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.
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
Why Fourier Transform
General Properties & Symmetry relations
Formula and steps
magnitude and phase spectra
Convergence Condition
mean-square convergence
Gibbs phenomenon
Direct Delta
Energy Density Spectrum
Sampling is a Simple method to convert analog signal into discrete Signal by using any one of its three methods
if the sampling frequency is twice or greater than twice then sampled signal can be convert back into analog signal easily......
The z-Transform is often time more convenient to use
Definition:
Compare to DTFT definition:
z is a complex variable that can be represented as z=r ej
Substituting z=ej will reduce the z-transform to DTFT
Region of Convergence for a discrete time signal x[n] is defined as a continuous region in z plane where the Z-Transform converges.
The roots of the equation P(z) = 0 correspond to the ’zeros’ of X(z)
The roots of the equation Q(z) = 0 correspond to the ’poles’ of X(z)
The RoC of the Z-transform depends on the convergence of the polynomials P(z) and Q(z),
Uses to analysis of digital filters.
Used to simulate the continuous systems.
Analyze the linear discrete system.
Used to finding frequency response.
Why Fourier Transform
General Properties & Symmetry relations
Formula and steps
magnitude and phase spectra
Convergence Condition
mean-square convergence
Gibbs phenomenon
Direct Delta
Energy Density Spectrum
Sampling is a Simple method to convert analog signal into discrete Signal by using any one of its three methods
if the sampling frequency is twice or greater than twice then sampled signal can be convert back into analog signal easily......
The z-Transform is often time more convenient to use
Definition:
Compare to DTFT definition:
z is a complex variable that can be represented as z=r ej
Substituting z=ej will reduce the z-transform to DTFT
Region of Convergence for a discrete time signal x[n] is defined as a continuous region in z plane where the Z-Transform converges.
The roots of the equation P(z) = 0 correspond to the ’zeros’ of X(z)
The roots of the equation Q(z) = 0 correspond to the ’poles’ of X(z)
The RoC of the Z-transform depends on the convergence of the polynomials P(z) and Q(z),
Uses to analysis of digital filters.
Used to simulate the continuous systems.
Analyze the linear discrete system.
Used to finding frequency response.
This presentation deals with the " Massive Open Online Course (MOOC ) which is an online course aimed at unlimited participation and open access via the web. In addition to traditional course materials such as filmed lectures, readings, and problem sets, many MOOCs provide interactive user forums to support community interactions between students, professors, and teaching assistants
An Efficient DSP Based Implementation of a Fast Convolution Approach with non...a3labdsp
"Finite impulse response convolution is one of the most widely used operation in digital signal processing field for filtering operations. In this context, low computationally demanding techniques become essential for calculating convolutions with low input/output latency in real scenarios, considering that the real time requirements are strictly related to the impulse response length. In this context, an efficient DSP implementation of a fast convolution approach is presented with the aim of lowering the workload required in applications like reverberation. It is based on a non uniform partitioning of the impulse response and a psychoacoustic technique derived from the human ear sensitivity. Several results are reported in order to prove the effectiveness of the proposed approach also introducing comparisons with the existing techniques of the state of the art."
Convolution techiniques for dummies like meKhan Nazir
I had difficulty in finding a detailed explanation on convolution techniques for large signals. So I did some research and came up with a technique which might make problem solving much easier, specially for those who have forgotten the basics.
Please mail me your comments.
These are notes I prepared for a course on control that I taught at King Abdulaziz University.
The notes did not go through the intended revision after the end of the semester due to my schedule so they remain a rough first draft.
They are largely (but not completely) inspired by a control course taught by Dr. Gregory Shaver at the ME dept. at Purdue.
Much of the information was gleaned through a variety of textbooks/papers/experiences, too many to mention here.
Although a reasonable attempt has been made to ensure all the facts are correct, these notes are as-is with no guarantee of accuracy. Use at your own risk.
