ss qp unit 1

8/6/2019 SS QP Unit 1

1/12

SIGNALS AND SYSTEMS

UNIT I -- CLASSIFICATION OF SIGNALS AND SYSTEMS

PART A

1. What are the major classifications of signals?

y Signals are classified as Continuous Time (CT) and Discrete Time(DT) signals.y Both CT and DT signals are further classified as

- Deterministic and Random signals- Even and Odd signals- Energy and Power signals

- Periodic and Aperiodic signals

2. With suitable examples distinguish a deterministic signal from a random

signal.Deterministic signal:

A signal which can be modeled ( represented) by a mathematical equation.

Example: cosine signal

Random signal:

A signal which cannot be modeled by a mathematical equation is called random

signal.

Example:

-8 -6 -4 -2 0 2 4 6 8-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

[ t

x(t)

x(t)=cos([ t)

-8 -6 -4 -2 0 2 4 6 8-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

t

x(t)

random signal

8/6/2019 SS QP Unit 1

2/12

3. What is an energy signal?

A signal x(t) is said to be energy signal if

y Energy is finite and

y Average power is zeroi.e.,

0

0

!

g

P

E

Where E = energy and P = Average power

4. What is power signal?

A signal x(t) is said to be power signal if

y power is finite and

y energy is infinite

g!

g

E

P0

Where E = energy and P = Average power

5. Give the mathematical and graphical representation of unit step sequence.

Mathematical representation:

00

01)(

!

u!

n

nnu

Graphical representation:

-3 -2 -1 0 1 2 3 4 5 60

0. 2

0. 4

0. 6

0. 8

1

n

u

(

n

)

unit step se quence

8/6/2019 SS QP Unit 1

3/12

6. Give the mathematical and graphical representation of unit ramp signal.


00

0)(

!

u!

t

tttr

Graphical representation

7. What are periodic signals? Give example.

A signal )( tx is said to be periodic if )()( Ttxtx ! for all t. The smallest

value of T for which the condition is satisfied is called the fundamental period.

Example: sinusoidal signals

8. What is even signal? Give example.

A signal )(tx is said to be even signal if )(tx = )( tx .

Example: tAtx [cos)( !

9. What is odd signal? Give example.

A signal )(tx is said to be odd signal if )(tx = )( tx .

Example: tAtx [sin)( !

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

r

(

t

)

unit ramp signal

8/6/2019 SS QP Unit 1

4/12

10. What is unit impulse function? How unit ramp function can be obtainedfrom a unit impulse function?


00

01)(

{!

!!

t

ttH

Graphical representation

functionrampunittr

functionstepunittufunctionimpulseunitt

trtutegrateegrate

!

!!

p p

)(

)()(

)()()(intint

H

H

11. Plot the signal x ( t ) = [ u ( t 1 ) u ( t 2 ) ] . What is the resultingsignal?

The resulting signal is a pulse.

-4 -3 -2 -1 0 1 2 3 40

0. 2

0. 4

0. 6

0. 8

1

t

H(

t

)

unit impulse function

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5 5. 50

0 . 5

1

t

u

(

t-1 u ( t - 1 )

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5 5. 50

0 . 5

1

t

u

(

t

-2

)

u ( t - 2 )

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 5 5. 50

0 . 5

1

t

u

(

t

-1

)-u

(t-2

)

u ( t - 1 ) - u ( t - 2 )

8/6/2019 SS QP Unit 1

5/12

12. Given the signal x[n] = { 1, 2, 3, -4, 6 } , plot the signal x(n)and x( n 2)

13. Draw the waveforms of u ( t + 2 ) and u ( - n +1 ).

14. Verify whether 0,)()( "! atuAetx at is an energy signal or not.

0,)()( "! atuAetx at

Joulesa

a

eltdt

eltdttxltEnergy

Tta

T

T

ta

T

T

TT 22

)(2

0

22

2

0

2

!

!!!

gp

gp

gp

Watta

e

Tltdttx

Tltpower

T

ta

T

T

TT

022

1)(21

0

22

2

!

