The document discusses the Z-transform, which is used in digital signal processing to characterize discrete-time signals and systems. It presents the basic theory of the Z-transform, including its formulation and properties like convergence. Examples are given to illustrate region of convergence concepts like stable, causal, and two-sided sequences. The inverse Z-transform and MATLAB commands for analysis are also covered.
This document discusses various properties and concepts related to the z-transform, which is used to analyze discrete-time signals. It defines the z-transform and region of convergence. It then provides examples of calculating the z-transform for different signals. Key properties discussed include time shifting, scaling, time reversal, differentiation, and convolution. Theorems regarding the initial value and final value are also covered. Worked examples are provided to demonstrate applying the various properties and concepts.
This document discusses various properties and concepts related to the z-transform, which is used to analyze discrete-time signals. It defines the z-transform and region of convergence. It then provides examples of calculating the z-transform for different signals. Key properties discussed include time shifting, scaling, time reversal, differentiation, and convolution. Theorems regarding the initial value and final value are also covered. Worked examples are provided to demonstrate applying the various properties and concepts.
Frequency Analysis using Z Transform.pptxDrPVIngole
This document provides an overview of frequency analysis using the Z-transform. It defines the Z-transform, discusses its properties and relationship to the Fourier transform. Examples are provided to demonstrate calculating the Z-transform of different signals and determining the region of convergence. Key topics covered include the definition of the Z-transform, its region of convergence, properties, inverse Z-transform, and analyzing discrete time linear time-invariant systems using the Z-transform.
The z-transform provides a third domain (besides time and frequency domains) to represent discrete-time signals and systems. It transforms a sequence into a complex function of z by taking the sum of the sequence multiplied by powers of z. The z-transform has useful properties like superposition and time-delay that allow convolutions to be represented as multiplications in the z-domain. Cascading systems in the time-domain corresponds to multiplying their z-transforms. Factoring z-transforms splits systems into subsystems, related to the roots of the z-transform polynomial.
This document discusses the z-transform, which is a mathematical tool used to analyze discrete-time control systems. It defines the one-sided and two-sided z-transform and provides examples of taking the z-transform of basic functions like unit step, ramp, polynomial and exponential functions. The document also covers important properties of the z-transform including linearity, shifting theorems, and the initial and final value theorems. It describes methods for finding the inverse z-transform including using tables, direct division, partial fraction expansion and inversion integrals.
The document discusses the z-transform, which is a mathematical tool used to analyze discrete-time control systems. The z-transform plays a similar role for discrete-time systems as the Laplace transform does for continuous-time systems. Some key properties of the z-transform discussed include the region of convergence, properties and theorems like the shifting theorem and initial/final value theorems, and applications to feedback control systems.
The document discusses the z-transform, which is the discrete-time equivalent of the Laplace transform. It defines the z-transform and provides examples of calculating the z-transform for various sequences, including the unit impulse, unit step function, sinusoids, and exponential sequences. It also discusses properties of the z-transform such as the region of convergence and relationship to the discrete-time Fourier transform.
The document discusses the Z-transform, which is used in digital signal processing to characterize discrete-time signals and systems. It presents the basic theory of the Z-transform, including its formulation and properties like convergence. Examples are given to illustrate region of convergence concepts like stable, causal, and two-sided sequences. The inverse Z-transform and MATLAB commands for analysis are also covered.
This document discusses various properties and concepts related to the z-transform, which is used to analyze discrete-time signals. It defines the z-transform and region of convergence. It then provides examples of calculating the z-transform for different signals. Key properties discussed include time shifting, scaling, time reversal, differentiation, and convolution. Theorems regarding the initial value and final value are also covered. Worked examples are provided to demonstrate applying the various properties and concepts.
This document discusses various properties and concepts related to the z-transform, which is used to analyze discrete-time signals. It defines the z-transform and region of convergence. It then provides examples of calculating the z-transform for different signals. Key properties discussed include time shifting, scaling, time reversal, differentiation, and convolution. Theorems regarding the initial value and final value are also covered. Worked examples are provided to demonstrate applying the various properties and concepts.
Frequency Analysis using Z Transform.pptxDrPVIngole
This document provides an overview of frequency analysis using the Z-transform. It defines the Z-transform, discusses its properties and relationship to the Fourier transform. Examples are provided to demonstrate calculating the Z-transform of different signals and determining the region of convergence. Key topics covered include the definition of the Z-transform, its region of convergence, properties, inverse Z-transform, and analyzing discrete time linear time-invariant systems using the Z-transform.
The z-transform provides a third domain (besides time and frequency domains) to represent discrete-time signals and systems. It transforms a sequence into a complex function of z by taking the sum of the sequence multiplied by powers of z. The z-transform has useful properties like superposition and time-delay that allow convolutions to be represented as multiplications in the z-domain. Cascading systems in the time-domain corresponds to multiplying their z-transforms. Factoring z-transforms splits systems into subsystems, related to the roots of the z-transform polynomial.
This document discusses the z-transform, which is a mathematical tool used to analyze discrete-time control systems. It defines the one-sided and two-sided z-transform and provides examples of taking the z-transform of basic functions like unit step, ramp, polynomial and exponential functions. The document also covers important properties of the z-transform including linearity, shifting theorems, and the initial and final value theorems. It describes methods for finding the inverse z-transform including using tables, direct division, partial fraction expansion and inversion integrals.
The document discusses the z-transform, which is a mathematical tool used to analyze discrete-time control systems. The z-transform plays a similar role for discrete-time systems as the Laplace transform does for continuous-time systems. Some key properties of the z-transform discussed include the region of convergence, properties and theorems like the shifting theorem and initial/final value theorems, and applications to feedback control systems.
The document discusses the z-transform, which is the discrete-time equivalent of the Laplace transform. It defines the z-transform and provides examples of calculating the z-transform for various sequences, including the unit impulse, unit step function, sinusoids, and exponential sequences. It also discusses properties of the z-transform such as the region of convergence and relationship to the discrete-time Fourier transform.
This document discusses the z-transform, which is a generalization of the discrete-time Fourier transform (DTFT) that can be used when the DTFT does not exist. The z-transform represents discrete-time signals and systems in the complex z-domain. It is defined as the sum of the sequence multiplied by powers of a complex variable z. Rational z-transforms that represent linear time-invariant systems are ratios of polynomials in z. The region of convergence (ROC) of a z-transform specifies the range of z values where the sum converges. The ROC depends on the location of poles and the type of sequence.
