make the transfer function infinite

) 1 )( 3 2 ( ) 2 )( 2 ( 3 2 4 ) (

2 2

− + − + = − + − = z z z z z z z z H

N N M M

z a z a a z b z b b z H

− − − −

+ + + + + + = L L

1 1 1 1

) ( ) ( ) ( ) ( z Q z P z H =

5

The Z Transform:

The Z plane (complex plane)

The z-plane is a complex plane with an

imaginary and real axis referring to the complex- valued variable z

Once the poles and zeros are found for the z

transform, they can be plotted into the z plane

The position on the complex plane is given in a

polar form by rejω

ω: the angle from the positive real axis around

the plane

H(z) is defined everywhere on this plane H(ejω) on the other hand is defined only where

|z| = 1 which is referred to as the unit circle

• This is useful because by representing the Fourier

transform as the z-transform on the unit circle, the periodicity of Fourier transform is easily seen

6

The Z Transform:

The Z plane (complex clane)

• Poles are denoted by “x” and zeros by “o”
• We use shaded regions to indicates the

Region of Convergence (ROC) for the z transform

7

The Z Transform:

The Z plane (complex plane)

• In MATLAB you can easily create pole/zero

plots, e.g.,

% Set up vector for zeros z = [j ; -j]; % Set up vector for poles p = [-1 ; .5+.5j ; .5-.5j]; figure(1); zplane(z,p); title('Pole/Zero Plot'); } 2 1 2 1 , 2 1 2 1 {-1, : are poles The i}

• {i,

: are zeros The )) 2 1 2 1 ( ))( 2 1 2 1 ( )( 1 ( ) )( ( ) ( i i i z i z z i z i z z H + − + − − − + + − =

8

Outline

Introduction to Z Transform Relationship to the Fourier transform Z Transform and Examples Region of Convergence of the Z Transform Inverse Z Transform and Examples Properties of Z Transform and Examples Analysis and characterization of LTI systems using z-transforms Geometric evaluation of the Fourier transform from the pole-zero

plot

Summary

9

Relationships: DT-FT and the ZT

We first express the complex variable z in

polar form as

z = rejω r is the magnitude of z and ω is the phase of z Representing z as such,

can be expressed as or equivalently,

10

Relationships: DT-FT and the ZT

By comparing equations

we can see that H(rejω) is essentially the FT of the sequence x[n] multiplied by a real exponential r−n

The exponential r−n may be decaying or

growing with increasing n depending on whether r is greater than or less than 1

If we let r = 1, then

which suggests that the ZT reduces to the FT on the unit circle (i.e., the contour in the complex z−plane corresponding to a circle with a radius

• f unity)

&

) ( | ) (

ω

ω

j e z

e H z H

j =

=

11

Relationships: DT-FT and the ZT

For convergence of the z−transform, we require

that the Fourier transform of h[n]r−n converge

For any specific sequence h[n], we would expect

this convergence for some values of r and not for

• thers (as in the Laplace transform)

The range of values for which the z−transform

converges is referred to as the region of convergence (ROC)

If the ROC includes the unit circle, then the

Fourier transform converges

12

Outline

Introduction to Z Transform Relationship to the Fourier transform Z Transform and Examples Region of Convergence of the Z Transform Inverse Z Transform and Examples Properties of Z Transform and Examples Analysis and characterization of LTI systems using z-transforms Geometric evaluation of the Fourier transform from the pole-zero

plot

Summary

13

The Z Transform: Examples

right-sided

14

The Z Transform: Examples

x[n] is right-sided; it decays when a<1 (e.g., a=0.5) It z−transform is a rational function with one zero at z = 0

and one pole at z = a

15

The Z Transform: ROC in the form |z| > |a|

0 < a < 1

• 1 < a < 0

a > 1 a < -1

16

The Z Transform: Examples

|a| |z| , a z z az z a X(z) a z| |a az az az az z a z a z n u a z n u a z n u a z X n n n u z n u a z X n u a n x

