[1.7 CLASSIFICATION OF](#page-7-0) SYSTEMS

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1.7 CLASSIFICATION OF SYSTEMS

Systems may be classified broadly in the following categories:

• 1. Linear and nonlinear systems
• 2. Constant-parameter and time-varying-parameter systems
• 3. Instantaneous (memoryless) and dynamic (with memory) systems
• 4. Causal and noncausal systems
• 5. Continuous-time and discrete-time systems
• 6. Analog and digital systems
• 7. Invertible and noninvertible systems
• 8. Stable and unstable systems

Other classifications, such as deterministic and probabilistic systems, are beyond the scope of this text and are not considered.

1.7-1 Linear and Nonlinear Systems

THE CONCEPT OF LINEARITY

A system whose output is proportional to its input is an example of a linear system. But linearity implies more than this; it also implies the additivity property: that is, if several inputs are acting on a system, then the total effect on the system due to all these inputs can be determined by considering one input at a time while assuming all the other inputs to be zero. The total effect is then the sum of all the component effects. This property may be expressed as follows: for a linear system, if an input x¹ acting alone has an effect y1, and if another input x2, also acting alone, has an effect y2, then, with both inputs acting on the system, the total effect will be y¹ +y2. Thus, if

$$x_1 \longrightarrow y_1$$

and $x_2 \longrightarrow y_2$

then for all x¹ and x²

x¹ +x² −→ y¹ +y² (1.20)

In addition, a linear system must satisfy the homogeneity or scaling property, which states that for arbitrary real or imaginary number k, if an input is increased k-fold, the effect also increases k-fold. Thus, if

x −→ y

^† Strictly speaking, this means independent inductor currents and capacitor voltages.

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then for all real or imaginary k

$$kx \longrightarrow ky \tag{1.21}$$

Thus, linearity implies two properties: homogeneity (scaling) and additivity.† Both these properties can be combined into one property (superposition), which is expressed as follows: If

$$x_1 \longrightarrow y_1$$

and $x_2 \longrightarrow y_2$

then for all inputs x¹ and x² and all constants k¹ and k2,

$$k_1x_1 + k_2x_2 \longrightarrow k_1y_1 + k_2y_2 \tag{1.22}$$

There is another useful way to view the linearity condition described in Eq. (1.22): the response of a linear system is unchanged whether the operations of summing and scaling precede the system (sum and scale act on inputs) or follow the system (sum and scale act on outputs). Thus, linearity implies commutability between a system and the operations of summing and scaling. It may appear that additivity implies homogeneity. Unfortunately, homogeneity does not always follow from additivity. Drill 1.11 demonstrates such a case.

DR ILL 1.11 Additivity but Not Homogeneity

Show that a system with the input x(t) and the output y(t) related by y(t) = Re{x(t)} satisfies the additivity property but violates the homogeneity property. Hence, such a system is not linear. [Hint: Show that Eq. (1.21) is not satisfied when k is complex.]

RESPONSE OF A LINEAR SYSTEM

For the sake of simplicity, we discuss only single-input, single-output (SISO) systems. But the discussion can be readily extended to multiple-input, multiple-output (MIMO) systems.

A system’s output for t ≥ 0 is the result of two independent causes: the initial conditions of the system (or the system state) at t = 0 and the input x(t) for t ≥ 0. If a system is to be linear, the output must be the sum of the two components resulting from these two causes: first, the zero-input response (ZIR) that results only from the initial conditions at t = 0 with the input x(t) = 0 for t ≥ 0, and then the zero-state response (ZSR) that results only from the input x(t) for t ≥ 0 when the initial conditions (at t = 0) are assumed to be zero. When all the appropriate initial conditions are zero, the system is said to be in zero state. The system output is zero when the input is zero only if the system is in zero state.