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Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
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Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
2. CONTENTS
• z-transform
• Region Of Convergence
• Properties Of Region Of Convergence
• z-transform Of Common Sequence
• Properties And Theorems
• Application
• Inverse z- Transform
• z-transform Implementation Using Matlab
2
3. Why z-Transform ?
• The z transform is a mathematical tool commonly used for the analysis and
synthesis of discrete-time control systems.
• The z transform in discrete-time systems play a similar role as the Laplace
transform in continuous-time systems
3
4. z-Transform
• The z-transform is the most general concept for the transformation of discrete-time series.
• The Laplace transform is the more general concept for the transformation of continuous
time processes.
4
TRANSFORM
5. The Transforms
5
The Laplace transform of a function f(t):
∫
∞
−
=
0
)()( dtetfsF st
The one-sided z-transform of a function x(n):
∑
∞
=
−
=
0
)()(
n
n
znxzX
The two-sided z-transform of a function x(n):
∑
∞
−∞=
−
=
n
n
znxzX )()(
6. • z-transform comes about same way as CTFT (Continuous Time Fourier Transform) in
which the variable is
Laplace Transform
►CTFT is generalized to Laplace Transform by going from
j σ + j
C.T.F.T L.T
Where , s = σ + j ; σ = real part
6
ω
ω generalized to
ω ω
ω
7. Motivation For C.T.F.T To L.T
j σ + j
CTFT L.T
Where , s = σ + j ;
σ = real part
This enlarges the capability of the transform to handle much wider class of
signal and system .This was the motivation for going to Laplace transform
from Continuous time fourier transform.
7
ω ω
ω
8. • In the similar manner z transform is generalized from D.T.F.T ( Discrete Time
Fourier Transform ) .
…………………(1)
• D.T.F.T of the sequence x(n) is given by eqation no . (1)
8
∑
∞
−∞=
−
n
nj
enx ω
)()(nx
D.T.F.T
∑
∞
−∞=
−
=
n
nj
enxX ω
ω )()(
9. • Here we notice that variable in this D.T.F.T obtained in eq. no. 1. is
9
ωj
e−
ωj
er.
ωj
e generalized to
Magnitude =1 Here we
introduce ‘r’
10. • Where r may be 1
If r = 1 : D.T.F.T
If r not equal to 1 : New transform : z-transform
• In Laplace transform we worked on the j realization to .We worked
on rectangular coordinates.
• Whereas in z- transform we work in polar coordinates. We specify complex
by its distance from the origin which is ‘r’ and angle it makes with the +ve
real axis i.e ‘ ‘.
10
ω σ + jω
ωj
e
ω
Z = ejω
ω
11. • In eq.1 if we give a new name say ‘z’.
• If we wish to generalize D.T.F.T into a new transform then is replaced by z
• Then
• The z- transform reduces to the Fourier transform when the magnitude of the
transform variable z is unity.
11
ωj
er.
ωj
e
∑
∞
−∞=
−
=
n
nj
enxX ω
ω )()(
∑
∞
−∞=
−
=
n
n
znxzX )()(
12. POLES AND ZEROS
12
• When X(z) is a rational function, i.e., a ration of polynomials in z, then:
The roots of the numerator polynomial are referred to as the zeros of X(z), and
The roots of the denominator polynomial are referred to as the poles of X(z).
• Note that no poles of X(z) can occur within the region of convergence since the z-
transform does not converge at a pole.
• Furthermore, the region of convergence is bounded by poles.
13. REGION OF CONVERGENCE
13
• The z-transform of x(n) can be viewed as the Fourier transform of x(n) multiplied by
an exponential sequence r-n
, and the z-transform may converge even when the
Fourier transform does not.
• By redefining convergence, it is possible that the Fourier transform may converge
when the z-transform does not.
• For the Fourier transform to converge, the sequence must have finite energy, or:
∞<∑
∞
−∞=
−
n
n
rnx )(
14. CONVERGENCE, CONTINUED
14
∑
∞
−∞=
−
=
n
n
znxzX )()(
• The power series for the z-transform is called a Laurent series:
• The Laurent series, and therefore the z-transform, represents an analytic function at
every point inside the region of convergence, and therefore the z-transform and all
its derivatives must be continuous functions of z inside the region of convergence.