• n

n n n n n n n n n n n n n n n n n n n n n n n n n n

< − = − = − − = > <

• −

= − ⇒ − = − = − = − − − − − − = − − − = < − − ∀ = − − − − − = − − − =

• −

− − ∞ = − ∞ = − ∞ = − ∞ = − − −∞ = − − − −∞ = − ∞ −∞ = ∞ = − − −∞ = − − ∞ −∞ = − > −

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

1 1 1 1 1 1 1

• 1

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 ly consequent , | | | z |

• r

1 if converges sum This ) ) (

• )

(az

• 1

(because ) ( 1 ) ( 1)

• by

n every g multiplyin (by ) ( n) power the combining (by ) ( ] 1 [ ] 1 [ ] 1 [ ) ( , 1 : ] 1 [ that Note ] 1 [ ) ( Then ] 1 [ ] [ Let 4 3 4 2 1 4 3 4 2 1 4 3 4 2 1

Note: The algebraic expression of X(z) for x[n]=anu[n] and x[n]=-anu[-n-1] are identical except for the ROCs left-sided

∑ ∑

∞ = = +

− = − − =

m n m n m m n m m n

z cz cz z ) z c(z cz 1 Formula Series Geometric Infinite 1 Formula Series Geometric Finite

2 1 2 1

1

17

The Z Transform: ROC in the form |z| < |a|

0 < a < 1

• 1 < a < 0

a > 1 a < 1

18

The Z Transform: Examples

Multiple Poles

19

The Z Transform: Examples

We first find the ROC for each

term individually and then find the ROC of both terms combined

(Similar to what we used to do in Laplace transform)

Provided that

|⅓z-1|<1 and |½z-1|<1

• r equivalently

|z| >⅓ and |z| > ½ The ROC is |z| > ½

20

Outline

Introduction to Z Transform Z Transform and Examples Region of Convergence of the Z Transform Inverse Z Transform and Examples Properties of Z Transform and Examples Analysis and characterization of LTI systems

using z-transforms

Geometric evaluation of the Fourier transform

from the pole-zero plot

Summary

21

The Region of Convergence for the Z Transform

Important properties of the ROC of the z−transform:

1.

The ROC of X(z) consists of a ring in the z−plane centered about the origin

2.

The ROC does not contain any poles

3.

If x[n] is of finite-duration, then the ROC is the entire z−plane, except possibly at z = 0 and/or z = ∞

• A finite-duration sequence is a sequence that is nonzero in a finite interval

n1<n<n2

• As long as each value of x [n] is finite then the sequence will be absolutely

summable

• When n2 > 0 there will be a z−1 term and thus the ROC will

not include z = 0

• When n1 < 0 then the sum will be finite and thus the ROC will not include |z| = ∞
• When n2 ≤ 0 then the ROC will include z = 0, and when n1 ≥ 0 the ROC will

include |z| = ∞

• With these constraints, the only signal with ROC as the entire z-plane is x [n] =

cδ[n]

22

The Region of Convergence for the Z Transform: Example

Example: Let x[n] = δ[n], Then

which suggests that the ROC is the entire z−plane, including z = 0 and z = ∞

Example: Now consider x[n] = δ[n − n0] where n0≠0

X(z) then becomes

If n0 > 0 then the ROC contains the entire z−plane except at z = 0 But if n0 < 0, the ROC contains the entire z−plane except at z = ∞

23

The Region of Convergence for the Z Transform: Example

• Since the system has a finite impulse response

and is zero for n<0, then we should expect according to Property 3 the ROC to include the entire z−plane except possibly at z = 0 and/or z = ∞

• X(z) has

N zeros at a pole at z = 0 of order N−1 a pole at z = a but there is also a zero at at

z=a the pole at z=a and zero at z = a (k=0) cancel out

What is left is a polynomial in the numerator

• f degree N −1, suggesting that there are

N−1 zeros Find X(z) and plot its poles and zeros

24

The Region of Convergence for the Z Transform

4.

