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CEN340 Signals and Systems by Dr. Abdulwadood Abdulwaheed

Lecture material courtesy of Dr. Anwar M. Mirza Lecture No. 03Date: June 16, 2012

1SIGNALS AND SYSTEMS

1.0. Introduction1.1. Continuous-Time and Discrete-Time Signals

1.1.1. Examples and Mathematical Representations1.1.2. Signal Power and Energy

1.2. Transformations of the Independent Variable1.2.1. Examples of the Transformations of the Independent Variable1.2.2. Periodic Signals1.2.3. Even and Odd Signals

1.3. Exponential and Sinusoidal Signals1.3.1. Continuous-Time Complex Exponential and Sinusoidal Signals1.3.2. Discrete-Time Complex Exponential and Sinusoidal Signals1.3.3. Periodicity Properties of Discrete-Time Complex Exponentials

1.4. The Unit Impulse and Unit Step Functions1.4.1. The Discrete-Time Unit Impulse and Unit Step Sequences1.4.2. The Continuous-Time Unit Step and Unit Impulse Functions

1.5. Continuous-Time and Discrete-Time Systems1.5.1. Simple Examples of Systems1.5.2. Interconnections of Systems

1.6. Basic System Properties1.6.1. Systems with and without Memory1.6.2. Invertibility and Inverse Systems1.6.3. Causality1.6.4. Stability1.6.5. Time Invariance1.6.6. Linearity

1.7. Summary

Department of Computer Engineering

College of Computer & Information Sciences, King Saud UniversityAr Riyadh, Kingdom of Saudi Arabia



1.3.2 Periodicity Property of Discrete-Time Complex Exponential (Recap)

There are many similarities between continuous-time and discrete-time signals. But also there are many important differences. One of them is related with the discrete-time exponential signal e jω0n.

The following properties were found with regard to the continuous-time exponential signal e jω0 t

:

1. The larger the magnitude of ω0, the higher is the rate of oscillations in the signal;2. e jω0 t is periodic for any value of ω0.

Let us now see how these properties are different in the discrete-time case:

1. To see the difference for the first property, consider the discrete-time complex exponential:

e j (ω¿¿0+2π )n=e j 2πn e j ω0n=e jω0 n¿ (1.51)

This shows that the exponential at ω0+2π is the same as that at frequency ω0. This is very different when compared with the continuous-time exponential case, in which the signals e jω0 t are all distinct for distinct values of ω0.

In discrete-time, these signals are not distinct. In fact, the signal with frequency ω0 is identical to signals with frequencies ω0±2π , ω0±4 π and so on. Therefore, in considering discrete-time complex exponentials, we need only consider a frequency interval of size 2π . The most commonly used 2π intervals are 0≤ω0≤2π or the interval −π ≤ω0≤ π.

Due to equation 1.51, as ω0 is gradually increased, the rate of oscillations in the discrete-time signal does not keep on increasing. If ω0 is increased from 0 to 2π , the rate of oscillations first increase and then decreases. This is shown in the figure 1.27 below.

Note in particular that for ω0=π or for any odd multiple of π,

e jπn=(e jπ )n=(−1 )n (1.52)so that the signal oscillates rapidly, changing sign at each point in time (see figure 1.27).





2. The second property is the periodicity of the discrete-time complex exponential signals.

e jω0 (n+N )=e j ω0n (1.53)Or equivalently,

e jω0 N=1 (1.54)Equation 1.54 could only be true if ω0N is a multiple of 2π

ω0N=2πm (1.55)Or

ω0

2π=m

N(1.56)

0 10 20 300

0.5

1x[n] = cos(0*n)

0 10 20 30-1

0

1x[n] = cos( n/8)

0 10 20 30-1

0

1x[n] = cos( n/4)

0 10 20 30-1

0

1x[n] = cos( n/2)

0 10 20 30-1

0

1x[n] = cos( n)

0 10 20 30-1

0

1x[n] = cos(3 n/2)

0 10 20 30-1

0

1x[n] = cos(7 n/4)

0 10 20 30-1

0

1x[n] = cos(15 n/8)

0 10 20 300

0.5

1x[n] = cos(2 n)

Figure 1.27: Discrete-time sinusoidal sequences for several different frequencies





This equation (1.56) means that the discrete-time signal e jω0n is periodic only when ω0

2π is a rational number (i.e. it could written as a fraction).

