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KLE Society's

KLE Technological University

SCHOOL OF ELECTRONICS ENGINEERING

FMTH0301/Rev.5.2

Course Plan

Semester: 4 - Semester Year: 2016-17

Course Title: Signals and Systems Course Code: 15EECC204

Total Contact Credits: 40 Duration of SEE: 3 Hours

SEE Marks: 50 CIA Marks: 50

Lesson Plan Author: Mrs. Ujwala Patil Mrs. Preeti S. Pillai Mrs. Shraddha Hiremath

Date: 08-12-2016

Checked By: Mrs.Nalini C Iyer Date: 14-12-2016

Course Outcomes (COs):

At the end of the course the student should be able to:

1. Identify different signals and systems and state their properties both in analog and discrete

domain.

2. Apply the concept of impulse response and perform convolution in both analog and discrete

domain to analyze the linear time invariant systems.

3. Perform spectral analysis of discrete time periodic and aperiodic signals using Fourier series,

transform and Z transform methods.

4. Perform the signal transformations on given signals and identify appropriate operations to be

performed.

Powered by www.ioncudos.com of 23.

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Course Articulation Matrix: Mapping of Course Outcomes (COs) with Program Outcomes (POs)

Course Title: Signals and Systems Semester: 4 - Semester

Course Code: 15EECC204 Year:2016-17

Course Outcomes (COs) / Program Outcomes (POs)

PO1

PO2

PO3

PO4

PO5

PO6

PO7

PO8

PO9

PO10

PO11

PO12

PO13

PO14

PO15

Identify different signals and systems and state their properties both analog and discrete domain.

M L

Apply the concept of impulse response and perform convolution in both analog and discrete domain to analyze the linear time invariant systems.

H L

Perform spectral analysis of discrete time

H M


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periodic and aperiodic signals using Fourier series, transform and Z transform methods.

Perform the signal transformations on given signals and identify appropriate operations to be performed.

H M

Degree of compliance L: Low M: Medium H: High


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Competency addressed in the Course and corresponding Performance Indicators

Competency Performance Indicators

PO1.1 - Demonstrate the competence in mathematical modeling.

PO1.1.3 - Ability to apply knowledge of Integral transforms

PO2.1 - Demonstrate an ability to identify and characterize an engineering problem

PO2.1.2 - Identify engineering systems, variables, and parameters to solve the problems

PO2.1.4 - Identify the mathematical, engineering and other relevant knowledge that applies to a given problem

Eg: 1.2.3: Represents Program Outcome ‘1’, Competency ‘2’ and Performance Indicators ‘3’.


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Course Content

Course Code: 15EECC204 Course Title: Signals and Systems

L-T-P-Self Study: 3-0-0-0 Credits: 3 Contact Hrs: 40

CIA Marks: 50 SEE Marks: 50 Total Marks: 100

Teaching Hrs: 40 Exam Duration: 3 hrs

Content Hrs

Unit - 1

Chapter No. 01: Signal Representation Definition of a signals and systems, classification of signals,(analog and discrete signal, periodic and aperiodic, deterministic and random signals, even and odd signals, energy and power) , basic operation on signals(independent variable, dependent variable , time scaling, multiplication, time reversal), elementary signals (Impulse, step, ramp, sinusoidal, complex exponential), Systems Interconnections(series, parallel and cascade), properties of linear systems. (homogeneity ,superposition, linearity and time invariance, stability, memory, causality)

8 hrs

Chapter No. 02 : LTI System Representation Impulse response representation and properties, Convolution, convolution sum and convolution integral. Differential and difference equation Representation, Block diagram representation

8 hrs

Unit - 2

Chapter No. 03:Fourier representation for signals Introduction, Discrete time Fourier series(derivation of series excluded) and their properties. Discrete Fourier transform (derivation of transform excluded) and properties

8 hrs

Chapter No. 04:Applications of Fourier transform Introduction, frequency response of LTI systems, Fourier transform representation of periodic signals, Fourier transform representation of discrete time signals. Sampling of continuous time signals.

