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SIGNALS & SYSTEMS

Unit:I Mr. M. S. PatilVIIT, Pune-48

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Elementary Signals

Exponential signals

Sinusoidal signals

Exponentially damped sinusoidal signals

( ) atx t Be

( ) cos( )x t A t

( ) cos( )atx t Ae t

Sinusoidal & Exponential Signals• Sinusoids and exponentials are important in signal

and system analysis because they arise naturally inthe solutions of the differential equations.

• Sinusoidal Signals can expressed in either of twoways :cyclic frequency form- A sin 2Пfot = A sin(2П/To)tradian frequency form- A sin ωot

ωo = 2Пfo = 2П/To

To = Time Period of the Sinusoidal Wave

Sinusoidal & Exponential Signals Contd.

x(t) = A sin (2Пfot+ θ)= A sin (ωot+ θ)

x(t) = Aeat Real Exponential

= Aejω̥t = A[cos (ωot) +j sin (ωot)] Complex Exponential

θ = Phase of sinusoidal waveA = amplitude of a sinusoidal or exponential signalfo = fundamental cyclic frequency of sinusoidal signalωo = radian frequency

Sinusoidal signal

x(t) = e-at

x(t) = eαt

Real Exponential Signals and damped Sinusoidal

Discrete Time Exponential and Sinusoidal Signals

• DT signals can be defined in a manner analogous to their continuous-time counter partx[n] = A sin (2Пn/No+θ)

= A sin (2ПFon+ θ)

x[n] = an

n = the discrete timeA = amplitudeθ = phase shifting radians, No = Discrete Period of the wave

1/N0 = Fo = Ωo/2 П = Discrete Frequency

Discrete Time Sinusoidal Signal

Discrete Time Exponential Signal

Discrete Time Sinusoidal Signals

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Real exponential sequence

nanx )(

1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

n

x(n)

. . .. . .

Unit Step Function

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Step function ( ) ( )x t u t

Unit Ramp Function

, 0ramp u u

0 , 0

tt tt d t t

t

•The unit ramp function is the integral of the unit step function.•It is called the unit ramp function because for positive t, its slope is one amplitude unit per time.

Discrete Time Unit Ramp Function

, 0ramp u 1

0 , 0

n

m

n nn m

n

Unit Impulse Function

As approaches zero, g approaches a unit

step andg approaches a unit impulse

a t

t

So unit impulse function is the derivative of the unit step function or unit step is the integral of the unit impulse function

Functions that approach unit step and unit impulse

Representation of Impulse Function

The area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. An impulse with a strength of one is called a unit impulse.

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Impulse function ( ) ( )x t t

(a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of strength a that results from allowing the duration Δ of a rectangular pulse of area a to approach zero.

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Unit impulse – the shape doesn’t matter

There is nothing special about rectangular pulse approx.to the unit impulse. A triangular pulse approximation is just as good. As far as our definition is concerned both the rectangular and triangular pulse are equally good approximations. Both can act as impulses

1/

t 2

1/

t

Area = 1

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3 important properties of unit impulse

Sampling properties

000 tttxtttx

Scaling properties

abt

abat 1

The Replication Property

g(t)⊗ δ(t) = g (t)

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Exercise:

dttt )10(2

5

0

2 )10( dttt

20

0

2 )10( dttt

Evaluate the following signals:

a)

b)

c)

= 100

= 0

= 100

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Integration of the unit impulse yields the unit step function

dtut

Conversely, the impulse is the derivative of the step function

t t

Unit impulse, (t) Unit step, u(t)

1 1integration

differentiation

)()( tudtdt

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Unit impulse – what do we need it for?

The unit impulse is a valuable idealization and is used widely in science and engineering. Impulses in time are useful idealizations

impulse of current in time delivers a unit charge instantaneously to a network.

impulse of force in time delivers an instantaneous momentum to a mechanical system.

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Unit-sample sequence (n)

0001

)(nn

n

1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

n

(n)1

Sometime call (n) a discrete-time impulse; or an impulse )1()()( nunun

Fact:

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(a) Rectangular pulse x(t) of amplitude A and duration of 1 s, symmetric about the origin. (b) Representation of x(t) as the difference of two step functions of amplitude A, with one step function shifted to the left by ½ and the other shifted to the right by ½; the two shifted signals are denoted by x1(t) and x2(t), respectively. Note that x(t) = x1(t) – x2(t).

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Why it’s called singularity?

A set of functions relating through integrals and derivatives to mathematically describe signals

• using combination of singularity functions, we can form more complicated function

Singularity functions

Unit step Unit rampintegration

differentiation

Unit ramp Unit parabolicdifferentiation

integration

Unit impulse Unit stepintegration

differentiation

Signum Function

1 , 0

sgn 0 , 0 2u 11 , 0

tt t t

t

Precise Graph Commonly-Used Graph

The signum function, is closely related to the unit-step function.

