handout_1_short.pdf

EE 312 Signals and Systems

Handout #1

Signals: Concepts & Properties

Prof. Mohamed Zribi

Updated 7 September 2015

EE312 Signals and Systems Dr. Mohamed Zribi 1

Outline

I. Definition of Signals

II. Classifications of Signals

III. Important CT Signals

IV. Important DT Signals

V. Operations on signals


I. Definition of Signals


o A signal is a function of time representing a physical

variable, e.g. voltage, current, spring displacement, share

market prices, number of student asleep in the Lab, cash in

the bank account.

o Typically we will use a mathematical function such as f(t),

u(t) or y(t) to describe a signal which is a continuous

function of time.

EE312 Signals and Systems Dr. Mohamed Zribi

What is a Signal?

4

Remarks about Signals

o Usually, a signal is a function of an independent variableExample 1: Daily high temperature measured over a month

o Continuous-time signals are functions of a real argumentx(t) where t can take any real valuex(t) may be 0 for a given range of values of t

o Discrete-time signals are functions of an argument that takes values from a discrete setx[n] where n {...-3,-2,-1,0,1,2,3...}Often, we use “index” instead of “time” for discrete-time

signals

o Values for x may be real or complex


II. Classification of Signals


Classification of Signals

1. One-dimensional and Multi-dimensional Signals2. Continuous-time and discrete-time Signals3. Analog and digital Signals4. Deterministic and Random Signals5. Periodic and Aperiodic Signals6. Causal and Anti-causal vs. Non-causal Signals7. Even and Odd Signals8. Finite and Infinite Length Signals9. Energy of a Signal 10. Power of a Signal11. Energy signals and power signals


1.One-dimensional vs. Multi-dimensional

A signal can be a function of a single variable or a function of multiple variables.

Example 2:

Speech varies as a function of time one-dimensional

Image intensity varies as a function of (x , y) coordinates

multi-dimensional

In this course, we will focus on one-dimensional

signals.


A signal is continuous time if it is defined for all

time, x(t).

A signal is discrete time if it is defined only at

discrete instants of time, x[n].

A discrete time signal can be derived from a

continuous time signal through sampling, i.e.:

s[ ] ( ), T is the sampling periodd c sx n x nT

2. Continuous-time vs. Discrete-time


Continuous-Time (CT) Signals

Most signals in the real world are continuous time.

E.g. voltage, velocity,

A CT signal is denoted by x (t), where the time interval

may be bounded (finite) or infinite


x(t)

t

10

Discrete-Time (DT) Signals

Some real world and many digital signals are discrete time, as

they are sampled. For example, pixels, daily stock price

(anything that a digital computer processes)

A DT signal is denoted by x[n], where n is an integer value that

varies discretely.


x[n]

n

11

Continuous-Time vs. Discrete-Time

Graphically,

It is meaningless to say 1.5 sample of a DT signal because it is

not defined.

t

)(tx

0 4-3 1 3

][nx

0

]2[x ]1[x]2[x ...

2 ...-1-2...

]0[x

]3[x

]1[x

...

n


Continuous time signal

xa(t)

Discrete time signal (sequence)

x[n]

x[n] = xa(nT ) T : sampling periodfs = 1/T : sampling rate


Examples of Signals

Graphical Representation of a DT is shown below:

Example 3:



Example 4:Graphical Representation of a DT is shown below:

15

A discrete-time signal is usually (not always) obtained from sampling acontinuous signal at a regular time period known as the sampling period,which we will represent by the parameter T.

Thus the discrete-time signal is equal in amplitude to the continuoussignal at the sampling instants.

A discrete-time signal is essentially a sequence of numbers (0,1,2, ...)where each of those numbers represents the amplitude of a continuous-time signal at a time equal to kT.

Note that often we neglect to put in the sampling period and write thesignal simple as a function of the sample number k.


Discrete-time Signals

kTk ff cd

16

Sampling of an analog signal is shown below:


Discrete-time Signals

17

Converting between a continuous and discrete signal.

Consider the continuous-time signal such as:

The sampling interval T = 0.1 seconds. Start sampling at t = 0


Example 5:

0

1*

2*

3*

0.1**

[0] 1.0000

[1] 0.9048

[2] 0.8187

[3] 0.7408

[ ]

dT

dT

dT

d

kk Td

x ex ex ex e

x k e e

tcx t e

18

We can also write:

Either method gives

i.e. a sequence of number, known as the discrete-time signal.

Note that the values of this sequence depend on the sampling period.

If the sampling interval T were doubled then :

Thus a single continuous function can yield an infinite number of sampled sequences, depending on the value chosen for T.


* 0.9048k Td

k kTx k e e

1.0000, 0.9048, 0.8187, 0.7408,dx k

1.0000, 0.8187,dx k

19

Continuous-Time vs. Discrete Time

To distinguish CT and DT signals, t is used to denote CT and the

independent variable in (.). Also, n is used to denote DT

independent variable in [.]

Continuous x(t), t is real

Discrete x[n], n is integer Signals can be represented in mathematical form. For example, x(t) = et, x[n] = n/2

Discrete signals can also be represented as sequences: {y[n]} = {…,1,0,1,0,1,0,1,0,1,0,…}

550

)( 2 ,t,t

tty


3. Analog vs. Digital Signals

The difference between analog and

digital signals is with respect to the

value of the function (y-axis).

An analog signal corresponds to a

continuous y-axis, while a digital

signal corresponds to a discrete y-

axis.


A signal whose amplitude can take on any value in a

continuous range is an analog signal.

A digital signal is one whose amplitude can take on only a

finite number of values. For example, Binary signals are

digital signals.

An analog signal can be converted into a digital signal

through quantization.

Analog vs. Digital Signals


1

-1

Analog vs. Digital Signals

The amplitude of an analog signal can take any real or

complex value at each time/sample

Analog

The amplitude of a digital signal takes values from a discrete

set

Digital


Digital vs. Analog Signals





Examples of analog technology photocopiers telephones audio tapes televisions (intensity and color info per scan line) VCRs (same as TV)

Examples of digital technology Digital computers!


Analog and Digital Signals


Deterministic signal is a signal in which each value of the

signal is fixed and can be determined by a mathematical

expression, rule, or table. Because of this the future values of the

signal can be calculated from past values with complete

confidence.

Random signal has a lot of uncertainty about its behavior. The

future values of a random signal cannot be accurately predicted

and can usually only be guessed based on the averages of sets of

signals

4. Deterministic vs. Random Signals


A deterministic signal is known for all time and can bepredicted in advance exactly, e.g. a sine-wave with knownphase.

A random signal cannot be predicted exactly, e.g. weathertemperatures.

Random signals are usually dealt with by statistical rather thananalytical techniques.

Noise is simply a signal we don't want. Sometime it iscompletely random, but on occasions it can be someone else'ssignal, such as cross-talk on a telephone line. So, since we don'twant it we call it noise.


