EC1305

SIGNALS & SYSTEMS

DEPT/ YEAR/ SEM: IT/ III/ V

PREPARED BY: Ms. S. THENMOZHI/ Lecturer/ECE

SYLLABUS

UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS

Continuous Time Signals (CT Signals) – Discrete Time Signals (DT Signals) – Step –Ramp –

Pulse – Impulse – Exponential – Classification of CT and DT Signals –Periodic and aperiodic –

Random Signals – CT systems and DT systems –Classification of systems – Linear time

invariant systems.

UNIT II ANALYSIS OF CT SIGNALS

Fourier series analysis – Spectrum of CT signals – Fourier transform and laplace transform in

signal analysis.

UNIT III LTI – CT SYSTEMS

Differential equation – Block diagram representation – Impulse response – Convolution

integral – Frequency response – Fourier methods and laplace transforms in analysis – State

equations and matrix.

UNIT IV ANALYSIS OF DT SIGNALS

Spectrum of DT signals – Discrete Time Fourier Transform (DTFT) – Discrete Fourier

Transform (DFT) – Properties of z transform in signal analysis.

UNIT V LTI – DT SYSTEMS

Difference equations – Block diagram representation – Impulse response – convolution SUM –

Frequency response – FFT and z - Transform analysis – State variable equation and matrix.

TEXT BOOK

1. Alan V. Oppenheim, Alan S. Willsky and S.Hamid Nawab, “Signals &

Systems”, Pearson / Prentice Hall of India, 2003.

REFERENCES

1. K.Lindner, “Signals and Systems”, Tata McGraw-Hill, 1999.

2. Simon Haykin and Barry Van Veen, “Signals and Systems”, John Wiley &

Sons, 1999.

UNIT I

CLASSIFICATION OF SIGNALS AND SYSTEMS

SIGNAL:

► Signal is a physical quantity that varies with respect to

time , space or any other independent variable

Eg x(t)= sin t.

► the major classifications of the signal are:

(i) Discrete time signal

(ii) Continuous time signal

UNIT STEP &UNIT IMPULSE

Discrete time Unit impulse is defined as

δ *n+= ,0, n≠ 0

{1, n=0

Unit impulse is also known as unit sample.

Discrete time unit step signal is defined by

U[n]={0,n=0

{1,n>= 0

Continuous time unit impulse is defined as

δ (t)=,1, t=0

,0, t ≠ 0

Continuous time Unit step signal is defined as

U(t)={0, t<0

,1, t≥0

Classification of CT signals.

The CT signals are classified as follows

(i) Periodic and non periodic signals (ii) Even and odd signals (iii) Energy and power signals (iv) Deterministic and random signals.

► Periodic Signal & Aperiodic Signal

A signal is said to be periodic, if it exhibits periodicity.

i.e., X(t +T)=x(t), for all values of t. Periodic signal has

the property that it is unchanged by a time shift of T. A

signal that does not satisfy the above periodicity

property is called an aperiodic signal

► Even and odd signal?

A discrete time signal is said to be even when,

x[-n]=x[n]. The continuous time signal is said to be

even when, x(-t)= x(t) For example, Cosωn is an even

signal.

ENERGY & POWER SIGNAL:

► A signal is said to be energy signal if it have finite energy and

zero power.

► A signal is said to be power signal if it have infinite energy

and finite power.

► If the above two conditions are not satisfied then the signal

is said to be neither energy nor power signal

SYSTEM

A system is a set of elements or functional block that are

connected together and produces an output in response to an input signal.

Eg: An audio amplifier, attenuator, TV set etc. Classification or characteristics of CT and DT systems.

The DT and CT systems are according to their characteristics as follows (i). Linear and Non-Linear systems (ii). Time invariant and Time varying system (iii). Causal and Non causal systems. (iv). Stable and unstable systems. (v). Static and dynamic systems. (vi). Inverse systems. linear and non-linear systems. A system is said to be linear if superposition theorem applies to that system. If it does not satisfy the superposition theorem, then it is said to be a non- linear system. Causal and non-Causal systems. A system is said to be a causal if its output at anytime depends upon present and past inputs only. A system is said to be non-causal system if its output depends upon future inputs also. Time invariant and time varying systems. A system is time invariant if the time shift in the input signal results in corresponding time shift in the output.

