lecture 2 : z-transform

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Lecture 2 : Z-Transform XILIANG LUO 2014/9

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Lecture 2 : Z-Transform. Xiliang Luo 2014/9. Fourier Transform. Convergence. A sufficient condition: absolutely summable it can be shown the DTFT of absolutely summable sequence converge uniformly to a continuous function. Square Summable. A sequence is square summable if: - PowerPoint PPT Presentation

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Page 1: Lecture 2 :  Z-Transform

Lecture 2: Z-TransformXILIANG LUO

2014/9

Page 2: Lecture 2 :  Z-Transform

Fourier Transform

Page 3: Lecture 2 :  Z-Transform

Convergence A sufficient condition: absolutely summable

it can be shown the DTFT of absolutely summable sequence converge uniformly to a continuous function

Page 4: Lecture 2 :  Z-Transform

Square Summable A sequence is square summable if:

For square summable sequence, we have mean-square convergence:

โˆ‘๐‘›=โˆ’โˆž

โˆž

|๐‘ฅ [๐‘› ]|2<โˆž

Page 5: Lecture 2 :  Z-Transform

Z-Transform

a function of the complex variable: z

If we replace the complex variable z by , we have the Fourier Transform!

Page 6: Lecture 2 :  Z-Transform

Z-Transform & Fourier Transform

Page 7: Lecture 2 :  Z-Transform

Complex z-plane

Page 8: Lecture 2 :  Z-Transform

Region of Convergence The set of z for which the z-transform converges is called ROC of the z-transform.

Absolutely summable criterion:

Page 9: Lecture 2 :  Z-Transform

ROC ROC consists of a ring in the z-plane

Page 10: Lecture 2 :  Z-Transform

Closed-Form in ROC When X(z) is a rational function inside ROC, i.e.

P(z), Q(z) are polynomials in z Zeros: values of z such that X(z) = 0 Poles: values of z such that X(z) = infinity

Page 11: Lecture 2 :  Z-Transform

Z-Transform Example: Right-Sided

Page 12: Lecture 2 :  Z-Transform

Z-Transform Example:Left-Sided

Page 13: Lecture 2 :  Z-Transform

Diff. Sum, Same Z-Transform? One is right-sided exponential sequence

One is left-sided exponential sequence

But they share the same algebraic expressions for their Z-Transforms

This emphasizes the importance of the region of convergence!!

Page 14: Lecture 2 :  Z-Transform

ROC Properties

Page 15: Lecture 2 :  Z-Transform

ROC Properties

Page 16: Lecture 2 :  Z-Transform

ROC Properties

Page 17: Lecture 2 :  Z-Transform

Inverse z-Transform From the z-Transform, we can recover the original sequence using the following complex contour integral:

๐‘ฅ [๐‘› ]= 12๐œ‹ ๐‘—โˆฎ๐ถ

โ‘

๐‘‹ (๐‘ง )๐‘ง๐‘›โˆ’1๐‘‘๐‘ง

C is a closed contour within the ROC of the z-transform

Page 18: Lecture 2 :  Z-Transform

Inverse z-Transform Methods Inspection

familiar with the common transform pairs

Partial Fraction Expansion

Power Series Expansion

Page 19: Lecture 2 :  Z-Transform

z-Transform Properties 1. Linearity

2. Time Shifting

3. Multiplication by an Exponential Sequence

Page 20: Lecture 2 :  Z-Transform

z-Transform Properties 4. Differentiation of X(z)

5. Conjugation of a Complex Sequence

7. Time Reversal

Page 21: Lecture 2 :  Z-Transform

z-Transform Properties 7. Convolution of Sequences

Page 22: Lecture 2 :  Z-Transform

z-Transform and LTI Systems LTI system is characterized by its impulse response h[n]

h[n]x[n] y[n]

๐‘ฆ [๐‘› ]=๐‘ฅ [๐‘› ]โ‹†h[๐‘›]

๐‘Œ (๐‘ง )=๐‘‹ (๐‘ง )ร—๐ป (๐‘ง )

H(z) is called the system function of this LTI system!

Page 23: Lecture 2 :  Z-Transform

Cauchy-Riemann Equations If function f(z) is differentiable at z0=x0+y0, then its component functions must satisfy the following conditions:

๐‘“ (๐‘ง )=๐‘ข (๐‘ฅ , ๐‘ฆ )+๐‘–๐‘ฃ (๐‘ฅ , ๐‘ฆ)

๐œ•๐‘ข๐œ•๐‘ฅ

=๐œ• ๐‘ฃ๐œ• ๐‘ฆ

๐œ•๐‘ข๐œ• ๐‘ฆ

=โˆ’๐œ• ๐‘ฃ๐œ• ๐‘ฅ

Page 24: Lecture 2 :  Z-Transform

Analytic Functions A function f(z) is analytic at a point z0 if it has a derivative at each point in some neighborhood of z0.

So, If f(z) is analytic at a point z0, it must be analytic at each point in some neighborhood of z0.

Page 25: Lecture 2 :  Z-Transform

Taylor Series Theorem: Suppose that a function f is analytic throughout a disk: |z-z0|<R0, centered at z0 and with radius R0, then f(z) has the power series representation:

๐‘“ (๐‘ง )=โˆ‘๐‘›=0

+โˆž

๐‘Ž๐‘› (๐‘ง โˆ’๐‘ง0 )๐‘› |๐‘งโˆ’ ๐‘ง0|<๐‘…0

๐‘Ž๐‘›=๐‘“ (๐‘› )(๐‘ง0)๐‘› !

Page 26: Lecture 2 :  Z-Transform

Laurent Series If a function is not analytic at a point z0, one cannot apply Taylorโ€™s theorem at that point!

Laurentโ€™s Theorem: Suppose a function f is analytic throughout an annular domain centered at z0:

Let C denote any positively oriented simple closed contour around z0 and lying in the domain, then, at each point in the domain, f(z) has the series representation:

๐‘…1<|๐‘งโˆ’๐‘ง 0|<๐‘…2

๐‘“ (๐‘ง )= โˆ‘๐‘›=โˆ’โˆž

+โˆž

๐‘๐‘› (๐‘งโˆ’๐‘ง 0 )๐‘›

Page 27: Lecture 2 :  Z-Transform

Laurent Series

๐‘๐‘›=12๐œ‹ ๐‘–โˆฎ๐ถ

โ‘ ๐‘“ (๐‘ง )๐‘‘๐‘ง

(๐‘งโˆ’ ๐‘ง0 )๐‘›+1

๐‘“ (๐‘ง )= โˆ‘๐‘›=โˆ’โˆž

+โˆž

๐‘๐‘› (๐‘งโˆ’๐‘ง 0 )๐‘›

Page 28: Lecture 2 :  Z-Transform

Homework Problems

3.59:

3.57:

3.52:

3.56:

Page 29: Lecture 2 :  Z-Transform

Next Sampling of Continuous-Time Signals

Please read the textbook Chapter 4 in advance!