Table 1.1 Comparison of the signals e jω0 t and e jω0n.

e jω0 t e jω0n

Distinct signals for distinct values of ω0. Identical signals for values of ω0 separated by multiples of 2π .

Periodic for any choice of ω0. Periodic only if ω0=2πm /N for some integer N>0 and m.

Fundamental frequency ω0. Fundamental frequency ω0/m.Fundamental period

ω0=0: undefined

ω0≠0 :2πω0

Fundamental periodω0=0: undefined

ω0≠0 :m( 2πω0 )

1.4 The Unit Impulse and Unit Step Functions1.4.1 The Discrete-Time Unit Impulse and Unit-Step

Sequences

One of the simplest discrete-time function is the unit impulse or unit sample, denoted by δ [n ]. It is defined as,

δ [n ]={0 ,1, n≠0n=0(1.63)

This is shown in the figure below,





Figure 1.28: Discrete-time Unit Impulse (sample)

0 n

1[n]

A second basic discrete-time signal is the discrete-time unit step, denoted by u [n ] and defined by

u [n ]={0 ,1 , n<0n≥0(1.64)

It is shown in the following figure.

Figure 1.29: Discrete-time Unit Step Sequence

0 n

1u[n]

There is a close relationship between the discrete-time unit impulse and unit step signals. The discrete-time unit impulse can be written as the first-difference of the discrete-time unit step

δ [n ]=u [n ]−u[n−1] (1.65)Conversely, the discrete-time unit step is the sunning sum of the unit sample.

u[n]= ∑m=−∞

n

δ [m ](1.66)





This is illustrated graphically in Figure 1.30 below. It is clear from the figure that the running sum of equation 1.66, is 0 for n<0 and 1 for n≥0.

By changing the variable (summation index) to k=n−m, equation 1.66 can be written as

u[n]=∑k=∞

0

δ [n−k ]

Or equivalently

u[n]=∑k=0

∞

δ [n−k ](1.67)

This is illustrated in figure 1.31. In this case, the nonzero value of δ [n−k ] is at the value of k equal to n. Therefore, from equation 1.67, we see that the summation is 0 for n<0 and 1 for n≥0.

Figure 1.30 Running Sum of Eq. 1,66

0 m

[m]

Case (a): n < 0

n

Interval of summation

0 m

[m]

n

Case (b): n > 0


Another interpretation of equation 1.67 is that the unit step is the superposition of delayed unit-impulses, i.e., we can view equation 1.67 as the sum of unit impulse δ [n ] at n=0, a unit impulse δ [n−1 ] at n=1, another, δ [n−2 ] at n=2, etc.



u(t)

1

0 t

Figure 1.32: Continuous-time unit step function



Figure 1.31 Relationship given in Eq. (1.67)

0 k

[n-k]

Case (a): n < 0

n


0 k

[n-k]

n

Case (b): n > 0


The unit impulse sequence can be used to sample the value of a signal at n=0. In particular, since δ [n ] is non-zero (and equal to 1) only for n=0, therefore

x [n ]δ [n ]=x [0 ]δ [n ] (1.68)More generally, if we consider a unit impulse δ [n−n0 ] at n=n0, then

x [n ]δ [ n−n0 ]=x [n0]δ [n−n0 ] (1.69)

1.4.2 The Continuous-Time Unit Step and Unit Impulse Functions

The continuous-time unit step function, denoted by u(t ) is defined by

u ( t )={0 ,1 , t<0t ≥0(1.70)

It is shown in the figure (1.322) below. It could be noticed that this function is discontinuous at t=0.



Figure 1.33: Continuous-time approximation to the unit step function,

1

0 t 0 t

Figure 1.34: Derivative of



The continuous-time unit impulse function δ (t) is related to the unit step in a manner similar to the way the discrete-time unit impulse is related to the unit step. In discrete-time case, the unit step is the running sum of the unit impulse. In the continuous case, it is the running integral instead of the running sum. So the unit step can be written as the running integral of the unit impulse,

u(t )=∫−∞

t

δ (τ )dτ (1.71)

Also analogous to equation (1.65), where the discrete-time unit impulse is represented as the first difference of unit step and its delayed version, we write

δ (t)=du (t)dt

(1.72)

that is, the unit impulse in the continuous-time can be written as the first derivative of the unit step in continuous time.

