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Proakis and Dimitris G Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 2007 Dr Deepa Kundur (University 



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[PDF] The z-Transform and Its Application - University of Toronto

Proakis and Dimitris G Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 2007 Dr Deepa Kundur (University 



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[PDF] The z-Transform and Its Application - University of Toronto

Thez-Transform and Its Application

Dr. Deepa Kundur

University of Toronto

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application1 / 36 Chapter 3: Thez-Transform and Its Application

Discrete-Time Signals and Systems

Reference:

Sections 3.1 - 3.4 of

John G. Proakis and Dimitris G. Manolakis,Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, 2007.

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application2 / 36 Chapter 3: Thez-Transform and Its Application

The Directz-Transform

Directz-Transform:

X(z) =1X

n=1x(n)zn

Notation:

X(z) Zfx(n)g

x(n)Z !X(z)

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application3 / 36 Chapter 3: Thez-Transform and Its Application

Region of Convergence

the region of convergence(ROC) ofX(z) is the set of all values ofzfor whichX(z) attains a nite value Thez-Transform is, therefore, uniquely characterized by: exp ressionfo rX(z)

ROC of X(z)

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application4 / 36

Chapter 3: Thez-Transform and Its Application

Power Series Convergence

For a power series,

f(z) =1X n=0a n(zc)n=a0+a1(zc) +a2(zc)2+ there exists a number 0r 1such that the series convergences forjzcjIdiverges forjzcj>r

Imay or may not converge for values onjzcj=r.

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application5 / 36 Chapter 3: Thez-Transform and Its Application

Power Series Convergence

For a power series,

f(z) =1X n=0a n(zc)n=a0+a1(zc)+a2(zc)2+ there exists a number 0r 1such that the series convergences forjzcj>r, and

Idiverges forjzcj

Imay or may not converge for values onjzcj=r.

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application6 / 36 Chapter 3: Thez-Transform and Its Application

Region of Convergence

Consider

X(z) =1X

n=1x(n)zn= n=1x(n)zn+1X n=0x(n)zn=

0=0x(n0)zn0

|{z}

ROC:jzj n=0x(n)zn |{z}

ROC:jzj>r2x(0)|{z}

ROC: allzDr. Deepa Kundur (University of Toronto)The z-Transform and Its Application7 / 36 Chapter 3: Thez-Transform and Its Application

Region of Convergence:r1>r2Dr. Deepa Kundur (University of Toronto)The z-Transform and Its Application8 / 36

Chapter 3: Thez-Transform and Its Application

Region of Convergence:r1

ROC Families: Finite Duration Signals

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application10 / 36 Chapter 3: Thez-Transform and Its Application

ROC Families: Innite Duration Signals

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application11 / 36 Chapter 3: Thez-Transform and Its Application z-Transform Properties Property Time Domainz-Domain ROCNotation:x(n)X(z) ROC:r21(n)X1(z) ROC1

2(n)X1(z) ROC2

Linearity:a1x1(n) +a2x2(n)a1X1(z) +a2X2(z) At least ROC1\ROC2

Time shifting:x(nk)zkX(z) ROC, except

z= 0 (ifk>0) andz=1(ifk<0) z-Scaling:anx(n)X(a1z)jajr2Time reversalx(n)X(z1)1r

1

2Conjugation:x(n)X(z) ROC

z-Dierentiation:n x(n)zdX(z)dz r2Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application12 / 36

Chapter 3: Thez-Transform and Its Application

Convolution Property

x(n) =x1(n)x2(n)()X(z) =X1(z)X2(z)

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application13 / 36 Chapter 3: Thez-Transform and Its Application

Convolution using thez-Transform

Basic Steps:

Compute z-Transform of each of the signals to convolve (time domain!z-domain):

1(z) =Zfx1(n)g

2(z) =Zfx2(n)g

Multiply the t woz-Transforms (inz-domain):

X(z) =X1(z)X2(z)

Find the inverse z-Transformof the product (z-domain!time domain): x(n) =Z1fX(z)g

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application14 / 36 Chapter 3: Thez-Transform and Its Application

Common Transform Pairs

Signal,x(n)z-Transform,X(z) ROC1(n) 1 Allz

2u(n)11z1jzj>1

3anu(n)11az1jzj>jaj

4nanu(n)az1(1az1)2jzj>jaj

5anu(n1)11az1jzj

6nanu(n1)az1(1az1)2jzj

7 cos(!0n)u(n)1z1cos!012z1cos!0+z2jzj>1

8 sin(!0n)u(n)z1sin!012z1cos!0+z2jzj>1

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application15 / 36 Chapter 3: Thez-Transform and Its Application

Common Transform Pairs

Signal,x(n)z-Transform,X(z) ROC1(n)1 All z

2u(n)11z1jzj>1

3anu(n)11az1jzj>jaj

4nanu(n)az1(1az1)2jzj>jaj

5anu(n1)11az1jzj

6nanu(n1)az1(1az1)2jzj

7 cos(!0n)u(n)1z1cos!012z1cos!0+z2jzj>1

8 sin(!0n)u(n)z1sin!012z1cos!0+z2jzj>1

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application16 / 36

Chapter 3: Thez-Transform and Its Application

Why Rational?

X(z) is a rationalfunction i it can be represented as the ratio of two polynomials inz1(orz):

X(z) =b0+b1z1+b2z2++bMzMa

0+a1z1+a2z2++aNzNI

For LTI systems that are represented by

LCCDEs

, the z-Transform of the unit sample responseh(n), denoted

H(z) =Zfh(n)g, isrational Dr. Deepa Kundur (University of Toronto)The z-Transform and Its Application17 / 36 Chapter 3: Thez-Transform and Its Application

Poles and Zeros

zerosofX(z): values ofzfor whichX(z) = 0 polesofX(z): values ofzfor whichX(z) =1

Dr. Deepa Kundur (University of Toronto)

The z-Transform and Its Application18 / 36 Chapter 3: Thez-Transform and Its Application

Poles and Zeros of the Rationalz-Transform

Leta0;b06= 0:

X(z) =B(z)A(z)=b0+b1z1+b2z2++bMzMa

0+a1z1+a2z2++aNzN=

b0zMa 0zN zM+ (b1=b0)zM1++bM=b0z

N+ (a1=a0)zN1++aN=a0=

b0a

0zM+N(zz1)(zz2)(zzM)(zp1)(zp2)(zpN)

=GzNMQ k=1(zzk)Q

k=1(zpk)Dr. Deepa Kundur (University of Toronto)The z-Transform and Its Application19 / 36 Chapter 3: Thez-Transform and Its Application

Poles and Zeros of the Rationalz-Transform

X(z) =GzNMQ

k=1(zzk)Q k=1(zpk)whereGb0a

0Note: \nite" does not include zero or1.I

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