In other words,

29

Example 6: Deterministic vs. Random Signals

Deterministic

Random


Deterministic signal (An example is sin wave, square wave)

Stochastic signal (An example is noise signal or human voice)

Example 7: Deterministic vs. Random Signals


Examples of Signals

DeterministicSignal

.

.

.

.RandomSignal



Signal with noise

Deterministic Signal

Signal with noise

Examples of Signals

A periodic signal x(t) is a function of time that satisfies

The smallest T, that satisfies this relationship is called the

fundamental period. is called the frequency of the signal (Hz).

Angular frequency, (radians/sec).

A signal is either periodic or aperiodic.

A periodic signal must continue forever.

A non-periodic signal is called aperiodic.

Tf 1

Tf 22

5. Periodic and Aperiodic Signals

0, )()( TtTtxtx


Example 8: The voltage at an AC power source is periodic.

Remark:0 0

0

( ) ( ) ( )a T b T

a b T

v t dt v t dt v t dt


Example 9: Periodic and Aperiodic signals

Periodic

Aperiodic


Periodic

Periodic

Example 10: Periodic and Aperiodic signals


Example 11: An example of a CT periodic signal is:

If x(t) is periodic with T then

Thus, x(t) is also periodic with 2T, 3T, 4T, ...

The fundamental period T0 of x(t) is the smallest value of T for which holds.

)( tx

0 TTT2 T2

ZnnTtxtx for )()(


Periodic

38

cos (t +2) = cos (t)

sin (t +2) = sin (t)

These two signals are both periodic with period 2

2


Periodic

Example 12: Periodic signals

39

5/2,....2,1,1,2...,25

)(5sin)(5cos)()(

5sin5cos)( 5

TkkT

TtTtjTtxtx

ttjjetx tj

This is periodic if there exists T > 0 such that:

This is true for sinusoidal signals:

Signal is periodic and fundamental period is

T = 5/2


Example 13: Periodic signals

40

Sum of Periodic Signals

x(t) = x1(t) + x2(t)

x(t+T) = x1(t+n1T1) + x2(t+n2T2)

n1T1=n2T2 = To =Fundamental period

Example 14:

cos(t/3)+sin(t/4)

T1=(2)/(/3)=6; T2 =(2)/(/4)=8;

T1/T2=6/8 = ¾

n1T1=n2T2 = 6*4 = 3*8 = 24 = ToEE312 Signals and Systems Dr. Mohamed Zribi 41

Example 15: Sum of periodic Signals

x1(t) = cos(3.5t)

x2(t) = sin(2t)

x3(t) = 2cos(7t/6)

Is v(t) = x1 (t) + x2(t) + x3(t) periodic?

What is the fundamental period of v(t)?


x1(t) = cos(3.5t) f1 = 3.5/2 T1 = 2 /3.5 x2(t) = sin(2t) f2 = 2/2 T2 = 2 /2 x3(t) = 2cos(t7/6) f3 = (7/6)/2 T3 = 2 /(7/6) T1/T2 = 4/7 Ratio or two integers T1/T3 = 1/3 Ratio or two integers Summation is periodic

n1T1 = n2T2 = n3T3 = To ; Hence we find To The question is how to choose m1, m2, m3 such that the above

relationship holds We know: 7(T1) = 4(T2) and 3(T1) = 1(T3) ; Hence:

21(T1) = 12(T2)= 7(T3); Thus, fundamental period: To = 21(T1) = 21(2 /3.5)=12(T2)=12


Example 16: Sum of periodic Signals – may not always be periodic!

T1=2/= 2; T2 =2/ T1/T2= Note: T1/T2 = is an irrational number

x(t) is aperiodic

tttxtxtx 2sincos)()()( 21

2

2

2


Example 17: Sum of periodic Signals – may not always be periodic!

T1=2/= 2; T2 = 1 There is no common factor between T1 and T2

x(t) is aperiodic

1 2( ) ( ) ( ) cos sin 2x t x t x t t t


Product of Periodic Signals

x(t) = xa(t) * xb(t)

It can be checked that

x(t) = sin(t/3)+sin(t/4)

Thus, To=24.


7If ( ) 2sin( ) cos( )24 24

x t t t

since 2sin( ) cos( ) sin( ) sin( )a b a b a b

46

Discrete Time Periodic Signals

A discrete time signal x[n] is periodic with period N if and only if

][][ Nnxnx for all n .

Definition:

N

][nx

n

Meaning: a periodic signal keeps repeating itself forever!


Periodic Signals for DT Signals

For DT we must have

Here the smallest N can be 1

The smallest positive value of N is N0 which is called the fundamental period

0, ][][ NnnxNnx

Period must beinteger!


A signal is periodic if repeats after T values:

x [n] = x [n+ N0] = x [n+2 N0] = x [n+3 N0] = …

N0 is the period of the signal

Periodic Signals for DT Signals


[ ] 2cos 0.2 0.9x n n

Consider the Sinusoid:

It is periodic with period since 10N

][29.02.0cos2

9.0)10(2.0cos2]10[nxn

nnx

for all n.


Example 18:

General Periodic Sinusoid

n

NkAnx 2cos][

Consider a Sinusoid of the form:

It is periodic with period N since

][22cos

)(2cos][

nxknNkA

NnNkANnx

for all n.

with k, N integers.


1.03.0cos5][ nnxConsider the sinusoid:

It is periodic with period since 20N

][231.03.0cos5

1.0)20(3.0cos5]20[nxn

nnx

for all n.

We can write it as:

1.0

2032cos5][ nnx


Example 19:

nN

kjAenx

2

][

Consider a Complex Exponential of the form:

for all n.

It is periodic with period N since

Periodic Complex Exponentials

][

][

22

)(2

nxeAe

AeNnx

jkn

Nkj

NnN

kj

1


njejnx 1.0)21(][

Consider the Complex Exponential:

We can write it as

njejnx

2012

)21(][

and it is periodic with period N = 20.


Example 20:

Example 21: Periodic Signals

727/2

,....2,1,1,2...,27

)(7sin)(7cos][][

)7sin()7cos(][ 7

kNkN

kkN

NnjNnNnxnx

njnenx nj

This is periodic if there exists N>0 such that :

This is true for sinusoidal signals:

Signal is periodic and fundamental period is N=2


Periodicity Properties of DT Signals

(*) :conditiony Periodicit 000

itymust be un

NjωnjωN)(njω eee

!!otherwise! periodicnot number, rational a is 2

when periodic is exp DT So

integers. bemust and that (**) and (*) from conditions thehave We

0

Nm

Nm

*)*(* 2

ly equivalentOr

2 havemust weminteger somefor s,other wordIn (**) .2 of multipleinteger an is if holds This

0

0

0

Nm

mNN


Periodicity Properties of DT Signals

mN02 then isfrequency lfundamenta The

outfactor common theTake

**)*(* 2 then is period lfundamenta The0

mN

!signals.)! sinusoidal DTfor validalso ist developmen same (The

*)*(*in as 2

express toneed weexp.complex a of freq. fund. thefind toTherefore 0


Example 22: Periodic Signals

2 nIf x[n] cos( )12

4 nIf x[n] cos( ) 12

00 0

2 12 n x[n] cos( ) cos( n) no factors in common,12 12 2 1212so by using (****) , N 1 12 periodic with fund period 12.1

00 0

4 2 14 n x[n] cos( ) cos( n) ,12 12 2 12 612then using (****) , N 1 6 is periodic with fundamental period 6.2


Comparison of Periodicity of CT and DT Signals

Consider x(t) and x[n]

x(t) is periodic with T=12,

x[n] is periodic with N=12.