The document discusses the Z-transform, which is a tool for analyzing and solving linear time-invariant difference equations. It defines the Z-transform, provides examples of common sequences and their corresponding Z-transforms, and discusses properties such as the region of convergence. Key topics covered include the difference between difference and differential equations, properties of linear time-invariant systems, and mapping between the s-plane and z-plane.
This document discusses the z-transform, which is a tool for analyzing discrete-time signals and systems analogous to the Laplace transform for continuous-time signals. It defines the z-transform, describes its region of convergence and pole-zero plot, lists some common z-transform pairs, and discusses how the location of poles and zeros in the z-plane relates to the stability of a system. Examples are provided to illustrate key concepts like finding the z-transform of sequences and determining the region of convergence.
This document discusses applying z-transforms to discrete systems to obtain the system's transfer function and output response. It begins by introducing z-transforms and their use in obtaining transfer functions for discrete systems. Transfer functions allow analyzing combinations of systems. The document provides examples of obtaining transfer functions for first and second order systems. It also covers obtaining the unit impulse and step responses from the transfer function. Finally, it discusses analyzing series combinations of systems by multiplying their individual transfer functions.
UNIT II DISCRETE TIME SYSTEM ANALYSIS 6+6
Z-transform and its properties, inverse z-transforms; difference equation – Solution by z transform,application to discrete systems - Stability analysis, frequency response –Convolution – Discrete Time Fourier transform , magnitude and phase representation.
Z TRANSFORM PROPERTIES AND INVERSE Z TRANSFORMTowfeeq Umar
The document discusses various methods for computing the inverse z-transform including inspection, partial fraction expansion, and power series expansion. It provides examples to illustrate each method. The inverse z-transform finds the original time domain sequence from its z-transform. Key properties like linearity, time shifting, and convolution are also covered.
The document defines z-transforms and provides properties of z-transforms. It defines the two-sided and one-sided z-transforms as infinite sums involving the input signal x[n] and complex variable z. Key properties include: convolution in the time domain is equivalent to multiplication in the z-domain; time shifts correspond to multiplication by powers of z; the region of convergence is determined by poles of the z-transform.
The document discusses the z-transform, which is the discrete-time equivalent of the Laplace transform. It defines the z-transform and describes its key properties, including:
- The region of convergence (ROC) which determines where the z-transform converges
- One-sided and two-sided z-transforms depending on if the signal is defined for all time or just positive/negative time
- Examples of z-transforms for basic signals like impulse, step, and exponential functions
- Properties like linearity, time-shifting, differentiation, and convolution
- How the ROC relates to stability and causality of systems described by the z-transform
Digital Signal Processing and the z-transformRowenaDulay1
The document discusses the z-transform, which is a generalization of the Fourier transform used to analyze discrete-time signals and systems. It defines the z-transform, explains why it is used instead of the Fourier transform, and discusses its region of convergence and properties of poles and zeros. Examples are provided to illustrate key concepts such as how the region of convergence is determined by the signal type (left-sided, right-sided, two-sided). Important z-transform pairs are also summarized.
The document discusses the z-transform, which is a generalization of the Fourier transform used to analyze discrete-time signals and systems. It defines the z-transform, explains why it is used instead of the Fourier transform, and discusses its representation in the z-plane. The key topics covered include the region of convergence, poles and zeros, important z-transform pairs like the unit step function, and properties related to system stability. Examples are provided to illustrate concepts like how the region of convergence is determined based on whether a sequence is right-sided, left-sided, or two-sided.
The z-transform is the discrete-time signal processing counterpart of the Laplace transform. It is a generalization of the Fourier transform that can be used for signals where the Fourier transform does not exist. The z-transform defines a signal in terms of complex variable z, rather than angular frequency like the discrete-time Fourier transform (DTFT). The region of convergence (ROC) of the z-transform determines whether it converges, and whether the DTFT exists for the signal. The ROC cannot contain any poles and determines properties like stability and causality of the system.
The document discusses the z-transform, which is useful for analyzing discrete-time signals and systems. It defines the z-transform as the sum of a discrete-time signal multiplied by z-n, where z is a complex variable. The region of convergence for a z-transform consists of the values of z where this sum converges. The z-transform has properties like linearity, time shifting, and convolution that allow discrete-time signals and systems to be analyzed in the complex z-domain. Key topics covered include the definition of the z-transform, properties like region of convergence and poles/zeros, and properties like linearity, time shifting, and the convolution theorem.
short course on Subsurface stochastic modelling and geostatisticsAmro Elfeki
This is a short course on Subsurface stochastic modelling and geo-statistics that has been held at Delft University of Technology, Delft The Netherlands.
This document discusses the Z-transform and its inverse. It begins by defining the Z-transform as a mapping from a discrete-time signal to a power series. The region of convergence (ROC) is introduced as the set of values where the Z-transform has a finite value. Common properties of the ROC are described. Rational Z-transforms containing poles and zeros are covered. The inverse Z-transform is discussed using inspection and partial fraction expansion methods. The relationship between the Z-transform of a system's impulse response and its transfer function is explained. Finally, the four-step process for analyzing discrete-time linear time-invariant systems in the transform domain using the Z-transform is outlined.
This document contains information about a student named Soumadip Ghara who is in the 6th semester of the Electrical Engineering department. It provides details of an exam on the topic of Z-transform that was held on February 2, 2023. The document defines the Z-transform and discusses its properties including linearity, multiplication by a constant, shifting theorems, and the region of convergence. It also lists some common Z-transform properties and theorems.
This document discusses methods for computing the inverse z-transform, including inspection of z-transform pairs and properties, partial fraction expansion, and power series expansion. The inverse z-transform can generally be computed as a contour integral, but these computational methods can be used for rational z-transforms. Examples are provided to illustrate each method.
Chapter 6 frequency domain transformation.pptxEyob Adugnaw
This document discusses the z-transform, which is a generalization of the Fourier transform that can be used to analyze discrete-time signals and systems. It introduces the z-transform and its properties such as the region of convergence, poles and zeros. Examples are provided to illustrate how to determine the region of convergence from the sequence and identify stable and unstable systems based on whether the unit circle is included in the region of convergence. Theorems and properties of the z-transform such as linearity, time shifting, and differentiation are also covered.
- The z-transform is a mathematical tool that converts discrete-time sequences into complex functions, analogous to how the Laplace transform handles continuous-time signals.
- Key properties and sequences that are transformed include the unit impulse δn, unit step un, and geometric sequences an.
- The z-transform is computed by taking the z-transform definition, which is an infinite summation, and obtaining closed-form expressions using properties like linearity and geometric series sums.