8 hrs

Unit - 3

Chapter No. 05: Z-transform Definition of z-transform, Properties of ROC, Properties of Z-transforms: Inverse z-transforms (Partial Fraction method, long division method), Unilateral Z-transform, Transform of LTI.

8 hrs


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Text Books (List of books as mentioned in the approved syllabus) 1. Simon Haykin and Barry Van Veen , Signals and Systems, Second, John Wiley & Sons, 20022. Alan V Oppenheim ,Alan S Willsky and S. Hamid Nawab , Signals and Systems, Second, PHI public, 1997

References1. H. P Hsu, R. Ranjan, Signals and Systems , TMH, 20062. Ganesh Rao and Satish Tunga, , Signals and Systems, Sanguine T, 20043. M.J. Roberts, Fundamentals of Signals and Systems, first Edition, TMH

Evaluation Scheme

CIA Scheme

Assessment Weight age in Marks

Minor 1 15

Minor 2 15

Assignment 20

SEE 50

Total 100


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Course Unitization for Minor Exams and Semester End Examination

Topics / Chapters Teaching Credits

No. of Questions in Minor 1

No. of Questions in Minor 2

No. of Questions

in Assignment

No. of Questions

in SEE

Unit IChapter 01: Signal Representation 8 1.50 -- -- 1.50

Chapter 02 : System Representation 8 1.50 -- -- 1.50

Unit IIChapter 03:Fourier representation for signals

8 -- 1.50 -- 1.50

Chapter 04:Applications of Fourier transform

8 -- 1.50 -- 1.50

Unit IIIChapter 05: Z-transform 8 -- -- -- 2.00

Note

1. Each Question carries 20 marks and may consists of sub-questions.2. Mixing of sub-questions from different chapters within a unit (only for Unit I and Unit II) is allowed in Minor I, II and SEE. 3. Answer 5 full questions of 20 marks each (two full questions from Unit I, II and one full question from Unit III) out of 8 questions in SEE.

Date: Head of Department


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Chapter wise Plan

Course Code and Title: 15EECC204 / Signals and Systems

Chapter Number and Title: Chapter 01: Signal Representation Planned Hours: 8 hrs

Learning Outcomes:-

At the end of the topic the student should be able to:

Topic Learning Outcomes COs BL CA Code

1. Distinguish between different kinds of signals. CO1 L2 PO1.1

2. Determine whether a signal has the following properties: discrete time, continuous time, power, energy, periodic, aperiodic, even, odd.

CO1 L3 PO2.1

3. Perform the following operations on signals, alone or in combination: amplitude scaling, addition, multiplication, differentiation, integration time scaling, reflection, time shifting.

CO1 L3 PO1.1

4. Determine whether an input/output description for a system has the following properties: stability, memory, memory less, causality, invariability (simple cases), and time invariance.

CO1 L3 PO1.1

5. Identify and use the following elementary signals: exponentials, sinusoids, complex exponentials, exponentially damped sinusoids step functions, impulses, sifting and time scaling properties of impulses

CO1 L2 PO1.1

Lesson Schedule Class No. - Portion covered per hour / per Class

1. Definition of a Signal and a system, Classification of Signals

2. Classification of Signals (cont)

3. Basic Operations on signals

4. Basic Operations on signals (contd)

5. Elementary signals

6. Elementary signals (contd)

7. Systems viewed as interconnections of operations

8. Properties of Systems (contd)


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Review Questions Sl.No. - Questions TLOs BL PI Code

1. Differentiate between continuous time and discrete time systems with examples.

TLO1 L1 1.1.3

2. Determine whether the following signals are periodic. If periodic, determine their fundamental period.

i) x(t)= cos(t+π4

)

ii) sin2t

TLO2 L2 1.1.3

3. Given a sequence x(n)=(6-n)(u[n]-u[n-6]) , make a sketch of: y[n]=x[n-4] TLO3 L2 1.1.3

4. Distinguish between different kinds of signals. TLO1 L2 1.1.3

5. Determine whether the following signal x(t) = cos(t) + sin √2 t x(n) = cos (nπ/12) + sin (nπ/18) is periodic or not.

TLO2 L2 1.1.3

6. Given a sequence x(n) = (6-n) [u(n) – u(n-6)], make a sketch of: i) y1(n) = x(4-n).ii) y2(n) = x(2n-3). iii)y3(n) = x(n+2).