2πt 2πn (t) cos ( ) [n] cos ( ) 12 12

x x


if and

x(t) is periodic with 31/4.

In DT, we can’t have fractional periods. Thus for x[n] we have

then N=31.

If and

x(t) is periodic with 12, but x[n] is not periodic, because there is no way to express it as in (***)

31t8cos)(

tx 318cos][ nnx

314

20

)61cos()( ttx )

61cos(][ nnx

121

20

Comparison of Periodicity of CT and DT Signals


6. Causal vs. Anti-causal vs. Non-causal

• Causal signals are signals that are zero for all negative time,

• Anticausal are signals that are zero for all positive time.

• Noncausal signals are signals that have nonzero values in both

positive and negative time


In other words: A signal that does not start before t=0 is a causal signal.

x(t)=0, t < 0

A signal that starts before t=0 is a noncausal signal.

A signal that is zero for t > 0 is called an anticausal signal.


Example 23: Causal vs. Anti-causal vs. Non-causal


causal signal

Anti-causal signal

Non-causal signal

0

63

Example 24:


7. Even and Odd Signals

If even signal (symmetric wrt y-axis)

If odd signal (symmetric wrt origin)

][][or )()( nxnxtxtx

][][or )()( nxnxtxtx

odd

t

x(t)

t

evenx(t)


Even and Odd Signals

)()( txtx


An even signal is identical to its time reversed signal, i.e. it can be reflected in the origin and is equal to the original:

Example 25:x(t) = cos(t)

x(t) = c

An odd signal is identical to its negated, time reversed signal, i.e. it is

equal to the negative reflected signal

Example 26:

x(t) = sin(t)

x(t) = t

Even and Odd Signals

)()( txtx


Example 27: Even and Odd Signals


even

odd

68

xe(t) = xe(-t) and xo(t) = - xo(-t)

evenodd

Example 28: Even and Odd Signals


Even and Odd CT FunctionsEven Functions Odd Functions

g t g t g t g t


g gg

2e

n nn

g g

g2o

n nn

g gn n g gn n

Even and Odd DT Functions


Combination of Even and Odd Functions

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

EE312 Signals and Systems Dr. Mohamed Zribi72

Even vs. Odd Signals

• Prove that product of two even signals is an even signal.

• Prove that product of two odd signals is an even signal.

• What is the product of an even signal and an odd signal?

)()()()()()(

)()()(

21

21

21

txtxtxtxtxtx

txtxtx

Eventxtxtxtx

txtxtxtxtxtx

)()()()(

)()()()()()(

21

21

21


Two Even Functions

Example 29: Products of Even and Odd Functions


An Even Function and an Odd Function






Two Odd Functions



Two Even Functions






Two Odd Functions



Function type and the types of

derivatives and integrals

Function type Derivative Integral

Even Odd Odd + constant

Odd Even Even

EE312 Signals and Systems Dr. Mohamed Zribi81

Integrals of Even and Odd Functions

0

g 2 ga a

a

t dt t dt

g 0a

a

t dt


Accumulation of Even and Odd Functions

1

g g 0 2 gN N

n N nn n

g 0N

n Nn


A signal can be even, odd or neither.

Any signal x(t) can be written as a combination of an even and odd signals.

1( ) ( ) ( )2

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

e

e o

o

x t x t x t

x t x t x t

x t x t x t


Decomposition of signals to Even and Odd Signals

84

Even and Odd Parts of Functions

1The of a signal is 2ex t x t x t even part

1The of a signal is 2ox t x t x t odd part

A signal whose even part is zero is odd and a signal

whose odd part is zero is even.


))()((21)( txtxtxo ))()((

21)( txtxtxe

Example 36:

Even part

Odd part


Given:

Example 37: Even-Odd Signals


Symmetric across the vertical axis Anti-symmetric across the vertical axis


Given x(t) find xe(t) and xo(t) x(t)

5

4___

5

2___

5

2___

-5-5



( )ex t( )ox t

89

Given x(t) find xe(t) and xo(t)4___

5

2___

5

2___

4e-0.5t

2___2e-0.5t

-2___

2___2e-0.5t2e+0.5t-2e+0.5t



( )ex t

( )ox t

( )x t

90



The even part of f(t) is as follows:

91 EE312 Signals and Systems Dr. Mohamed Zribi

The odd part of f(t) is as follows:

92


The addition of the even part and the odd part gives f(t)

93

• x(t) is a finite length signal if it is nonzero over a

finite interval a < t < b

• x(t) is infinite length signal if it is nonzero over all

real numbers.

• Remark: Periodic signals are infinite length.

8. Finite vs. Infinite Length Signals


In other words,

• f (t) is a finite-length signal if it is nonzero over a finite

interval t1 < f (t) < t2

• Infinite-length signal, f (t), is defined as nonzero over all real numbers:


9. Energy of Signals

2x xE t dt

The energy of a signal x 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.

2xand, xE n


9a. Energy of Signals (over finite time intervals)

Total energy of a CT signal x(t) over [t1, t2] is:

where |.| denote the magnitude of the (complex) number/signal.

Total energy of a DT signal x[n] over [n1, n2]:

2

1

2)(t

tdttxE

2

1

2][n

nnnxE


9b. Energy of Signals (over infinite time intervals)

For many signals, we’re interested in examining the energy over

an infinite time interval (-∞, ∞):

If the sums or integrals do not converge, the signal energy is

infinite.

dttxdttxET

TT22 )()(lim

n

N

NnN nxnxE 22 ][][lim


For DT signal

For CT signal

98

Example 41: Energy of Signals

2

2 2 2

4 4

00

( ) ( )

| ( ) | ( ( ))

1/ 4

1/ 4

t

t

t t

x t e u t

E x t dt e u t dt

e dt e


1 12 2 3

00

( ) [ ( ) ( 1)]

| ( ) | 1/ 3 1/ 3

x t t u t u t

E x t dt t dt t


Example 42: Energy of Signals

100

Example 43: Determine the energy of this signal?

