1 signals and systems. chapter 1 school of computer and communication engineering, unimap puan...

1

Signals Signals and and

Systems.Systems.

CHAPTER CHAPTER 11

School of Computer and Communication School of Computer and Communication Engineering, UniMAPEngineering, UniMAP

Puan Nordiana Binti Mohamad SaaidPuan Nordiana Binti Mohamad [email protected]@unimap.edu.my

EKT 230 EKT 230

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1.1 1.1 What is a Signal ? What is a Signal ? 1.2 1.2 Classification of a Signals.Classification of a Signals.

1.2.1 Continuous-Time and Discrete-Time 1.2.1 Continuous-Time and Discrete-Time Signals Signals 1.2.2 Even and Odd Signals.1.2.2 Even and Odd Signals.1.2.3 Periodic and Non-periodic Signals.1.2.3 Periodic and Non-periodic Signals.1.2.4 Deterministic and Random Signals.1.2.4 Deterministic and Random Signals.1.2.5 Energy and Power Signals.1.2.5 Energy and Power Signals.

1.3 1.3 Basic Operation of the Signal.Basic Operation of the Signal.1.4 1.4 Elementary Signals.Elementary Signals.

1.4.1 Exponential Signals.1.4.1 Exponential Signals.1.4.2 Sinusoidal Signal.1.4.2 Sinusoidal Signal.1.4.3 Sinusoidal and Complex Exponential 1.4.3 Sinusoidal and Complex Exponential Signals.Signals.1.4.4 Exponential Damped Sinusoidal 1.4.4 Exponential Damped Sinusoidal Signals.Signals.1.4.5 Step Function.1.4.5 Step Function.1.4.6 Impulse Function.1.4.6 Impulse Function.1.4.7 Ramped Function.1.4.7 Ramped Function.

Signals and Systems.Signals and Systems.

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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.

Cont’d…Cont’d…

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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 or objects around us.

Real life examplesReal life examples of signals;(i) Doctor listening to the heartbeat, blood pressure and temperature of the patient. These represent signals that conveys information about the state of health of the patient.(ii) Weather forecast provides information on the temperature, humidity, and the speed and direction of the prevailing wind. The signals represented by these quantities help us decide whether to stay indoor or doing some outdoor activity.

Indeed , the list of what constitutes a Indeed , the list of what constitutes a signal is almost endless.

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

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By definition, signal is a function of one or signal is a function of one or more variable that conveys information on the more variable that conveys information on the nature of a physical phenomenonnature of a physical phenomenon..

When the function depends on a single variable, the signal is said to be one dimensional. Example of one dimensional signal: A speech signal whose amplitude varies with time, depending on the spoken word and who speaks it.

When the function depends on two or more variables, the signal is said to be multidimensional. Example of multidimensional signal: An image with horizontal and vertical coordinates of the image representing the two dimensions.


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There are five typesfive types of signals;(i) Continuous-Time and Discrete-Time (i) Continuous-Time and Discrete-Time Signals Signals (ii) Even and Odd Signals.(ii) Even and Odd Signals.(iii) Periodic and Non-periodic Signals.(iii) Periodic and Non-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 1.2 Classifications of a Signal.Signal.

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Continuous-Time (CT) SignalsContinuous-Time (CT) Signals Continuous-Time (CT) SignalsContinuous-Time (CT) Signals are functions

whose amplitude or value varies continuously with time, x(t).

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

Example: microphone converts variation in sound pressure (e.g speech) into corresponding variation in voltage and current.

Figure 1.1: Continuous-Time Signal, x(t).

1.2.1 Continuous-Time 1.2.1 Continuous-Time and Discrete-Time and Discrete-Time Signals.Signals.

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Discrete-Time SignalsDiscrete-Time Signals Discrete-Time SignalDiscrete-Time Signal is defined only at discrete

instants of time. Thus, the independent variable has discrete values only, which are usually uniformly spaced.

It is often derived from continuous-time signal by sampling it at a uniform rate. Let Ts denote the sampling period and n denote an integer. The symbol n denotes time for discrete time signal and [ ] is used to denote discrete-value quantities.