- Common transforms include U(z) for the unit step, 1/1-az^-1 for geometric sequences an, and expressions involving z, sinh/cosh, and sin/cos for exponential and trigonometric sequences.
The document discusses using z-transforms to solve difference equations. It provides examples of first and second order linear difference equations and explains how to solve them using z-transforms. The process involves taking the z-transform of each term in the difference equation, resulting in an algebraic equation that can be solved for the z-transform of the solution sequence. The inverse z-transform is then found to obtain the solution sequence. Partial fractions and residues can be used to invert z-transforms.
This document discusses the z-transform, which is a generalization of the discrete-time Fourier transform (DTFT) that can be used when the DTFT does not exist. The z-transform represents discrete-time signals and systems in the complex z-domain. It is defined as the sum of the sequence multiplied by powers of a complex variable z. Rational z-transforms that represent linear time-invariant systems are ratios of polynomials in z. The region of convergence (ROC) of a z-transform specifies the range of z values where the sum converges. The ROC depends on the location of poles and the type of sequence.
The document discusses the Z-transform, which is a tool for analyzing and solving linear time-invariant difference equations. It defines the Z-transform, provides examples of common sequences and their corresponding Z-transforms, and discusses properties such as the region of convergence. Key topics covered include the difference between difference and differential equations, properties of linear time-invariant systems, and mapping between the s-plane and z-plane.
This document discusses the z-transform, which is a tool for analyzing discrete-time signals and systems analogous to the Laplace transform for continuous-time signals. It defines the z-transform, describes its region of convergence and pole-zero plot, lists some common z-transform pairs, and discusses how the location of poles and zeros in the z-plane relates to the stability of a system. Examples are provided to illustrate key concepts like finding the z-transform of sequences and determining the region of convergence.
This document discusses applying z-transforms to discrete systems to obtain the system's transfer function and output response. It begins by introducing z-transforms and their use in obtaining transfer functions for discrete systems. Transfer functions allow analyzing combinations of systems. The document provides examples of obtaining transfer functions for first and second order systems. It also covers obtaining the unit impulse and step responses from the transfer function. Finally, it discusses analyzing series combinations of systems by multiplying their individual transfer functions.
UNIT II DISCRETE TIME SYSTEM ANALYSIS 6+6
Z-transform and its properties, inverse z-transforms; difference equation – Solution by z transform,application to discrete systems - Stability analysis, frequency response –Convolution – Discrete Time Fourier transform , magnitude and phase representation.
Z TRANSFORM PROPERTIES AND INVERSE Z TRANSFORMTowfeeq Umar
The document discusses various methods for computing the inverse z-transform including inspection, partial fraction expansion, and power series expansion. It provides examples to illustrate each method. The inverse z-transform finds the original time domain sequence from its z-transform. Key properties like linearity, time shifting, and convolution are also covered.
The document defines z-transforms and provides properties of z-transforms. It defines the two-sided and one-sided z-transforms as infinite sums involving the input signal x[n] and complex variable z. Key properties include: convolution in the time domain is equivalent to multiplication in the z-domain; time shifts correspond to multiplication by powers of z; the region of convergence is determined by poles of the z-transform.
The document discusses the z-transform, which is the discrete-time equivalent of the Laplace transform. It defines the z-transform and describes its key properties, including:
- The region of convergence (ROC) which determines where the z-transform converges
- One-sided and two-sided z-transforms depending on if the signal is defined for all time or just positive/negative time
- Examples of z-transforms for basic signals like impulse, step, and exponential functions
- Properties like linearity, time-shifting, differentiation, and convolution
- How the ROC relates to stability and causality of systems described by the z-transform
Digital Signal Processing and the z-transformRowenaDulay1
The document discusses the z-transform, which is a generalization of the Fourier transform used to analyze discrete-time signals and systems. It defines the z-transform, explains why it is used instead of the Fourier transform, and discusses its region of convergence and properties of poles and zeros. Examples are provided to illustrate key concepts such as how the region of convergence is determined by the signal type (left-sided, right-sided, two-sided). Important z-transform pairs are also summarized.
The document discusses the z-transform, which is a generalization of the Fourier transform used to analyze discrete-time signals and systems. It defines the z-transform, explains why it is used instead of the Fourier transform, and discusses its representation in the z-plane. The key topics covered include the region of convergence, poles and zeros, important z-transform pairs like the unit step function, and properties related to system stability. Examples are provided to illustrate concepts like how the region of convergence is determined based on whether a sequence is right-sided, left-sided, or two-sided.
The z-transform is the discrete-time signal processing counterpart of the Laplace transform. It is a generalization of the Fourier transform that can be used for signals where the Fourier transform does not exist. The z-transform defines a signal in terms of complex variable z, rather than angular frequency like the discrete-time Fourier transform (DTFT). The region of convergence (ROC) of the z-transform determines whether it converges, and whether the DTFT exists for the signal. The ROC cannot contain any poles and determines properties like stability and causality of the system.
The document discusses the z-transform, which is useful for analyzing discrete-time signals and systems. It defines the z-transform as the sum of a discrete-time signal multiplied by z-n, where z is a complex variable. The region of convergence for a z-transform consists of the values of z where this sum converges. The z-transform has properties like linearity, time shifting, and convolution that allow discrete-time signals and systems to be analyzed in the complex z-domain. Key topics covered include the definition of the z-transform, properties like region of convergence and poles/zeros, and properties like linearity, time shifting, and the convolution theorem.
short course on Subsurface stochastic modelling and geostatisticsAmro Elfeki
This is a short course on Subsurface stochastic modelling and geo-statistics that has been held at Delft University of Technology, Delft The Netherlands.
This document discusses the Z-transform and its inverse. It begins by defining the Z-transform as a mapping from a discrete-time signal to a power series. The region of convergence (ROC) is introduced as the set of values where the Z-transform has a finite value. Common properties of the ROC are described. Rational Z-transforms containing poles and zeros are covered. The inverse Z-transform is discussed using inspection and partial fraction expansion methods. The relationship between the Z-transform of a system's impulse response and its transfer function is explained. Finally, the four-step process for analyzing discrete-time linear time-invariant systems in the transform domain using the Z-transform is outlined.
This document contains information about a student named Soumadip Ghara who is in the 6th semester of the Electrical Engineering department. It provides details of an exam on the topic of Z-transform that was held on February 2, 2023. The document defines the Z-transform and discusses its properties including linearity, multiplication by a constant, shifting theorems, and the region of convergence. It also lists some common Z-transform properties and theorems.