EE312 Signals and Systems Dr. Mohamed Zribi 101 EE312 Signals and Systems Dr. Mohamed Zribi

Example 44: Energy of a Signal

102



103


2

2

0

[ ] (1/ 2) [ ]

| [ ] |

((1/ 2) [ ])

(1/ 4)

1/ (1 1/ 4)4 / 3

n

n

nn

nn

x n u n

E x n

u n


By dividing the energy by (t2-t1) and (n2-n1+1), respectively, gives

the average power over finite time intervals,

2

112

2

112

2)1(

1

2)(

1

][

)(

n

nnnn

t

ttt

nxP

dttxP


For DT signal

For CT signal

105

10. Power of SignalsPower of Signals

(over infinite time intervals)

The corresponding power is:

TEdttx

TP

T

T

TT 2

lim)(21lim 2

12lim][

121lim 2

NEnx

NP

N

N

NnN


For CT signal

For DT signal

106

Example 47: Energy and Power of Signals

2

2 2 2

4 4

00

212

1 12 4

( ) ( )

| ( ) | ( ( ))

1/ 4

1/ 4

lim | ( ) |

lim *0

t

t

t t

T

T T T

T T

x t e u t

E x t dt e u t dt

e dt e

P x t dt


Example 48: Determine the suitable measure for this signal?

The signal (b) does not approach 0 as |t| and it is a periodic wave, therefore use the power equation where g2 is replaced with t2.

.


Example 49: Determine the power

•Periodic signal with• Suitable measure of size is power

0 2 / oT

- First term on the right hand side equals C2/2- Second term is zero –integral appearing in this term is area under a sinusoid.-Area is at most the area of half cycle – positive and negative portion cancels-A sinusoid of amplitude C has a power of C2/2 regardless of angular frequency


g (t) = C1 cos (1t + 1) + C2 cos (2t + 2) 1 ≠ 2

This signal is the sum of two sinusoid signals. Therefore, use the power equation. Therefore, Pg = (C1

2 / 2) + (C22 / 2)

This Can be generalized

Example 50: Determine the power


Example 51: What is the suitable measure for this signal?

g (t) = Dejt

The signal is complex and periodic. Therefore, use the power equation averaged over T0.

|ejt| = 1 so that |Dejt|2 = |D|2 and


Example 52: Power of a Signal

212

212

212

( ) 5 10cos(100 / 3)

lim | ( ) |

lim (5 10cos(100 / 3))

1 1lim 25 100cos (100 / 3) 100[ cos(200 2 / 3)]2 2

25 50 75

T

T T TT

T T T

T

T T T

x t t

P x t dt

t dt

t t dt



21

)]2cos(21

21[lim

)(coslim

)cos()cos(lim

|)(|lim

)cos()(

21

221

21

221

T

T oTT

T

T oTT

T

T otj

otj

TT

T

TTT

otj

dtt

dtt

dttete

dttxP

tetx



0*lim

|][|lim

3/4)4/11/(1

)4/1(

])[)2/1((

|][|

][)2/1(][

34

121

212

1

0

2

2

NN

N

NnNN

nn

nn

n

n

nxP

nu

nxE

nunx


Power in a Sine wave:

where P is period of the sine wave.

The period can be taken:

Usually we take the former as it often makes the mathematics easier.

2sin ty t AT

2/2

/2

/2/22 2 2

/2/2

1/ 2 / 2

41 cos2 2 2

2sin

4sin4

T

T

TT

TT

P dtT T

A t A AP dtT T T T

tAT

T ttT

1 2 1 2from / 2 to / 2 or from 0 to t T t T t t T



Example 56: Exponential and Sinusoidal Signals

Complex periodic exponential and sinusoidal signals are of

infinite total energy but finite average power

As the upper limit of the integrand is increased as

However, always Thus,

0

02

TT

T

tjperiod dteE

periodE

1periodP

T

T

tj

Tdte

TP 1

21lim

20

Finite average power!

1)(

1

0

periodperiod ETTT

P

0

1TT

T

dt 00 )( TTTT


• A signal with finite signal energy (0< E <∞) is

called an energy signal.

• A signal with infinite signal energy and finite

average signal power (0< P< ∞) is called a

power signal.

•An energy signal has zero power.

•A power signal has infinite energy.

11. Energy Signals and Power Signals


Energy Signals and Power Signals

• Energy signals have finite energy. All energy signals

decay to zero as |t| .

• Power signals have finite and non-zero power. All

periodic signals are power signals.


• Periodic signals and random signals are usually

power signals.

• Signals that are both deterministic and aperiodic

are usually energy signals.

• Finite length and finite amplitude signals are

energy signals.



Examples for signals with finite energy (a) and finite power (b):

Remark: The terms energy and power are not used in their conventional sense as electrical energy or power, but only as a measure for the signal size.

Example 57: Energy signals and power signals


Energy signal iff 0<E<, and so P=0

Example 58:

Power signal iff 0 < P < , and so E=

Example 59:

)()( 2 tuetx t

nnx )1(][


0 and 25.0 PE

11212lim][

121lim 2

NNnx

NP

N

N

NnN

12lim)1(lim][lim22 NnxE N

N

Nnn

NN

NnN

121

Neither energy nor power, when both E and P are infinite

Example 60:

)()( tuetx t


2 2 2

00

2 2

0

lim ( ) 0.5

1 1lim ( ) lim2 2

T t tT T

T Tt

T TT

E x t dt e dt e

P x t dt e dtT T

122

• Are all energy signals also power signals? • No. Any signal with finite energy will have zero power.

• Are all power signals also energy signals? • No. Any signal with non-zero power will have infinite

energy.

• Are all signals either energy signals or power signals? • No. Any infinite-duration, increasing-magnitude

function will not be either. (For example, the signal x(t) =tis neither.)



III. Important CT Signals

124

Building-block Signals

1. We will represent signal as sums of building-block signals.

2. Important families of building-block signals are the unit

step, unit ramp, unit parabolic, unit impulse, and

complex exponentials functions.


1. The Unit Step Function

1 , 0

u 1/ 2 , 00 , 0

tt t

t

Precise Graph Commonly-Used Graph


A definition of the unit step function is as follows:

2. The 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.


Ramp FunctionsA shifted ram function with slop B is defined as

Unit ramp function being at t=0 by making B=1 and t0=0 and multiplying by u(t), giving

)()( 0ttBtg

.0,

0,0)()(

ttt

ttutr

Time, t

f(t)

r(t)=tu(t)


The unit parabolic function

t

)(!2

2

tut

02/00

2 tifttif

tp


3. The Unit Parabolic Function 4. The 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.


5. The Rectangular Pulse Function

Rectangular pulse, 1/ , / 2

0 , / 2a

a t at

t a


The Unit Rectangle Function

1 , 1/ 2

rect 1/ 2 , 1/ 2 u 1/ 2 u 1/ 2

0 , 1/ 2

t

t t t t

t

The signal “turned on” at time t = -1/2 and “turned back off” at time t = +1/2.