A system which does not satisfy the above condition is time variant system. Stable and unstable systems. When the system produces bounded output for bounded input, then the system is called bounded input, bounded output stable. A system which does not satisfy the above condition is called a unstable system. Static and Dynamic system.

A system is said to be static or memoryless if its output depends upon the present input only. The system is said to be dynamic with memory if its output depends upon the present and past input values.

IMPORTANT QUESTIONS

1. Define Signal.

2. Define System.

3. Define CT signals.

4. Define DT signal.

5. Give few examples for CT signals.

6. Give few examples of DT signals.

7. Define unit step,ramp and delta functions for CT.

8. State the relation between step, ramp and delta functions(CT).

9. State the classification of CT signals.

10. Define deterministic and random signals.

11. Define power and energy signals.

12. Compare power and energy signals.

13.Define odd and even signal.

14. Define periodic and Aperiodic signals.

15. State the classification or characteristics of CT and DT systems.

16. Define linear and non-linear systems.

17. Define Causal and non-Causal systems.

18. Define time invariant and time varying systems.

19. Define stable and unstable systems.

20. Define Static and Dynamic system.

PART-B

1. Discuss the classification of DT and CT signals with examples.

2. Discuss the classification of DT and CT systems with examples.

3. Problems on the properties & classifications of signals & systems

Find whether the following signals are periodic or not

a. x(t)=2cos(10t+1)-sin(4t-1)

Ans:Periodic signal.

b. x(t)=3cos4t+2sinπt

Ans:Non periodic signal

Check whether the following system is

1. Static or dynamic

2. Linear or non-linear

3. Causal or non-causal

4. Time invariant or variant

y(n)=sgn[x(n]

UNIT II

ANALYSIS OF CT SIGNALS

FOURIER SERIES:

The Fourier series represents a periodic signal in terms of

frequency components:

We get the Fourier series coefficients as follows:

The complex exponential Fourier coefficients are a sequence of

complex numbers representing the frequency component ω0k.

Fourier series: a complicated waveform analyzed into a number of

harmonically related sine and cosine functions

A continuous periodic signal x(t) with a period T0 may be

represented by:

X(t)=Σ∞k=1 (Ak cos kω t + Bk sin kω t)+ A0

Dirichlet conditions must be placed on x(t) for the series to be

valid: the integral of the magnitude of x(t) over a complete period

must be finite, and the signal can only have a finite number of

discontinuities in any finite interval

TRIGONOMETRIC FORM OF FOURIER SERIES

If the two fundamental components of a periodic signal

areB1cosω0t and C1sinω0t, then their sum is expressed by

trigonometric identities:

1

0

0)(p

k

nik

keXnx

k

tik

keXtx 0)(

1

0

0)(1 p

n

nik

k enxp

X

p

tik

k dtetxp

X0

0)(1

X(t)= A0 + Σ∞k=1 ( Bk

2+ Ak 2)1/2 (Ck cos kω t- φk) or

X(t)= A0 + Σ∞k=1 ( Bk

2+ Ak 2)1/2 (Ck sin kω t+ φk)

FOURIER TRANSFORM:

► Viewed periodic functions in terms of frequency components

(Fourier series) as well as ordinary functions of time

► Viewed LTI systems in terms of what they do to frequency

components (frequency response)

► Viewed LTI systems in terms of what they do to time-domain

signals (convolution with impulse response)

► View aperiodic functions in terms of frequency components

via Fourier transform

► Define (continuous-time) Fourier transform and DTFT

► Gain insight into the meaning of Fourier transform through

comparison with Fourier series

► A transform takes one function (or signal) and turns it into

another function (or signal)

► Continuous Fourier Transform:

CONTINUOUS TIME FOURIER TRANSFORM

We can extend the formula for continuous-time Fourier

series coefficients for a periodic signal to aperiodic signals as well.

