signal & system unit-1

8/2/2019 Signal & System Unit-1

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Signal & System

Signal & system



2Signal & system

Unit-1Signals



3

What is a Signal?

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Signal & system



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What is a Signal?Anything which carries information is a signal.

human voice,

chirping of birds,

smoke signals,

gestures (sign language)fragrances of the Àowers

Signals: The Mathematical Way

A signal is a real (or complex) valued function of one or more real variable(s).

one-dimensional signal :- when the function depend on one variable.

Multi ±dimensional signal :- when the function depend on more than one variable.

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Signal & system



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What is a System?

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Signal & system



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What is a system?

Examples of systems include :

1. An oscilloscope: takes a voltage signal as i/p , produce a 2-dimensional image of the voltage signal.

2. A computer monitor: takes voltage pulses from the CPU as inputs and outputs a time varying display.

1.1. Systems are physical models which process the information signalSystems are physical models which process the information signal

2.2. System gives out an output signal in response to an input signalsSystem gives out an output signal in response to an input signals

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Signal & system



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What is a Signal processing ?

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Signal & system



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What is Signal processing?

Processing means operating in some fashion on a signal to extract some useful

informatione.g. we use our ears as input device and then auditory pathways in the brain to extract the

inf ormation. The signal is pr ocessed by a system

The signal processor may be an:-

electronic system, a mechanical system or even it might be a computer program

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Signal & system



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Difference between

Analog Signal processing & Digital Signal processing ?

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Signal & system



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Analog signal processingo analog signal pr ocessing uses elements such as resistors, capacitors, transistors,

diodes etc

o Based on natural ability of the analog system to solve differential equations that

describe a physical system

o The solutions are obtained in real time...

Digital signal processing

Digital signal pr ocessing means that the pr ocessing is done either by a digital hardware or

by a digital computer .

o Relies

on numerical calculati

ons

o The method may or may not give results in real time..

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Signal & system



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Why we use Digital Signal processing ?

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Signal & system



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The advantages of digital approach over

analog approach

o Flexibility: Same hardware can be used to do various kind of signal

processing operation,

o Repeatability: The same signal processing operation can be

repeated again and again giving same results

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Signal & system



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Classification of signals:-

Signal

Continuoustime signal

Discretetime signal

Defined only at discrete times and the independent

variable takes on only a discrete set of values

Defined for a continuum of values

of the independent variable, t.

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Signal & system



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Elementary Signals

There are ten elementary signals that occur

prominently in the study of :-

Digital signals and


1. Unit Step signal

2. Unit Impulse signal or Dirac Delta signal

3. Unit Ramp signal

4. Exponential signal

5. Sine signal

Elementary Signals

There are ten elementary signals that occur

prominently in the study of :-

Digital signals and


1. Unit Step signal

2. Unit Impulse signal or Dirac Delta signal

3. Unit Ramp signal

4. Exponential signal

5. Sine signal

6. Cosine signal

7. DC signal

8. Sinc signal

9. Rectangular signal

10. Signum signal

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Signal & system



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Continuous time Discrete time

Defined by:- Defined by:

1. Unit Step Signal

°¯®

u!

0for 0

0for 1

n

nnu

°¯®

u!

0for 0

0for 1

t

t t u

Graphically it as shown below Graphically it as shown below

The unit step function is discontinuous at the origin

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Signal & system



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Graphically it as shown below

2. Unit Impulse Signal or Dirac Delta Signal

°¯®

{

!!(

0for 0

0for 1

n

nn

´

g

g

!1dt t H

? A 0!t H 0{t

Note:-

The Impulse function is not an ordinary function

Ordinary function defined for all values of t.

The impulse function is 0 everywhere except at t = 0

where it is undefined

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Signal & system



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Relationship Between D.T. unit impulse & D.T. unit step signals

)1()()( ! nununH

)()(0

k nnu ! §

g

H

We can obtain a D.T. unit step signals u(n) by taking the sum of unit samples

We can obtain a D.T. unit impulse signals from a D.T. unit step signal .

Signal & system



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Important properties of unit impulse signal

P1). Sampling (or shifting) property:- Area under the pr oduct of a function x(t) with deltafunction is equal to the value of that function at that instant where the unit impulse is located

´g

g

! )()()(00t xdt t t t x H

P2). Replication property:- Convolution of any function with delta function yields the same function

0),(1

)( {! EHE

EH t t

P3). Scaling property:-

)()()( t xdt t t x !H

Signal & system



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Example :

1

Signal & system



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3. Unit Ramp Signal

°¯®

u!