• In general, the Laurent series will converge in an annular region of the z-plane.
15. EXAMPLE
15
)()( nuanx n
=
The z-transform is given by:
∑∑
∞
=
−
∞
−∞=
−
==
0
1
)()()(
n
n
n
nn
azznuazX
Which converges to:
azfor
az
z
az
zX >
−
=
−
= −1
1
1
)(
Clearly, X(z) has a zero at z = 0 and a pole at z = a.
×
a
Region of convergence
16. 16
• Stable System : A system is said to be stable if
∞<∑
∞
−∞=k
kh )(
Which means that a bounded input will not yield an unbounded output.
•Causal System : A causal system is one in which changes in output do not
precede changes in input. In other words,
[ ] [ ] .for)()(then
for)()(If
021
021
nnnxTnxT
nnnxnx
<=
≤=
Linear, shift-invariant systems are causal if h(n) = 0 for n < 0.
17. PROPERTIES OF ROC
17
• The region of convergence (ROC) for a given DT transfer function is a disk or annulus which
contains no poles. In general, the ROC is not unique, and the particular ROC in any given
case depends on whether the system is causal or anti-causal.
• If the ROC includes the unit circle, then the system is bounded-input, bounded-output
(BIBO) stable.
• If the ROC extends outward from the pole with the largest (but not infinite) magnitude,
then the system has a right-sided impulse response.
18. • If the ROC extends outward from the pole with the largest magnitude and there is
no pole at infinity, then the system is causal.
• If the ROC extends inward from the pole with the smallest (nonzero) magnitude,
then the system is anti-causal.
18
19. • If you need stability then the ROC must contain the unit circle.
• If you need a causal system then the ROC must contain infinity and
the system function will be a right-sided sequence.
19
20. • If you need an anti-causal system then the ROC must contain the origin and the system
function will be a left-sided sequence.
20
21. • If you need both, stability and causality, all the poles of the system function
must be inside the unit circle. The unique x[n] can then be found
21
23. 23
PROPERTIES AND THEOREMS
Multiplication by a Constant
Linearity of the z Transform
Multiplication by ak
Shifting Theorem
Complex Translation Theorem
Initial Value Theorem
Final Value Theorem
24. Multiplication by a constant
If X(z) is the z transform of x(n), then
Z [ax(n)] = a Z[x(n)] = a X(z)
where a is a constant.
To prove this, note that by definition
Z [ax(n)] =
24
∑∑
∞
=
−
∞
=
−
==
00
)()()(
n
n
n
n
zaXznxaznax
25. Linearity of the z transform
The z transform possesses an important property: linearity. This means
that, if f(n) and g(n) are z-transformable and α and β are scalars, then
x(n) formed by a linear combination
x(n) = αf(n) + βg(n)
has the z transform
X(z) = αF(z) + βG(z)
where F(z) and G(z) are the z transforms of f(n) and g(n), respectively.
25
26. 26
Multiplication By Ak
If X(z) is the z transform of x(k), then the z transform of ak
x(n) can be given by X(a-1
z):
Z[ak
x(n)] = X(a-1
z)
This can be proved as follows:
Z[an
x(n)] =
= X(a-1
z)
∑∑
∞
=
−−
∞
=
−
=
0
1
0
))(()(
n
n
n
nn
zanxznxa
27. 27
Shifting Theorem
Also call real translation theorem. If x(t) = 0 for t < 0 and x(t) has the z transform X(z), then
Z[x(t-nT)] = z-n
X(z)
and
Z[x(t+nT)] =
n = zero or a positive integer
− ∑
−
=
−
1
0
)()(
n
k
kn
zkTxzXz
28. 28
EXAMPLE
Q. Find the z transforms of unit-step functions that are delayed by 1 sampling period and 4
sampling periods, respectively, as shown in figure (a) and (b) below
29. 29
SOLUTION
Using the shifting theorem ,we have
Z [1(t-T)] = z-1
Z [1(t)] =
Also,
Z [1(t-4T)] = z-4
Z [1(t)] =
(Note that z-1
represents a delay of 1 sampling period T, regardless of the value of T.)
