signals and systems( chapter 1)

Signals and Systems:

A computer without

signals - without

networking, audio

and video.

Terms

• Signal• voltage over time

• State• the variables of a differential equation

• System• linear time invariant transfer function

Chapter 1 : Signals And Systems

1.0 Introduction

Signals and Systems subject is focusing on a signal involving dependent

variable (i.e : time even though it can be others such as a distance , position ,

temperature , pressure and others)

Signals & Systems

• Signal

– physical form of a waveform

– e.g. sound, electrical current, radio wave

• System

– a channel that changes a signal that passes through it

– e.g. a telephone connection, a room, a vocal tract

Input Signal System Output Signal

systemInput

signal

Output

signal


1.1 Signals and Systems

Definition

a) Signal

• A function of one/more variable which convey information on the natural of a

physical phenomenon.

• Examples : human speech, sound, light, temperature, current etc

b) Systems

• An entity that processes of manipulates one or more signals to accomplish a

function, thereby yielding new signal.

• Example: telephone connection


e) Energy and power signals

1.2 Classification of Signals

There are several classes of signals

a) Continuous time and discrete time signals

c) Real and Complex signals

b) Analog and digital signals

f) Periodic and aperiodic signals

d) Even and Odd Signals


a) Continuous time and discrete time signals

• Continuous signals : signal that is specified for a continuum (ALL) values

time t

: can be described mathematically by continuous

function of time as :

x(t) = A sin (ω0 t + ɸ)

where A : Amplitude

ω : Radian freq in rad / sec

ɸ : phase angle in rad / degree

• Discrete time signals : signal that is specified only at discrete values of t


b) Analog and digital signals

• Analog signals : signal whose amplitude can take on any value in a

continuous range

• Digital signals : signal whose amplitude can take only a finite number

of values

(signal which associated with computer since involve

binary 1 / 0 )


Examples of signals

• Example 1:

• Example 2:


c) Deterministic and probabilistic signals

• Deterministic signals : a signal whose physical description is known

completely either in a mathematical form or a

graphical form and its future value can be determined.

• Probabilistic signals : a signal whose values cannot be predicted precisely

but are known only in terms of probabilistic value such

as mean value / mean-squared value and therefore

the signal cannot be expressed in mathematical form.


d) Energy and power signals

• Energy signals : a signal with finite energy signal

• Power signals : a signal with finite and nonzero power

Finite energy signal Infinite energy signal


1.9 Energy and Power Signals

For an arbitrary signal x(t) , the total energy , E is defined as

The average power , P is defined as



Based on the definition , the following classes of signals are defined :

a) x(t) is energy signal if and only if 0 < E < ∞ so that P = 0.

b) x(t) is a power signal if and only if 0 < P < ∞ thus implying that E = ∞.

c) Signals that satisfy neither property are therefore neither energy nor power

signals.

Exercise

1) Calculate the total energy of the

rectangular pulse shown in figure.

2) Given a signal as listed below, determine

whether x(t) is energy, power or neither

signal. Justify the answer.

a. x(t) = cos t

b. x(t) = 3 e-4t u(t)


e) Periodic and aperiodic signal

• Periodic signals : signal that repeats itself within a specific time or in

other words, any function that satisfies :

where T is a constant and is called the fundamental period of the function.

• Aperiodic signals : signal that does not repeats itself and therefore does

not have the fundamental period.

( ) ( )f t f t T= +

0

2

ω

π=T


Examples of signals

Periodic signal Aperiodic signal

• Example:

Find the period for

3cos)(

ttf =


e) Periodic and aperiodic signal (continue)

• Any continuous time signal x(t) is classified as periodic if the signal satisfies

the condition :

x(t) = x(t + nT) where n = 1 , 2 , 3 ....

• The sum of two or more signals is periodic if the ratio (evaluation of two

values) of their periods can be expressed as rational number. The new

fundamental period and frequency can be obtained from a periodic signal.

• The sum of two or more signals is aperiodic if the ratio (evaluation of two

values) of their periods is expressed as irrational number and no new

fundamental period can be obtained.

A rational number is a number that can be written as a simple fraction (i.e. as a ratio).