The continuous-time Fourier series is not defined for aperiodic

dfefHth

dtethfH

ift

ift

2

2

signals, but we call the formula the (continuous time) Fourier

transform.

INVERSE TRANSFORMS

If we have the full sequence of Fourier coefficients for a

periodic signal, we can reconstruct it by multiplying the complex

sinusoids of frequency ω0k by the weights Xk and summing:

We can perform a similar reconstruction for aperiodic signals

These are called the inverse transforms.

FOURIER TRANSFORM OF IMPULSE FUNCTIONS:

Find the Fourier transform of the Dirac delta function:

Find the DTFT of the Kronecker delta function:

2/

2/0

00 )(1

)(1

p

p

tik

p

tik

k dtetxp

dtetxp

X

dtetxX ti)()(

1

0

0)(p

k

nik

keXnxk

tik

keXtx 0)(

deXnx ni)(2

1)( deXtx ti)(

2

1)(

1)()()( 0ititi edtetdtetxX

1)()()( 0i

n

ni

n

ni eenenxX

The delta functions contain all frequencies at equal amplitudes.

Roughly speaking, that’s why the system response to an impulse

input is important: it tests the system at all frequencies.

LAPLACE TRANSFORM

► Lapalce transform is a generalization of the Fourier

transform in the sense that it allows “complex frequency”

whereas Fourier analysis can only handle “real frequency”.

Like Fourier transform, Lapalce transform allows us to

analyze a “linear circuit” problem, no matter how

complicated the circuit is, in the frequency domain in stead

of in he time domain.

► Mathematically, it produces the benefit of converting a set

of differential equations into a corresponding set of

algebraic equations, which are much easier to solve.

Physically, it produces more insight of the circuit and allows

us to know the bandwidth, phase, and transfer

characteristics important for circuit analysis and design.

► Most importantly, Laplace transform lifts the limit of Fourier

analysis to allow us to find both the steady-state and

“transient” responses of a linear circuit. Using Fourier

transform, one can only deal with he steady state behavior

(i.e. circuit response under indefinite sinusoidal excitation).

► Using Laplace transform, one can find the response under

any types of excitation (e.g. switching on and off at any

given time(s), sinusoidal, impulse, square wave excitations,

etc.

APPLICATION OF LAPLACE TRANSFORM TO CIRCUIT ANALYSIS

CONVOLUTION

► The input and output signals for LTI systems have special

relationship in terms of convolution sum and integrals.

► Y(t)=x(t)*h(t) Y[n]=x[n]*h[n]

IMPORTANT QUESTIONS

1.Define CT signal

2. Compare double sided and single sided spectrums.

3. Define Quadrature Fourier Series.

4.Define polar Fourier Series.

5.Define exponential fourier series.

6. State Dirichlets conditions.

7. State Parsevals power theorem.

8.Define Fourier Transform.

9. State the conditions for the existence of fourier series. 10. Find the Fourier transform of function x(t)=δ(t)

11. State Rayleigh’s energy theorem.

12.Define laplace transform.

13. Obtain the laplace transform of ramp function.

14. What are the methods for evaluating inverse Laplace transform.

15. State initial value theorem.

16. State final value theorem.

17. State the convolution property of fourier transform.

18.What is the relationship between Fourier transform and Laplace transform.

19.Find the fourier transform of sgn function.

20. Find out the laplace transform of f(t)=eat

PART- B

1.State and prove properties of fourier transform.

2. State the properties of Fourier Series.

3. State the properties of Laplace transform.

4.Problems on fourier series, Fourier transform and laplace transform.

a. Find the fourier series of of the periodic signal x(t)=t 0<=t<=1

b. Find the fourier transform of x(t)=e-at

u(t)

c. Find the laplace transform of the signal x(t)= e-at

u(t)+ e-bt

u(-t)

5. State and prove parsevals power theorem and Rayleigh’s energy theorem.

UNIT III LTI-CT SYSTEMS

SAMPLING THEORY

► The theory of taking discrete sample values (grid of color

pixels) from functions defined over continuous domains

(incident radiance defined over the film plane) and then

using those samples to reconstruct new functions that are

similar to the original (reconstruction).