However, we notice that u(t ) is discontinuous at t=0 (and consequently cannot be differentiated at t=0), therefore, there is some formal difficulty with this equation.

However, we interpret equation (1.72) by considering an approximation to the unit step u∆ (t) (see figure 1.33 below), in which the function rises from 0 to 1 in a short time interval of length ∆. The step function u(t ) can be considered as an idealization of u∆ (t) for ∆ so short that its duration doesn’t matter for any practical purpose. More formally, u(t ) is the limit of u∆ (t) as ∆→0. Now the derivative of equation (1.72) can be written as

δ∆(t )=du∆(t)dt

(1.73)

as shown in figure (1.34).



Figure 1.35: Continuous-time unit impulse

1

0 t

Figure 1.36: Continuous-time scaled impulse

k

k

0 t

(b)Figure 1.37: Running Integral given in equation 1.71. (a) ; (b) .

1

0

Interval of integration

1

0

Interval of integration



It could be noticed that δ∆(t ) is short pulse of duration ∆ and with unit area for any value of ∆. If we gradually decrease the value of ∆, the pulse will become narrower and the height will increase (to maintain the area to unity). Therefore, in the limiting case, we can write

δ (t )=lim∆→0

δ∆ ( t ) (1.73)Since, δ (t ) represents a pulse with no duration and unit area, we adopt the graphical representation as given in figure 1.35.

In general, a scaled impulse k δ ( t ) will have an area , k , and

∫τ=−∞

τ=t

kδ (τ )dτ=ku(t)

Also, the continuous-time step function can be written as the running integral

u(t )=∫0

∞

δ ( τ−σ )dσ (1.75)



Continuous-timesystem

Discrete-timesystem



Also for any continuous-time signal x (t), we can write

x (t ) δ (t−t 0 )=x (t 0 )δ( t−t0) (1.76)This is called the sampling property of the continuous-time impulse function.

1.5 Continuous-Time and Discrete-Time Systems

A continuous-time system is a system in which continuous-time input signals are applied and result in continuous-time output signals. The input-output relation of such systems can be represented by the notation:

x (t)→y ( t) (1.78)This is shown in the figure below,

A discrete-time system is a system in which discrete-time input signals are applied and result in discrete-time output signals. The input-output relation of such systems can be represented by the notation:

x [n]→y [n] (1.79)This is shown in the figure below,



Figure 1.1: A simple RC circuit with source VS and capacitor voltage VC.

R

C VCVS

Figure 1.2: An automobile responding to an applied force f from the engine and to a retarding force v proportional to the automobile’s velocity v.

fv



1.5.1 Simple Examples of Systems

It has been noticed that systems from many different applications have very similar mathematical descriptions. This serves as an important motivation for developing general tools for systems analysis and design. We illustrate this point with some simple examples.

Example 1.8Consider the RC circuit of Figure 1.1 (shown in the following).

According to Ohm’s law, the current i(t) through the resistor R is proportional (with proportionality constant 1/R) to the voltage drop (vs ( t )−vc (t)) across the resistor, i.e.,

i(t)=vs (t )−vc (t)

R(1.80)

Similarly, the current is related with the rate of change with time of the voltage across the capacitor, i.e.,

i(t)=Cd vc( t)dt

(1.81)

Equating the right-hand sides of equations (1.80) and (1.81), we obtain the differential equation giving a relationship between the input vs ( t ) and the output vc ( t ),

d vc (t )dt

+ 1RC

vc (t )= 1RC

v s(t )(1.82)

Example 1.9Consider the RC circuit of Figure 1.1 (shown in the following).





Here we regard the force f (t) as the input and velocity v (t) as the output. If we let m denote the mass of the automobile and mρv the resistance due to friction, then equating acceleration i.e. the time derivative of velocity, with net force divided by mass, we get

dv (t)dt

= 1m [ f ( t )−ρv (t) ] (1.83)

i.e.,dv (t)dt

+ ρm

v (t)= 1m

f (t ) (1.84)

Examining and comparing equations (1.82) and (1.84) in the above examples, we notice that the input-output relationship of these entirely different systems are basically the same. In fact, both of them are example of the first-order linear differential equations of the form

dy (t)dt

+ay ( t )=bx (t) (1.85)

where x (t) is the input, y (t ) is the output, and a and b are constants. Therefore, if we could develop methods for analyzing general classes of systems (such as represented by equation 1.85), we will be able to use them in a wide variety of applications.