Figure 1.12 (a) Continuous-time signal x(t) . b) Representation of x(t) as a discrete-time signal.


,....2,1,0, nnTxnx s

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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

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.

tallfortxtx

tallfortxtx

1.2.2 Even and Odd 1.2.2 Even and Odd Signals.Signals.

Suppose we are given an arbitrary signal x(t).x(t) is a sum of two components of , which is

even function and , which is odd function.For even signal,

For odd signal, Putting t = -t in the expression for x(t), we may

write,

Solving for and , we thus obtain,

and 10

Example : Even and Odd SignalsExample : Even and Odd Signals

)()( txtx ee

)()( txtx oo

)()(

)()()(

txtx

txtxtx

oe

oe

)()(2

1)( txtxtxe )()(

2

1)( txtxtxo

)(txe

)(txo

)(txe )(txo

11


12

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

Find the even and odd components of each of the following signals:

(a) x(t) = 4cos(3πt)

Answer: Answer: ge(t) = 4cos(3πt)go(t) = 0

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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.

Figure 1.6: Aperiodic Signal. Figure 1.7: Periodic Signal.

,tallforTtxtx

1.2.3 Periodic and Non-1.2.3 Periodic and Non-Periodic Signals.Periodic Signals.

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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.

Figure 1.8: Deterministic Signal; Square Wave.

1.2.4 Deterministic and 1.2.4 Deterministic and Random Signals.Random Signals.

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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.

Figure 1.9: Random Signal


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1.2.5 Energy Signal and 1.2.5 Energy Signal and Power Signals.Power Signals.Energy Signal.Energy Signal.

A signal is refer to energy signal if and only if the total energy satisfy the condition;

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

n

nxE 2

E0

P0

17

Figure 1.10: Bounded and Unbounded Signal

1.2.6 Bounded and 1.2.6 Bounded and Unbounded Signals.Unbounded Signals.

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1.3 Basic Operation of 1.3 Basic Operation of the Signals.the Signals.1.3.1 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 1.3.4 Precedence Rule for Time Shifting and Time Scaling.Time Scaling.

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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:

Figure 1.11: 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.

atxty

1.3.1 Time Scaling.1.3.1 Time Scaling.

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In the discrete time,

It is defined for integer value of k, k > 1. Figure below for k = 2, sample for n = +-1,

Figure 1.12: 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.

,knxny


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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 reflected version of x(t) about t = 0.

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

txty

1.3.2 Reflection and 1.3.2 Reflection and Folding.Folding.

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Example 1.2:Example 1.2: Reflection.Reflection.Given the triangular pulse Given the triangular pulse xx((tt), find the reflected ), find the reflected version of version of xx((tt) about the amplitude axis (origin).) about the amplitude axis (origin).Solution:Solution:Replace the variable t with –t, so we get y(t) = x(-t) as in figure below.

Figure 1.13: 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.

.

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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 time axis.

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

Example:

Figure 1.14: Shift to the Left. Figure 1.15: Shift to the Right.

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

1.3.3 Time Shifting.1.3.3 Time Shifting.

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

and unit duration. Find and unit duration. Find yy((tt)=)=x (t - 2)x (t - 2)

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

time units.

Figure 1.16: 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. .

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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,

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.

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

batxty

atvty

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Example 1.4:Example 1.4: Continuous Signal. Continuous Signal. A CT signal is shown in Figure 1.17 below, sketch A CT signal is shown in Figure 1.17 below, sketch and label each of this signal;and label each of this signal;

a) a) x(tx(t -1) -1) b) b) x(2t)x(2t)c) c) x(-t)x(-t)

Figure 1.17

-1 3

2

t

x(t)

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

(c) x(-t)

-3 1

2

t

x(-t)

0 4

t

x(t-1)

2

-1/2 3/2

2

t

x(t)

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Example 1.5:Example 1.5: Discrete Time Signal. Discrete Time Signal. A discrete-time signal x[n] is shown below, A discrete-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 – 2](a) x[n – 2] (b) x[2n](b) x[2n]

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

x[n]

n

4

2

0 1 2 3

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

A delay by 2

4

2

0 1 2 3 4 5 n

x[n-2]


30

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

Time scaling by a factor of 2.