1
1
1
1
11
1
−
−
−
−
−
=
− z
z
z
z
1
4
1
4
11
1
−
−
−
−
−
=
− z
z
z
z
30. 30
Q. Obtain the z transform of
Solution: Referring to Equation (2. 18), we have
Z [x(k-1)] = z-1
X(z)
The z transform of ak
is
Z[ak
] =
and so
Z [f(a)] = Z[ak-1
] =
where k = 1,2,3, ....
≤
=
=
−
0,0
....,3,2,1,
)(
1
k
ka
af
k
1
1
1
−
− az
1
1
1
1
11
1
−
−
−
−
−
=
− az
z
az
z
31. 31
Complex Translation Theorem
If x(t) has the z transform X(z), then the z transform of e-at
x(t) can be given by X(z eat
).
To prove
Z [e-at
x(t)]
Thus, we see that replacing z in X(z) by z eat
gives the z transform of e- at
x(t).
)(
))((
)(
0
0
aT
k
kaT
k
kakT
zeX
zekTx
zekTx
=
=
=
∑
∑
∞
=
−
∞
=
−−
32. 32
EXAMPLE
Q. By using the complex translation theorem, obtain the z transforms of:
1. e-at
sin ωt and
2. e-at
cosωt, respectively,
33. 33
Solution
1. We know that
Z [sin ωt ] =
Using the complex translation theorem
Z
21
1
cos21
sin
−−
−
+− zTz
Tz
ω
ω
221
1
cos21
sin
]sin[ −−−−
−−
−
+−
=
zeTze
Tze
te aTaT
aT
at
ω
ω
ω
34. 34
Solution
2. We know that
Z [cos ωt ]
Using the complex translation theorem
Z[e-at
cos ωt ]
21
1
cos21
cos1
−−
−
+−
−
=
zTz
Tz
ω
ω
221
1
cos21
cos1
−−−−
−−
+−
−
=
zeTze
Tze
aTaT
aT
ω
ω
35. 35
Initial Value Theorem
• If x(t) has the z transform X(z) and if exists, then the initial value x(0) of x(t) or x(k) is
given by
• The initial value theorem is convenient for checking z transform calculations for possible
errors. Since x(0) is usually known, a check of the initial value by can easily spot
errors in X(z), if any exist.
)(lim zX
z ∞→
)(lim)0( zX
z
x
∞→
=
)(lim zX
z ∞→
36. 36
EXAMPLE
Q. Determine the initial value x(0) if the z transform of x(t) is given by
By using the initial value theorem, we find
Referring to Example 2-2, notice that this X(z) was the z transform of
and thus x(0) = 0, which agrees with the result obtained earlier.
)1)(1(
)1(
)( 11
1
−−−
−−
−−
−
=
zez
ze
zX T
T
0
)1)(1(
)1(
lim)0( 11
1
=
−−
−
= −−−
−−
∞→ zez
ze
x T
T
z
t
etx −
−= 1)(
37. 37
Final Value Theorem
• The final value of x(n), that is, the value of x(n) as approaches infinity, can be given by
(2. 27)
)]()1[(lim)(lim 1
1
zXznx
zn
−
→∞→
−=
38. 38
EXAMPLE
Q. Determine the final value of
by using the final value theorem.
• By applying the final value theorem to the given X(z), we obtain
)(∞x
0,
1
1
1
1
)( 11
>
−
−
−
= −−−
a
zez
zX aT
[ ])()1(lim)( 1
1
zXzx
z
−
→
−=∞
−
−
−
−= −−−
−
→ 11
1
1 1
1
1
1
)1(lim
zez
z aTz
1
1
1
1lim 1
1
1
=
−
−
−= −−
−
→ ze
z
aTz
39. APPLICATION
39
• A closed-loop (or feedback) control system is shown in Figure.