TLO3 L2 1.1.3

7. For the following system T [x(n)] = g(n) x(n),determine whether the system is i. Stable. ii. Causal. iii. Linear. iv. Time-invariant v. Memory less

TLO4 L2 1.1.3

8. Given the signal x(t) as shown in figure , Sketch the following. i) x(-2t+3) ii) x(t/2 – 2)

TLO3 L2 1.1.3


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Chapter Number and Title: Chapter 02 : System Representation Planned Hours: 8 hrs

Learning Outcomes:-



1. Evaluate the convolution sum and integral on given input and impulse response.

CO2 L3 PO1.1

2. Predict the behavior of system. CO2 L3 PO2.1

3. Define the properties of convolution. CO2 L1 PO1.1

4. Determine whether a system described by an impulse response has the following properties: memory less, causal, stable.

CO2 L2 PO1.1

5. Represent the given system using block diagram. CO2 L2 PO1.1

6. Characterize the natural response, forced response, and complete response for systems described by difference or differential equations.

CO2 L3 PO1.1


1. Convolution sum and properties

2. Convolution integral and properties

3. Examples on convolution sum and intergral

4. Properties of Impulse response representations

5. Differential and Difference equation representations

6. Differential and Difference equation representations (contd)

7. Differential and Difference equation representations (contdd)

8. Block diagram representations



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1. Sequences x(n) = u(n) and y(m) = ( 12)m

are convolved. Determine the

result of convolution sum if x(n) exists for n=0 to 5 and y(m) for m =-3 to -2. x(n) and y(m) are of equal length.

TLO1 L3 1.1.3

2. Prove that the response of an LTI system is governed by the convolution sum.

TLO3 L1 1.1.3

3. A linear time invariant continuous time signal is characterized by its unit impulse response h(t). What are the conditions on h(t) such that the system is memory less, causal and stable?

TLO4 L2 2.1.2

4. Draw the direct form I and II implementation for the following differential

equation y(n)=14 y(n-1)-

18y(n-2)=2x(n)+3x(n-1) .

TLO5 L2 2.1.2

5.Obtain the difference equation description for the system having impulse

response h (n )=u (n )+2( 12 )n

.u (n )+¿

TLO6 L3 1.1.3

6. Describe in detail every step of the Reflect and Shift Convolution Integral evaluation procedure.

TLO2 L2 1.1.3

7. Let the impulse response of an LTI system be given by h (n) = u (n-1) – u (n-4). Find the output of this system in response to the input x (n) = u (n) + u (n-1) -2u (n-2).

TLO4 L2 1.1.3

8. Evaluate the following discrete-time convolution sums: 1. y [n] = u [n] * u [n-3] 2. y [n] = (½)n u [n-2] * u [n]

TLO4 L2 1.1.3


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Chapter Number and Title: Chapter 03:Fourier representation for signals Planned Hours: 8 hrs

Learning Outcomes:-



1. Determine whether the DTFS representation is appropriate for a given signal.

CO3 L3 PO2.1

2. Use the tables of representations and properties to find the appropriate representation for time signal.

CO3 L2 PO2.1

3. List the different properties of DTFS. CO3 L1 PO1.1

4. Determine whether the DTFT representation is appropriate for a given signal.

CO3 L3 PO2.1

5. Evaluate the DTFT representations of time signals by using the defining equations.

CO3 L3 PO1.1


1. Introduction.

2. Fourier representations for two signal classes.

3. Fourier representations for two signal classes (Contd).

4. Orthogonality of complex sinusoidal signals.

5. Definition, Problems & Properties of DTFS representations.

6. Definition, Problems & Properties of DTFS representations (contd)

7. Definition, Problems & Properties of DTFT representations.

8. Definition, Problems & Properties of DTFT representations (contd)



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1. Determine the appropriate Fourier representation for the following time domain signals using the defining equations.

