elec361: signals and systems topic 10: the z transformamer/teach/elec361/slides/topic10-zt.pdf · 1...

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o Introduction to Z Transformo Relationship to the Fourier transformo Z Transform and Exampleso Region of Convergence of the Z Transformo Inverse Z Transform and Exampleso Properties of Z Transform and Examples o Analysis and characterization of LTI systems using z-transformso Geometric evaluation of the Fourier transform from the pole-zero ploto Summary

Dr. Aishy AmerConcordia UniversityElectrical 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 TransformThe 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 transformLet 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.)

;][ nznx =

3

The Z TransformThe Fourier transform of h[n] can be obtained by evaluating the z−transform at z = ejω with ω real

ZT: z = rejω

ω realH(z) is defined for a region in z – called the ROC- for which the sum existsSince the Z-Transform is a power series, it converge when h[n]z-n is absolutely summable, i.e.,

DT-FTz = ejω (r=1)

ω realRecall:

h[n] is the impulse response of an LTI systemH(ei ω) is the frequency responseH(z) is the transfer function

∞<∑∞

−∞=

−

n

nznh |][|

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 orderThe rational function is undefined when the denominator equals zero, i.e., we have a discontinuity in the functionThe z−transform is characterized by its zeros and polesZeros: The value(s) for z where P (z) = 0, i.e., the complex frequencies that make the transfer function zeroPoles: The value(s) for z where Q(z) = 0, i.e., the complex frequencies that make the transfer function infinite

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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 zOnce the poles and zeros are found for the z transform, they can be plotted into the z planeThe position on the complex plane is given in a polar form by rejω

ω: the angle from the positive real axis around the planeH(z) is defined everywhere on this planeH(ejω) on the other hand is defined only where |z| = 1 which is referred to as the unit circleThis 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 zerosz = [j ; -j];% Set up vector for polesp = [-1 ; .5+.5j ; .5-.5j];figure(1);zplane(z,p);title('Pole/Zero Plot');

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8

Outline

Introduction to Z TransformRelationship to the Fourier transformZ Transform and ExamplesRegion of Convergence of the Z TransformInverse Z Transform and ExamplesProperties of Z Transform and Examples Analysis and characterization of LTI systems using z-transformsGeometric evaluation of the Fourier transform from the pole-zero plotSummary

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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,

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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, thenwhich suggests that the ZT reduces to the FT on the unit circle(i.e., the contour in the complex z−planecorresponding to a circle with a radius of unity)

&

)(|)( ωω

jez

eHzH j ==

11


For convergence of the z−transform, we require that the Fourier transform of h[n]r−n convergeFor any specific sequence h[n], we would expect this convergence for some values of r and not for others (as in the Laplace transform)The range of values for which the z−transformconverges is referred to as the region of convergence (ROC)If the ROC includes the unit circle, then the Fourier transform converges

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Outline

Introduction to Z TransformRelationship to the Fourier transformZ Transform and ExamplesRegion of Convergence of the Z TransformInverse Z Transform and ExamplesProperties of Z Transform and Examples Analysis and characterization of LTI systems using z-transformsGeometric evaluation of the Fourier transform from the pole-zero plotSummary

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The Z Transform: Examplesright-sided

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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

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The Z Transform:Examples

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17

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

0 < a < 1 -1 < a < 0

a > 1 a < 1

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The Z Transform: ExamplesMultiple Poles

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The Z Transform: ExamplesWe 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|<1or equivalently |z| >⅓ and |z| > ½The ROC is |z| > ½

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Outline

Introduction to Z TransformZ Transform and ExamplesRegion of Convergence of the Z TransformInverse Z Transform and ExamplesProperties of Z Transform and Examples Analysis and characterization of LTI systems using z-transformsGeometric evaluation of the Fourier transform from the pole-zero plotSummary

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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 origin2. The ROC does not contain any poles3. 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 summableWhen n2 > 0 there will be a z−1 term and thus the ROC will not include z = 0When 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]

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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≠0X(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 = ∞