Example 1.10As a simple example of a discrete-time system, consider a simple model for the balance in a bank account from month to month. Specifically, let y [n] denote the balance at the end of the n-th month, and suppose that y [n] evolves from month to month according to the equation,

y [n ]= y [n−1 ]+0.1 y [n−1]+x [n ]=1.01 y [n−1]+x [n] (1.86)Or equivalently

y [n ]−1.01 y [n−1 ]=x [n] (1.87)Where x [n] represents the net deposit (i.e., deposits minus withdrawals) during the month and the term 1.01 y [n−1] is the 1% profit each month.

Example 1.11As a second example, consider the digital simulation of the differential equation in equation (1.84) in which we resolve the time into discrete intervals of length ∆ and approximate the derivative dv (t )/dt at t=n∆ by the first backward difference, i.e.,



For example, a modern digital telephone system involves the interconnections of a microphone receiver, audio to digital converter, a transmitter, a receiver, a digital to audio convertor and one or more speakers (apart from several other sub-systems).

Series (cascade) interconnection

System 1Input OutputSystem 2



v (n∆ )−v ((n−1)∆)∆

In this case, if we let

v [n ]=v (n∆)

and

f [n ]=f (n∆)

We obtain the following discrete-time model relating the sampled signals f [n ] and v [n ]:

v [n ]− m(m+ ρ∆)

v [n−1]= ∆(m+ ρ∆ )

f [n] (1.88)

Comparing equations (1.87) and (1.88) we see that they both are examples of the same first-order linear difference equation, namely,

y [n ]+ay [n−1]=bx [n] (1.89)

1.5.2 Interconnections of Systems

Many real systems are built as interconnections of several subsystems.

One can construct a variety of system interconnections. However, there are several basic system interconnections that are encountered more frequently. They are shown in the following figures.



Parallel Interconnection

Input Output

System 1

System 2

Series-Parallel Interconnection

OutputSystem 4System 2

System 3

System 1

Input

System 1

System 2

Input Output



1.6 Basic System Properties1.6.1 Systems with and without Memory

A system is said to be memoryless if its output for each value of the independent variable at a given time is dependent on the input at only that same time.

Feedback Interconnection





For example, the system specified by the relationship

y [n ]=(2 x [n ]− x2 [n ] )2 (1.90)is memoryless, as the value of y [n ] at any particular time n0 depends on the value of x [n ] only at that time, i.e. x [n0 ].

As a particular case, a resistor can be considered as a memoryless system: with the input x (t) taken as the current and with voltage taken as the output y (t ), the input-output relationship for a resistor is,

y (t )=Rx(t ) (1.91)where R is the resistance.

Another particular, simple memoryless system is the identity system, whose output is identical with the input. That is, the input-output relationship for the continuous-time identity system is

y (t )=x (t)and the corresponding relationship in discrete-time is

y [n]=x [n ]

An example of a discrete-time system with memory is an accumulator or summer

y [n]= ∑k=−∞

n

x [k ] (1.92)

and a second example is a delay

y [n]=x [n−1] (1.93) A capacitor is an example of a continuous-time system with memory; since if the

input is taken to be the current and the output is the voltage, then

y (t )= 1C ∫

−∞

t

x (τ )dτ (1.94)

where C is the capacitance.

The concept of memory in a system corresponds to the presence of a mechanism in the system that retains or stores information about the input values at times other than the current time.

The delay of equation (1.93), retains or stores the preceding value of the input. The accumulator of equation (1.92), “remembers” or stores information about

the past inputs.Equation (1.92) can also be written as:



Inversesystem

System

Figure 1.45 (a): Concept of an inverse system for a general invertible system.



y [n ]= ∑k=−∞

n−1

x [k ]+x [n] (1.95)

or equivalently,

y [n ]= y [n−1]+ x [n ] (1.96)This shows that to obtain the output at the current time n, the accumulator must remember the running sum of previous input values (which is exactly the preceding value of the accumulator output).