► Sampler: selects sample points on the image plane

► Filter: blends multiple samples together

► For band limited function, we can just increase the sampling

rate

► However, few of interesting functions in computer graphics

are band limited, in particular, functions with

discontinuities.

► It is because the discontinuity always falls between two

samples and the samples provides no information of the

discontinuity.

ALIASING

Z-TRANSFORMS

► For discrete-time systems, z-transforms play the same role of

Laplace transforms do in continuous-time systems

Bilateral Forward z-transform Bilateral Inverse z-transform

► As with the Laplace transform, we compute forward and

inverse z-transforms by use of transforms pairs and

properties

n

nznhzH ][R

n dzzzHj

nh 1 ][ 2

1][

REGION OF CONVERGENCE

► Region of the complex z-plane for which

forward z-transform converges

► Four possibilities (z=0 is a special case and may or may not

be included)

Z-TRANSFORM PAIRS

► h[n] = d[n]

Region of convergence: entire z-plane

► h[n] = d[n-1]

Region of convergence: entire z-plane

h[n-1] z-1 H[z]

► h[n] = an u[n]

Region of convergence: |z| > |a| which is the complement of a

disk

STABILITY

► Rule #1: For a causal sequence, poles are inside the unit

circle (applies to z-transform functions that are ratios of two

polynomials)

► Rule #2: More generally, unit circle is included in region of

convergence. (In continuous-time, the imaginary axis would

be in the region of convergence of the Laplace transform.)

1 ][0

0n

n

n

n znznzH

11

1

1 1][ zznznzHn

n

n

n

This is stable if |a| < 1 by rule #1.

It is stable if |z| > |a| and |a| < 1 by rule #2.

INVERSE Z-TRANSFORM

► Yuk! Using the definition requires a contour integration in

the complex z-plane.

► Fortunately, we tend to be interested in only a few basic

signals (pulse, step, etc.)

Virtually all of the signals we’ll see can be built up from

these basic signals.

For these common signals, the z-transform pairs have

been tabulated (see Lathi, Table 5.1)

dzzzFj

nf n

jc

jc

1

2

1

EXAMPLE

► Ratio of polynomial z-domain

functions

► Divide through by the highest

power of z

Factor denominator into first-

order factors

Use partial fraction

decomposition to get first-order

terms

► Find B0 by polynomial division

► Express in terms of B0

► Solve for A1 and A2

2

1

2

3

12][

2

2

zz

zzzX

21

21

2

1

2

31

21][

zz

zzzX

11

21

12

11

21][

zz

zzzX

1

2

1

10

1

2

11

][z

A

z

ABzX

15

23

2

1212

3

2

1

1

12

1212

z

zz

zzzz

11

1

12

11

512][

zz

zzX

8

2

1

121

2

11

21

921

441

1

21

1

1

21

2

2

1

21

1

1

1

z

z

z

zzA

z

zzA

► Express X[z] in terms of B0, A1, and A2

► Use table to obtain inverse z-transform

► With the unilateral z-transform, or the bilateral z-transform

with region of convergence, the inverse z-transform is

unique

11 1

8

2

11

92][

zz

zX

nununnx

n

82

1 9 2

► Convolution definition

► Take z-transform

► Z-transform definition

► Interchange summation

► Substitute r = n - m

► Z-transform definition

zFzF

zrfzmf

zrfmf

zmnfmf

zmnfmf

mnfmfZnfnfZ

mnfmfnfnf

r

rm

m

m r

mr

m n

n

n

n

m

m

m

21

21

21

21

21

2121

2121

IMPORTANT QUESTIONS

1. Define LTI-CT systems.

2. What are the tools used for analysis of LTI-CT systems?

3.Define convolution integral.

4.List the properties of convolution integral. 5.State commutative property of convolution.

6.State the associative property of convolution.

7.State distributive property of convolution.

8. When the LTI-CT system is said to be dynamic?

9. When the LTI-CT system is said to be causal?

10. When the LTI-CT system is said to be stable?

11. Define natural response.

12. Define forced response. 13. Define complete response.

14. Draw the direct form I implementation of CT systems.

15. Draw the direct form II implementation of CT systems. 16. Mention the advantages of direct form II structure over direct form I structure.