Precise graph Commonly-used graph


6. The Unit Triangle Function

1 , 1

tri0 , 1

t tt

t

The unit triangle is related to the unit rectangle through an operation called convolution. It is called a unit triangle because its height and area are both one (but its base width is not).


7. The Unit Sinc Function

sinsinc

tt

t

The unit sinc function is related to the unit rectangle function through the Fourier transform.


The CT Impulse Function or the dirac delta function

Relationship

0

1/

(t)

Δ 0

l imt t

dt

tdut

0

dttu

8. The CT Impulse Function


The Dirac delta function ( )t

( ) 0 , 0 .(0 )

( ) 1

t fo r t

t d t

Normalization

Representation:

t

• This “unit impulse” function (sometimes called the Dirac Delta Function) is defined by the conditions:


The Delta Function

The value of delta function can also be defined in the sense

of generalized function:

(t): Test Function

we talk about the values of integrals involving (t).

)0()()(

dttt


The Unit Step and Unit Impulse Function

As approaches zero, g approaches a unit

step andg approaches a unit impulse

a t

t

The unit step is the integral of the unit impulse and

the unit impulse is the generalized derivative of the unit step

Functions that approach unit step and unit impulse


CT Unit Impulse FunctionThe continuous unit impulse signal is defined:

Note that it is discontinuous at t=0The arrow is used to denote area (1), rather than

actual value ()

The continuous unit step signal is defined:

000

)(tt

t

tdtu )()(

0100

)(tt

tu

The step function is discontinuous at time t=0

dttdut )()(

1)(

dtt


Derivative of the Unit Step Function

0 t

u(t)

Derivative

0 t

(t)


( )du tdt

0 0( )

1 0if t

u tif t

Remark: The value of u(t) at t = 0 is not well defined, we generally adopt one by convention.

( ) ( )t

u t d

CT Unit Impulse Function (t)


Example 61: Negative steps and deltas

0 0( )

1 0if t

u tif t

( )d u

tdt

t

-u

-1

t

(-1)Representation:

t

(-1)

or


Graphical Representation of the Impulse Function

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


Example 62: integrate ( ) ( ) ( )f t t u t

( ) ( ) [ ( ) ( ) ] ( ) ( )t t t

g t f d u d u t u d

t

g

1

1

f

0 0( ) ( )

0

t tif tu d u d

t if t

( ) ( )dg t u tdt

Due tojump

Standard derivativeaway from t = 0.

( )t u t

( ) ( ) 1g t u t t


Translations and flips of steps and deltas0

( )1

if tu t

if t

t

1

t( ) ( )t t

A pulse function:

( ) ( ) ( ) ( )1 [ , ]0

u t a u t b u t a u b tif t a botherwise

ta b

11( )

0if t

u tif t


Example 63:( ) ( ) ( 2)

( ) (2 )f t u t u t

u t u t

t

f

0 2

t0 2

( ) 2df t tdt

dfdt

Using the first expression,

Using the second,

( ) (2 ) ( ) 2 ( 1)

( ) (2 ) ( ) 2

df t u t u t tdt

t u t u t t

Different answer?


A basic property of the delta

• If f(t) is continuous at 0, ( ) ( ) 0 ( )t f t t f t

• If f(t) continuous at , ( ) ( ) ( )t f t t f t

Let f(t) be a standard function.

t

(1)

f

t

( )f

=

1 1

( ) (2 ) ( ) 2 ( ) (2) (2) 2

( ) 2

df t u t u t t t u u tdt

t t

Apply to the previous example:

Consistent with previous answerEE312 Signals and Systems Dr. Mohamed Zribi 147

Consequence of our basic property

( ) ( ) (0) ( ) (0) ( ) (0)f t t dt f t dt f t dt f

( ) ( ) ( )f t t dt f

( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( )

0

b

af t t dt u t a u t b f t t dt

f if a bf u a u b

if a or b

Similarly,

t

(1)

f

a b


( )( ) ( )

0

b

a

f if a bf t t dt

if a or b

t

(1)

f

a bWhat if falls exactly in one of the limits of integration?In that case, for the integral to be well defined we must specify whether that point is included or not, as follows:

( ) ( ) ( ), ( ) ( ) 0.b b

a af t t a dt f a f t t a dt

( ) ( ) ( ), ( ) ( ) 0,b b

a af t t b dt f b f t t b dt


Properties of Unit Impulse Function

dtttt )()( 0

Write t as t + t0

dtttt )()( 0 )( 0t

)()()( 00 tdtttt

Proof:

P1: The Sampling Property


The sampling property “extracts” the value of a function at a point.

Properties of the Impulse Function

0 01a t t t ta

P2: The Scaling Property

This property illustrates that the impulse is different from

ordinary mathematical functions.



dttat )()( )0(

||1

a

dttt

a)()(

||1

dttt

a)()(

||1

)(||

1)( ta

at

Proof:

P3:


Proof:

dttat )()(

Write t as t/a

Consider a>0

dt

att

a)(1

)0(||

1

a

dttat )()(

Consider a<0

dt

att

a)(1

)0(||

1

a

)0(||

1)()( a

dttat

P4:



dttttf )()]()([

dtttft )]()()[(

)0()0( f

dtttf )()()0(

dtttf )()]()0([

)()0()()( tfttf

Proof:

P5:

)()()()(or 000 tttftttf



Example 64:

)()0()()( tfttf

0)( tt

)(||

1)( ta

at

)()( tt


0

)()0(' ,)()('

tdttd

dttdt

0

)()( )()0( ,)()(

tn

nn

n

nn

dttd

dttdt

)0(')(')()()('

dtttdttt

)0()1()()( )()( nnn dttt

P6:

P7:



dttttf )(')]()([

dttttf )(')]()([

dtttft )](')()[(

dtttfttft )}()(')]'()(){[(

dtttftdtttft )]()'()[()]'()()[(

dtttftdtttft )]()'()[()]()()[('

dtttfttft )()](')()()('[

)(')()()(')]'()([ ttfttfttf

P8:

Proof:



)()'()]'()([)(')( ttfttfttf

)]'()0([ tf )()0(' tf

)()0(')(')0()(')( tftfttf

P9:

Proof:



Summary of important properties of unit impulseSifting properties

2

1

00 0

21t

t otherwisettttf

dttttf

Sampling properties

000 tttxtttx

Scaling properties

abt

abat 1

2010 ,12

1

ttttxdttttx nnt

t

n

Extension of Sifting properties


Example 65:

dttt )10(2

5

0

2 )10( dttt

20

0

2 )10( dttt

20

0

'2 )10( dttt

a)

b)

c)

d)

= 100

= 0

= 100

= - 20)(.1 2tdtd


Successive integration of the unit impulse1. Successive integration of the unit impulse yields a

family of functions.