In many physical systems, the memory is directly associated with the storage of energy.

The capacitor of equation (1.94) stores energy by accumulating electrical charge (represented in the form of integral of the current).

In discrete-time systems implemented with computers or digital microprocessors, memory is typically directly associated with the storage registers that retain values between clock ticks.

Memory dependent on the future values of the input and the output:

Our formal definition of memory-less systems leads us to the conclusion that a system can be considered to have memory, even if the current value of the output is dependent upon the future values of the input and output.

1.6.2 Invertibility and Inverse Systems

A system is said to be invertible if distinct inputs lead to distinct outputs.

As depicted in figure 1.45(a) above, if a system is invertible, than an inverse system exists that, when cascaded with the original system, yields an output w [n ] equal to the input x [n ] to the first system. The overall input-output relationship of the system shown in figure 1.45(a) is that of an identity system.

An example of an invertible continuous-time system is

y (t )=2x (t ) (1.97)



Figure 1.45 (b): Concept of an inverse system for the system described by equation (1.98)

Figure 1.45 (c): Concept of an inverse system for the system described by equation (1.92)



For which the inverse system is,

x (t)=12y ( t) (1.98)

This system is shown in figure 1.45(b) below.

Another example of an invertible system is the accumulator of equation (1.92). The inverse system in this case is shown in figure 1.45(c).

Examples of noninvertible systems are:

A system that produces a zero output sequence for any input sequence.y (t )=0 (1.100)

A system where the output is the square of the inputy (t )=x2( t)

Here we cannot find out about the sign of the input from the knowledge of the output.

(1.101)

The concept of invertibility is important in many applications. One particular example is that of systems used for encoding in a variety of communication systems.





1.7 Basic System Properties1.7.1 Systems with and without Memory

A system is said to be memoryless if its output for each value of the independent variable at a given time is dependent on the input at only that same time.

1.7.2 Invertibility and Inverse Systems

A system is said to be invertible if distinct inputs lead to distinct outputs.

1.7.3 Causality

A system is causal if the output at any time depends on values of the input at only the present and past times.

Such systems are also referred to as non-anticipative, as the system output does not anticipate future values of the input.

The RC circuit of figure 1.1, is causal, since the capacitor voltage depends only on the present and past values of the source voltage.

The motion of an automobile is causal, as it does not anticipate future actions of the driver.



Figure 1.1: A simple RC circuit with source VS and capacitor voltage VC.

R

C VCVS



The systems described by equations (1.92) – (1.94) are also causal. All memoryless systems are causal, since the output responds only to the current

value of the input.

The following systems are not causal:

y [n ]=x [n ]−x [n+1] (1.102)And

y (t )=x (t+1) (1.103)

Causality is not an essential consideration in applications where the independent variable is not time, such as in image processing.

In processing data that have been collected previously, as often is the case with speech, geophysical or meteorological signals etc., we are by no means constrained to causal processing.

An example of noncausal averaging system is

y [n ]= 12M+1 ∑

k=−M

+M

x [n−k ] (1.104)

Example 1.12

It is important to carefully analyze the input-output relationship of a system, while checking the causality. The following two systems can be used as examples.

1. Consider the following system first

y [n ]=x [−n ] (1.105)We note that the output y [n0 ] at a positive time n0 depends only on the value of the input signal x [−n0 ] at time (−n0), which is negative and therefore in the past of n0. This tempts us to infer that the system is causal. But, if we check carefully, we come to know that the system is not causal. Checking for the negative time, e.g. n=−4, we see that y [−4 ]=x [4], so that the output at this time depends on a future value of the input. Hence the system is not causal.





2. It is also important to distinguish carefully the effects of the input from those of any other functions used in the definition of the system. For example

y (t )=x (t ) cos ( t+1) (1.106)In this system, the output at any time equals the input at that same time multiplied with a number that fluctuate with time. Specifically, we can re-write

y (t )=x ( t ) g (t) (1.107)where g(t ) is a time-varying function, namely g (t )=cos (t+1). Thus, only the current value of the input influences the current value of the output, and we conclude that this system is causal (and, also memoryless).

1.6.4 Stability

A system is said to be stable if a small input leads to a response that does not diverge.

