Mr. Mr. MohdMohd RashidiRashidi Bin Bin CheChe BesonBesonCoECoE ACEACE--SCCE, SCCE, UniMAPUniMAP

EE--Mail: Mail:

Signal & SystemsSignal & Systems

EE--Mail: Mail: mrashidi.unimap@gmail.commrashidi.unimap@gmail.com

Mobile: 019Mobile: 019--59888705988870

EKT 232/230 EKT 232/230

Chapter 1Chapter 1

2

11..11 WhatWhat isis aa SignalSignal ??

11..22 ClassificationClassification ofof aa SignalsSignals..

11..22..11 ContinuousContinuous--TimeTime andand DiscreteDiscrete--TimeTime SignalsSignals

11..22..22 EvenEven andand OddOdd SignalsSignals..

11..22..33 PeriodicPeriodic andand NonNon--periodicperiodic SignalsSignals..

11..22..44 DeterministicDeterministic andand RandomRandom SignalsSignals..

11..22..55 EnergyEnergy andand PowerPower SignalsSignals..

11..33 BasicBasic OperationOperation ofof thethe SignalSignal..

11..44 ElementaryElementary SignalsSignals..11..44 ElementaryElementary SignalsSignals..

11..44..11 ExponentialExponential SignalsSignals..

11..44..22 SinusoidalSinusoidal SignalSignal..

11..44..33 SinusoidalSinusoidal andand ComplexComplex ExponentialExponential SignalsSignals..

11..44..44 ExponentialExponential DampedDamped SinusoidalSinusoidal SignalsSignals..

11..44..55 StepStep FunctionFunction..

11..44..66 ImpulseImpulse FunctionFunction..

11..44..77 RampedRamped FunctionFunction..

1.5 What is a System ?1.5 What is a System ?

1.5.1 System Block Diagram.1.5.1 System Block Diagram.

1.6 Properties of the System.1.6 Properties of the System.

1.6.1 Stability.1.6.1 Stability.

1.6.2 Memory.1.6.2 Memory.

1.6.3 Causality.1.6.3 Causality.

1.6.4 Inevitability.1.6.4 Inevitability.

1.6.5 Time Invariance.1.6.5 Time Invariance.

1.6.6 Linearity.1.6.6 Linearity.

� A common form of human communication;

(i) use of speechspeech signal, face to face or telephone channel.

(ii) use of visualvisual, signal taking the form of images of people orobjects around us.

�� RealReal lifelife exampleexample of signals;

(i) Doctor listening to the heartbeat, blood pressure and

1.1 What is a Signal ?1.1 What is a Signal ?

5

(i) Doctor listening to the heartbeat, blood pressure andtemperature of the patient. These indicate the state of health of thepatient.

(ii) Daily fluctuations in the price of stock market will convey aninformation on the how the share for a company is doing.

(iii) Weather forecast provides information on the temperature,humidity, and the speed and direction of the prevailing wind.

� By definition, signalsignal isis aa functionfunction ofof oneone oror moremore variable,variable,whichwhich conveysconveys informationinformation onon thethe naturenature ofof aa physicalphysicalphenomenonphenomenon..

� A function of time representing a physical or mathematical

quantities.

e.g; Velocity, acceleration of a car, voltage/current of a circuit.

An example of signal; the electrical activity of the heart recorded

with electrodes on the surface of the chest-the electrocardiogram

Cont’d…Cont’d…

with electrodes on the surface of the chest-the electrocardiogram

(ECG) in the figure below.

τ

� There are five typesfive types of signals;

((ii) Continuous) Continuous--Time and DiscreteTime and Discrete--Time Signals Time Signals

(ii) Even and Odd Signals.(ii) Even and Odd Signals.

(iii) Periodic and Non(iii) Periodic and Non--periodic Signals.periodic Signals.

(iv) Deterministic and Random Signals.(iv) Deterministic and Random Signals.

(v) Energy and Power Signals.(v) Energy and Power Signals.

1.2 Classifications of a Signal1.2 Classifications of a Signal

(v) Energy and Power Signals.(v) Energy and Power Signals.

ContinuousContinuous--Time (CT) SignalsTime (CT) Signals

�� ContinuousContinuous--TimeTime (CT)(CT) SignalsSignals are functions whose amplitudeor value varies continuously with time, x(t).

� The symbol t denotes time for continuous-time signal and ( ) usedto denote continuous-time value quantities.

� Example, speed of car, converting acoustic or light wave intoelectrical signal and microphone converts variation in sound

1.2.1 Continuous1.2.1 Continuous--Time and DiscreteTime and Discrete--Time SignalsTime Signals

electrical signal and microphone converts variation in soundpressure into correspond variation in voltage and current.

Figure 1.2: CT Signal

DiscreteDiscrete--Time SignalsTime Signals

�� DiscreteDiscrete--TimeTime SignalsSignals are function of discrete variable, i.e. theyare defined only at discrete instants of time.

� It is often derived from continuous-time signal by sampling atuniform rate. Ts denotes sampling period and n denotes integer.