2. Later we will talk about the successive derivatives of (t)


Remark:

Unit step Unit rampintegration

Unit ramp Unit parabolicintegration

Unit impulse Unit stepintegration

differentiation

differentiation

differentiation


9. The Unit Periodic Impulse

The unit periodic impulse/impulse train is defined by

, an integerTn

t t nT n

The periodic impulse is a sum of infinitely many uniformly-spaced impulses.


10. Exponential Signals

These signals occur frequently and serve as building blocks to construct many other signals

CT Complex Exponential:where a and C are in general complex.

Depending on the values of these parameters, the complex exponential can exhibit several different characteristics

atCetx )(

x(t) x(t)

C Ctt

a < 0a > 0 Real Exponential (C and a are real)

growing Exponential

Decaying Exponential


atCetx )(

164

Real Exponential SignalsExponential signals are characterized by exponential

functions

Where e is the Naperian constant 2.718… and C and are real constants.

( ) tf t Ce

f(t)

Time, t

( ) <0tf t Ce


Example 66: Exponential Signals

166

11. Sinusoidal Signals

x(t) = A cos(ωt + Φ)

A is the maximum amplitude of the sinusoidal signal

ω is the radian frequency

Φ is the phase shift


Sinusoidal SignalsA sinusoidal function

frequency in hertz or cycles per second, the phase shift in radians, the radian frequency is rad/s and the period

is (sec)Exponential functions, as in

Where B is the amplitude, is angular frequency in radians/second, and is the phase shift in radians.

0

2cos)cos()2cos()(T

tAtAftAtf

( ) ( )( ) cos( )2 2

j t j tB Bg t e e B t

f.2 fT /10

f.2


Exponentially Modulated Sinusoidal Functions1. If s sinusoid is multiplied by a real exponential, we have an

exponentially modulated sinusoidthat also can arise as a sum of complex exponentials, as in

)2cos()( ftAetf t

( ) ( )1( ) cos( )

2 2t j t t j tB Bf t e e e e B t

Example 67:)1cos(3)( 2.0 tetf t

Turned on at t = +1by multiplying shiftedunit step u(t-1)


Example 68:

Complex exponentials and unit steps can be combined to produce

causal and anti-causal decaying exponentials.


12. Complex Exponentials

The complex exponential signals have the form:

1. for all t

2. where C and s, are complex numbers.

3. If s is complex then it can be written as s = + j,

where and are the real and imaginary parts of s.

( ) stx t Ce


Complex ExponentialsCase 1: real sIf s = is real and C is real then

,And we get the family of real exponential functions.

Case 2: imaginary sIf s = j is imaginary and C is real then

and we get the family of sinusoidal functions.

( ) tx t Ce

( ) (cos sin ),j tx t Ce C t j t


( ) stx t Ce

172

Case 3: s complexIf s = + j is complex and C is real then

and we get the family of damped

sinusoidal functions.

( )( ) (cos sin ),j t tx t Ce Ce t j t


Complex Exponentials

173

For is plotted for

different values of s superimposed on the complex s-

plane.

( ) , { ( )} coss t tx t Ce x t Ce t



174

1. For is plotted for different values of s superimposed on the complex s-plane.

( ) , Im{ ( )} sins t tx t Ce x t Ce t



175

General Complex Exponential Signals

Here, C and s are general complex numbers

Say,

(Real and imaginary parts) Growing and damping sinusoids for r>0 and r<0

and s ( )j stC C e j x t Ce

Re{ ( )}x t

, 0t

( ) cos( )trx t Ce t

, 0t

envelope

( )Then ( ) st t j tx t Ce C e e cos( ) sin( )t tC e t j C e t


( ) stx t Ce

Re{ ( )}x t

General Complex Exponential Signals


( ) cos( )trx t Ce t

Example 69: Examples of General Complex

Exponential Signals2 3

2.5 0.5 0.5

2.5

( )

( ) ( )

( ) 2 cos(0.5 )

( ) 2 cos(0.5 )

j t j t

j t j t j t

j t

x t e e

x t e e e

x t e t

x t t


Example 70:

Superposition of sinusoidal signals ejt to

produce x(t)x(t) = sin(t) + 0.2cos(2t) + 0.1sin(5t)

x(t)

tt

sin(t)

sin(2t)

sin(5t)


IV. Important DT Signals

180

1. The DT Impulse Function

Unit-sample sequence [n]

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

n

[n]

[n] is sometimes called a discrete-time impulse; or an impulse

0001

][nn

n


The DT Impulse Function

)(n )3( n1

n

… …

1

n

… …

0m = n m = n-3

Plots of Unite Sample Sequences

0 1 2 3


The DT Impulse Function

1 , 00 , 0

nn

n

The discrete-time unit impulse (also known as the “Kronecker delta function”) is a function in the ordinary sense (in contrast with the continuous-time unit impulse). It has a sampling property,

0 0x xn

A n n n A n

but no scaling property. That is,

for any non-zero, finite integer .n an a


2. The Unit Step SequenceThe unit step sequence is the discrete-time version of the unit step

in CT situations.

Definition of unit step sequence:

The unit step sequence u(n) is related to unit sample sequence by

.0,0

0,1)(

nn

nu

1

n

…

Plot of Unit Step Sequence

0 1 2 3

.)()(

n

mmnu U(n)

step sequence


00

0

,A shifted step is: [ ]

0,B n n

Bu n nn n

The DT Unit Impulse and Step SignalsThe discrete unit impulse signal is defined:

Critical in convolution as a basis for analyzing other DT signals

The discrete unit step signal is defined:

Note that the unit impulse is the first difference of the step signal

Similarly, the unit step is the running sum of the unit impulse.

0100

][][nn

nnx

0100

][][nn

nunx

]1[][][ nunun


DT Unit Impulse and Unit Step Functions

0

k]-[n][k

nu

][][][][ 000 nnnxnnnx

][]0[][][ nxnnx

(DT step is therunning sum of DT unit sample)

More generally for a unit impulse [n-n0] at n0 :

[n-k]- - -- - -

nk

0

Interval of summation

n>0

[n-k]- - -- - -

n k0

Interval of summation

n<0

Sampling property


Since a sequence of Discrete-Time Signals can be represented in term of

Shifted Unit impulse as defined below :

∞

x[n] = Σx[k]δ[n-k]k=-∞

Then, the unit step sequence can be defined in term of Shifted Unit Impulse

as shown below :

∞

u[n] =Σδ[n-k]k= 0


Remark:

187

3. The DT Periodic Impulse Function

Nm

n n mN


4. The DT Unit Ramp Function

, 0ramp u 1

0 , 0

n

m

n nn m

n


r[n]=nu[n]

The Shifted Ramp FunctionRamp SequenceA shifted ramp sequence with slop of B is defined by:The unit ramp sequence and shifted ramp sequences

Example 71: g[n]= 2(n-10)u[n].