0for 0

0for

n

nnnr

°¯®

u!

0for 0

0for

t

t t t r

Graphically it as shown belowGraphically it as shown below

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Signal & system



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4.Exponential Signal

at Cet x !

Defined by:-

Here

The constants C and a are real constants

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Signal & system



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5. Sine Signal

)sin(wt At x ! )sin(wn An x !

Here ,

A = Amplitude

W= Angular frequency


Here ,

A = AmplitudeW= Angular frequency


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Signal & system



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6. Cosine Signal

)cos(wt At x ! )cos(wn An x !

Here ,

A = Amplitude



Here ,

A = Amplitude



Signal & system



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7. DC Signal

At x ! At x !

Here ,

A = Amplitude


Here ,

A = Amplitude


Signal & system



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Defined by:-

8. Sinc Signal

t

t At c

T

T )sin(sin !

Sinc (t) =1 at t=0

Sinc (t) =0 at t=±1 , ±2, ±3 ««.

Note:- Sinc signal has the shape of sine wave .

amplitude (A) decrease as the value of t increase.

0sin pt c gpt When

Signal & system



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9) C.T.Rectangular Signal

±°

±¯® ee!¹¹

º

¸©©ª

¨¼½

»¬-

«

otherwise 022

Tfor

T t A

T

t r ect A

Defined by:-

Here ,

A=Amplitude

2w= width/duration of the pulse

f= variable

°̄

® ee

!¹¹ º

¸

©©ª

¨

¼½

»

¬-

«otherwise 0

w-for

2

wt A

W

f

r ect A

Time Domain

Frequency Domain

Here ,

A=AmplitudeT= width/duration of the pulse

t= variable

Signal & system



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9) D.T. Rectangular Signal

±°

±¯® ee!¹¹

º

¸©©ª

¨¼½

»¬-

«

otherwise 022

N-for

N t A

N

nr ect A

Defined by:-

Time Domain

Here ,

A=AmplitudeN= width/duration of the pulse

n= variable

Signal & system



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10. Signum Signal

Defined by

:-

°¯®

"!

0for 1

0for 1sgn

t

t t


Signum signal is an

Odd

Anti-symmetric function

Signal & system



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Relationship Between Step , Ramp , & Delta signals

)()()(

)()()( intint

t t ut r

t r t ut

atediffer entiatediffer enti

eg r ateeg r ate

H

H

p p

p p

Signal & system



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1. Deterministic and non-deterministic signals

2. Periodic and non-periodic signals

3. Even and odd signals

4. Energy and power signals

Classification of C.T. & D.T. signalsClassification of C.T. & D.T. signals

Signal & system



31Signal & system

Deterministic

Deterministicandand

Non Non--DeterministicDeterministicSignalsSignals



32Signal & system

Deterministic signals Non-Deterministic signals

Described by mathematical

Expression , l

ookup table ,

or some well defined Rule

Can¶t be Described by

mathematical Expressio

n , look

up table , or some well defined

Rule

Can be periodic & Non-periodic Non-Periodic

It¶s possible to determine Amplitude of such signals at a

given instant of Time.

It¶s not possible to determine Amplitude of such signals at a

given instant of Time



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Periodic and NonPeriodic and Non--periodic Signals periodic Signals

Signal & system



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CT signal is said to be periodic if it repeat itself after a fixed time period T0.

0T t xt x !

Periodic signalPeriodic signal

T0 = fundamental period

? A ? A0

N n xn x !

DT signal is said to be periodic if it repeat itself after a fixed time period N0

N0 = fundamental period

Signal & system



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Periodic Condition for C.T. & D.T.

)().1 1 t xt x !

)()().3 21 t xt xt x ! )()().4 21 n xn xn x !

)().2 1 n xn x !

wT

T2!

11

2

wT

T

!

2

1

N

N N !

w N

T2!

2

2

2

w

T T

!1

1

2

w

N T

!2

2

2

w

N T

!

2

1

T

T T !

Signal & system

pnI ntegerso f Ratio

pnI ntegerso f Ratio

Time period=LCM (T1,T2 ) or (N1,N2 )



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Problem based on periodicity

)500cos(25 t t t x T!

)50sin(5)100cos(2 nnn x ! T

t t t x 3sin3)3sin(4 !