1. x(n) = e−3n . cosπn .u (n )

2. x(n) =cos ( πn5

) + j sin ( πn5

), n< 10

TLO1 L3 2.1.2

2. Prove the following properties of DTFS i. Time shift property ii. Frequency shift property.

TLO3 L1 1.1.3

3. Determine the signal x(n) given its Fourier representation as:

TLO2 L2 2.1.4

4. Find the DTFT representation for the periodic signal x(n) = cos ( π3)n. Also

draw the spectrum.

TLO4 L3 1.1.3

5. Use the method of inspection to determine the DTFS coefficients for the following signals: a) x[n]=1+sin(nπ/12+3π/8) b) x[n]= cos(nπ/30)+2sin(nπ/90).

TLO2 L3 1.1.3

6. Use the defining equation for the DTFS coefficients to evaluate the DTFS representation of the following signals:

TLO1 L3 1.1.3

7. Find the DTFS coefficients of the N-periodic impulse train: TLO1 L3 1.1.3


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Chapter Number and Title: Chapter 04:Applications of Fourier transform Planned Hours: 8 hrs

Learning Outcomes:-



1. Apply frequency response to solve for the input, output, or impulse response of a system given the other two signals.

CO3 L2 PO2.1

2. Determine the frequency response of systems described by differential and difference equations.

CO3 L3 PO1.1

3. Define sampling theorem. CO3 L2 PO1.1

4. Determine the conditions on the sampling rate or interval that guarantee a band limited signal can be uniquely reconstructed from its samples

CO3 L3 PO2.1


1. Frequency response of LTI systems

2. Frequency response of LTI systems (contd)

3. Solution of Differential equations using system function

4. Solution of Differential equations using system function(contd)

5. Solution of Difference equations using system function

6. Solution of Difference equations using system function(contd)

7. Fourier transform representations for periodic signals

8. Sampling of continuous time signals


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1. Obtain the difference equation description for the system having impulse

response h (n )=u (n )+2( 12 )n

.u (n )+¿

TLO1 L2 1.1.3

2. The input and output of a causal LTI system are related by differential

equationd2 y (t )dt2

+6 dy (t)dt

+8 y ( t )=2x (t). Find the impulse response of

the system. What is the response of this system if x(t)=e-2t u(t)?

TLO2 L3 2.1.2

3. State and prove Sampling theorem for a band limited signal. TLO3 L2 2.1.4

4. Specify the Nyquist rate and nyquist interval for each of the following signals. G(t)=sinc(200t), G(t)=sinc(200t)+sinc2(200t).

TLO4 L2 1.1.3


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Chapter Number and Title: Chapter 05: Z-transform Planned Hours: 8 hrs

Learning Outcomes:-



1. Define Z transform. CO4 L1 PO1.1

2. Prove the properties of Z transform. CO4 L2 PO1.1

3. Define ROC and explain the properties of ROC CO4 L2 PO1.1

4. Compute the Z transform of a given sequence and estimate the ROC. CO4 L3 PO1.1

5. Identify the unilateral Z transform CO4 L1 PO1.1

6. Define inverse Z transform CO4 L1 PO1.1

7. Identify and apply different methods to calculate inverse Z transform CO4 L3 PO1.1


1. Introduction and Definition of Z transform

2. Properties of ROC and Z transform

3. Numericals on Z transform and the estimation of ROC.

4. Unilateral Z transform of LTI system.

5. Computation and estimation of ROC for unilateral Z transforms.

6. Definition of inverse Z transform

7. Partial fraction and Long division methods to compute the inverse Z transforms and numericals .

8. Solving difference equation


1. State and prove initial value and final value theorem of Z transform TLO2 L1 1.1.3

2. What is ROC explain the significance of estimating the same. TLO3 L1 1.1.3

3. Explain the different properties of ROC. TLO4 L2 1.1.3


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4. Find Z transform of following sequences and estimate the ROC

i.x (n )=2n .u (n )+3( 12 )n

.u (n)

ii.x (n )=3(−12 )n

. u (n )−2 (3 )n . u(−n−1)