If x[n] is a right-sided sequence, and if the circle |z| = r0 is in the ROC, then all finite values of z for which |z| > r0 will also be in the ROC

• A right-sided sequence is a sequence where x[n]=0 for n < n1 < ∞

5.

If x[n] is a left-sided sequence, and if the circle |z| = r0 is in the ROC, then all finite values of z for which 0 < |z| < r0 will also be in the ROC

• A left-sided sequence is a sequence where x[n]=0 for n > n2 > -∞
• Properties 4 and 5 above parallel the corresponding properties for LT

6.

If x[n] is a two-sided sequence, and if the circle |z| = r0 is in the ROC, then all finite values of z for which |z| = r0 will also be in the ROC

• Since the sequence is two sided, then it can decomposed into at least one

left-sided sequence and one right-sided sequence

• For the right-sided sequence, the ROC is bounded on the inside by a

circle and extending outward to infinity

• For the left-sided sequence, the ROC is bounded on the outside by a

circle and extending to the origin

• The ROC for both sequences combined is the intersection of both

individual ROCs

25

The Region of Convergence for the Z Transform

7.

If the z−transform X(z) of x[n] is rational, then its ROC is bounded by the poles or extends to infinity

8.

If the z−transform X(z) of x[n] is rational, and if x[n] is right-sided, then the ROC is the region in the z−plane outside the outermost pole (i.e.,

• utside the circle of radius equal to the largest magnitude of the poles
• f X(z)
• Also, if x[n] is causal (i.e., if it is right-sided and equal to 0 for n < 0),

then the ROC also includes z = ∞

9.

If the z−transform X(z) of x[n] is rational, and if x[n] is left-sided, then the ROC is the region in the z−plane inside the innermost pole (i.e.,

• utside the circle of radius equal to the smallest magnitude of the poles
• f X(z) other than any at z = 0 and extending inward to and possibly

including z = 0

• In particular, if x[n] is anticausal (i.e., if it is left-sided and equal to 0 for

n > 0), then the ROC also includes z = 0

26

The Region of Convergence for the Z Transform: Example

• Consider
• Since there are two poles; then there

are three possibilities for the ROC:

1)

ROC: |z| >2

the ROC is extending outward from the outermost pole, suggesting that the sequence x[n] is right-sided

2)

ROC: 1/3 < |z| <2

the ROC is bounded between two poles, suggesting that the sequence x[n] is two-sided

3)

ROC: |z| < 1/3

the ROC is inward from the innermost pole, suggesting that the sequence x[n] is left-sided

27

Outline

Introduction to Z Transform Z Transform and Examples Region of Convergence of the Z Transform Inverse Z Transform and Examples Properties of Z Transform and Examples Analysis and characterization of LTI systems

using z-transforms

Geometric evaluation of the Fourier transform

from the pole-zero plot

Summary

28

The inverse Z transform

• When using the z-transform

it is often useful to be able to find x [n] given X (z) (inverse transform)

• There are at least 4 different methods to do this:

1.

Inspection

2.

Partial-Fraction Expansion

3.

Power Series Expansion

4.

Long Division

29

The inverse Z transform

The equation for the inverse z-transform is a contour integration in

the z-plane (counterclockwise) as follows:

This equation can be obtained by writing the z-transform of x[n] as

the Fourier transform of the signal x[n]r −n as was discussed before, and taking the inverse Fourier transform

In general, one can find the partial fraction expansion for the

z-transform expressions that are rational functions of z We will have:

One can then take the inverse z-transform of each individual term

very easily

30

The inverse Z transform: Inspection Method

This "method" is to basically become familiar with the

z-transform pair tables and then "reverse engineer"

