basic signals

Introduction to Signals & Systems

Satwik Patnaik

July 30, 2014

Satwik Patnaik

Course Structure

Lectures @ 3/week @ Wednesday & Thursday

Practicals @ 1/week

Total Credits @ 4

Theory Examination @ 100 marks

Internal Continuous Assessment @ 50 marks

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More Info...

Pre-requisite: Knowledge of Basic ElectricalEngineering & Engineering Mathematics.

Objectives: To provide knowledge of various typesof signals & systems for time & frequency domainanalysis.

To study various continuous & discrete timetransforms.

Outcomes: Recognize the various types of signals& systems.

Apply different transforms to analyze signals &systems.

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What are Signals ?

A signal is a pattern of some kind which is used to convey amessage or information.

Examples: Light, Speech, Image, Current in circuit etc.

A speech signal is an example of one-dimensional or singledimensional signal.

The reason being the signals amplitude varies with timedepending on the spoken word and the person who is speakingit.

Any examples of a multi-dimensional signal ??

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Contd...

Image is an example of a multi-dimensional signal.

The reason being we require two-coordinates, i.e., the verticaland the horizontal coordinate to describe the image explicitly.

Hence, whenever a signal is dependent on one or moreterms, we can conclude it to be a multi-dimensional signal.

Video is an another example for a multi-dimensional signal.

For this subject, we will be restricting ourselves strictly toone-dimensional signals only.

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Single-Valued Signals

As the name suggests, single-valued signals are those signalsthat have an unique value for a particular instant of time.

That value can either be Real or Complex which means thatthe definition of real valued signal and complex valued signalsfollows it.

In either case, the independent variable, time (t) is a realvalued quantity only !!

We will now look at the classification of the signals in thenext slide...

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Classification of Signals...

Continuous & Discrete Signals.

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Analog & Digital Signals.

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Periodic & Non-periodic Signals.

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Even & Odd Signals.

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Even & Odd Signals.

Energy & Power Signals.

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Even & Odd Signals.

Energy & Power Signals.

Deterministic & Random Signals.

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Continuous & Discrete Signals

A signal x(t) is a continuous-time signal if t is a continuousvariable.

If t is a discrete variable, that is, x(t) is defined at discretetime intervals, then x(t) is a discrete-time signal.

Since, a discrete-time signal is defined only at discrete times,a discrete-time signal is often identified as a sequence ofnumbers, denoted by x[n], where n = integer.

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A Continuous Signal

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A Discrete Signal

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Contd...

Throughout this subject, the letter t will be used forcontinuous signals & n will be used for discrete signals.

Also, parantheses (.) will be used to denote continuous signalsand brackets [.] will be used to denote discrete signals.

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Even & Odd Signals

A signal x(t) or x[n] is referred to as an EVEN signal if:

x(-t) = x(t) & x[-n] = x[n]

A signal x(t) or x[n] is referred to as an ODD signal if:

x(-t) = - x(t) & x[-n] = - x[n]

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Contd...

Even signals are symmetric about the vertical axis ortime origin & Odd signals are anti-symmetric about thetime origin.

One can develop the even/odd decomposition for any generalsignal x(t) or x[n] by using some general decompositionrelations.

The relations are shown in the next slide...

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Contd...

Any continuous or discrete time signal can be expressed as thesummation of even and odd parts.

x(t) = xe(t) + xo(t) (1)

Substituting t as -t in above equation, it follows,

x(t) = xe(t) + xo(t) (2)

Adding (1) & (2), we get,

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Contd...

x(t) + x(t) = xe(t) + xo(t) + xe(t) + xo(t) (3)

As per definition, for an even signal;

xe(t) = xe(t) (4)

As per definition, for an odd signal;

xo(t) = xo(t) (5)

Substituting (4) & (5) in (3);

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Contd...

x(t) + x(t) = xe(t) + xo(t) + xe(t) xo(t) (6)

x(t) + x(t) = 2xe(t) (7)

xe(t) =1

2[x(t) + x(t)] (8)

Similarly, the expression for odd part of the signal can becalculated.

xo(t) =1

2[x(t) x(t)] (9)

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Illustration of an Even Signal

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Illustration of an Odd Signal

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Some Results & Points

When a signal is even, then its odd part is zero.