There are several examples of stable systems. “Stability of physical systems generally results from the presence of mechanisms that dissipate energy”. For example, in the RC circuit of Example 1.8, the resistor dissipates energy and this circuit is a stable system. The system in Example 1.9 is also stable, because of the dissipation of energy through friction.

More specifically,

If the input to a stable system is bounded (i.e., if its magnitude does not grow without bounds), then the output must also be bounded, and therefore cannot diverge.

This definition of stability, we will use throughout this course.

As an example, consider a system whose input-output relationship is given by Equation (1.104), i.e.,

y [n ]= 12M+1 ∑

k=−M

+M

x [n−k ] (1.104)

If the input x [n] to the system is bounded (say by a number B), for all values of n, then according to Equation (1.104) above, the output y [n] of the system is also bounded by B. This





is because the output y [n] is the average of a finite set of values of the input. Therefore, the output y [n] is bounded and the system is stable.

On the other hand, consider the accumulator described by Equation (1.92).

y [n]= ∑k=−∞

n

x [k ](1.92)

This systems sums all of the past values of the input rather than just a finite set of values, and the system is unstable, since the sun can grow even if the input x [n] is bounded.

Example 1.13

Suppose we suspect that a particular system is unstable, then a useful strategy is to look for a specific bounded input that leads to an unbounded output for that system. To illustrate, let us consider the following two systems

S1 : y (t )=tx (t) (1.109)and

S2 : y ( t )=ex(t) (1.110)Now, for system S1, a constant input x (t)=1 yields y ( t )=t, which is unbounded: since no matter what finite constant input we pick, |y (t )| will exceed that constant for some t . Therefore, the system S1 is unstable.

For system S2, let us the input x (t) be bounded by a positive number B, i.e.

|x (t )<B| (1.111)or

−B<x (t )<B (1.112)for all t . Using the definition of S2 from Equation (1.110), we can write

e−B<|y ( t )|<eB (1.113)The system S2 is therefore, stable.

1.6.5 Time Invariance

A system is said to be time invariant if a time shift in the input signal leads to an identical time shift in the output signal.

That is, if y [n ] is the output of a discrete-time, time-invariant system when x [n ] is the input, then y [n−n0 ] is the output when x [n−n0 ] is applied as an input. In continuous-time when y (t )





is the output corresponding to the input x (t), a time-invariant system will have y (t−t 0) as the output when x (t−t 0) is the input.

Example 1.14

Consider the continuous-time system defined by,

y ( t )=sin [ x (t)] (1.114)To check this system is time invariant, we must determine whether the time-invariance property holds for any input and any time shift t 0. Thus, let x (t ) be an arbitrary input to this system, and let

y1 ( t )=sin [ x1(t )] (1.115)to be the corresponding output. Then, consider a second input obtained by shifting x1(t) in time:

x2 ( t )=x1 ( t+t 0 ) (1.116)The corresponding output to this new input (according to input-output relationship given by Equation (1.114)) is:

y2 (t )=sin [ x2 (t ) ]=sin [ x1 ( t+t 0 ) ] (1.117)Similarly, from Equation (1.115),

y1 (t )=sin [ x1 (t+t 0 ) ] (1.118)

From Equations (1.117) and (1.118), we see that y2 ( t )= y1 (t+t 0 ), and therefore, the system is time invariant.

Example 1.15

Consider now the discrete-time system defined by,

y [n ]=n x [n ] (1.119)Suppose, we consider the input signal x1 [n ]=δ [n], which yields an output y1 [n ]=0 (since nδ [n ]=0¿. However, the input x2 [n ]=δ [n−1] yields the output y2 [n ]=nδ [n−1 ]=δ [n−1]. Thus, while x2 [n ] is a shifted version of x1 [n ], y2 [n ] is not a shifted version of y112is not a shifted versionof t signal variant .cording ¿ input−output relationship givenby Equation(1.114 )¿ is :d any time shi¿ [n ].

1.6.6 Linearity





A linear system, in continuous-time or discrete-time, is a system that possesses the important property of superposition: If an input consists of the weighted sum of several signals, then the output is the superposition – that is, the weighted sum – of the responses of the system to each of those signals.