)()( 0nnBng

MATLAB Code:n=-10:1:20;f=2*(n-10);stem(n,f);


5. The DT Rectangle Function

1 ,

rect , 0 , an integer0 ,w

wN w w

w

n Nn N N

n N


Exponential function

n=0 n

x [n]

x[n] = A n 0 < < 1

5. The DT Exponential Function


Sinusoidal function

n=0 n

x [n]

x[n] = A cos(n + )

1

6. The DT Sinusoidal Function


The Sinusoidal DT SequenceA sinusoidal sequence may be described as:

where A is positive real number (amplitude), N is the period, and alpha is the phase.

Example 72:A = 5, N = 16 and

MATLAB Code:n=-20:1:20;f=5*[cos(n*pi/8+pi/4)]; stem(n,f);

NnAnf 2cos)(

.4/


Summary of Basic DT Sequences

Unit impulse Unit step

Exponential

Periodic

Sinusoidal

Random


7. The DT Real Exponential Function

)0for n alternatioSign (1,01,10,1

of instead use tocustomary and convenient more isIt numbers real are and C where

for

eαCnx

nr

n

1 10 01 1


DT Real Exponential Signals

(a) 1 0

(b)0 1 C > 0

(c) 1 0

(d) 1

C

real are and C :signals lexponentia Real][ nCnx

trivial0are or 1 when cases 197

C is constant and аlpha is a real number.

Example 73: Real Exponential DT Signal


nCnx ][

198


0 2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y(k)=0.7k

y(11)=0.0198


199

Real exponential sequence is defined as:Example for C = 10 and alpha = 0.9, as n goes to infinity the sequence

approaches zero and as n goes to minus infinity the sequence approaches plus infinity.

nCnf )()(

)()()( nuaAnf nComposite sequence:

MATLAB Code:n=-10:1:10;f =10*(.9).^n;stem(n,f);axis([-10 10 0 30]);



8. Complex Exponential and Sinusoidal DT Signals

When C and are real numbers real exponential DT signals

When =ej0 complex exponential DT signal

n nx n C x n C

00 0Note that: cos sinj ne n j n


DT Complex Exponential Signals

Unlike CT complex exponential signals ej0n is not distinct for distinct values of 0.

DT complex exponential signals are periodic with period N such that

njnj ee 200 0, 0±2, 0±4, … are identical. We need only consider a frequency interval of 2, e.g. 0≤ 0 ≤ 2.

number rational 2

21

0

00

00

Nm

mNeee

Nj

njNnj

If N and m have no factors in common then N is the fundamental period of x(t).


DT Complex Sinusoidal Signals

]}[Im{]}[Re{ )sin()cos(][

then ,CC

as formpolar in are and C If

00

0

nxjnxnCjnCCnx

ee

nnn

jj

)1 decaying and ,1 (growing ssinusoidal are expcomplex general DT of partsimaginary and Real

1

1


complex are and C :signals lexponentiaComplex ][ nCnx

]}[Im{or ]}[Re{

nxnx

]}[Im{or ]}[Re{

nxnx

203

DT Complex Exponential Signals

0

0 0

;

cos( ) sin( ) Re{ [ ]} Im{ [ ]}

jwj

n nn

C C e e

C C w n j C w nx n j x n

][nx

1

1

]}[{mor ]}[Re{ nxInx

]}[{mor ]}[Re{ nxInx


Sinusoidal sequence

n

x [n]

)cos(]}[Re{ 0 nnCnx


1

Example 76: Complex Exponential

205

0 2 4 6 8 10 12 14 16 18-1

-0.5

0

0.5

1

sin(k*pi/6)

0 2 4 6 8 10 12 14 16 18-1

-0.5

0

0.5

1

cos(k*pi/6)

1


]}[Re{ nx

]}[{m nxI


x[n] = Cеjωn; ω frequency of complex exponential sinusoid, C is a constant



])[Im(

])[Re(

nx

nx1

207

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

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

u ( k ) = c o s ( k * p i / 6 ) * 0 . 9 k


])[Re( nx

1


Exponentialy Modulated DT Sinusoidal SequenceBy multiplying an exponential sequence by sinusoidal sequence, we

obtain an exponentially modulated sequence described by:

Example 80:A = 10, N = 16, a = 0.9AndMATLAB Code:n=-20:1:20;f=10*[0.9 .^n];g=[cos(2*n*pi/16+pi/4)];h=f .*g;stem(n,h);axis([-20 20 -30 70]);

NnaAng n 2cos)()(

.4/


Example 81:

0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

1.5

2

)3

kπcos(0.8)3

kπsin(0.82y(k) kk


Example 82:

The dash line are the CT function. The CT function are obviously different but the DT function are not.


Comparison of the Signals

0 0 and jw t jw ne e



V. Operation on Signals

213

Basic operations on signals

Operation on Signals


g t Ag t Example 83: Amplitude Scaling,


g t Ag t Example 84: Amplitude Scaling,


Example 85: Amplitude Scaling


Operation on Signals

Sometimes we need to change the independent variable axis for

theoretical analysis or for just practical purposes (both in CT

and DT signals). Some of these operations are:

Time shift

[ Delay x(t-2); Advance x(t+2) ]

Time reversal

Time scaling

)()( ottxtx

)()( txtx

)2/()( txtx


Example 86: Time Shifting

t0 < 0 x(t-t0) is an advanced version of x(t)

x(t)

x(t-t0)

t

tt0

Time shift




Replace t by t- to.

If to>0, shift to the right. If to <0, shift to the left. e.g. to = -2 :

Time shift



221

• The original signal x(t) is shifted by an amount t0 > 0 .

• Given x(t) = u(t+2) - u(t-2), find x(t-t0) and x(t+t0).



222

Shifting the function to the right or left by t0



If n0 > 0 x[n-n0] is the delayed version of x[n] (Each point in x[n] occurs later in x[n-n0])

x[n]

x[n-n0]

. . . . . .

. . . . . . . . . . .

n

nn0

Time shift



224

n

nx

n

3nx

t

tx

4 t

4tx

8

Time shift



225

Determine x(t) + x(2-t) , where x(t) = u(t+1) - u(t-2)

find x(2-t): Advance, then reverse in time.

Add the two functions: x(t) + x(2-t)

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

t=0



226

0 0 , an integern n n n Time shifting



Time Reversal

Reflection about t=0

x(t)

x(-t)

t

t

Time reversal


Example 95: Time reversal




Replace t by – to.

Referred the reflected signal as folding (view it from behind the paper. e.g to = 6:

Time reversal



231

n

nx

n

nx

t

tx

t

tx

Time reversal



232

Time Scaling

Example 99: Given x(t), find y(t) = x(2t). This speeds up x(t) (the graph is shrinking)

What happens to the period T?The period of x(t) is 2 and the period of y(t) is 1.


x(t)

t

compressed!

stretched!

x(2t)

t

x(t/2)

t

Time scaling


Time Scaling

234

Multiply t by a constant , say, β .