When a signal is odd, then its even part is zero.

Product of two odd signals will be an even signal.

Product of two even signals will be an even signal.

Product of an even & odd signal will be an odd signal.

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Periodic & Non-periodic Signals

A signal that has a definite pattern and repeats itself againand again over a certain period of time is defined as aperiodic signal.

A signal can be termed as periodic if it satisfies the givenrelation:

x (t+T)= x(t) (10)

T denotes the fundamental time period of the signal.

Inverse of T is called the fundamental frequency F, unitsbeing cycles/seconds or Hz.

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Contd...

Non-periodic signals are those signals that do not have aspecific pattern or do not repeat themselves over a given timeinterval.

One can identify the nature of periodicity of any signal eitherby plotting the signal or by mathematical analysis.

Examples of periodic signals and non-periodic signals ??

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Illustration of a Periodic Signal

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Calculate time period & fundamental frequency ?

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Continuous Periodic Signal

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Discrete Periodic Signal

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Contd...

When a continuous time signal is a mixture of two periodicsignals with fundamental time periods T1 & T2, then theresulting signal is periodic only if the ratio of T1/T2 is arational number.

Lets check a few functions and comment about its periodicity.

2 cos(T

4) (11)

on comparing with the general form,

A cos(2f0T ) (12)

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Contd...

2f0 =1

4(13)

f0 =1

8(14)

period = T =1

f0= 8 (15)

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Contd...

x(t) = 3 cos(5t +

6) (16)

x(t + T ) = 3 cos(5(t + T ) +

6) (17)

3 cos(5t + 5T +

6) (18)

3 cos((5t +

6) + 5T ) (19)

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Energy and Power signals

Signals having finite energy are called energy signals.

Non-periodic signals like exponential signals will have constantenergy and hence, non-periodic signals are energy signals.

Signals having finite average power are called power signals.

Periodic signals like sinusoidal waves/signals have constantpower and hence, periodic signals are power signals.

In electrical systems, a signal may represent a voltage or acurrent. Now, consider a voltage v(t) is developed across aresisitor R carrying current i(t).

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Contd...

Instantaneous power developed across the resistor is:

p(t) =v2(t)

R(20)

Or equivalently,p(t) = R .i2(t) (21)

Assuming R= 1, both equations converge to the type asfollows:

p(t) = x2(t) (22)

It is important to note that, in signal analysis, it is customaryto keep R=1, so that the regardless whether the signal x(t)represents a voltage or current signal, the power can beexpressed as the form given above.

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Contd...

We now define the total energy of a continuous signal x(t)as:

E = limT

T

2

T

2

x2(t)dt (23)

E =

+

x2(t)dt (24)

Average power is given by:

P = limT

1

T

T

2

T

2

x2(t)dt (25)

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Contd...

Equation (25) can be simplified to:

P =1

T

T

2

T

2

x2(t)dt (26)

The square root of the average power P, gives the root meansquare (RMS) value of signal x(t).

In case for discrete time signals, x[n], the integrals in theabove equations are replaced by corresoponding sums.

Total energy of x[n] is:

E =

n=+

n=

x2[n] (27)

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Contd...

Average power for discrete signals x[n] is defined as:

P= limN1

2N

N

n=Nx2[n] (28)

If a signal x[n] is periodic with fundamental period N, then:

P =1

N

N1

n=0

x2[n] (29)

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Contd...

A signal can be defined as Energy signal if it satisfies:

0 < E < (30)

A signal can be defined as Power signal if it satisfies:

0 < P < (31)

A signal can never have both finite energy & finite power.