This document discusses methods for computing the inverse z-transform, including inspection of z-transform pairs and properties, partial fraction expansion, and power series expansion. The inverse z-transform can generally be computed as a contour integral, but these computational methods can be used for rational z-transforms. Examples are provided to illustrate each method.
Chapter 6 frequency domain transformation.pptxEyob Adugnaw
This document discusses the z-transform, which is a generalization of the Fourier transform that can be used to analyze discrete-time signals and systems. It introduces the z-transform and its properties such as the region of convergence, poles and zeros. Examples are provided to illustrate how to determine the region of convergence from the sequence and identify stable and unstable systems based on whether the unit circle is included in the region of convergence. Theorems and properties of the z-transform such as linearity, time shifting, and differentiation are also covered.
- The z-transform is a mathematical tool that converts discrete-time sequences into complex functions, analogous to how the Laplace transform handles continuous-time signals.
- Key properties and sequences that are transformed include the unit impulse δn, unit step un, and geometric sequences an.
- The z-transform is computed by taking the z-transform definition, which is an infinite summation, and obtaining closed-form expressions using properties like linearity and geometric series sums.
- Common transforms include U(z) for the unit step, 1/1-az^-1 for geometric sequences an, and expressions involving z, sinh/cosh, and sin/cos for exponential and trigonometric sequences.
The document discusses using z-transforms to solve difference equations. It provides examples of first and second order linear difference equations and explains how to solve them using z-transforms. The process involves taking the z-transform of each term in the difference equation, resulting in an algebraic equation that can be solved for the z-transform of the solution sequence. The inverse z-transform is then found to obtain the solution sequence. Partial fractions and residues can be used to invert z-transforms.
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Understanding Inductive Bias in Machine LearningSUTEJAS
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1. Digital Signal & Image
Processing
Lecture-6
Dr Muhammad Arif
m.arif@faculty.muet.edu.pk
https://sites.google.com/site/mdotarif/teaching/dsip
2. Overview
• Z Transform
• Properties of z-transform
• Transfer Function
• Transfer Function & Difference Equation
• Transfer Function & Impulse Response
• Inverse Z Transform
• Transfer Function & System Stability
• Difference Equation & System Stability
• Impulse & Step Responses
• Steady State Output
3. Z Transform
• The z transform is an important digital signal processing tool
for describing and analyzing digital systems.
• It also supports the techniques for digital filter design and
frequency analysis of digital signals.
• It takes a signal from the time domain to a frequency domain
called the z domain.
3
4. Z Transform
• The z transform for a digital signal x[n] is defined as
𝑋 𝑧 = 𝒁 𝑥[𝑛]
𝑋 𝑧 =
𝑛=−∞
∞
𝑥 𝑛 𝑧−𝑛
where z is the complex variable.
4
5. Z Transform
• The z transform for causal signals is
𝑋 𝑧 =
𝑛=0
∞
𝑥 𝑛 𝑧−𝑛
It is referred to as a one-sided z-transform or a unilateral transform.
5
6. Z Transform Table
6
# Signal x[n] Z Transform X(z) Region of Convergence
1 [n] 1 All z
2 u[n] Z/(Z-1) Z> 1
3 nu[n] Z/(Z-) Z>
4 nu[n] Z/(Z-1)2 Z> 1
5 nn u[n] Z-1/(1-Z-1)2 Z>
6 Cos(nΩ)u[n] ZsinΩ/(Z2 - 2zcosΩ + β) Z> 1
8. Region of Convergence (ROC)
• The z transform for every signal has an associated Region of
Convergence (ROC), the region of the z domain for which the
transform exists.
• Since the z-transform is an infinite series, it exists only for
those values of z for which this series converges.
• All the values of z that make the summation exist form a
Region of Convergence (ROC) in the z-transform domain.
• While all other values of z outside the ROC will cause the
summation to diverge.
8
9. Z Transform
Example-1: Determine the z-transform of the following signals.
a) x[n] = δ[n]
solution
𝑋 𝑧 =
𝑛=0
∞
𝛿 𝑛 𝑧−𝑛
= 𝛿 0 = 1
ROC: entier 𝑧 plane
9
10. Z Transform
Example-1: Determine the z-transform of the following signals.
b) x[n] = δ[n-1]
solution
𝑋 𝑧 =
𝑛=0
∞
𝛿 𝑛 − 1 𝑧−𝑛
= 𝛿 0 𝑧−1
= 𝑧−1
ROC: entire 𝑧 plane except z = 0.
10
11. Z Transform
Example-1: Determine the z-transform of the following signals.
c) x[n] = u[n]
Solution 𝑋 𝑧 = 𝑛=0
∞
𝑢 𝑛 𝑧−𝑛
= 𝑛=0
∞
𝑧−𝑛
𝑋 𝑧 = 1 + 𝑧−1
+ 𝑧−2
+ 𝑧−3
+……
• This is a geometric series of the form a+ ar + ar2 +…. With initial
term a equal to 1 and multiplier r equal to z-1.
• The sum of infinite geometric series is 𝑆∞ =
𝑎
1−𝑟
• So X(z)=
1
1−𝑧−1 =
𝑧
𝑧−1
ROC: 𝑧 > 1 11
12. Z Transform
Example-1: Determine the z-transform of the following signals.
d) x[n] = u[n-1]
Solution
X z = 𝑧−1 1
1−𝑧−1 = 𝑧−1 𝑧
𝑧−1
=
1
𝑧−1
ROC: 𝑧 > 1
12
13. Z Transform
Example-1: Determine the z-transform of the following signals.
e)
Solution
x[n] = δ[n] + 2δ[n-1] + 5δ[n-2] + 7δ[n-3] + δ[n-5]
ROC: entire 𝑧 plane except 𝑧 = 0 and z =
13
14. Z Transform
Example-1: Determine the z-transform of the following signals.
f)
Solution
x[n] = δ[n+2] + 2δ[n+1] + 5δ[n] + 7δ[n-1] + δ[n-3]
ROC: entire 𝑧 plane except 𝑧=0
14
17. Z Transform
Example-2: Find the z transform of the signal x[n] depicted in
the figure.