31

Common Z Transform Pairs

32

Common Z Transform Pairs

33

The inverse Z transform Partial Fraction Expansion: Examples

34

The inverse Z transform Partial Fraction Expansion: Examples

35

The inverse Z transform Partial Fraction Expansion: Examples

36

The inverse Z transform: Power Series Expansion

• One can find the inverse z-transform of non-rational expressions of z, by writing that

expression as a power series (for example using Taylor expansion)

• The z-transform is defined as a power series in the form
• Then each term of the sequence x [n] can be determined by looking at the coefficients
• f the respective power of z−n
• One of the advantages of the power series expansion method is that many functions

encountered in engineering problems have their power series' tabulated Thus functions such as log, sin, exponent, sinh, etc, can be easily inverted

37

The inverse Z transform: Power Series Expansion: Examples

38

The inverse Z transform: Power Series Expansion: Examples

39

The inverse Z transform: Long Division Method

In algebra, polynomial long division is an algorithm for

dividing a polynomial by another polynomial of lower degree

A generalized version of the familiar arithmetic technique

called long division

It can be done by hand, because it separates an otherwise

complex division problem into smaller ones

For any polynomials F(z) and G(z), where the degree of

F(z) is greater than or equal to the degree of G(z), there exist unique polynomials Q(z) and R(z) such that with R(z) having smaller degree than G(z)

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( z R z Q z F z G z G z R z Q z G z F ⇔ + =

40

The inverse Z transform: Long Division Method: Examples

41

The inverse Z transform: Long Division Method: Examples

42

Outline

Introduction to Z Transform Z Transform and Examples Region of Convergence of the Z Transform Inverse Z Transform and Examples Properties of Z Transform and Examples Analysis and characterization of LTI systems

using z-transforms

Geometric evaluation of the Fourier transform

from the pole-zero plot

Summary

43

Properties of the Z Transform

The properties of the z-transform are very

similar to those of the Laplace transform

The properties can easily been shown using

the definition of the z-transform

Linearity

44

Properties of the Z Transform

Time shifting For example, the z-transform of the unit impulse δ [n]

is equal to 1 and the ROC is the entire z-plane

The z-transform of δ [n −1] is equal to z and the ROC

is the entire z-plane except for the infinity

The z-transform of δ [n +1] is equal to z−1 and the

ROC is the entire z-plane except for the origin

45

Properties of the Z Transform

46

Properties of the Z Transform

} 1 | {| ) ( 1 ) ( 1 1 ] [ R ROC X(z) x[n] ) n" integratio " t to counterpar (DT n Accumlatio

1 k

> = − = − ⎯→ ← = ⎯→ ←

− −∞ =

∑

z R ROC z X z z z X z n x

n

I

2 1 2 1 2 1 2 2 2 1 1 1

ROC and ROC

• f

n intersctio least the at ROC ) ( ) ( ] [ x ] [ x ROC ) ( ] [ x ROC ) ( ] [ x n Convolutio z X z X n n z X n z X n ⎯→ ← ∗ ⎯→ ← ⎯→ ←

47

Properties of the Z Transform

48

Properties of the Z Transform: Summary

49

Outline

Introduction to Z Transform Z Transform and Examples Region of Convergence of the Z Transform Inverse Z Transform and Examples Properties of Z Transform and Examples Analysis and characterization of LTI systems

using z-transforms

Geometric evaluation of the Fourier transform from

the pole-zero plot

Summary

50

Analysis and characterization of LTI systems using z-transforms

• Consider a discrete-time LTI system with the impulse response h[n] and

the transfer function (system function) H(z)

• From the convolution property we have:

Causality:

• For a causal DT-LTI system, h[n]=0 for n <0 h[n] right-sided
• For an anticausal DT-LTI system, h[n]=0 for n≥0 h[n] left-sided
• A discrete-time LTI system with rational transfer function H(z) is causal if

and only if:

a) The ROC is the exterior of a circle outside the outermost pole b) With H(z) expressed as a rational function, the order of the

numerator cannot be greater than the order of the denominator

• A discrete-time LTI system is causal if and only if the ROC of its transfer

function is the exterior of a circle, including infinity

51

Analysis and characterization of LTI systems using z-transforms

Stability:

A DT-LTI system is BIBO stable iff A discrete-time LTI system is stable if and only if the

ROC of its transfer function H(z) includes the unit circle r = 1 Causality & Stability:

• A causal discrete-time LTI system with rational

transfer function H(z) is stable iff all of the poles of H(z) lie inside the unit circle (i.e., they must all have magnitude smaller than 1)

∑

∞ −∞ =

∞ <

n

n h

2

| ] [ |

52

Outline

Introduction to Z Transform Z Transform and Examples Region of Convergence of the Z Transform Inverse Z Transform and Examples Properties of Z Transform and Examples Analysis and characterization of LTI systems using z-

transforms

LTI systems characterized by Difference

Equations

Geometric evaluation of the Fourier transform from

the pole-zero plot

Summary

53

LTI systems characterized by Difference Equations

• Note that y [n − k] represents the outputs and x [n − k] represents the Inputs
• The value of N represents the order of the difference equation

(N corresponds to the memory of the system)

• Because this equation relies on past values of the output, to compute a

numerical solution, certain past outputs (called “initial conditions”) must be known

54

LTI systems characterized by Difference Equations

• One of the most important concepts of Signals&Systems is to be able to

properly represent the input/output relationship to a given LTI system

• A linear constant-coefficient difference equation (LCCDE) serves as a way to

express this relationship in a discrete-time system

• A difference Equation shows the relationship between consecutive values of a

sequence and the difference among them which are often rearranged as a recursive formula so that a systems output can be computed from the input signal and past outputs

• LTI systems can be

characterized by LCCDE

55

LTI systems characterized by Difference Equations

Consider an LTI system whose input and output are

related through the general form of the LCCDE

We can also write the general form to easily express a

recursive output:

Taking the z-transform of both sides of this equation, we

will have:

The ROC of H(z) is not specified but must be inferred

with additional requirements on the system (.e.g, stable)

56

LTI systems characterized by Difference Equations: Conversion to Frequency Response

Once the z-transform has been calculated from the difference

equation, we can go one step further to define the frequency response of the system (or filter) represented by the difference equation

The objective is to aid in system/filter design A LCCDE is one of the easiest ways to represent filters By finding the frequency response, we will be able to look at the

basic properties of any filter represented by a simple LCCDE

The general formula for the frequency response of a z-transform The conversion is simple a matter of taking the z-transform

formula, H (z), and replacing every instance of z with eiw

57

LTI systems characterized by Difference Equations: Solving an LCCDE

In order for a LCCDE to be useful in analyzing a LTI system, we

must be able to find the systems output based upon a known input, x[n], and a set of initial conditions

Two common methods exist for solving a LCCDE:

• 1. Direct Method

The final solution to the output based on the direct method is the

sum of two parts, expressed in the equation: y[n] = yh[n] + yp[n]

The first part, yh[n], is referred to as the homogeneous solution The second part, yp[n] is referred to as particular solution

• 2. Indirect Method

use the relationship between the difference equation and z-

transform

convert the difference equation into a z-transform to get the Y(z) inverse transform Y(z) using partial-fraction expansion to gey y[n]

(y[n] is the solution of the LCCDE)