� The symbol n denotes time for discrete time signal and [ ] isused to denote discrete-value quantities.

� Example: the value of stock at the end of the month.

[ ] ( ) ,....2,1,0, ±±== nnTxnx s

� Example: the value of stock at the end of the month.

Figure 1.3: Discrete-Time Signal.

� A continuous-time signal x(t) is said to be an eveneven signal if

� The signal x(t) is said to be an oddodd signal if

( ) ( ) tallfortxtx =−

( ) ( ) tallfortxtx −=−

1.2.2 Even and Odd Signals1.2.2 Even and Odd Signals

� In summary, an even signal are symmetric about the vertical axis

(time origin) whereas an odd signal are antisymetric about the

origin.

Figure 1.4: Even Signal Figure 1.5: Odd Signal.

11

Example 1.1Example 1.1: Even and Odd Signals: Even and Odd Signals

Find the even and odd components of each of the followingsignals:

(a) x(t) = 4cos(3πt)

12

Periodic Signal.Periodic Signal.

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

condition

where T is a positive constant.

� The smallest value of T that satisfy the definition is called a period.

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

1.2.3 Periodic and Non1.2.3 Periodic and Non--Periodic SignalsPeriodic Signals

13

The smallest value of T that satisfy the definition is called a period.

Figure 1.6 (a) Square wave with amplitude A=1 and period T=0.2 s. (b) Rectangular

pulse of amplitude A and duration T1

Deterministic Signal.Deterministic Signal.

� A deterministic signaldeterministic signal is a signal that is no uncertainty with

respect to its value at any time.

� The deterministic signal can be modeled as completely specified

function of time.

11..22..44 DeterministicDeterministic andand RandomRandom

SignalsSignals

14

Random Signal.Random Signal.

� A random signal is a signal about which there is uncertainty

before it occurs. The signal may be viewed as belonging to an

ensemble or a group of signals which each signal in the ensemble

having a different waveform.

� The signal amplitude fluctuates between positive and negative in a

randomly fashion.

� Example; noise generated by amplifier of a radio or television.

15

� Example; noise generated by amplifier of a radio or television.

Figure 1.7: Random Signal

A signal with finite signal energy is called anenergy signal.

A signal with infinite signal energy and finiteaverage signal power is called a power signal.

Energy Energy SignalSignal and and PowerPower Signals.Signals.

16

average signal power is called a power signal.

1.2.5 Energy Signal and Power Signals1.2.5 Energy Signal and Power Signals

Energy Signal.Energy Signal.

� A signal is refer to energy signal if and only if the total energy

satisfy the condition;

[ ]∑∞

−∞=

=n

nxE2

∞<< E0

17

Power Signal.Power Signal.

� A signal is refer to power signal if and only if the average power

of signal satisfy the condition;

[ ]∑−

=

=1

0

21 N

n

nxN

P

∞<< E0

∞<< P0

1.3 Basic Operation of the Signals1.3 Basic Operation of the Signals1.3.1 Time Scaling.1.3.1 Time Scaling.

1.3.2 Reflection and Folding.1.3.2 Reflection and Folding.

1.3.3 Time Shifting.1.3.3 Time Shifting.

1.3.4 Precedence Rule for Time Shifting and Time Scaling.1.3.4 Precedence Rule for Time Shifting and Time Scaling.

18

� Time scaling refers to the multiplicationmultiplication of the variable by a real

positive constant.

� If aa > 1 the signal y(t) is a compressedcompressed version of x(t).

� If 0 < aa < 1 the signal y(t) is an expandedexpanded version of x(t).

� Example:

( ) ( )atxty =

1.3.1 Time Scaling1.3.1 Time Scaling

19

Figure 1.8: Time-scaling operation; continuous-time signal x(t),

(b) version of x(t) compressed by a factor of 2, and

(c) version of x(t) expanded by a factor of 2.

� In the discrete time,

� It is defined for integer value of k, k > 1. Figure below for k = 2,

sample for n = +-1,

[ ] [ ],knxny =

20

Figure 1.9: Effect of time scaling on a discrete-time signal:

(a) discrete-time signal x[n] and (b) version of x[n] compressed by a factor of 2, with some values of the original x[n] lost as a result of the compression.

� Let x(t) denote a continuous-time signal and y(t) is the signal

obtained by replacingreplacing time t with –t;

� y(t) is the signal represents a refracted version of x(t) about t = 0.

� Two special casesspecial cases for continuous and discrete-time signal;

( ) ( )txty −=

1.3.2 Reflection and Folding1.3.2 Reflection and Folding

21

(i) Even signal; x(-t) = x(t) an even signal is same as reflected

version.

(ii) Odd signal; x(-t) = -x(t) an odd signal is the negative of its

reflected version.

Example 1.2:Example 1.2: Reflection.Reflection.Given the triangular pulse Given the triangular pulse xx((tt), find the reflected version of ), find the reflected version of xx((tt) about ) about

the amplitude axis (origin).the amplitude axis (origin).

Solution:Solution:Replace the variable t with –t, so we get y(t) = x(-t) as in figure below.