Solution
The signal x[n] is described as:
x[n] = 2δ[n] + δ[n-1] + 0.5δ[n-2]
The z transform of the signal is
• 𝑋 𝑧 = 𝑛=0
∞
𝑥 𝑛 𝑧−𝑛
• 𝑋 𝑧 = 𝑥 0 + 𝑥 1 𝑧−1 + 𝑥[2]𝑧−2
• 𝑋 𝑧 = 2 + 𝑧−1
+ 0.5𝑧−2
17
22. Properties of z-transform
Time Shifting/Shift Theorem
• A one-sample delay in the time domain appears in the z
domain as a z-1 factor. That is,
Z{x[n-1]} = z-1X(z)
More generally,
Z{x[n-k]} = z-kX(z)
22
25. Properties of z-transform
Time Shifting/Shift Theorem
25
Example-6: Find the z-transform of the signal x[n] defined by
Solution
Applying the time shifting property of the z-transform, we
have
27. Properties of z-transform
Time Reversal
27
Example-7: Find the z-transform of the signal x[n] = u[-n]
Solution
Applying the time reversal theorem of the z-transform, we have
33. Difference Equation Diagram using z–1 Notation
• Time shifting property of the z transform suggests a notation
change for difference equation diagram.
• The delay blocks can be replaced by z-1 bocks.
• This convention mixes the time and z domain notations.
33
34. Difference Equation Diagram using z–1 Notation
• The general form of the non-recursive difference equation is
y[n] = b0x[n] + b1x[n-1] + b2x[n-2] + … + bMx[n-M]
• Re-expressing the non-recursive difference equation diagram
using the z-1 notation.
34
36. Transfer Function
• In the z domain, the transfer function of a filter can be
defined.
• The transfer function is the ratio of the output to the input in
the z domain:
𝐻 𝑧 =
𝑌(𝑧)
𝑋(𝑧)
In this equation
Y(z) is the z transform of the output y[n]
X(z) is the z transform of the input x[n]
H(z) is the transfer function of the filter
36
37. Transfer Function & Difference Equation
• The general form of a difference equation is
a0y[n] + a1y[n-1] + a2y[n-2] + … + aNy[n-N]
= b0x[n] + b1x[n-1] + b2x[n-2] + … + bMx[n-M]
Taking the z transform of the above equation
a0Y(z)+ a1z-1Y(z)+ a2z-2Y(z) + … + aNz-NY(z)
= b0X(z) + b1z-1X(z) + b2z-2X(z) + … + bMz-MX(z)
Taking Y(Z) and X(Z) common and then cross multiply to get TF.
37
38. Transfer Function & Difference Equation
Example-10: Find the transfer function described by the
difference equation.
2y[n] + y[n-1] + 0.9y[n-2] = x[n-1] + x[n-4]
Solution: Taking z transforms term by term we get,
2Y(z) + z-1Y(z) + 0.9z-2Y(z) = z-1X(z) + z-4X(z)
Factoring out Y(z) on the left side and X(z) on the right side:
(2 + z-1 + 0.9z-2)Y(z) = (z-1 + z-4)X(z)
The transfer function (TF) is
H 𝑧 =
𝑌(𝑧)
𝑋(𝑧)
=
𝑧−1+𝑧−4
2+𝑧−1+0.9𝑧−2 38
39. Transfer Function & Difference Equation
Example-11: Find the transfer function described by the
difference equation.
y[n] – 0.2y[n-1] = x[n] + 0.8x[n-1]
Solution: Taking z transforms term by term we get,
Y(z) – 0.2z-1Y(z) = X(z) + 0.8z-1X(z)
Factoring out Y(z) on the left side and X(z) on the right side:
(1 – 0.2z-1)Y(z) = (1 + 0.8z-1)X(z)
The transfer function (TF) is
H 𝑧 =
𝑌(𝑧)
𝑋(𝑧)
=
1 + 0.8𝑧−1
1 − 0.2𝑧−1 39
40. Transfer Function & Difference Equation
Example-12: Find the transfer function described by the
difference equation.
y[n] = 0.75x[n] - 0.3x[n-2] – 0.01x[n-3]
Solution: Taking z transforms term by term we get,
Y(z) = 0.75X(z) - 0.3z-2X(z) – 0.01z-3X(z)
Factoring out Y(z) on the left side and X(z) on the right side:
Y(z) = (0.75 - 0.3z-2 - 0.01z-3)X(z)
The transfer function (TF) is
H 𝑧 =
𝑌(𝑧)
𝑋(𝑧)
= 0.75 − 0.3𝑍−2 − 0.01𝑍−3
40
41. Transfer Function & Difference Equation
Example-13: Find the difference equation that correspond to
transfer function.
𝐇 𝒛 =
𝟏 + 𝟎. 𝟓𝒛−𝟏
𝟏 − 𝟎. 𝟓𝒛−𝟏
Solution: Since H(z) = Y(z)/X(z), do the cross multiply to get
(1 – 0.5z-1)Y(z) = (1 + 0.5z-1)X(z)
then
Y(z) – 0.5z-1Y(z) = X(z) + 0.5z-1X(z)
Finally taking the inverse z transform term by term to get
y[n] – 0.5y[n-1] = x[n] + 0.5x[n-1]
41
42. Transfer Function & Difference Equation
Example-14: Find the difference equation that correspond to
transfer function.
𝐇 𝒛 =
𝟏 + 𝟎. 𝟖𝒛−𝟏
𝟏 − 𝟎. 𝟐𝒛−𝟏 + 𝟎. 𝟕𝒛−𝟐
Solution: Since H(z) = Y(z)/X(z), do the cross multiply to get
(1 – 0.2z-1 + 0.7z-2)Y(z) = (1 + 0.8z-1)X(z)
then
Y(z) – 0.5z-1Y(z) + 0.7z-2Y(z)= X(z) + 0.8z-1X(z)
Finally taking the inverse z transform term by term to get
y[n] – 0.2y[n-1] + 0.7y[n-2]= x[n] + 0.8x[n-1]
42
43. Transfer Function & Difference Equation
Example-15: Find the difference equation that correspond
to transfer function.
𝐇 𝒛 =
𝒛
(𝟐𝒛 − 𝟏)(𝟒𝒛 − 𝟏)
Solution: H 𝑧 =
𝑧
8𝑧2−6𝑧+1
Since H(z) = Y(z)/X(z), do the cross multiply to get
(8𝑧2
− 6𝑧 + 1 )Y(z) = (z)X(z)
Then 8z2Y(z) – 6zY(z) + y(z) = zX(z)
Finally taking the inverse z transform term by term to get
8y[n] – 6y[n-1] + y[n-2] = x[n-1]
43
44. Transfer Function & Impulse Response
• The relationship between the transfer function and the
impulse response of a system is also straightforward.
• the transfer function H(z) is the z transform of the impulse
response h[n].
𝐻 𝑧 = 𝒁 ℎ[𝑛]
𝐻 𝑧 =
𝑛=0
∞
ℎ[𝑛]𝑧−1
• Similarly Impulse response h[n] is inverse z transform of the
transfer function H(z).