TLO5 L2 1.1.3

5. Calculate the inverse Z transform of the following.

X (Z )= z (z2−4 z+5)(z−3 ) (z−2 )(z−1)

ROCs: 2<|z|<3, |z|>3, |z|<1,

TLO7 L2 1.1.3

6. Calculate the inverse Z transform of the following.

i. X (Z )= z(2 z2−3 z+1 ) ROC: |z|<1/2

ii. X (Z )= z(2 z2−3 z+1 ) ROC: |z|>1

TLO7 L2 1.1.3


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Question Paper Title: Model Question Paper for Minor Exam I

Total Duration (H:M):1:15 Course :Signals & Systems (15EECC204)

Maximum Marks :60

Note :Answer any two full questions

Q.No. Questions Marks CO BL PO PI Code

1a Explain the following properties of the systems with example for each.

i. Stability.ii. Time invariance.iii. Linearity.

4 CO1

L2 1 1.1.3

1b Obtain the response of the system for the signals x(n)=αn

and h(n)=u[n] ,0<α<1. Also sketch the output. 8 CO

1 L3 2 2.1.2

1c Determine whether the system y(n)=n3is memory less, stable, causal, linear and time variant.

8 CO1

L3 2 2.1.2

2a The output of a system is related to its input as y(n)=5x(n)+4x(n-1)+ 5x(n-2)+ 7x(n-3). Realize a suitable hardware structure using minimum number of components.

6 CO1

L3 2 2.1.2

2b Differentiate the following with examplei. Periodic and non-periodic signals ii. Even and odd signals iii. Energy and power signals

8 CO2

L1 1 1.1.3

2c A staircase waveform x(t) defined from t=0 to t=5 with step rise of 2 V per second. Starting with rectangular pulse g(t) ,construct this waveform and express x(t) in terms of g(t). Where g(t) =1 for |t|≤1.

6 CO1

L3 2 2.1.2

3a Given a system below, find the interaction of the system with the step signal as an input. Check whether the system is stable, causal and memory less.

12 CO5 L3 2 2.1.2


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3b Sketch the signal r(t+2)-r(t+1)-r(t-2)+r(t-3) and determine their even and odd components.

8 CO2 L2 1 1.1.3


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Question Paper Title: Model Question Paper for Minor Exam IITotal Duration (H:M):1:15 Course :Signals & Systems

(15EECC204)Maximum Marks :60

Note: Answer any two full questions.

Q.No. Questions Marks CO BL PO PI Code

1a Prove the following properties of DTFS i. Time shift property ii. Frequency shift property.

10 CO3 L2 1 1.1.3

1b Determine the appropriate Fourier representation for the following time domain signals.

i. cos ( 2πn/5 ) + j sin ( 3πn/5 ), |n|< 10. x[n]= 0, Otherwise.

ii. x(n) = 3( 1+n ) . u(−n+2)

10 CO3 L3 2 2.1.2

2a Use the method of inspection to determine the DTFS coefficients for the signal x[n] =1+sin(nπ/12+3π/8) .

6 CO3 L3 2 2.1.2

2b The input and output of a causal LTI system are related by differential equation

d2 y (t )dt2

+6 dy (t)dt

+8 y ( t )=2x (t).

Find the impulse response of this system. What is the response of this system if x(t)=e-2t u(t)?

10 CO3 L3 2 2.1.2

2c Find the DTFS of x[n]=1 for 0≤ n≤4. Plot the magnitude and phase spectra.

4 CO3 L3 1 1.1.3

3a Using convolution theorem ,find the inverse FT of X(w) =1/(a+jw)2

10 CO3 L3 1 1.1.3

3b Find the Fourier transform of x(t) = sgn(t) and hence u(t) where sgn(t)= 1 , for t >0 , sgn(t)=-1 ,for t<0 What are the conditions for existence of Fourier transform.