4

2

0 1 2 3 n

x(2n)


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

Time shifting and reflection

4

2

-1 0 1 2 n

x(-n+2)


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

Reflection

4

2

-3 -2 -1 0 1 n

x(-n)


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

(a) (a) x(t – 2)x(t – 2) (b) (b) x(2t)x(2t) (c.) (c.) x(t/2)x(t/2) (d) (d) x(-t)x(-t)

x(t)

t

4

0 4

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1.4 Elementary 1.4 Elementary Signals.Signals. There are many types of signalstypes of signals prominently used

in the study of signals and systems.1.4.1 Exponential Signals.1.4.2 Exponential Damped Sinusoidal Signals.1.4.3 Step Function.1.4.4 Impulse Function.1.4.5 Ramp Function.

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A real exponential signal, is written as x(t) = Beat. Where both B and a are real parameters. B is the

amplitude of the exponential signal measured at time t = 0.(i) Decaying exponential, for which a < 0.(ii) Growing exponential, for which a > 0.

Figure 1.18: (a) Decaying exponential form of continuous-time signal. (b) Growing exponential form of

continuous-time signal.

Figure 1.19: (a) Decaying exponential form of discrete-time signal.

(b) Growing exponential form of discrete-time signal.

1.4.1 Exponential Signals.1.4.1 Exponential Signals.

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Continuous-Time. Continuous-Time. Case a = 0: Constant signal x(t) =C. Case a > 0: The exponential tends to infinity as t infinity.

Case a > 0 Case a < 0

Case a < 0: The exponential tend to zero as t infinity

(here C > 0).


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Discrete-Time. Discrete-Time.

where B and are real.There are six cases to consider apart from = 0. Case 1 (= 0): Constant signal x[n]=B.Case 2 (> 1): positive signal that grows

exponentially.

Case 3 (0 < < 1): The signal is positive and decays exponentially.

n Benx


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Case 4 (a < 1): The signal alternates between positive and negative values and grows exponentially.

Case 5 (a = -1): The signal alternates between +C and -C.

Case 6 (-1 < a <0): The signal alternates between positive and negative values and decays exponentially.


39

A general form of sinusoidal signal is

where A is the amplitude, is the frequency in radians per second, and is the phase angle in radians.

Figure 1.20: Continuous-Time Sinusoidal signal A cos(ω0t + θ).

tAtx ocos

1.4.2 Sinusoidal Signals.1.4.2 Sinusoidal Signals.

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Discrete time version of sinusoidal signal, written as

Figure 1.21: Discrete-Time Sinusoidal Signal A cos(ωt + Φ).


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Complex exponential, Euler’s Identity, Complex exponential signal, Where, Hence,

Thus, in terms of real and imaginary parts;

1.4.3 Sinusoidal and 1.4.3 Sinusoidal and Complex Exponential Complex Exponential Signals.Signals.

sincos je j

je

tjBe

jAeB

tjAtAcos

Ae

eAeBetj

tjjtj

sin

tj

tj

BetA

BetA

Imsin

Recos

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Continuous time sinusoidal signals,

In the discrete time case,

tj

tj

BetASin

BetACos

tASintx

Im

Re

1.4.3 Sinusoidal and 1.4.3 Sinusoidal and Complex Exponential Complex Exponential Signals.Signals.

tj

tj

BenASin

BenACos

Im

Re

43

Figure 1.22: Complex plane, showing eight points uniformly distributed on the unit circle.


44

Multiplication of a sinusoidal signal by a real-value decaying exponential signal result in an exponential damped sinusoidal signal.

Where ASin(t + ) is the continuous signal and e-t is the exponential

Figure 1.23: Exponentially damped sinusoidal signal Ae-at sin(ωt), with A = 60 and α = 6.

Observe that in Figure 1.23, an increased in time t, the amplitude of the sinusoidal oscillation decrease in an exponential fashion and finally approaching zero for infinite time.