ℎ[𝑛] = 𝒁−1 𝐻(𝑧) 44
45. Transfer Function & Impulse Response
Example-16: Find the transfer function of the system whose
impulse response is
h[n] = δ[n] + 0.4 δ[n-1] + 0.2 δ[n-2] + 0.05 δ[n-3]
Solution
The transfer function H(z) of the system is the z transform of the
impulse response h[n]. Taking z transform term by term we get
H(z) = 1 + 0.4z-1 + 0.2z-2 + 0.05z-3
Note that we can also get the difference equation from the TF.
y[n] = x[n] + 0.4x[n-1] + 0.2x[n-2]+ 0.05x[n-3]
45
46. System Outputs in Time & Z Domains
• The system output can be find using three different ways.
46
47. System Output using TF
• The definition of the transfer function (TF) provides a means
of calculating filter outputs. That is,
Y(z) = H(z)X(z)
• To determine the time domain output y[n], the inverse z
transform of Y(z) must be taken.
47
49. Inverse Z Transform
• To convert a function in the z domain into a function in the
time domain requires an inverse z transform.
• This conversion is necessary, for example, to find the time
domain functions like
x[n] that correspond to the z transforms X(z)
y[n] that correspond to the z transforms Y(z)
h[n] impulse response from a transfer function H(z)
49
50. Inverse Z Transform
There are several ways of finding inverse z transforms:
A: Formal Method
• Contour Integration
B: Informal Methods
1- Inspection method using Z Transform Tables
2- Long Division (Synthetic Division or Power Series Expansion)
3- Partial Fraction Expansion
50
51. Inverse Z Transform
A: Formal Method
• Contour Integration:
where C represents a closed contour within the ROC of the z-
transform.
The most fundamental method for the inversion of z transform is
the general inversion method which is based on the Laurent
theorem.
The contour integral of the above equation can be evaluated using
the residue theorem.
51
52. Inspection Method using Z Transform Tables
Example-17: Find the x[n] that corresponds to the z transform
𝑿 𝒛 =
𝒛
𝒛 − 𝟎. 𝟖
Solution
Using z transform table, the inverse z transform is
𝑥 𝑛 = 𝑍−1 𝑋(𝑧)
𝑥 𝑛 = (0.8)𝑛
𝑢[𝑛]
52
53. Inspection Method using Z Transform Tables
Example-18: Find the inverse z transform of the function
𝑿 𝒛 =
𝒛𝟐 − 𝟎. 𝟗𝒛
𝒛𝟐 − 𝟏. 𝟖𝒛 + 𝟏
Using z transform table, the inverse z transform is
𝑥 𝑛 = 𝒁−1
𝑋(𝑧)
𝑋 𝑧 =
𝑧2 − 0.9𝑧
𝑧2 − 1.8𝑧 + 1
cosΩ = 0.9
Ω = cos-1(0.9) = 0.451
𝑥 𝑛 = cos(𝑛Ω)𝑢[𝑛]
𝑥 𝑛 = cos(0.451Ω)𝑢[𝑛]
53
54. Long Division Method
ADVANTGES
• Relatively straight forward method
• Applicable to any rational function
• Can be use to convert improper rational function into proper
rational function
DISADVANTAGES
• Sometimes will run to infinity
• General close-form solution cannot be found
54
55. Transfer Function & System Stability
• Transfer function can be expressed as a rational function
consist of numerator polynomial divided by denominator
polynomial.
• The highest power in a polynomial is called its degree.
• In a proper rational function, the degree of the numerator is
less than or equal to the degree of the denominator.
• In a strictly proper rational function, the degree of the
numerator is less than or the degree of the denominator.
• In an improper rational function, the degree of the numerator
is greater than the degree of the denominator. 55
57. Long Division Method
57
Example-19: Using long division method, determine the inverse z-transform of
The inverse Z transform is h[n] = δ[n] – 0.5δ[n-1] – 0.6δ[n-2] + 0.64δ[n-3] + …
H(z) = 1 – 0.5z-1 - 0.6z-2 + 0.64z-3 + …
58. Long Division Method
58
Example-20: Using long division method, determine the inverse z-transform of
The inverse Z transform is x[n] = 5δ[n-2] – δ[n-3] + 0.2δ[n-4] – 0.04 δ[n-5] + …
X(z) = 5z-2 – z-3 + 0.2z-4 – 0.04z-5 + …
59. Long Division Method
Example-21: Using long division method, determine the inverse z-
transform of
Solution: First arranged in descending powers of Z
then dividing the numerator of 𝑋(𝑧) by its denominator we obtain
power series
59
60. Long Division Method
60
The inverse Z transform is x[n] = δ[n+2] + 3δ[n] + δ[n] + δ[n-2] + δ[n-3] + δ[n-4] + …
61. Long Division Method
Example-22: Using long division method, determine the inverse z-
transform of
Solution: By dividing the numerator of 𝑋(𝑧) by its denominator we
obtain power series
Using z-transform table
or
61
62. Long Division Method
Example-23: Using long division method, determine the inverse z-
transform of
Solution: By dividing the numerator of 𝑋(𝑧) by its denominator we
obtain power series
Using z-transform table
or
62
63. Partial Fraction Method
ADVANTGES
• It decompose the higher order system into sum of lower order
system
• General close-form solution can be found
DISADVANTAGES
• Applicable to strictly proper rational function in standard form
• Getting complex by handling 3 different types of roots for a
polynomial function of z, i.e.,
1. Distinct Real Roots
2. Repeated Real Roots
3. Complex Conjugate Roots
63
64. Partial Fraction Method
Example-24: Using partial fraction method find the inverse z-
transform of the signal Y(z), if x[n] = u[n-1], h[n] = (-0.25)nu[n].
Solution
As we know that Y(z) = X(z)H(z)
where
𝑋 𝑧 =
1
𝑧 − 1
𝐻 𝑧 =
𝑧
𝑧 + 0.25
So,
𝑌(z) =
𝑧
(z + 0.25)(𝑧−1)
64
65. Partial Fraction Method
𝑌(𝑧) =
𝑧
(z + 0.25)(𝑧−1)
𝑌 𝑧 =
𝐴
𝑧 + 0.25
+
𝐵
𝑧 − 1
• The coefficient A and B can be found using the cover-up method.