17. Define Eigen function and Eigen value.

18. Define Causality and stability using poles.

19. Find the impulse response of the system y(t)=x(t-t0) using laplace transform.

20. The impulse response of the LTI CT system is given as h(t)=e-t

u(t).

Determine transfer function and check whether the system is causal and stable.

PART B

1.Derive convolution integral and also state and prove the properties of the same.

2. Explain the properties of LTICT system interms of impulse response.

3.Problems on properties of LTI CT systems.

4. Problems on differential equation.

5. Realization of LTI CT system using direct form I and II structures.

6. Finding frequency response using Fourier methods.

7. Solving differential equations using Fourier methods

8. Solving Differential Equations using Laplace transforms.

9. Obtaining state variable description.

10. Obtaining frequency response and transfer functions using state variable.

UNIT – IV

ANALYSIS OF DISCRETE TIME SIGNALS

INTRODUCTION

► Impulse response h[n] can fully characterize a LTI system, and we can have the

output of LTI system as

► The z-transform of impulse response is called transfer or system function H(z).

► Frequency response at is valid if ROC includes

and

FREQUENCY RESPONSE OF LIT SYSTEM

► Consider and , then

magnitude

phase

► We will model and analyze LTI systems based on the magnitude and phase

responses.

SYSTEM FUNCTION

► General form of LCCDE

)()()(jeXjjj eeXeX

)()()( jjj eHeXeY

)()()( jjj eHeXeY

nhnxny

.zHzXzY

1z

j zHeH ,1z

jjj eHeXeY

)()()(jeHjjj eeHeH

knxbknyaM

k

k

N

k

k

00

zXzbzYza kM

k

k

N

k

k

k

00

)(

► Compute the z-transform

SYSTEM FUNCTION: POLE/ZERO FACTORIZATION

► Stability requirement can be verified.

► Choice of ROC determines causality.

► Location of zeros and poles determines the frequency

response and phase

SECOND-ORDER SYSTEM ► Suppose the system function of a LTI system is

N

k

k

k

kM

k

k

za

zb

zX

zYzH

0

0

N

k

k

M

k

k

zd

zc

a

bzH

1

1

1

1

0

0

1

1 .,...,,:zeros 21 Mccc

.,...,,:poles 21 Nddd

.

)4

31)(

2

11(

)1()(

11

21

zz

zzH

► To find the difference equation that is satisfied by the input and out of this

system

► Can we know the impulse response?

System Function: Stability

► Stability of LTI system:

► This condition is identical to the condition that

The stability condition is equivalent to the condition that the ROC of H(z)

includes the unit circle.

System Function: Causality

► If the system is causal, it follows that h[n] must be a right-sided sequence. The

ROC of H(z) must be outside the outermost pole.

► If the system is anti-causal, it follows that h[n] must be a left-sided sequence.

The ROC of H(z) must be inside the innermost pole.

)(

)(

8

3

4

11

21

)4

31)(

2

11(

)1()(

21

21

11

21

zX

zY

zz

zz

zz

zzH

]2[2]1[2][]2[8

3]1[

4

1][ nxnxnxnynyny

n

nh ][

.1 when][ zznhn

n

DETERMINING THE ROC

► Consider the LTI system

► The system function is obtained as

SYSTEM FUNCTION: INVERSE SYSTEMS

► is an inverse system for , if

► The ROCs of must overlap.

► Useful for canceling the effects of another system

► See the discussion in Sec.5.2.2 regarding ROC

][]2[]1[2

5][ nxnynyny

)21)(2

11(

1

2

51

1)(

11

21

zz

zz

zH

zHzH i

1)()()( zHzHzG innhnhng i

)(

1)(

zHzH i

)(

1)(

j

j

ieH

eH

)( and )( zHzH i

ALL-PASS SYSTEM

► A system of the form (or cascade of these)

ALL-PASS SYSTEM: GENERAL FORM

► In general, all pass systems have form

real poles complex poles

Causal/stable:

1

1

1 az

azZH Ap j

j

era

rea

1*/1 :zero

:pole

j

jj

j

jj

Apae

eae

ae

aeeH

1

*1

1

1j

Ap eH

cr M

k kk

kk

M

k k

kAp

zeze

ezez

zd

dzzH

11*1

1*1

11

1

)1)(1(

))((

1

1, kk de

MINIMUM-PHASE SYSTEM

► Minimum-phase system: all zeros and all poles are inside the unit circle.