A signal can either be an energy signal or a power signal.

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An Important Result

For an Energy signal, the Power is infinity. ()

The energy of a signal is defined as:

E = limT

T

T

|x(t)|2dt (32)

The power of a signal is defined as:

P = limT

1

2T

+T

T

|x(t)|2dt (33)

limT

1

2Tlim

T

+T

T

|x(t)|2dt (34)

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Contd...

P = limT

1

2T.E (35)

Since, for energy signals, E= constant, equation (35)becomes;

P = E . limT

1

2T(36)

E .1

2.(37)

P = E .0 = 0 (38)

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Contd...

Prove that for a Power signal, Energy = !! (Due nextclass !!)

There are certain signals that are neither energy nor powersignals, example being Ramp Signal.

All practical signals have finite energies and are thereforeenergy signals.

It is impossible to generate a true power signal in practice assuch a signal would have infinte duration & infinite energy.

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Deterministic & Random Signals

A signal whose physical description is known completely,either in a mathematical form or graphical form is adeterministic signal.

A deterministic signal has no uncertainity w.r.t. its value atany insant of time.

They can be modelled in terms of equations or mathematicalrelations.

On the other hand, random signals cannot be predictedprecisely but are known only in terms of probabilisticdescriptions.

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Contd...

We will exclusively deal in deterministic signals, howeverrandom signals evoke a special interest in many real timesystems and its study is very important.

Noise is an example of a random signal.

An EEG signal is an another example.

Next, we will see some useful signal models followed by basicoperations on signals...

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Signal Models

In the area of signals and systems, the step, impulse and theexponential functions are very useful.

They serve as basic building block or help in representingcomplex signals.

These signals may also be used to model many pther physicalsystems.

We will first look at the simplest of them, being the unit stepsignal...

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Unit Step Signal u(t)

The unit step function (signal) is denoted as u(t) and alsoknown as Heaviside unit function and defined as:

u(t) =

{

1 t > 00 t < 0

(39)

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Contd...

The function is discontinuous at t=0 and hence the value ofsignal at t=0 instant is not defined.

Shifted unit step function u(t-t0) is hence defined as:

u(t t0) =

{

1 t > t00 t < t0

(40)

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Contd...

u(t 2) =

{

1 t 2 > 00 t 2 < 0

(41)

u(t 2) =

{

1 t > 20 t < 2

(42)

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Some Useful Signal Operations

Time Reversal y(t) = x(-t).

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Time Shifting y(t) = x(t-td ).

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Amplitude Scaling y(t) = Bx(t).

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Addition y(t) = x1(t) + x2(t).

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Multiplication y(t) = x1(t) . x2(t).

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Multiplication y(t) = x1(t) . x2(t).

Time scaling y(t) = x(at).

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Time Reversal

It flips the signal along the y-axis.

y(t) = x(-t)

Let x(t) = u(t) and lets perform its time reversal.

u(a) =

{

1 a 00 a < 0

(43)

Putting a = -t,

u(t) =

{

1 t 00 t > 0

(44)

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Time Reversal of u(t)

Hence, the plot of u(-t) is represented as:

u(t) =

{

1 t 00 t > 0

(45)

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Time Shifting/Delay

y(t) = x(t-td )

Shifts the signal either left or right.

Shifts the origin of the signal to td .

Rule Set (t-td) = 0 (set the argument equal to zero)

Then move the origin of x(t) to td Effectively, y(t) equalswhat x(t) was td seconds ago.

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Time Shifting/Delay Contd...

eg. Sketch y(t) = u(t-2)

y(a) =

{

1 a 00 a < 0

(46)

Substituting a with t-2;

y(t 2) =

{

1 t 2 00 t 2 < 0

(47)

y(t 2) =

{

1 t 20 t < 2

(48)

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Time Shifting Contd..