0,sin tAetx t

1.4.4 Exponential Damped 1.4.4 Exponential Damped Sinusoidal Signals.Sinusoidal Signals.

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The discrete-time version of the unit-step function is defined by,

Figure 1.24: Discrete–time of Step Function of Unit Amplitude.

0

0

,0

,1

n

nnu

1.4.5 Step Function.1.4.5 Step Function.

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The continuous-time version of the unit-step function is defined by,

Figure 1.25: Continuous-time of step function of unit amplitude.

The discontinuity exhibit at t = 0 and the value of u(t) changes instantaneously from 0 to 1 when t = 0. That is the reason why u(0) is undefined.

0

0

,0

,1

t

ttu


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The discrete-timediscrete-time version of the unit impulse is defined by,

Figure 1.26: Discrete-Time form of Impulse.

Figure 1.41 is a graphical description of the unit impulse (t).

The continuous-time version of the unit impulse is defined by the following pair,

The (t) is also refer as the Dirac Delta function.

0,0

0,1

n

nn

1

00

dtt

and

tfort

1.4.6 Impulse Function.1.4.6 Impulse Function.

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Figure 1.27 is a graphical description of the continuous-time unit impulse (t).

Figure 1.27: (a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of

strength a that results from allowing the duration Δ of a rectangular pulse of area a to approach zero.

The duration of the pulse, (t) decreased and its amplitude is increased. The area under the pulse is maintained constant at unity.


49


50

The integral of the step function u(t) is a ramp function of unit slope.

or

Figure 1.28: Ramp Function of Unite Slope.

The discrete-time version of the ramp function,

Figure 1.29: Discrete-Time Version of the Ramp Function.

0,0

0,

t

tttr

ttutr

0,0

0,

n

nnnr

1.4.7 Ramp Function.1.4.7 Ramp Function.

nnunr

51

Successive Integration of Unit Impulse Successive Integration of Unit Impulse Function.Function.

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A system can be viewed as an interconnection of interconnection of operationoperation that transfer an input input signal into an outputoutput signal with properties different from those of the input signal.

y(t) is the impulse response of the continuous-time system and y[n] is the impulse response of the discrete-time system.

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

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Real life example of system;(i) In automatic speaker recognition system; the system is to extract the information from an incoming speech signal for the purpose of recognizing and identifying the speaker. (ii) In communication system; the system will transport the the information contained in the message over a communication channel and deliver that information to the destination.

Figure 1.30: Elements of a communication system.

Figure 1.31: Block diagram representation of a system.


54

By definition, a system is an entity that a system is an entity that manipulates one or more signals to manipulates one or more signals to accomplish a function, thereby yielding new accomplish a function, thereby yielding new signals.signals.

A physical process or a mathematical model of the physical process that relates a set of input signals to yield another set of output signal.

Process input signals to produce output signals System representation of the systems.


55

Mechanical free-body diagram.Physically divergent systems can have similar dynamic

properties.

(I) Dynamic Analogies.(I) Dynamic Analogies.

56

The mechanical and electrical systems are dynamically analogous.

Thus, understanding one of these systems gives insights into the other.

(II) Circuit Sum Element (II) Circuit Sum Element Current.Current.

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Electronic synthesis of block diagram The integrator, adder, and gain blocks are other examples of functional descriptions of systems. We can produce a structural modelstructural model of each of these blocks. For example, the gain block is easily synthesized with an op-amp circuit.

(III) Block Diagram Using (III) Block Diagram Using Integrators, Adders and Gain.Integrators, Adders and Gain.

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System may be interconnectionsinterconnections of other system.

CascadeCascade interconnection.

ParallelParallel interconnection.

FeedbackFeedback interconnection.

Eg. Car cruise control system.

1.5.1 System Block 1.5.1 System Block Diagram.Diagram.

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The properties of a system describe the characteristics of the operator H representing the system.

Basic 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 Inevertibility.1.6.4 Inevertibility.