► The name minimum-phase comes from a property of the phase response

(minimum phase-lag/group-delay).

► Minimum-phase systems have some special properties.

► When we design a filter, we may have multiple choices to satisfy the certain

requirements. Usually, we prefer the minimum phase which is unique.

► All systems can be represented as a minimum-phase system and an all-pass

system.

IMPORTANT QUESTIONS

PART-A

1. Define DTFT.

2. State the condition for existence of DTFT?

3. List the properties of DTFT.

4. What is the DTFT of unit sample?

5. Define DFT.

6. Define Twiddle factor. 7. Define Zero padding.

8. Define circularly even sequence.

9. Define circularly odd sequence.

10. Define circularly folded sequences.

11. State circular convolution.

12. State parseval’s theorem.

13. Define Z transform.

14. Define ROC.

15. Find Z transform of x(n)={1,2,3,4}

16. State the convolution property of Z transform.

17. What z transform of (n-m)?

18. State initial value theorem.

19. List the methods of obtaining inverse Z transform.

20. Obtain the inverse z transform of X(z)=1/z-

PART – B

1. State and prove properties of DTFT

2. State and prove the properties of DFT.

3. State and prove the properties of z transform.

4.Find the DFT of x(n)={1,1,1,1,1,1,0,0}

5. Find the circular convolution of x1(n)={1,2,0,1} X2(n)={2,2,1,1}

6. Problems on z transform and inverse z transform.

UNIT – V

LTI – DT SYSTEM

Example

► Block diagram representation of

BLOCK DIAGRAM REPRESENTATION

► LTI systems with rational system function can be represented as constant-

coefficient difference equation

► The implementation of difference equations requires delayed values of the

input

output

intermediate results

► The requirement of delayed elements implies need for storage

► We also need means of

addition

multiplication

nxbnyanyany 021 21

DIRECT FORM I

► General form of difference equation

► Alternative equivalent form

ALTERNATIVE REPRESENTATION

M

k

k

N

k

k knxbknya00

ˆˆ

M

k

k

N

k

k knxbknyany01

► Replace order of cascade LTI systems

ALTERNATIVE BLOCK DIAGRAM

► We can change the order of the cascade systems

zWzbzWzHzY

zX

za

zXzHzW

za

zbzHzHzH

M

k

k

k

N

k

k

k

N

k

k

k

M

k

k

k

0

1

1

2

1

0

21

1

1

1

1

M

k

k

N

k

k

knwbny

nxknwanw

0

1

M

k

k

N

k

k

knwbny

nxknwanw

0

1

DIRECT FORM II

► No need to store the same data twice in previous system ► So we can collapse the delay elements into one chain ► This is called Direct Form II or the Canonical Form ► Theoretically no difference between Direct Form I and II ► Implementation wise

Less memory in Direct II Difference when using finite-precision arithmetic

SIGNAL FLOW GRAPH REPRESENTATION

► Similar to block diagram representation Notational differences

► A network of directed branches connected at nodes ► Example representation of a difference equation

EXAMPLE

► Representation of Direct Form II with signal flow graphs

nwny

nwnw

nwbnwbnw

nwnw

nxnawnw

3

24

41203

12

41

1

1

1

1110

11

nwbnwbny

nxnawnw

DETERMINATION OF SYSTEM FUNCTION FROM

FLOW GRAPH

nwnwny

nwnw

nxnwnw

nwnw

nxnwnw

42

34

23

12

41

1

zWzWzY

zzWzW

zXzWzW

zWzW

zXzWzW

42

1

34

23

12

41

zWzWzY

z

zzXzW

z

zzXzW

42

1

1

4

1

1

2

1

1

1

1

nununh

z

z

zX

zYzH

nn 11

1

1

1

1

BASIC STRUCTURES FOR IIR SYSTEMS: DIRECT

FORM I

BASIC

STRUCTUR

ES FOR IIR

SYSTEMS:

DIRECT

FORM II

BASIC STRUCTURES FOR IIR SYSTEMS: CASCADE FORM

General form for cascade implementation More practical form in 2nd order systems

EXAMPLE

21

21

1

11

1

1

1

11

1

1

111

111

N

k

kk

N

k

k

M

k

kk

M

k

k

zdzdzc

zgzgzf

AzH

1

12

2

1

1

2

2

1

10

1

M

k kk

kkk

zaza

zbzbbzH

1

1

1

1

11

11

21

21

25.01

1

5.01

1

25.015.01

11

125.075.01

21

z

z

z

z

zz

zz

zz

zzzH

CASCADE OF DIRECT FORM I SUBSECTIONS

Cascade of Direct Form II subsections

BASIC STRUCTURES FOR IIR SYSTEMS:

PARALLEL FORM

REPRESENT SYSTEM FUNCTION USING PARTIAL FRACTION EXPANSION

Or by pairing the real poles

P PP N

k

N

k kk

kk

k

k

N

k

k

kzdzd

zeB

zc

AzCzH

1 111

1

10 11

1

1

SP N

k kk

kk

N

k

k

kzaza

zeezCzH

12

2

1

1

1

10

0 1

EXAMPLE

Partial Fraction Expansion

Combine poles to get

BASIC STRUCTURES FOR FIR SYSTEMS: DIRECT FORM

1121

21

25.01

25

5.01

188

125.075.01

21

zzzz

zzzH

21

1

125.075.01

878

zz

zzH

► Special cases of IIR direct form structures

► Transpose of direct form I gives direct form II

► Both forms are equal for FIR systems

► Tapped delay line

BASIC STRUCTURES FOR FIR SYSTEMS: CASCADE FORM

Obtained by factoring the polynomial system function

STRUCTURES FOR LINEAR-PHASE FIR SYSTEMS

► Causal FIR system with generalized linear phase are symmetric:

M

n

M

k

kkk

nS

zbzbbznhzH0 1

2

2

1

10

IV)or II (type M0,1,...,n

III)or I (type M0,1,...,n

nhnMh

nhnMh

► Symmetry means we can half the number of multiplications

► Example: For even M and type I or type III systems:

STRUCTURES FOR LINEAR-PHASE FIR SYSTEMS

► Structure for even M

► Structure for odd M

2/2/

2/2/

2/2/

12/

0

12/

0

12/

0

12/

12/

00

MnxMhkMnxknxkh

kMnxkMhMnxMhknxkh

knxkhMnxMhknxkhknxkhny

M

k

M

k

M

k

M

Mk

M

k

M

k

IMPORTANT QUESTIONS

PART-A

1. Define convolution sum?

2. List the steps involved in finding convolution sum? 3. List the properties of convolution?

4. Define LTI causal system?

5. Define LTI stable system?

6. Define FIR system?

7. Define IIR system?

8. Define non recursive and recursive systems?

9. State the relation between fourier transform and z transform?

10. Define system function?

11. What is the advantage of direct form 2 over direct form 1 structure?

12. Define butterfly computation?

13.What is an advantage of FFT over DFT?

14. List the applications of FFT? 15. How unit sample response of discrete time system is defined?

16.A causal DT system is BIBO stable only if its transfer function has .

17. If u(n) is the impulse response of the system, What is its step response? 18.Convolve the two sequences x(n)={1,2,3} and h(n)={5,4,6,2}

19. State the maximum memory requirement of N point DFT including twiddle factors?

20. Determine the range of values of the parameter ‘a’ for which the linear time invariant system with

impulse response h(n)=an

u(n) is stable?

PART-B

1. State and prove the properties of convolution sum?

2. Determine the convolution of x(n)={1,1,2} h(n)=u(n) graphically?

3. Determine the forced response for the following system

4. Compute the response of the system

5. Derive the 8 point DIT and DIF algorithms

UNIVERSITY QUESTIONS