• If you can describe your plant and your controller using linear difference equations, and if
the coefficients of the equations don't change from sample to sample, then your controller
and plant are linear and shift-invariant, and you can use the z transform.
40. 40
• Suppose xn = output of the plant at sample time n
un = command to the DAC at sample time n
a and b = constants set by the design of the plant
• You can solve the behavior equation of the plant over time.
• Furthermore you can also investigate what happens when you add feedback to
the system.
• The z transform allows you to do both of these things.
41. • Deals with many common feedback control problems using continuous-time control.
• Also used in sampled-time control situations to deal with linear shift-invariant difference
equations.
42. TRANSFER FUNCTION
• The function H(Z) is called the “Transfer Function" of the system – it shows how the input
signal is transformed into the output signal.
H(Z)=Y(Z)/X(Z)
• In Z domain, the Transfer Function of a system isn't affected by the nature of the input
signal, nor does it vary with time.
• We can predict the behavior of the motor using H(Z).
• Let's say we want to see what the motor will do if x goes from 0 to 1 at time n = 0, and
stays there forever. This is called the ‘unit step function’ and the Z-Transform of the unit
step response is H(Z)=Z/(Z-1).
• Thus we can know everything about the system behavior and avoid undesirable
situations.
43. SOFTWARE
• You can write software from the Z-Transform with utter ease.
• Like, if you have a Transfer Function of a system, then the software turns it into
a Z-domain equation which can then be converted into a difference
equation which in turn can be turned into a software very quickly.
• This saves the manual work and a software for a plant can be produced within
seconds.
44. INVERSE Z-TRANSFORM
44
The inverse z-transform can be derived by using Cauchy’s integral theorem. Start with
the z-transform
∑
∞
−∞=
−
=
n
n
znxzX )()(
Multiply both sides by zk-1
and integrate with a contour integral for which the contour
of integration encloses the origin and lies entirely within the region of convergence of
X(z):
transform.-zinversetheis)()(
2
1
2
1
)(
)(
2
1
)(
2
1
1
1
11
nxdzzzX
i
dzz
i
nx
dzznx
i
dzzzX
i
C
k
n C
kn
C n
kn
C
k
=
=
=
∫
∑ ∫
∫ ∑∫
−
∞
−∞=
−+−
∞
−∞=
−+−−
π
π
ππ
46. MATLAB Code For Finding z-TRANSFORM
• MATLAB Symbolic Toolbox gives the z-transform of a function .
• MATLAB program for Z-Transform
Program Code
% z-Transform %
clc;
close all;
clear all;
syms 'n';
syms 'z';
x=input('Input the sequence to be converted');
a=symsum((x*(z^(-n))),n,0,2);
disp(‘z-Transform of the given sequence is ');
disp(a);
46
47. Example of Output
•Input the sequence to be converted [1 2 3 4]
•z-Transform of the given sequence is
[ 1+1/z+1/z^2, 2+2/z+2/z^2, 3+3/z+3/z^2, 4+4/z+4/z^2]
47
50. MATLAB Code For Pole And Zero Plot
• Pole-zero Diagram The MATLAB function “z-plane” can display the pole-zero
• Program Code
%Plotting zeros and poles of z-transform
clc;
close all;
clear all;
disp('For plotting poles and zeros');
b=input('Input the numerator polynomial coefficients');
a=input('Input the denominator polynomial coefficients');
[b,a]=eqtflength(b,a);
[z,p,k]=tf2zp(b,a);
zplane(z,p);
disp('zeros');
disp(z);
disp('poles');
disp(p);
disp('k');
disp(k);
50
51. • Example of Output
For plotting poles and zeros
Input the numerator polynomial coefficients[1 2 3 4]
Input the denominator polynomial coefficients[1 2 3]
zeros
-1.6506
-0.1747 + 1.5469i
-0.1747 - 1.5469i
poles
0
-1.0000 + 1.4142i
-1.0000 - 1.4142i
k
1
51