!!

gp!

gp

y Energy is finitey Power is zero

The signal is energy signal

0 1 2 3 4 5 6-5

0

5

n

x

(n

)

x ( n )

0 1 2 3 4 5 6-5

0

5

n

x

(n

-2

) x ( n-2)

- 2 - 1 0 1 2 30

0 .5

1

t

u

(

t+

2

u ( t+ 2 )

- 4 - 3 - 2 - 1 0 1 20

0 .5

1

n

u

(

-n

+

1

)

u ( - n + 1 )

8/6/2019 SS QP Unit 1

6/12

15. Show that the complex exponential signal tj oetx [!)( is periodic and that

the fundamental period iso[

T2.

0

0

)(

2

1

int2,.

1)()(

)(

)(

0

000

0

[

T

T[

[

[[[

[

!

!

!!

!!

!!

!

periodlfundamenta

mforvaluesmallest

egermwheremwhenei

ewhenTtxtx

eeeTtx

etx

Tj

Tjtj

Ttj

tj

16. Determine the power and RMS value of the signal tetx ojat

[cos)( ! .

2

1

2

1)2(

4

1

1sin2cos14

1

cos2

1

0

2

0

!

!!

!!

!

gp

gp

gp

valueRMS

wattTT

lt

ecedttT

lt

dtteTltpower

T

jat

T

TT

T

T

taj

T

[

[

17.Find the average power of the signal ).()( Nnunu

Average power of a DT signal x(n) is

wattN

ltnxN

ltPowerN

nN

N

NnN

0)1()12(

1)(

)12(

1

0

2

2

!

!

! !

gp!

gp

18. If the discrete time signal x[n]={0,0,0,3,2,1,-1,-7,6}, then findy[n]=x[2n].

y[0]=x[0]y[1]=x[2]

y[2]=x[4]y[3]=x[6]y[4]=x[8]

y[n]={0,0,2,-1,6}

8/6/2019 SS QP Unit 1

7/12

19. Determine whether the signal )1.0cos()( nnx T! is periodic or not.

20,1

int,

int,201.0

22

1.0

0

0

!!

v!!!

!

Nmwhen

valueegerpossiblelowestthetakesNmofvaluesomefor

egerareNmwheremm

N

mN

T

T

T[

T[

x(n) is periodic with period N=20

20. Define symmetric and anti symmetric signal.

Symmetric signal:

It is a even signal.A signal )(tx is said to be symmetric signal if )(tx = )( tx .

Example: tAtx [cos)( !

Anti symmetric signal:

A signal )(tx is said to be anti symmetric signal if )(tx = )( tx .

Example: tAtx [sin)( !

21. Give the formula for decomposing an arbitrary signal x(t) in to odd andeven part.

CT signal:

? A

? A)()(2

1)(

)()(2

1)(

)(&)(

)()()(

txtxtx

txtxtx

componentoddtxcomponenteventx

txtxtx

o

e

oe

oe

!

!!

!

DT signal:

? A

? A)()(2

1)(

)()(2

1)(

)(&)()()()(

nxnxnx

nxnxnx

componentoddnxcomponentevennx

nx

nx

nx

o

e

oe

oe

!

!!

!

8/6/2019 SS QP Unit 1

8/12

22.Give the precedence rule for combined time scaling and time shiftingoperation.

For combined time scaling and time shifting operation,y time shifting is carried out first

y next, time scaling is carried out

23. How are systems classified?

y Systems are classified as continuous time (CT) and discrete time (DT) systems.

y Both CT and DT systems are classified as,

- static and dynamic system

- causal and non causal system

- linear and non linear system

- time variant and time invariant system- stable and unstable system

- lumped parameter and distributed parameter system

24. Define a causal system.

y A causal system is a non-anticipating system

y Output of the system depend only on past and present inputs and not on future

inputs.

25.Distinguish static system from dynamic system.

Static system:

y Static system is a system with no memory or energy storage element.

y Output of a static system at any specific time depends on the input at thatparticular time.