• Periodic and aperiodic signal

(continue)

x(t) = x(t + nT)

Tοοοο

= 2ππππ / ωωωωοοοο

ΩοΩοΩοΩο = m

2ππππ N


Exercise 1

• Determine whether listed x(t) below is periodic or aperiodic signal. If a signal is

periodic, determine its fundamental period.

a) x(t) = sin 3t

b) x(t) = 2 cos 8πt

c) x(t) = 3 cos (5πt + π/2)

d) x(t) = cos t + sin √2 t

e) x(t) = sin2t

f) x(t) = ej[(π/2)t-1]

0

2

ω

π=T

EXERCISE 2

a) x[n]=ej(π/4)n

b) x[n]=cos1/4n

c) x[n]= cos π/3 n + sin π/4n

d) x[n]=cos2π/8n


Exercise 2

• Determine whether the following signals are periodic or aperiodic. Find the

new fundamental period if necessary.

a) x3(t) = 6 x1(t) + 2 x2(t)

b) x5(t) = 6 x1(t) + 2 x2(t) + x4(t)

where : x1(t) = sin 13t

x2(t) = 5 sin (3000t + π/4)

x4(t) = 2 cos (600πt – π/3)


Exercise 3

• Given x1(t) = 2 sin (5t) , x2(t) = 5 sin (3t) and x3(t) = 5 sin (2t + 25o) .

• If x(t) = x1(t) – 3x2(t) + 2x3(t) , determine whether x(t) is periodic or aperiodic

signal .

• If it is periodic, determine it’s period and frequency

• Even and Odd A signal x ( t ) or x[n] is referred to as an even signal if

x ( - t ) = x ( r )

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 ]

Examples of even and odd signals are shown in Fig. 1-2.



is an important and unique sub-class of aperiodic signals

they are either discontinuous or continuous derivatives

they are basic signals to represent other signals

0 ; t < 0

1 ; t ≥ 0u(t) =

u(t)

1

t

a) Unit Step, u(t)

1.4) Singularity Functions


b) Unit Ramp , r(t)

0 ; t < 0

t ; t ≥ 0r(t) =

r(t)

1

t1


c) Unit impulse, δ(t)

1 ; t = 0

0 ; t ≠ 0δ(t) =

δ(t)

1

t

1.5) Representation of Signals

A deterministic signal can be represented in

terms of:

1. sum of singularity functions

2. sum of steps functions and

3. piece-wise continuous functions


Sum Of Singularity Function

Express signal in term of sum of singularity function

Note : the ramp function r(t) can be described by step function as :

r(t) = t u(t)

r(t±a) = (t±a) u(t±a)

Example

Express the following signal in term of sum of singularity function.

-1 1

-1

1

x(t)

t

Answer

x(t) = u(t+1) – r(t+1) + r(t-1) + u(t-1)

= u(t+1) – (t+1)u(t+1) + (t-1)u(t-1) + u(t-1)


Exercise

Express the following signals in term of sum of singularity function.

1

-1

1

x(t)

t2 3 4

-1

2

x(t)

t1 2-2

1

Answer

x(t)= r(t) – 2r(t-1) + 2r(t+3) +r(t-4)

= r(t) – 2(t-1)u(t-1) +2(t+3)u(t+3) +(t-4)u(t-4)

Answer

x(t) = 2δ(t+2) - u(t+2) + r(t+1) – r(t-1) – u(t-1) + δ(t+2)


Exercise

Sketch the following signal if the sum of singularity function of the signal is given as :

a) x(t) = r(t) + r (t-1) - u (t-1)

b) y(t) = u(t+1)-r(t+1)+r(t-1)+u(t-1)

c) x(t) = r(t) + r(t+1) + 2u(t+1) – r(t+1) + 2r(t) – r(t-1) + u(t-2) – 2u(t-3)


Piece – wise continous function

Description of signal from a general form of y = mx + c

Example

Given the signal x(t) as shown below , express the signals x(t) in terms of piece

wise continuous function

x(t)

-1t

1 2

1

0

Solution

- t - 1 ; -1 < t < 0

t ; 0 < t < 1

1 ; 1 < t < 2

0 ; elsewhere

x(t) =


Exercise

Example


wise continuous function .

x(t)

