On Z-transform and Its Applications
some kinds of linear difference equations applications in digital signal In studying discrete-time signals and systems
Introduction to the z-transform
z-transforms and applications today in the areas of applied mathematics digital signal processing
The z-Transform and Its Application
Chapter 3: The z-Transform and Its Application. Discrete-Time Signals and Systems John G. Proakis and Dimitris G. Manolakis Digital Signal Processing:.
Digital signal processing: Roles of Z-transform & Digital Filters
Time-domain and frequency domain representation methods offer alternative clear insights into a system and depending on the application
SIGNALS AND SYSTEMS For SIGNALS AND SYSTEMS For
12 juin 2020 Fourier transform. • Discrete systems : Z-transform generalization of DTFT
The ?-transform and its application for solving linear time-invariant
The z-transform of common discrete-time signals have been determined in closed form Just as linear continuous-time systems are described by differential ...
Discrete - Time Signals and Systems Z-Transform-FIR filters
n output y n by applying inverse Z transform to Y yn. Y z z. -. ?--?. - ! -. Calculating the output of a FIR filter using Z transforms.
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application to LTI differential systems state-space systems
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 ApplicationDiscrete-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 ApplicationThe Directz-Transform
Directz-Transform:
X(z) =1X
n=1x(n)znNotation:
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 ApplicationRegion 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 forjzcjImay 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 ApplicationPower 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, andIdiverges 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)
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 ApplicationRegion 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
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:r1ROC 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:r22(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)jajr2
1 2Conjugation:x(n)X(z) ROC
z-Dierentiation:n x(n)zdX(z)dz r2Dr. Deepa Kundur (University of Toronto)
2Conjugation:x(n)X(z) ROC
z-Dierentiation:n x(n)zdX(z)dz r2The 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 ApplicationConvolution 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)gDr. Deepa Kundur (University of Toronto)
The z-Transform and Its Application14 / 36 Chapter 3: Thez-Transform and Its ApplicationCommon 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
quotesdbs_dbs2.pdfusesText_4
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
quotesdbs_dbs2.pdfusesText_4
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 ApplicationCommon 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
quotesdbs_dbs2.pdfusesText_4
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
quotesdbs_dbs2.pdfusesText_4
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), denotedH(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) =1Dr. Deepa Kundur (University of Toronto)
The z-Transform and Its Application18 / 36 Chapter 3: Thez-Transform and Its ApplicationPoles 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=b0zN+ (a1=a0)zN1++aN=a0=
b0a0zM+N(zz1)(zz2)(zzM)(zp1)(zp2)(zpN)
=GzNMQ k=1(zzk)Qk=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)whereGb0a0Note: \nite" does not include zero or1.I
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