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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 ata pole at z = 0 of order N−1a pole at z = a but there is also a zero at atz=a the pole at z=a and zero at z = a (k=0) cancel outWhat is left is a polynomial in the numerator of degree N −1, suggesting that there are N−1 zeros

Find X(z) and plot its poles and zeros

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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 sequenceFor the right-sided sequence, the ROC is bounded on the inside by a circle and extending outward to infinityFor the left-sided sequence, the ROC is bounded on the outside by a circle and extending to the originThe ROC for both sequences combined is the intersection of both individual ROCs

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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., outside the circle of radius equal to the largest magnitude of the poles of 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., outside the circle of radius equal to the smallest magnitude of the poles of 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

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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

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Outline


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. Inspection2. Partial-Fraction Expansion3. Power Series Expansion 4. Long Division

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The inverse Z transformThe 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 transformIn general, one can find the partial fraction expansion for the z-transform expressions that are rational functions of zWe will have:

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

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Common Z Transform Pairs

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The inverse Z transformPartial Fraction Expansion: Examples

34


35


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 of 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

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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 divisionIt can be done by hand, because it separates an otherwise complex division problem into smaller onesFor 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))(

)()()(

)()()(

)()(

zR

zQzFzG

zGzRzQ

zGzF

⇔+=

40

The inverse Z transform: Long Division Method: Examples

41

The inverse Z transform:Long Division Method: Examples

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Outline

Introduction to Z TransformZ Transform and ExamplesRegion of Convergence of the Z TransformInverse Z Transform and ExamplesProperties of Z Transform and ExamplesAnalysis and characterization of LTI systems using z-transformsGeometric evaluation of the Fourier transform from the pole-zero plotSummary

43

Properties of the Z Transform

The properties of the z-transform are very similar to those of the Laplace transformThe properties can easily been shown using the definition of the z-transformLinearity

44


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 infinityThe z-transform of δ [n +1] is equal to z−1 and the ROC is the entire z-plane except for the origin

45


46


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47


48

Properties of the Z Transform: Summary

49

Outline


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-sidedFor an anticausal DT-LTI system, h[n]=0 for n≥0 h[n] left-sidedA 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 poleb) With H(z) expressed as a rational function, the order of the

numerator cannot be greater than the order of the denominatorA discrete-time LTI system is causal if and only if the ROC of its transferfunction 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 iffA 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

nh 2|][|

52

Outline

Introduction to Z TransformZ Transform and ExamplesRegion of Convergence of the Z TransformInverse Z Transform and ExamplesProperties of Z Transform and Examples Analysis and characterization of LTI systems using z-transformsLTI systems characterized by Difference EquationsGeometric evaluation of the Fourier transform from the pole-zero plotSummary

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


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 systemA 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


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)

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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 differenceequation The objective is to aid in system/filter design A LCCDE is one of the easiest ways to represent filtersBy finding the frequency response, we will be able to look at the basic properties of any filter represented by a simple LCCDEThe 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 conditionsTwo common methods exist for solving a LCCDE:

1. Direct MethodThe 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 Methoduse the relationship between the difference equation and z-transformconvert 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

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60

Outline


61

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

Recall: h[n] is the impulse response of an LTI systemH(ei ω) is the frequency responseH(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 circleAs 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


63


Using the distances from the unit circle to the poles and zeros, we can plot the frequency response of the systemAs 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


Consider: H (z) = z + 1 H (ejω) = ejω+1Some of the vectors represented by |ejω +1|, for random values of ω, are explicitly drawn onto the complex planeThese 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 amplitudeAs 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


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 polesSimilarly, the phase of H (ejω) ω is equal to the summation of the angles of all vectors associated with the zeros minus the summation of 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 polesConsider 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 plotFrom this plot, it can be concluded that the z-transform of the impulse response is:

67


The amplitude and angle of the vectors v1, v2and v3 depend on the frequency ωThe 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 zerosSimilarly, 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


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

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69

Summary

elec361: signals and systems topic 10: the z transformamer/teach/elec361/slides/topic10-zt.pdf · 1...

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