In summary, a linear system response can be expressed as the sum of the zero-input and zero-state responses:

total response = zero-input response + zero-state response

^† A linear system must also satisfy the additional condition of smoothness, where small changes in the system’s inputs must result in small changes in its outputs [3].

This property of linear systems, which permits the separation of an output into components resulting from the initial conditions and from the input, is called the decomposition property. For the RC circuit of Fig. 1.26, the response y(t) was found to be [see Eq. (1.19) with t⁰ = 0]

$$y(t) = \underbrace{v_C(0)}_{\text{ZIR}} + \underbrace{Rx(t) + \frac{1}{C} \int_0^t x(\tau) d\tau}_{\text{ZSR}} \tag{1.23}$$

From Eq. (1.23), it is clear that if the input x(t) = 0 for t ≥ 0, the output y(t) = vC(0). Hence vC(0) is the zero-input response of the response y(t). Similarly, if the system state (the voltage vC in this case) is zero at t = 0, the output is given by the second component on the right-hand side of Eq. (1.23). Clearly this is the zero-state response of the response y(t).

In addition to the decomposition property, linearity implies that both the zero-input and zero-state components must obey the principle of superposition with respect to each of their respective causes. For example, if we increase the initial condition k-fold, the zero-input response must also increase k-fold. Similarly, if we increase the input k-fold, the zero-state response must also increase k-fold. These facts can be readily verified from Eq. (1.23) for the RC circuit in Fig. 1.26. For instance, if we double the initial condition vC(0), the zero-input response doubles; if we double the input x(t), the zero-state response doubles.

EXAMPLE 1.10 Linearity of Constant-Coefficient Linear Differential Equations

Show that the system described by the equation

$$\frac{dy(t)}{dt} + 3y(t) = x(t)$$

(1.24)

is linear.

Let the system response to the inputs x1(t) and x2(t) be y1(t) and y2(t), respectively. Then

$$\frac{dy_1(t)}{dt} + 3y_1(t) = x_1(t) \qquad \text{and} \qquad \frac{dy_2(t)}{dt} + 3y_2(t) = x_2(t)$$

Multiplying the first equation by k1, the second by k2, and adding them yield

$$\frac{d}{dt}[k_1y_1(t) + k_2y_2(t)] + 3[k_1y_1(t) + k_2y_2(t)] = k_1x_1(t) + k_2x_2(t)$$

But this equation is the system equation [Eq. (1.24)] with

x(t) = k1x1(t)+k2x2(t) and y(t) = k1y1(t)+k2y2(t)

Therefore, when the input is k1x1(t) + k2x2(t), the system response is k1y1(t) + k2y2(t). Consequently, the system is linear. Using this argument, we can readily generalize the result to

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show that a system described by a differential equation of the form

$$a_0 \frac{d^N y(t)}{dt^N} + a_1 \frac{d^{N-1} y(t)}{dt^{N-1}} + \dots + a_N y(t) = b_{N-M} \frac{d^M x(t)}{dt^M} + \dots + b_{N-1} \frac{dx(t)}{dt} + b_N x(t) \quad (1.25)$$

is a linear system. The coefficients ai and bi in this equation can be constants or functions of time. Although here we proved only zero-state linearity, it can be shown that such systems are also zero-input linear and have the decomposition property.

DR ILL 1.12 Linearity of a Differential Equation with Time-Varying Parameters

Show that the system described by the following equation is linear:

$$\frac{dy(t)}{dt} + t^2 y(t) = (2t + 3)x(t)$$

DR ILL 1.13 A Nonlinear Differential Equation

Show that the system described by the following equation is nonlinear:

$$y(t)\frac{dy(t)}{dt} + 3y(t) = x(t)$$

MORE COMMENTS ON LINEAR SYSTEMS

Almost all systems observed in practice become nonlinear when large enough signals are applied to them. However, it is possible to approximate most of the nonlinear systems by linear systems for small-signal analysis. The analysis of nonlinear systems is generally difficult. Nonlinearities can arise in so many ways that describing them with a common mathematical form is impossible. Not only is each system a category in itself, but even for a given system, changes in initial conditions or input amplitudes may change the nature of the problem. On the other hand, the superposition property of linear systems is a powerful unifying principle that allows for a general solution. The superposition property (linearity) greatly simplifies the analysis of linear systems. Because of the decomposition property, we can evaluate separately the two components of the output. The zero-input response can be computed by assuming the input to be zero, and the zero-state response can be computed by assuming zero initial conditions. Moreover, if we express an input x(t) as a sum of simpler functions,

$$x(t) = a_1 x_1(t) + a_2 x_2(t) + \cdots + a_m x_m(t)$$

then, by virtue of linearity, the response y(t) is given by

$$y(t) = a_1 y_1(t) + a_2 y_2(t) + \cdots + a_m y_m(t)$$

where yk(t) is the zero-state response to an input xk(t). This apparently trivial observation has profound implications. As we shall see repeatedly in later chapters, it proves extremely useful and opens new avenues for analyzing linear systems.

For example, consider an arbitrary input x(t) such as the one shown in Fig. 1.27a. We can approximate x(t) with a sum of rectangular pulses of width t and of varying heights. The approximation improves as t → 0, when the rectangular pulses become impulses spaced t seconds apart (with t → 0).† Thus, an arbitrary input can be replaced by a weighted sum of impulses spaced t (t → 0) seconds apart. Therefore, if we know the system response to a unit impulse, we can immediately determine the system response to an arbitrary input x(t) by adding the system response to each impulse component of x(t). A similar situation is depicted in Fig. 1.27b, where x(t) is approximated by a sum of step functions of varying magnitude and spaced t seconds apart. The approximation improves as t becomes smaller. Therefore, if we know the system response to a unit step input, we can compute the system response to any arbitrary input x(t) with relative ease. Time-domain analysis of linear systems (discussed in Ch. 2) uses this approach.

Chapters 4, 5, 6, and 7 employ the same approach but instead use sinusoids or exponentials as the basic signal components. We show that any arbitrary input signal can be expressed as a weighted sum of sinusoids (or exponentials) having various frequencies. Thus a knowledge of the system response to a sinusoid enables us to determine the system response to an arbitrary input x(t).

Figure 1.27 Signal representation in terms of impulse and step components.

^† Here, the discussion of a rectangular pulse approaching an impulse at ^{t →} 0 is somewhat imprecise. It is explained in Sec. 2.4 with more rigor.

1.7-2 Time-Invariant and Time-Varying Systems

Systems whose parameters do not change with time are time-invariant (also constant-parameter) systems. For such a system, if the input is delayed by T seconds, the output is the same as before but delayed by T (assuming initial conditions are also delayed by T). This property is expressed graphically in Fig. 1.28. We can also illustrate this property, as shown in Fig. 1.29. We can delay the output y(t) of a system S by applying the output y(t) to a T second delay (Fig. 1.29a). If the system is time invariant, then the delayed output y(t−T) can also be obtained by first delaying the input x(t) before applying it to the system, as shown in Fig. 1.29b. In other words, the system S and the time delay commute if the system S is time invariant. This would not be true for time-varying systems. Consider, for instance, a time-varying system specified by y(t) = e−^t x(t). The output for such a system in Fig. 1.29a is e−(t−T) x(t − T). In contrast, the output for the system in Fig. 1.29b is e−^t x(t −T).

Figure 1.28 Time-invariance property.

Figure 1.29 Illustration of timeinvariance property.

It is possible to verify that the system in Fig. 1.26 is a time-invariant system. Networks composed of RLC elements and other commonly used active elements such as transistors are time-invariant systems. A system with an input–output relationship described by a linear differential equation of the form given in Ex. 1.10 [Eq. (1.25)] is a linear time-invariant (LTI) system when the coefficients ai and bi of such equation are constants. If these coefficients are functions of time, then the system is a linear time-varying system.