If β > 0, the signal is compressed. If β <0, the signal is expanded. e.g β = 2:

Time scaling

Time Scaling


Time Scaling

Given y(t), find w(t) = y(3t) v(t) = y(t/3).


Expands the function horizontally by a factor of |a|

Example 100: Time Scaling

/t t a


/t t a

If a < 0, the function is also time inverted. The time inversionmeans flipping the curve 1800 with the g axis as the rotation axis of the flip.



t

x(t)

T20

10

t

(t) = x(t/2)

0

t

(t) = x(2t)

Original signal

0

T1

Expanded signal(a = 0.5)

Compressed signal(a = 2)

2T22T1

2T1

2T2



239




n Kn

K an integer > 1



/ , 1n n K K Time expansion

For all such that / is an integer, g / is defined.

For all such that / is not an integer, g / is not defined.

n n K n K

n n K n K

Remark : Time Expansion for DT Signals


Combined Operations on Signals

We can use various combinations of the three operations just

covered: time shifting, time scaling, and time reversal. The

operations can often be applied in different orders, but care must

be taken.



To form x(at - b) from x(t) we could use two approaches:

1) Time-shift then time-scaleA. Time-shift x(t) by b to obtain x(t - b). i.e., replace every t by t

- b.B. Time-scale x(t - b) by a (i.e., replace t by at) to form x(at - b)

2) Time-scale then time-shiftA. Time-scale x(t) by a to obtain x(at).B. Time-shift x(at) by b/a (i.e., replace t with t – b/a) to yield

x(a[t – b/a]) = x(at – b)


0g g t tt Aa

0

amplitudescaling, / 0g g g gt t tA t t a t ttt A t A A

a a

A multiple transformation can be done in steps

0

amplitudescaling, / 0

0 0g g g g gt t tA t t a t ttt A t A t t A t Aa a

The order of the changes is important. For example, if weexchange the order of the time-scaling and time-shifting operations, we get:

Amplitude scaling, time scaling and time shifting can be appliedsimultaneously.



g t Agt t0

a

Example 106: Operation on Signals


Find y(t) = x(2t + 3).




x(t+1)

t10-1

1

x(1.5t+1)

t2/30-2/3

1

Find x(3t/2+1)

t

x(t)

1

1 20



Given the signal x(t):

Let us find x(t+1):

Let us find x(-t+1):

t

x(t)

1

1 20

(time reversal of x(t+1)) t

1

10-1

(It is a time shift to the left)

x(t+1)

t

x(-t+1)

10-1

1



0 1 2 3 43 2 1

0 1 2 3 4

tx

t

tx

2/2 tx

t

03 2 1 1 2 3 4 t

3 2 1

1

1



x(2t

)



Given x2(t), find y(t) = 1 - x2(t).

Remember: This is x(t) =1


y(t) = 1 - x2(t)

252


Multiplication of two signals: x2(t)u(t)

Step unit function



254

Properties of the CT Unit Step Function

kttuttuttu )]([)]([)( 02

00

0),/()( 00 aattutatu

)(1)( tutu


CT Unit Ramp Function

)()()()( 000

0

0

ttuttddtutft

t

t

Unit ramp function can be achieved by:

to to+1

1

Non-zero only for t>t0

0)()()( dttdttut

Remark:


Note: u(-t+3)=1-u(t-3)

Example 115:

Example 116:


Signals can be combined to make a rich population of signals

Unit steps and ramps can he combined to produce pulse signals.


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

1

1

t t1 2

1

Example 117:


Example 118:Describe analytically the signal shown in

Solution: Signal is (A/2)t at , turn on this signal at t = 0 and turn it off again at t = 2. This gives,

)].2()([2

)( tututAtf

20 t

A

0 2

t

f(t)


Example 119:

)2(5)1(]1[)()(3)( tututttututf


Example 120:

Plot

• t<-2 f(t)=0• -2<t<-1 f(t)=3[t+2]• -1<t<1 f(t)=-3t• 1<t<3 f(t)=-3• 3<t f(t)=0

)3(3)1(]1[3)1(]1[6)2(]2[3)( tututtuttuttf


Example 121:

rect(t/T)

Can be expressed as u(T/2-t)-u(-T/2-t) Draw u(t+T/2) first; then reverse it!

Can be expressed as u(t+T/2)-u(t-T/2)

Can be expressed as u(t+T/2).u(T/2-t)

-T/t T/t

1

-T/t T/t

1

-T/t T/t

1

-T/t T/t


Example 122:

Write the expression of the plotted function

Answer:f(t)=0 for t<-2f(t)=3(t+2) for -2<t<-1f(t)=-3t for -1<t<1f(t)=-3 for 1<t<3f(t)=0 for 3<t


)3(3)1(]1[3 )1(]1[6)2(]2[3

)]3()1([3 )]1()1([3)]1()2()[2(3)(

tututtuttut

tututututtututtf

264

Representation of DT Signals using Sequences

Discrete-time system theory is concerned with processing

signals that are represented by sequences.

1 2

3 4 5 6 7

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

n

x [n]


It is possible to re-generate an arbitrary signal by sampling it with shifted unit impulse:

This is called the sifting property:

[ ] [ ] [ ]k

x n x k n k

Sifting property of DT impulseweights

shifted impulse

x[n] = - - - + x[-4] δ[n+4] + x[-3] δ[n+3] + x[-2] δ[n+2] + + x[-1] δ[n+1] + x[0] δ[n] + x[1] δ[n-1] + - - -



266

The Unit Impulse can be shifted or delayed. The shifted Unit Impulse is denoted as :-δ[n-k] => The unit impulse is shifted to right by kδ[n+k] => The unit impulse is shifted to left by k

Example 123:The sequence, p[n] is expressed as :p[n] = a-3 δ[n + 3] + a1 δ[n – 1] + a2 δ[n – 2] + a7 δ[n – 7] Graphical representation of Shifted unit Impulse of p[n]:



267

1

2

3 4 5 6

7

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

n

x [n]

a1

a2 a7

a-3

k

knkxnx ][][][

]7[]3[]1[]3[][ 7213 nananananx


Example 124:

268

Think of a DT signal as a sequence of individual impulses

Consider x[n]

4-3 13

0...2-1

-2... n-4


Example 125:

269

x[n] is actually a sequence of time-shifted and scaled impulses

-2

x[-2]δ[n+2]-1x[-1]δ[n+1]

0

x[0]δ[n]

1

x[1]δ[n-1]

0

2

x[2]δ[n-2]

10


Example 126:

Describe analytically the following sequence.

1

n

A Pulse Sequence

0 1 2 3

f[n]

-1-2

This pulse sequence can be describe by

f[n] = u[n]– u[n-3]

The first step sequence turn on the pulse at n = 0, and second step turns it off at n = 3.


Remember These Geometric Series


handout_1_short.pdf

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