Dynamic system:

y Dynamic systems have memory or energy storage elements.

y Ouput of a dynamic system at any specific time depends on the inputs at thatspecific time and at other times( past or future values of inputs)

8/6/2019 SS QP Unit 1

9/12

26.Define a time invariant system.

y A system is said to be time invariant if its input-output characteristics do not

change with time.

y Let ? A)()( txty +! outputtyinputtx !! )(&)(

)()( txonoperationtiontransformasomedenotes+

Let ),( Tty denote the output due to delayed input )( Ttx

i.e, ? A)(),( TtxTty +!

let )( Tty be the out put delayed by T

if ),()( TtyTty ! then the system is time invariant

27.Define a continuous time LTI system.

A continuous time system which posses two properties

i) linearity

ii) Time invariance

is a CT LTI system.

28.Define a non causal system.

A non causal system is one whose output depends on future inputs also .

29. Define a non linear system

y A system which does not obey superposition principle is a non linear system.

y Superposition principle states that the output of the system to a weighted sum ofinputs is equal to the corresponding weighted sum of the outputs of the systemto each of the individual inputs.

8/6/2019 SS QP Unit 1

10/12

30.Determine whether the system described by the following input-outputrelationship is linear and causal y(t) = x(-t)

inputtxoutputty

iprelationshoutputinputtxty

!!

p!

)()(

)()(

Checking for linearity:

y For an input )(1 tx , the output )(1 ty is,

)()( 11 txty !

y For an input )(2 tx , the output )(2 ty is,

)()( 22 txty !

y For an input )()( 21 txbtxa , the output )(3 ty is,

)()()(213

txbtxaty !

y )()()(213tybtyaty !

y The system obeys superposition principle. Therefore the system is linear

Checking for causality:For 1!t , )1()1( xy !

For negative values of time t , the output depends on the future input.Therefore the system is non-causal.

8/6/2019 SS QP Unit 1

11/12

PART B

1. i) State the precedence rule for combined time scaling and time shifting operation.ii)Given x(t) . Sketch the following signals.

(1)x ( 3 t + 2)(2)x ( - t / 2 1)

2. i) Sketch the signal x (t) = u (t) u ( t - 15). Determine the energy and power inthe signal x(t).

ii) Determine the energy and power in the signal )(2

1)( nunx

n

!

3. (i) A continuous time signal x(t) is shown in figure. Sketch and label carefully eachof the following signals:

)12()4(

]2

3

2

3)[()3(

)2()2()1()1(

tx

tttx

txtx

HH

(ii) Determine the value of power and energy for each of the signals

)(

2

1)()2(

)()1( 82

nunx

enx

n

nj

!

!

TT

4. i) Explain real exponential and complex exponential signal.

ii) Find the periodicity of the signal2

cos3

2sin)(

nnnx

TT

! .

X t

1

2

0 1/2 1-1/2-1t

8/6/2019 SS QP Unit 1

12/12

5.i) Sketch 22)( ee! nwhereanx n for the two cases

i) a = - (2) a = - 4ii) Determine and sketch the even and odd part of the signal }3,1,2,1,2{)( !

onx

6. i) sketch the signal )2()3( nunu .

ii) (1) Define symmetric and anti symmetric signal.(2) Determine the even and odd part of the given signal .

x(t) = cos t + sin t + cos t sin t

7.What are the classification of signals? Define and sketch all the signals.

8.i) Derive expressions for decomposing an arbitrary signal x(t) in to odd and even part.ii) Discuss on the relationship between u(t) , r (t) and (t)

9. i) Test whether the system described by the equation )()( nxnny ! is

i) Linear ii) shift Invariant .

ii) Given )2(3

1)1(

8

1)()( ! nxnxnxny . Find whether the system is stable or not.

10 i) Define LTI system.ii) Verify the Linearity, Causality and Time invariance of the system described by the

equation )3()1()2( ! nxbnxany .

11. })({)( txoddty ! . Check for Linearity, Time Invariance and Causalilty.

12.Explain briefly abouti) Linear and Non linear systemii) Time Invariant and Time Variant Systemiii) Causal and Non Causal systemiv) Stable and unstable system

13.Determine whether the following system described by the differential equation islinear.

i) )()(2 txtydt

dy!

ii) )(4)()(

txtydt

tdy!

ss qp unit 1

Documents