𝐴 = lim
𝑧→−0.25
𝑧 + 0.25 𝑧
(z + 0.25)(𝑧 − 1)
=
−0.25
−0.25 − 1
= 0.2
𝐵 = lim
𝑧→1
𝑧 − 1 𝑧
(z + 0.25)(𝑧 − 1)
=
1
1 + 0.25
= 0.8
𝑌 𝑧 =
0.2
𝑧 + 0.25
+
0.8
𝑧 − 1
= 𝑧−1
0.2𝑧
𝑧 + 0.25
+
0.8𝑧
𝑧 − 1
65
• The partial fraction expansion is
66. Partial Fraction Method
𝑌 𝑧 = 𝑧−1
0.2𝑧
𝑧 + 0.25
+
0.8𝑧
𝑧 − 1
• The portion inside the brackets has a inverse z transform is
0.2(-0.25)nu[n] + 0.8u[n]
• The z-1 term outside the brackets indicates a time shift by one step.
• Thus, the final inverse transform is
X[n] = 0.2(-0.25)n-1u[n-1] + 0.8u[n-1]
66
67. Partial Fraction Method
Example-25: Using partial fraction method find the inverse z-
transform of the signal
X(z) =
5
𝑧2 + 0.2𝑧
Solution
X 𝑧 =
𝐴
𝑧
+
𝐵
𝑧 + 0.2
=
25
𝑧
+ −25
𝑧 + 0.2
= 𝑧−1
(25 − 25
𝑧
𝑧 + 0.2
)
• Thus, the final inverse transform is
X[n] = 25δ[n-1] – 25(−0.2)𝑛−1𝑢[𝑛 − 1] 67
• The denominator of X(z) can be factored to give
• The partial fraction expansion is
X(z) =
5
𝑧(𝑧 + 0.2)
68. Partial Fraction Method
Example-26: Using partial fraction method find the inverse z-
transform of the signal
Y z =
0.5
𝑧(𝑧 − 1)(𝑧 − 0.6)
Solution
• The denominator is already factored into simple factors. The partial fraction
expression of Y(z) has three terms, one for each of the roots in the
denominator;
𝑌(𝑧) =
𝐴
𝑧
+
𝐵
𝑧 − 1
+
𝐶
𝑧 − 0.6
• Covering up the z term in the denominator and evaluating Y(z) at z = 0,
A =
0.5
(0 − 1)(0 − 0.6)
=
5
6 68
69. Partial Fraction Method
• Covering up the (z - 1) term in the denominator and evaluating at t = 1,
𝑩 =
𝟎. 𝟓
(𝟏)(𝟎 − 𝟎. 𝟔)
=
𝟓
𝟒
• Covering up the (z - 0.6) term and evaluating at t = 0.6,
•
𝑪 =
𝟎. 𝟓
(𝟎. 𝟔)(𝟎. 𝟔 − 𝟏)
= −
𝟐𝟓
𝟏𝟐
• Hence
𝒀 𝒛 =
𝟓
𝟔
𝒛
+
𝟓
𝟒
𝒛 − 𝟏
+
−
𝟐𝟓
𝟏𝟐
𝒛 − 𝟎.𝟔
= 𝒛−𝟏 𝟓
𝟔
+
𝟓
𝟒
𝒛
𝒛 − 𝟏
+
−
𝟐𝟓
𝟏𝟐
𝒛
𝒛 − 𝟎.𝟔
• The inverse z transform using the Table is
y[n] =
𝟓
𝟔
δ[n - 1] +
𝟓
𝟒
𝒖 𝒏 − 𝟏 −
𝟐𝟓
𝟏𝟐
(0.6)n-1 u[n - 1]
69
70. Partial Fraction Method
Example-27: Using partial fraction method find the impulse response
of the system
𝐻 𝑧 =
𝑧−2
1+0.25𝑧−1
Solution
• Changing to standard from, the transfer function becomes;
𝐻(𝑧) =
1
𝑧2 + 0.25𝑧
• Its partial fraction expansion is
𝐻 𝑧 =
1
𝑧 𝑧 + 0.25
=
𝐴
𝑧
+
𝐵
𝑧 + 0.25
70
71. Partial Fraction Method
𝐻 𝑧 =
4
𝑧
+
−4
𝑧 + 0.25
𝐻 𝑧 = 𝑧−1 4 −
4𝑧
𝑧 + 0.25
The portion within the brackets gives the inverse transform
4δ[n] - 4(-0.25)n u[n], so the final inverse transform is
h[n] = 4δ[n - 1] - 4(-0.25)n-1u[n - 1]
71
72. Partial Fraction Method
Example-28: Using partial fraction method find the inverse z-
transform of the signal
𝑋(𝑧) =
5
𝑧2 + 0.2𝑧
Solution
• The denominator of X(z) can be factored to give;
𝑋 𝑧 =
5
𝑧 𝑧 + 0.2
• Its partial fraction expansion is
𝑋 𝑧 =
5
𝑧 𝑧 + 0.2
=
𝐴
𝑧
+
𝐵
𝑧 + 0.2
72
74. Partial Fraction Method
Example-29: Using partial fraction method find the inverse z-
transform of the signal
Solution
74
• Eliminating the negative power of 𝑧 by multiplying the numerator and
denominator by 𝑧2 yields
• Dividing both sides by 𝑧 leads to
77. Partial Fraction Method
Example-30: Using partial fraction method find the inverse z-
transform of the signal
Solution
• Dividing both sides by 𝑧 leads to
• Using partial fraction method
• Multiplying 𝑧 on both sides gives
• From table of z-transform pairs
77
78. Partial Fraction Method
Example-31: Using partial fraction method find the inverse z-
transform of the signal
Solution
• Eliminating the negative power of 𝑧 by multiplying the numerator and
denominator by 𝑧3 yields
• Coefficient of highest power in denominator should be 1. Therefore
78
79. Partial Fraction Method
• Dividing both sides by 𝑧 leads to
• Using partial fraction method
• Multiplying 𝑧 on both sides gives
• From table of z-transform pairs
79
81. Transfer Function & System Stability
• The poles and zeros of a system can be determined easily
from the system’s transfer function.
• The poles and zeros of a system can provide a great deal of
information about the behavior of the system.
• In a standard form, TF can be expressed as a rational function
consist of numerator polynomial divided by denominator
polynomial.
81
82. Transfer Function & System Stability
• It is easiest to identify the poles and zeros if the rational
transfer function
is converted to the form
which has only positive exponents.
82
83. Transfer Function & System Stability
The zeros or roots of the numerator polynomial are the zeros of
the system.
The roots of the denominator polynomial are the poles of the
system.
83
85. Transfer Function & System Stability
• Poles are the values of 𝑧 that make the denominator of a transfer
function zero.
• Zeros are the values of 𝑧 that make the numerator of a transfer function
zero.