58

LTI systems characterized by Difference Equations: Example

59

LTI systems characterized by Difference Equations: Example

) ( ) 3 1 1 ( ) ( ) 2 1 1 (

1 1

z X z z Y z

− −

+ = −

60

Outline

Introduction to Z Transform Z Transform and Examples Region of Convergence of the Z Transform Inverse Z Transform and Examples Properties of Z Transform and Examples Analysis and characterization of LTI systems using

z-transforms

Geometric evaluation of the Fourier transform

from the pole-zero plot

Summary

61

Geometric evaluation of the Fourier transform from the pole-zero plot

Recall: h[n] is the impulse response of an LTI system H(ei ω) is the frequency response H(z) is the transfer function; z=rejω , |z|=r , z=ω We know that the z-transform reduces to the Fourier

transform for |z| = 1 (i.e., for the values of the complex variable z on the unit circle, provided that the ROC of the z-transform includes the unit circle

As a result, one can use the pole-zero plot of a

transfer function to find the frequency response of the system by evaluating the magnitude and phase of the system on the unit circle in the complex plane

62

Geometric evaluation of the Fourier transform from the pole-zero plot

63

Geometric evaluation of the Fourier transform from the pole-zero plot

Using the distances from the unit circle to the

poles and zeros, we can plot the frequency response of the system

As ω goes from 0 to 2π, the following two

properties specify how one should draw |H(ei ω)|

While moving around the unit circle:

• 1. if close to a zero, then the magnitude is small. If

a zero is on the unit circle, then the frequency response is zero at that point.

• 2. if close to a pole, then the magnitude is large. If a

pole is on the unit circle, then the frequency response goes to infinity at that point.

64

Geometric evaluation of the Fourier transform from the pole-zero plot

Consider: H (z) = z + 1

H (ejω) = ejω+1

Some of the vectors represented by |ejω +1|,

for random values of ω, are explicitly drawn

• nto the complex plane

These vectors show how the amplitude of

H(ejω) changes as ω goes from 0 to 2π, and also show the physical meaning of the terms in H(ejω)

When ω = 0, the vector is the longest and

thus the frequency response will have its largest amplitude

As ω approaches π , the length of the vectors

decrease as does the amplitude of |H(ejω)|

Since there are no poles in the transform,

there is only this one vector term

65

Geometric evaluation of the Fourier transform from the pole-zero plot

We will use this result to find the frequency response of any discrete-

time LTI system with rational transfer function given by

At any frequency ω , find the magnitude and phase of the vectors

drawn from the poles and zeros to the point e jω (a point on the unit circle with angle ω )

The magnitude of H (ejω) at ω is equal to the product of the

magnitudes of all vectors associated with the zeros divided by the product of the magnitudes of all vectors associated with the poles

Similarly, the phase of H (ejω) ω is equal to the summation of the

angles of all vectors associated with the zeros minus the summation

• f the angles of all vectors associated with the poles

66

Geometric evaluation of the Fourier transform from the pole-zero plot: Example

One zero, two poles

Consider an LTI system with a

impulse response h[n]

Assume that H(z) is a rational

function of z whose pole-zero configuration is given in the plot

From this plot, it can be

concluded that the z-transform

• f the impulse response is:

67

Geometric evaluation of the Fourier transform from the pole-zero plot: Example

• The amplitude and angle of the vectors v1, v2

and v3 depend on the frequency ω

• The magnitude of the frequency response of

this system is proportional to

• The phase of the frequency response of this

system is equal to

• The magnitude of the frequency response is

large at those frequencies that correspond to the points on the unit circle which are close to the poles and far from the zeros

• Similarly, the magnitude of the frequency

response is small at those frequencies that correspond to the points on the unit circle which are close to the zeros and far from the poles

68

Geometric evaluation of the Fourier transform from the pole-zero plot: Example

The following transfer function

is analyzed in order to represent the system's frequency response |H(ejω)|

We can see that when ω = 0 the

frequency response |H(ejω)| will peak since it is at this value of ω that the pole is closest to the unit circle

As ω moves from 0 to π, we see how the

zero begins to mask the effects of the pole and thus force the frequency response |H(ejω)| closer to 0

ω ω j j

e e H z z z z H

− −

− = ⇒ − = − = 2 1 1 1 ) ( 2 1 1 1 2 1 ) (

1

69

Summary