The system described in Drill 1.12 is linear time varying. Another familiar example of a time-varying system is the carbon microphone, in which the resistance R is a function of the mechanical pressure generated by sound waves on the carbon granules of the microphone. The output current from the microphone is thus modulated by the sound waves, as desired.

EXAMPLE 1.11 Assessing System Time Invariance

Determine the time invariance of the following systems: (a) ^y(t)=x(t)u(t) and (b) ^y(t)^{= d} dt x(t).

(a) In this case, the output equals the input for t ≥ 0 and is otherwise zero. Clearly, the input is being modified by a time-dependent function, so the system is likely time variant. We can prove that the system is not time invariant through a counterexample. Letting x1(t) = δ(t+1), we see that y1(t) = 0. However, x2(t) = x1(t−2) = δ(t−1) produces an output of y2(t) = δ(t − 1), which does equal y1(t − 2) = 0 as time-invariance would require. Thus, y(t) = x(t)u(t) is a time variant system.

(b) Although it appears that x(t) is being modified by a time-dependent function, this is not the case. The output of this system is simply the slope of the input. If the input is delayed, so too is the output. Applying input ^x(t) to the system produces output ^y(t) ^{= d} dt x(t); delaying this output by ^T produces ^y(^{t − T}) ^{= d d}(t−T) ^x(^{t − T}) ^{= d} dt x(t − T). This is just the output of the system to a delayed input x(t − T). Since the T-delayed output of the system to input x(t) equals the output of the system to the T-delayed input x(t −T), the system is time invariant.

DR ILL 1.14 A Time-Variant System

Show that a system described by the following equation is a time-varying-parameter system:

$$y(t) = (\sin t)x(t-2)$$

[Hint: Show that the system fails to satisfy the time-invariance property.]

1.7-3 Instantaneous and Dynamic Systems

As observed earlier, a system’s output at any instant t generally depends on the entire past input. However, in a special class of systems, the output at any instant t depends only on its input at that

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instant. In resistive networks, for example, any output of the network at some instant t depends only on the input at the instant t. In these systems, past history is irrelevant in determining the response. Such systems are said to be instantaneous or memoryless systems. More precisely, a system is said to be instantaneous (or memoryless) if its output at any instant t depends, at most, on the strength of its input(s) at the same instant t, and not on any past or future values of the input(s). Otherwise, the system is said to be dynamic (or a system with memory). A system whose response at t is completely determined by the input signals over the past T seconds [interval from (t−T) to t] is a finite-memory system with a memory of T seconds. Networks containing inductive and capacitive elements generally have infinite memory because the response of such networks at any instant t is determined by their inputs over the entire past (−∞,t). This is true for the RC circuit of Fig. 1.26.

EXAMPLE 1.12 Assessing System Memory

Determine whether the following systems are memoryless: (a) y(t − 1) = 2x(t − 1), (b) ^y(t) ^{= d} dt x(t), and (c) y(t) = (t −1)x(t).

(a) In this case, the output at time t −1 is just twice the input at the same time t −1. Since the output at a particular time depends only on the strength of the input at the same time, the system is memoryless.

(b) Although it appears that the output y(t) at time t depends on the input x(t) at the same time t, we know that the slope (derivative) of x(t) cannot be determined solely from a single point. There must be some memory, even if infinitesimally small, involved. This is confirmed by using the fundamental theorem of calculus to express the system as

$$y(t) = \lim_{T \to 0} \frac{x(t) - x(t - T)}{T}$$

Since the output at a particular time depends on more than just the input at the same time, the system is not memoryless.

(c) The output y(t) at time t is just the input x(t) at the same time t multiplied by the (time-dependent) coefficient t − 1. Since the output at a particular time depends only on the strength of the input at the same time, the system is memoryless.