Sinc Function

sinsinc

tt

t

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Basic Operations on Signals

•Operations performed on dependent signals

•Operations performed on the independent signals

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Operations performed on dependent signals

Amplitude scaling

Addition

Multiplication

Differentiation

Integration

( ) ( )y t cx t

1 2( ) ( ) ( )y t x t x t

1 2( ) ( ) ( )y t x t x t

( ) ( )dy t x tdx

( ) ( )t

y t x d

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Operations performed on the independent signals

Time scalinga>1 : compressed0<a<1 : expanded

( ) ( )y t x at

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Operations performed on the independent signals

Reflection or folding

( ) ( )y t x t

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Operations performed on the independent signals

Time shifting/ Origin Shifting

Precedence Rule for time shifting & time scaling

0( ) ( )y t x t t

( ) ( ) ( ( ))by t x at b x a ta

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The incorrect way of applying the precedence rule. (a) Signal x(t). (b) Time-scaled signal v(t) = x(2t). (c) Signal y(t) obtained by shifting v(t) = x(2t) by 3 time units, which yields y(t) = x(2(t + 3)).

The proper order in which the operations of time scaling and time shifting (a) Rectangular pulse x(t) of amplitude 1.0 and duration 2.0, symmetric about the origin. (b) Intermediate pulse v(t), representing a time-shifted version of x(t). (c) Desired signal y(t), resulting from the compression of v(t) by a factor of 2.

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• All these singularity signals can be combined to produce pulse signals

u(t) - u(t-1) r(t) - 2r(t-1)+r(t-2)

1

1

t t1 2

1

To do the above operation, you need to understand the operation on signals !

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Classification of Signals

Continuous and discrete-time signals Continuous and discrete-valued signals Even and odd signals Periodic signals, non-periodic signals Deterministic signals, random signals Causal and anticausal signals Right-handed and left-handed signals Finite and infinite length

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Classification of signals

CT or DT

Deterministic Random/stochastic

Periodic Aperiodic

Complex Transient Almostperiodic

Sinusoid

Power, Energy

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Deterministic and Random Signals

Deterministic signal: Signal which can be completely modeled as function of time.

Random signal: Signal which is processed using probability theory.

e.g. x(t) = sin(20πt) for all t

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Deterministic signals, random signals

Deterministic signals -There is no uncertainty with respect to its value

at any time. (ex) sin(3t)

Random signals - There is uncertainty before its actual

occurrence.

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Even and odd signals

Even signals : x(-t)=x(t)Odd signals : x(-t)=-x(t)

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Even and Odd CT Functions

Even Functions Odd Functions

g t g t g t g t

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Even and Odd Parts of Functions

g gThe of a function is g

2e

t tt

even part

g gThe of a function is g

2o

t tt

odd part

A function whose even part is zero is odd and a functionwhose odd part is zero is even.

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Combination of even and odd function

Function type

Sum Difference Product Quotient

Both even Even Even Even Even

Both odd Odd Odd Even Even

Even and odd

Neither Neither Odd Odd

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Two Even Functions

Products of Even and Odd Functions

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Cont…An Even Function and an Odd Function

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An Even Function and an Odd Function

Cont…

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Two Odd Functions

Cont…

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Integrals of Even and Odd Functions

0

g 2 ga a

a

t dt t dt

g 0a

a

t dt

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Periodic signals, non-periodic signals

Periodic signals - A function that satisfies the condition

x(t)=x(t+T) for all t

- Fundamental frequency : f=1/T

- Angular frequency : = 2/T

Non-periodic signals

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Periodicity Condition for DT Signal

A sequence x(n) is defined to be periodic with period N if

NNnxnx allfor )()( Example: consider njenx 0)(

)()( 0000 )( Nnxeeeenx njNjNnjnj

kN 20 0

2

k

N0

2 must be a rational

number

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How to prove Periodic or Aperiodic

To prove periodic: - for a simple sinusoid signal, use

x(t+To)=x(t)- for a sum or a product of 2 signals, find the

ratio of their periods that can be expressed as rational number. Use least common multiple method if possible.

Try these:1. x(t)=10 sin(12πt) + 4 cos (18πt)2. x(t)=10 sin(12πt) + 4 cos (18t)3. x(t)=10 cos(πt) + sin (4πt)

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Signal Energy and Power

How to express size of a signal by as single value??

Integration : Area under the signalPower: Time Average of Energy and MS

value of a signalUse: Comm-SNR, Image: MSE

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Signal Energy and Power

2x xE t dt

The signal energy of a signal x(t) is

All physical activity is mediated by a transfer of energy.

No real physical system can respond to an excitation unless it has energy.Signal energy of a signal is defined as the area under the square of the magnitude of the signal.

The units of signal energy depends on the unit of the signal.

If the signal unit is volt (V), the energy of that signal is expressed in V2.s.

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Signal Energy and PowerSome signals have infinite signal energy. In that caseit is more convenient to deal with average signal power.

/ 2

2x

/ 2

1lim xT

TT

P t dtT

The average signal power of a signal x(t) is

For a periodic signal x(t) the average signal power is

2x

1 xT

P t dtT

where T is any period of the signal.

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Signal Energy and Power

A signal with finite signal energy is called an energy signal.

A signal with infinite signal energy and finite average signal power is called a power signal.

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Energy and Power Signals

• x(t) : energy signal iif 0 < E < , so that P = 0• x(t) : power signal iif 0 < P < , so that E = • If cannot satisfy both properties: neither energy nor power signal

00

0

2

0

1 Tt

tave dttx

TPFor periodic signal :

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Energy of a Sequence Energy of a sequence is defined by

|)(| 2

n

nnxE