Hence, the plot of y(t-2) is represented as:

y(t 2) =

{

1 t 20 t < 2

(49)

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Amplitude Scaling

Multiply the entire signal by a constant value.

y(t) = Bx(t)

eg. Sketch y(t) = 5u(t)

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Addition of two signals

Point-by-point systematic addition of multiple signals.

Move from left to right (or vice-versa) and add the value ofeach signal together to achieve the final signal.

y(t) = x1(t) + x2(t)

Graphical solution Plot each individual portion of thesignal (break into parts) and then Add the signals point bypoint.

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Addition of two signals Contd...

eg. y(t) = u(t) - u(t - 2)

First, plot each of the portions of this signal separately...

x1(t) = u(t) Simply a step signal.

x2(t) = -u(t-2) Delayed step signal, multiplied by -1.

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Addition of two signals Contd...

The signal when added with each other properly looks like:

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Multiplication of two signals

Point-by-point multiplication of the values of each signal.

y(t) = x1(t).x2(t)

Graphical solution Plot each individual portion of thesignal (break into parts) and then Multiply the signalsvalue point by point.

eg. y(t) = u(t) . u(t - 2)

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Multiplication of two signals Contd...

The plot is:

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Time Scaling

Speed up or slow down a signal.

Multiply the time in the argument by a constant.

y(t) = x(at)

|a| > 1 Speed up x(t) by a factor of a

|a| < 1 Slow down x(t) by a factor of a

Key Replace all instances of t with at.

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Time Scaling Contd...

eg. Let x(t) = u(t) - u(t-2), Sketch y(t) = x(2t)

First plot x(t);

Replace all t with 2t

y(t) = x(2t) = u(2t)-u(2t-2)

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Time Scaling Contd...

u(2t) turns on at 2t 0 t 0 No change !!

u(2t-2) turns on at 2t-2 0 t 1

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Sketch y(t) = x(t/2)

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Combinations of Operations

Easier to determine the final signal in stages.

Create intermediary signals in which one operation isperformed.

We will now see an example wherein time shifting and timescaling are performed concurrently...

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Combinations of Operations Contd...

Let x(t) = u(t + 2) - u(t-4), then Sketch y(t) = x(2t- 2)

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Combinations of Operations Contd...

Method 2: (Scale & then Shift)

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Unit Impulse Signal (t)

The unit impulse is most commonly denoted as (t)

It possesses the following properties:

(t) =

{

0 t 6= 0 t = 0

(50)

+

(t) = 1 (51)

Area/strength of impulse signal is unity.

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Key Properties of Impulse function

Very important properties are:

(t) = (t) (52)

(at) =1

|a|(t) (53)

x(t)(t t0) = x(t0)(t t0) (54)

t2

t1

x(t).(t t0)dt = x(t0); t1 < t0 < t2 (55)

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Contd...

Figure: Unit Impulse function

Figure: Unit Impulse delayed function

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Contd...

Figure: Unit Impulse function (Discrete)

Figure: Unit Impulse delayed function (Discrete)

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Contd...

Figure: Step function (Discrete)

Figure: Step delayed function (Discrete)

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Exponentially Increasing Signal

Figure: Exponentially Increasing Signal

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Exponentially Decreasing Signal

Figure: Exponentially Decreasing Signal

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Exponentially Increasing Signal (Discrete)

Figure: Exponentially Increasing Signal (Discrete)

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Exponentially Decreasing Signal (Discrete)

Figure: Exponentially Decreasing Signal (Discrete)

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A Question !!

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Contd...

For the signal shown in the above slide, draw the followingsignals:

x(t-2)

x(2t)

x(t/2)

x(-t)

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x(t-2)

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x(2t)

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x(t/2)

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x(-t)

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Consider the 2 signals given below:

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x1[n] + x2[n]

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2x1[n]

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x1[n].x2[n]

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Consider the signal given below and sketch its even & odd

part ?

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Answer

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part ?

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Answer

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part ?

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Answer

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End of Module 1

Thank You.

Any Questions ???

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basic signals

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