)2cos(6)sin( nnt x T!

)4sin(3 t t x !

Signal & system



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Problem based on periodicity

)18sin(412sin10 t t t x TT !

n j

en x¼½»

¬-«

! 4T

)sin(481nn x !

)4cos(6)3sin( nnn x T!

)7cos(45sin t t t x !

Signal & system



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Energy and Power SignalsEnergy and Power Signals

Signal & system



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Energy signalEnergy signal

The total Energy of a Continuous time signal x(t) over an interval is defined as:-

´!2

1

2t

t

dt t x E

21 t t t ee

21 t t t ee

For discrete-time sequence x[n]? A§!

2

1

2

n

n

n x E

A signal is called ³energy signal´ if

? A§´

g!

g!

g

g

g!

g!n

n

n x E

dt t x E

2

2

f or C.T.

f or D.T.

g E 0

Signal & system



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Power signalPower signal

g

!́

gp

N

N

N

n x N

P 2

121lim

g P 0

The total power of a Continuous time signal x(t) is defined as:-

For discrete-time sequence x[n]

g!́

gp

T

T

T

t xT

P 2

2

1lim g! ´

gp

2/

2/

21lim

T

T

T

t xT

P

g! ´

gp

2/

2/

21lim

N

N N

n x N

P

Signal & system



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Energy signals Power signals

0<E< 0<P<

Non-periodic Periodic

Time limited Exist over an infinite time

P=0 E=

Difference b/w Energy signal & power signal

gpp t a s A 0 gp{ t a s A 0

Signal & system



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Problem based on Energy & Power

gg! t f or t t x sin10

Identify wheat her it is an E nergy signal or Pow er S ignal

0"!

a f or et xt a

)(2 t uet x t !

Signal & system



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Even and Odd SignalsEven and Odd Signals

Signal & system



t xt x !

Even Signal or Symmetrical signal

C.T. Even signal

D. T. Even signal

? A ? An xn x !

Where,

X(t)=value of the signal for positive ³ t ´.

X(-t)=value of the signal for negative ³ t ´.

Where,

X(n)=value of the signal for positive ³ n ´.

X(-n)=value of the signal for negative ³ n´.

Signal & system



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Odd Signal or Antisymmetricalsignal

Odd Signal or Antisymmetricalsignal

C.T. Odd signal

C.T. Odd signal

D. T. Odd signalD. T.

Odd signal

)()( n xn x !

Where,

X(t)=value of the signal for positive ³ t ´.

X(-t)=value of the signal for negative ³ t ´.

Where,

X(n)=value of the signal for positive ³ n ´.

X(-n)=value of the signal for negative ³ n´.

t xt x !

Signal & system



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? At xt xt xo !2

1

Decomposing a C.T. signal intoDecomposing a C.T. signal into Even and Odd PartEven and Odd Part

? At xt xt xe !2

1Even part is given by

:-

Odd part is given by:-

? At xt xt x oe ! A given signal can be

expressed as the

sum of Odd part & Even part

Signal & system



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? An xn xn xo!

2

1

? An xn xn xe !2

1

Even part is given by:-

Odd part is given by:-

? An xn xn x oe ! A given signal can be

expressed as the

sum of Odd part & Even part

Decomposing a D.T. signal intoDecomposing a D.T. signal into Even and Odd PartEven and Odd Part

Signal & system



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1.) Identify Even & Odd Signals:-

Problem based on Even & odd

Signal & system



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2. Function x(t) is shown draw Even & Odd part.

Problem based on Even & odd

Signal & system

X(t)



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Solution :-

Even Part

Signal & system

X(t)

X(-t)

X(t) + X(-t)

1 /2 [X(t) + X(-t)]

X(t)



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Odd Part

Solution :-

Signal & system

X(t) - X(-t)

X(-t)

X(t)

1 /2 [X(t) + X(-t)]



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3. Even & odd Part of a signal are shown in fig draw the signal X(t)

Signal & system



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Solution:-

Signal & system



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4. Find Even & Odd parts of signal

Signal & system

X(t)



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Solution:-

Even part

Signal & system

X(t)

X(- t)

X(t) + X(-t )

1 /2 [X(t) + X(-t)]

X(t)



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Solution:-

Odd part

Signal & system

( )

X(- t)

X(t) + X(-t )

1 /2 [X(t) + X(-t)]

signal & system unit-1

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