-1t

1 2

1

0


Exercise

Example


wise continuous function .

x(t)

-1t

1 2

1

0

Solution

t + 1 ; -1 < t < 0

-1 ; 1 < t < 2

0 ; elsewherex(t) =


1.6 Properties of Signals

There are 4 properties of signals

a) Magnitude scaling

b) Time reflection

c) Time scaling

d) Time shifting


a) Magnitude scaling : Any arbitrary real constant is multiplied to a signal and

the result is, for a unit step the amplitude changes, for a unit ramp, the slope

changes and for a unit impulse, the area changes

A

3

t

3u(t)

-A

-2

t

-2u(t)

A

2

t

2r(t)

1

A

-2

t

-2r(t)

A

3

t

3δ(t)

0

A

-0.5

t

-0.5δ(t)

0

slope = 2

slope = -2


b) Time reflection : The mirror image of the signal with respect to the y-axis

u(t)

1

t

u(-t)

r(t)

t

r(-t)

δ(t)

1

t

δ(-t)

00

slope = -1


c) Time scaling : The expansion or compression of the signal with

respect to time t axis

x(kt)

t

x(kt)

b

a

x(0.5t)

t

a

2b

x(2t)

t

x(2t)

0.5b

ak > 1 compressionk < 1 expansion


f) Time shifting : The shifting of the signal with respect to the x - axis

u(t)

1

t

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

r(t)

t

r(t-1)

1

r(t)

t

r(t+1)

δ(t)

3

t

3δ(t-2)

0

-0.5

t

-0.5δ(t+2)

1

u(t)

1

t-1

slope = 1

slope = 1

2

-2

-1


Example 1

Given the signal x(t) as shown below , sketch y(t) = 3x (1- t/2) .

x(t)

-1

t

1 2

1

0


Example 2

Given the signal x(t) as shown below , sketch y(t) = 2x (-0.5t+1) using both

graphical and analytical method.

x(t)

-1 t

1 2

-1

1


Exercise

Given the signal x(t) as shown below , sketch y(t) = -2x (2-0.5t) + 1

using both graphical and analytical method.

x(t)

-1

t

1 2

-1

1

3-2



For an arbitrary signal x(t) , the total energy , E is defined as

The average power , P is defined as



Based on the definition , the following classes of signals are defined :

a) x(t) is energy signal if and only if 0 < E < ∞ so that P = 0.

b) x(t) is a power signal if and only if 0 < P < ∞ thus implying that E = ∞.

c) Signals that satisfy neither property are therefore neither energy nor power

signals.


2.0 Classification Of System

i) a) With memory(dynamic) : the present output depends on past

and/or future input.

b) Without memory(static) : the present output depends only on

present input.

ii) a) Causal : the output does not depends on future but can

depends on the past or the present input..

b) Non-causal : the output depends on future input



iii) a) Time variant : y2 (t) ≠ y1 (t- to)

: same input produces different output at different time

b) Time invariant : y2 (t) = y1 (t- to)

: same input produces same output at different time

where :

y1 (t- to) is the output corresponding to the time shifting , (t- to) at y1 (t)

y2(t) is the output corresponding to the input x2(t) where x2(t) = x1 (t- to)



4) a) Linear : y(t) = ay1(t) + by2(t) (superposition applied)

b) Non linear : y(t) ≠ ay1(t) + by2(t) (superposition not applied )

where :

If an excitation x1[t] causes a response y1[t] and an excitation x2 [t] causes a

response y2[n] , then an excitation :

x [t = ax1[t] + bx2[t] (to be presented as y(t) in solution)

y [t] = ay1[t] + by2[t]

will cause the response


Example

The following system is defined by the input – output relationship where x(t) is the

input and y(t) is the output.

a) y(t) = 10x2 (t+1) e) y’(t) + y(t) = x(t)

b) y(t) = 10x(t) + 5 f) y’(t) + 10y(t) + 5 = x(t)

c) y(t) = cos (t) x(t) + 5 g) y’(t) + 3y(t) = x(t) + 2x2(t)

Determine whether the system is :

i. Static or dynamic

ii. Causal or non-causal

iii. Time - variant or time invariant

iv. Linear or non-linear

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