Lety1 ( t ) be the response of a continuous-time system to an inputx1(t), and lety2 (t ) be the response of a continuous-time system to an inputx2(t). Then the system is linear if,

1. The response tox1 ( t )+x2 ( t ) is y1 ( t )+ y2 ( t ).2. The response toax1 (t ) is ay1 (t ), where a is any complex constant.

The first of these two properties is called the additivity property and the second is known as the scaling or homogeneity property.

The two properties defining a linear system can be combined into a single statement:

Continuous-time: a x1 ( t )+b x2 ( t )=a y1 (t )+b y2 (t ) (1.121)Discrete-time: a x1 [n ]+b x2 [n ]=a y1 [n ]+b y2 [n ] (1.122)

Here a and b are complex constants.

Furthermore, it is straightforward to show from the definition of linearity that if xk [n ], k=1,2,3…, are a set of inputs to a discrete-time linear system with corresponding outputs yk [n ], k=1,2,3…, then the response to a linear combination of these inputs given by

x [n ]=∑kak xk [n ]=a1 x1 [n ]+a2 x2 [n ]+a3 x3 [n ]+… (1.123)

is

y [n ]=∑kak y k [n ]=a1 y1 [n ]+a2 y2 [n ]+a3 y3 [n ]+… (1.124)

This very important fact is known as the superposition property, which holds for linear systems in both continuous and discrete time.

Example 1.17

Consider a system S whose input x (t ) and output y (t ) are related by

y (t )=tx (t )

To determine whether or not S is linear, we consider two arbitrary inputs x1 ( t ) andx2 (t ),

x1 ( t )→ y1 (t )= t x1 ( t )

x2 (t )→ y2 ( t )=t x2 (t )





Letx3 ( t ) be a linear combination ofx1 ( t ) andx2 (t ). That is,

x3 ( t )=ax1 (t )+b x2 ( t )

wherea and b are arbitrary scalars. Ifx3 ( t ) is the input toS, then the corresponding output y3 ( t ) may be expressed as:

y3 ( t )=t x3 ( t )

¿ t (a x1 (t )+b x2 ( t ))

¿at x1 ( t )+bt x2 (t )

¿a y1 ( t )+b y2 ( t )

We conclude that the system S is linear.

Example 1.18

Let now consider another system S whose input x (t ) and output y (t ) are related by

y ( t )=x2 (t )

Like the previous example, to determine whether or not S is linear, we consider two arbitrary inputs x1 ( t ) andx2 (t ),

x1 (t )→ y1 (t )=x12 ( t )

x2 (t )→ y2 (t )=x22 ( t )

Letx3 ( t ) be a linear combination ofx1 ( t ) andx2 (t ). That is,

x3 ( t )=ax1 (t )+b x2 ( t )

wherea and b are arbitrary scalars. Ifx3 ( t ) is the input toS, then the corresponding output y3 ( t ) may be expressed as:

y3 ( t )=x32 ( t )

¿ (ax1 (t )+b x2 (t ) )2

¿a2 x12 ( t )+b2 x2

2 ( t )+2abx1 (t ) x2 (t )

¿a2 y1 (t )+b2 y2 (t )+2ab x1 ( t ) x2 ( t )





which clearly shows that the systemS is not linear.

Example 1.19

While checking the linearity of a system, it is important to keep in mind that the system must satisfy both the additivity and homogeneity properties and that the signals as well the scaling constants are allowed to be complex.

To illustrate these points, let us consider a system S whose input x (t ) and output y ( t ) are related by

y [n ]=Re {x [n ] }

We can show that this system is additive; however, it does not satisfy the homogeneity condition. Let us assume

x1 [n ]=r [n ]+ js [n ]

where r [n ] and s [n ] are the real and imaginary parts of the complex signal x [n ], and that the corresponding output is given by

y1 [n ]=r [n ]

Now we consider the scaling of the complex input with a complex number say a= j, i.e.

x2 [n ]= j x1 [n ]= j (r [n ]+ js [n ] )= jr [n ]−s [n ]

Therefore, the corresponding output y2 [n ] is given by

y2 [n ]=Re {x2 [n ] }=−s

which is not equal to the scaled version of the y1 [n ]

a y1 [n ]= jr [n ]

We conclude that the system violates the homogeneity property, therefore it is not linear.



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