10 CO3 L3 1 1.1.3


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Question Paper Title: Model Question Paper for Semester End Examination

Total Duration (H:M):3:00 Course :Signals & Systems (15EECC204)

Maximum Marks :100

Note: Answer 5 full questions of 20 marks each, two full questions from Unit I, Unit II, and 1 full question from Unit III out of 8.

Unit-I

Q.No

QuestionsMark

sCLO

PI Code

B L

1aFind the periodicity of x(t)=cos[2πn/5] +cos[2πn/5] 4 1 1.1.3 L3

b Given the signal sketch the following x (−2 t+3) +x ( t2−1)

8 1 1.1.3 L1

c

Determine whether the system shown below is memory less, stable, causal, linear and time variant

y[n]=ddt

¿ [e-t x(t) ] )

8 1 2.1.2 L3

2a

Differentiate the following with example,

(i) Stable and unstable system(ii) Linear and nonlinear system(iii) Causal and non causal system

6 1 1.1.3 L2

b

Obtain the response of the system for given signal x[n] = 2{u[n+2]-u[n-12] } and h[n]= 2n{u[n-2]-u[n-13]}. Sketch the output.

8 2 2.1.2 L3

c

A staircase waveform x(t) defined from t=0 to t=5 with step rise of 2 V per second. Starting with rectangular pulse g(t) ,construct this waveform and express x(t) in terms of g(t). Where g(t) =1 for |t|≤1.

6 1 2.1.2 L3

3a The output of a system is related to its input as y(n)=5x(n)+4x(n-1)+5x(n-2)+7x(n-3).Realize a suitable hardware structure using minimum number of components.

10 2 2.1.2 L3


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b

Determine whether the following signals are periodic. If periodic, determine their fundamental period.

i. x(t) = e j(πt-1)

ii. x[n] = cos ( π

8n2)

10 1 1.1.3 L3

Unit 2

4a Prove the following properties of DTFS i. Time shift property ii. Frequency shift property.

10 3 1.1.3 L2

b

Use the frequency-shift property to find the time-domain signals corresponding to the following Fourier representations:

cos ( n/5 ) + j sin ( n/5 ), n< 10

10 3 2.1.4 L3

5a

Use the method of inspection to determine the DTFS coefficients for the signal x[n]=1+sin(nπ/12+3π/8) .

6 3 1.1.3 L2

b

The input and output of a causal LTI system are related by differential equation d2 y (t )dt2

+6 dy (t)dt

+8 y (t )=2x (t).

Find the impulse response of this system.What is the response of this system if x(t)=t e-2t u(t)?

10 3 2.1.2 L3

cFind the DTFS of x[n]=2 for 0≤ n≤5. Plot the magnitude and phase spectra.

4 3 1.1.3 L3

6aUsing convolution theorem find the inverse FT of X(w)=a/(a+jw)2 10 3 1.1.3 L3

b

Find the Fourier transform of x(t) = sgn(t) and hence u(t) where

sgn(t)= 1 , for t >0 , sgn(t)=-1 ,for t<0

What are the conditions for existence of Fourier transform.

10 3 2.1.2 L3


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Unit 3

7aState and prove the following

i. Final value theoremii. Time shifting propertyiii. Convolution property

10 3 1.1.3 L2

b

Find Z transform of following sequence and estimate the ROC

1.x (n )=2n .u (n )+3( 12 )n

.u (n)

2.x (n )=3(−12 )n

. u (n )−2 (3 )n . u(−n−1)

10 3 1.1.3 L3

8a

Calculate the inverse Z transform of the following.

X (Z )= z (z2−4 z+5)(z−3 ) (z−2 )(z−1)

ROCs: 2<|z|<3, |z|>3, |z|<1

10 3 1.1.3 L3

bExplain following properties of Z-transform with a suitable example

i. Casuality ii. Stabilty

10 3 1.1.3 L2


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