• Of the two, poles have the biggest effect on the behavior of a digital
system (digital filter).
• Zeros tend to modulate, to a greater or lesser degree depending on their
position relative to the poles.
• The poles of digital filter can be found if its transfer function is known.
• Both zeros and poles are in general complex numbers.
85
86. Transfer Function & System Stability
• A very powerful tool for the digital system analysis and design is
a complex plane called z plane, on which poles and zeros of the
transfer function are plotted.
• On the z plane,
poles are plotted as crosses (X)
zeros are plotted as circles (O)
• A plot showing pole and zero locations is called a pole-zero plot.
86
87. Transfer Function & System Stability
Example-32: for a first order system the poles and zeros are
𝐻 𝑧 =
2
1+0.4𝑧−1
• Poles: at 𝑧 = -0.4
• Zeros: at 𝑧 = 0
87
88. Transfer Function & System Stability
• The position of the poles and zeros on the z plane can give
clue about the way a digital filter will behave.
• One reason the poles of a system are so useful is that they
determine whether or not the filter is stable.
• The system is stable as long as the poles lie inside the unit
circle, which is a circle of unit radius on the z plane.
• Since poles are complex numbers, this requires that their
magnitudes be less than one.
• Mathematically, the region of stability can be described as
88
89. Transfer Function & System Stability
• If the magnitude of each pole is less than one, the poles are
less than one unit’s distance from the center of the unit circle,
and the filter is stable.
• If any of the poles of a system lie outside the unit circle, the
filter is unstable.
• If the outermost pole lies on the unit circle, the filter is
described as being marginally stable.
89
90. Transfer Function & System Stability
Example-33: Find the poles and zeros and stability for the
digital filter whose transfer function is
Solution
Eliminating negative exponents yields
• Poles: at 𝑧 = 0.25 and 𝑧 = 2
• Zeros: at 𝑧 = 0
• As one pole lie outside the unit circle at z = 2, hence the
system is unstable.
90
91. Transfer Function & System Stability
Example-34: The transfer function of a digital system is
𝐻 𝑧 =
1 − 𝑧−2
1 + 0.7𝑧−1 + 0.9𝑧−2
Is this system stable?
The poles are located at −0.35 ± 𝑗0.8818
For these poles the distance from the center of the unit circle is
𝑧 = −0.35 2 + 0.8818 2 = 0.9487
As both poles lie inside the unit circle,
So the system is stable.
91
92. Transfer Function & System Stability
Example-35: Determine the stability of the following system.
Solution: Eliminating negative exponents yields
As all poles lie inside the unit circle,
hence the system is stable.
92
93. Difference Equation & System Stability
93
Example-36: Find the stability of the filter if the difference equation
of the filter is
Y[n] + 0.8y[n-1] – 0.9y[n-2] = x[n-2]
Solution:
96. Impulse & Step Responses
96
For a step input, we can determine step response assuming zero
initial conditions. Letting
the step response can be found as
97. Impulse & Step Responses
• The z-transform of the general system response is given by
• We can determine the output 𝑦(𝑛) in time domain as
97
98. Impulse & Step Responses
98
Example-37: The transfer function of a digital system is
a) Determine the difference equation of the system.
b) Find the pole-zero plot and evaluate stability.
c) Find and plot the impulse response.
Solution
a) The difference equation is
y[n] – 0.4y[n – 1] = 2x[n]
𝐻 𝑧 =
2
1 − 0.4𝑧−1
99. Impulse & Step Responses
99
b) The poles and zeros are found from
𝐻 𝑧 =
2
1 − 0.4𝑧−1
=
2𝑧
𝑧 − 0.4
There is single zero at z = 0 and a single pole at z = 0.4. as shown
in the figure.
The pole is within the unit circle
So the system is stable.
100. Impulse & Step Responses
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c) The impulse response of the system is
h[n] = 2(0.4)nu[n]
The impulse response is plotted in the figure.
101. Impulse & Step Responses
Example-38: Given a transfer function depicting a DSP system
Determine
a) the Impulse response ℎ(𝑛)
b) the step response 𝑦(𝑛)
c) system response 𝑦(𝑛) if the input is given as 𝑥(𝑛) = (0.5)𝑛𝑢(𝑛)
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102. Impulse & Step Responses
Solution
a) the Impulse response ℎ(𝑛)
• The transfer function can be rewritten as
• We get
• Taking inverse z transform yields
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2
103. Impulse & Step Responses
b) the Step response s(n) or y(𝑛)
• the z-transform of the step response is
or
• We get
• Taking inverse z transform yields
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104. Impulse & Step Responses
c) system response 𝑦(𝑛) if the input is given as 𝑥(𝑛) = (0.5)𝑛𝑢(𝑛)
• the z-transform of the step response is
or
• We get
• Taking inverse z transform yields
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106. Impulse & Step Responses
• The impulse response of a stable system always settles to
zero.
• The step response of a stable system always settles to a
constant value.
• For unstable systems, on the other hand, these responses
grow without bound.
• Marginally stable systems produce cycling or oscillating
behavior.
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6
109. Impulse & Step Responses
• Among the stable systems, the closer the poles are to the unit
circle, the longer the impulse and step responses take to
settle to their final values.
• When all poles are extremely close to the origin of the z
plane, the responses reach their final values almost
immediately.
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9
110. Impulse & Step Responses
Stable and unstable impulse responses on the z plane
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115. Steady State Output
• The steady state output for the step response of a stable
system may be computed using the system’s difference
equation, by replacing all outputs y with ySS and all inputs x
with one (1).
For example, the difference equation
y[n] + Ay[n-1] + By[n-2] = x[n]
produces
ySS + AySS + BySS = 1
which gives a steady state output
ySS = 1/(1+A+B)
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116. Steady State Output
• The steady state output for the impulse response of a stable
system is always zero.
• Replacing the outputs y with ySS and the inputs x with zero (0)
For example, the difference equation
y[n] + Ay[n-1] + By[n-2] = x[n]
produces
ySS + AySS + BySS = 0
which gives a steady state output
ySS = 0 11
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117. Steady State Output
• The zeros of a system do not have as great an impact on the
system’s behavior as do the poles.
• In fact, when zeros occur far away from the poles, they have a
negligible effect.
• When a zero lies close to a pole, however, it effectively
cancels the behavior due to the pole.
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7
118. Impulse & Step Responses
Effect of Zero Position on Impulse Response
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8
119. Impulse & Step Responses
Effect of Zero Position on Impulse Response
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9
120. Impulse & Step Responses
Effect of Zero Position on Impulse Response
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