22

Figure 1.10: Operation of reflection: (a) continuous-time signal x(t) and (b) reflected version of x(t) about the origin

x(t) = 0 for t < -T1 and t > T2.

y(t) = 0 for t > T1 and t < -T2.

�.

� A time shift delaydelay or advancesadvances the signal in time by a time interval

+t0 or –t0, without changing its shape.

y(t) = x(t - t0)

� If t0 > 0 the waveform of y(t) is obtained by shifting x(t)toward the rightright, relative to the tie axis.

� If t0< 0, x(t) is shifted to the leftleft.

Example:

1.3.3 Time Shifting1.3.3 Time Shifting

23

Example:

Figure 1.11: Shift to the Left. Figure 1.12: Shift to the Right.

Q: How does the x(t) signal looks like?

Example 1.3: Example 1.3: Time Shifting.Time Shifting.Given the rectangular pulse Given the rectangular pulse xx((tt) of unit amplitude and unit duration. ) of unit amplitude and unit duration.

Find Find yy((tt)=)=x (t x (t -- 2)2)

Solution:Solution:t0 is equal to 2 time units. Shift x(t) to the right by 2 time units.

24

Figure 1.13: Time-shifting operation:

(a) continuous-time signal in the form of a rectangular pulse of amplitude 1.0 and

duration 1.0, symmetric about the origin; and

(b) time-shifted version of x(t) by 2 time shifts.

� .

� Time shiftingshifting operation is performed first on x(t), which results in

� Time shift has replace t in x(t) by t - b.

� Time scalingscaling operation is performed on v(t), replacing t by at and resulting in,

11..33..44 PrecedencePrecedence RuleRule forfor TimeTime ShiftingShiftingandand TimeTime ScalingScaling

25

resulting in,

� Example in real-life: Voice signal recorded on a tape recorder;

� (a > 1) tape is played faster than the recording rate, resulted in

compression.

� (a < 1) tape is played slower than the recording rate, resulted

in expansion.

( ) ( )( ) ( )batxty

atvty

−=

=

Example 1.4:Example 1.4: Continuous Signal. Continuous Signal. A CT signal is shown in Figure 1.14 below, sketch and label each of A CT signal is shown in Figure 1.14 below, sketch and label each of

this signal;this signal;

a) a) x(tx(t --1) 1)

b) b) x(2t)x(2t)

c) c) x(x(--t)t)

2

x(t)

26

Figure 1.14

-1 3

2

t

Solution:Solution:(a) x(t -1) (b) x(2t)

0 4

t

x(t-1)

2

-1/2 3/2

2

t

x(t)

27

(c) x(-t)

-3 1

2

t

x(-t)

Example 1.5:Example 1.5: Continuous Signal. Continuous Signal. AA continuouscontinuous signalsignal xx((tt)) isis shownshown inin FigureFigure 11..1414aa.. SketchSketch andand labellabel eacheach

ofof thethe followingfollowing signalssignals..

a)a) x(t)=x(t)= u(tu(t --11))

b) b) x(t)= [u(t)x(t)= [u(t)--u(tu(t--1)]1)]

c) c) x(t)= x(t)= δδ(t (t -- 3/2)3/2)

Solution:Solution:Figure 1.14a

28

Figure 1.14a

(a) x(t)= u(t(a) x(t)= u(t --1) 1) ((b) b) x(t)= [u(t)x(t)= [u(t)--u(tu(t--1)] (1)] (c) c) x(t)=d(t x(t)=d(t -- 3/2)3/2)

Example 1.5:Example 1.5: Discrete Time Signal. Discrete Time Signal.

A discreteA discrete--time signal x[n] is shown below, time signal x[n] is shown below,

Sketch and label each of the following signal.Sketch and label each of the following signal.

(a) x[n (a) x[n –– 2]2] (b) x[2n](b) x[2n]

(c.) x[(c.) x[--n+2]n+2] (d) x[(d) x[--n]n]

x[n]

29

x[n]

n

4

2

0 1 2 3

(a) A discrete-time signal, x[n-2].

�A delay by 2

x[n-2]

Cont’d…Cont’d…

30

4

2

0 1 2 3 4 5 n

(b) A discrete-time signal, x[2n].

Down-sampling by a factor of 2.

x(2n)

Cont’d…Cont’d…

31

4

2

0 1 2 3 n

(c) A discrete-time signal, x[-n+2].

Time reversal and shifting

x(-n+2)

Cont’d…Cont’d…

32

4

2

-1 0 1 2 n

(d) A discrete-time signal, x[-n].

�Time reversal

4

x(-n)

Cont’d…Cont’d…

33

4

2

-3 -2 -1 0 1 n

In Class ExercisesIn Class Exercises ..A continuousA continuous--time signal time signal x(t)x(t) is shown below, Sketch and label each is shown below, Sketch and label each

of the following signalof the following signal

(a) (a) x(t x(t –– 2)2) (b) (b) x(2t)x(2t) (c) (c) x(x(--t) (d) t) (d) xx((--t+3)t+3)

x(t)

4

34

t0 4

� Continue for the following week

35