1.6.5 Time Invariance.1.6.5 Time Invariance.

1.6.6 Linearity.1.6.6 Linearity.

1.6 Properties of Systems.1.6 Properties of Systems.

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A system is said to be bounded-input bounded-output (BIBO) stable if and only if all bounded inputs result in bounded outputs. The output of the system does not diverge if the input does not diverge.

For the resistor, if i(t) is bounded then so is v(t), but for the capacitance this is not true. Consider i(t) = u(t) then v(t) = tu(t) which is unbounded.

1.6.1 Stability.1.6.1 Stability.

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A system is said to possess memory if its output signal depend on pass or future values of the input signal.

Note that v(t) depends not just on i(t) at one point in time t. Therefore, the system that relates v to i exhibits memory.

The system is said to be memoryless if its output signal depends only on the present value of the input signal.

Example: The resistive divider network

Therefore, vo(to) depends upon the value of vi(to) and not on vi(t) for t = to.

1.6.2 Memory.1.6.2 Memory.

62

Example 1.6:Example 1.6: Memory and Memoryless Memory and Memoryless System.System.

Below is the moving-average system described by the Below is the moving-average system described by the input-output relation. Does it has memory or not?input-output relation. Does it has memory or not?

(a)

(b)

Solution:Solution:(a)It has memory, the value of the output signal y[n]

at time n depends on the present and two pass values of x[n].

(b)It is memoryless, because the value of the output signal y[n] depends only on the present value of the input signal x[n].

.

213

1 nxnxnxny

nxny 2

63

Causal.Causal. A system is said to be casual if the present value of the output

signal depends only on the present or the past values of the input signal. The system cannot anticipate the input.

Noncausal.Noncausal. In contrast, the output signal of a noncausal system depends on

one or more future values of the input signal.

1.6.3 Causality.1.6.3 Causality.

64

Example 1.7: Example 1.7: Causal and Noncausal.Causal and Noncausal.

Causal or noncausal?Causal or noncausal?

Solution:Solution:Noncausal; the output signal y[n] depends on a future value of

the input signal, x[n+1]

Causality is required for a system to be capable of operating in real time.

.

113

1 nxnxnxny

113

1 nxnxnxny

65

A system is said to be invertible if the input of the system can be recovered from the output.

Figure 1.32: The notion of system inevitability. The second operator Hinv is the inverse of the first operator H. Hence, the input x(t) is passed through the cascade correction of H

and H-1 completely unchanged.

txHH

txHHtyH

inv

invinv

1.6.4 Inevitability.1.6.4 Inevitability.

66

A system is said to be time invariant if the time delay or time advance of the input signal leads to an identical time shift in the output signal.

The Time invariance system responds identically no mater when the input signal is applied.

Figure 1.33: (a) Time-shift operator St0 preceding operator H. (b) Time-shift operator St0 following operator H. These two situations are equivalent, provided that H is time invariant

HSHS tt 00

1.6.5 Time Invariance.1.6.5 Time Invariance.

67

A system is said to be linear in term of the system input (excitation) x(t) and the system output (response) y(t) if it satisfies the following two two properties.properties.

1. 1. SuperpositionSuperposition The system is initially at rest. The input is

x(t)=x1(t), the output y(t)=y1(t). So x(t)=x1(t)+x2(t) the corresponding output y(t)=y1(t)+y2(t).

2. 2. Homogeneity/ScalingHomogeneity/Scaling The system is initially at rest. Input x(t) result

in y(t). The system exhibit the property of homogeneity if x(t) scaled by constant factor a result in output y(t) is scaled by exact constant a.

1.6.6 Linearity.1.6.6 Linearity.

68

Figure 1.34: The linearity property of a system. (a) The combined operation of amplitude scaling and summation

precedes the operator H for multiple inputs.(b) The operator H precedes amplitude scaling for each input; the resulting outputs are summed to produce the overall output y(t). If these two configurations produce

the same output y(t), the operator H is linear.

If the system violates either of the properties the system is said to be nonlinear.


69


Example 1.8:Example 1.8: Linearity. Linearity.

70

Cont’d…Cont’d…Solution:Solution:

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