1.7-4 Causal and Noncausal Systems

A causal (also known as a physical or nonanticipative) system is one for which the output at any instant t⁰ depends only on the value of the input x(t) for t ≤ t0. In other words, the value of the output at the present instant depends only on the past and present values of the input x(t), not on its future values. To put it simply, in a causal system the output cannot start before the input is applied. If the response starts before the input, it means that the system knows the input in the

Figure 1.30 Input–output of a noncausal system and the causal output achieved by delay.

future and acts on this knowledge before the input is applied. A system that violates the condition of causality is called a noncausal (or anticipative) system.

Any practical system that operates in real time† must necessarily be causal. We do not yet know how to build a system that can respond to future inputs (inputs not yet applied). A noncausal system is a prophetic system that knows the future input and acts on it in the present. Thus, if we apply an input starting at t = 0 to a noncausal system, the output would begin even before t = 0. For example, consider the system specified by

$$y(t) = x(t-2) + x(t+2)$$

\n(1.26)

For the input x(t) illustrated in Fig. 1.30a, the output y(t), as computed from Eq. (1.26) (shown in Fig. 1.30b), starts even before the input is applied. Equation (1.26) shows that y(t), the output at t, is given by the sum of the input values 2 seconds before and 2 seconds after t (at t − 2 and t + 2, respectively). But if we are operating the system in real time at t, we do not know what the value of the input will be 2 seconds later. Thus it is impossible to implement this system in real time. For this reason, noncausal systems are unrealizable in real time.

EXAMPLE 1.13 Assessing System Causality

Determine whether the following systems are causal: (a) y(t) = x(−t), (b) y(t) = x(t + 1), and (c) y(t +1) = x(t).

^† In real-time operations, the response to an input is essentially simultaneous (contemporaneous) with the input itself.

(a) Here, the output is a reflection of the input. We can easily use a counterexample to disprove the causality of this system. The input x(t) = δ(t − 1), which is nonzero at t = 1, produces an output y(t) = δ(t + 1), which is nonzero at t = −1, a time 2 seconds earlier than the input! Clearly the system is not causal.

(b) In this case, the output at time t depends on the input at future time of t + 1. Clearly the system is not causal.

(c) In this case, the output at time t + 1 depends on the input one second in the past, at time t. Since the output does not depend on future values of the input, the system is causal.

WHY STUDY NONCAUSAL SYSTEMS?

The foregoing discussion may suggest that noncausal systems have no practical purpose. This is not the case; they are valuable in the study of systems for several reasons. First, noncausal systems are realizable when the independent variable is other than “time” (e.g., space). Consider, for example, an electric charge of density q(x) placed along the x axis for x ≥ 0. This charge density produces an electric field E(x) that is present at every point on the x axis from x = −∞ to ∞. In this case the input [i.e., the charge density q(x)] starts at x = 0, but its output [the electric field E(x)] begins before x = 0. Clearly, this space-charge system is noncausal. This discussion shows that only temporal systems (systems with time as independent variable) must be causal to be realizable. The terms “before” and “after” have a special connection to causality only when the independent variable is time. This connection is lost for variables other than time. Nontemporal systems, such as those occurring in optics, can be noncausal and still realizable.

Moreover, even for temporal systems, such as those used for signal processing, the study of noncausal systems is important. In such systems we may have all input data prerecorded. This often happens with speech, geophysical, and meteorological signals, and with space probes. In such cases, the input’s future values are available to us. For example, suppose we had a set of input signal records available for the system described by Eq. (1.26). We can then compute y(t) since, for any t, we need only refer to the records to find the input’s value 2 seconds before and 2 seconds after t. Thus, noncausal systems can be realized, although not in real time. We may therefore be able to realize a noncausal system, provided we are willing to accept a time delay in the output. Consider a system whose output yˆ(t) is the same as y(t) in Eq. (1.26) delayed by 2 seconds (Fig. 1.30c), so that

$$\hat{y}(t) = y(t-2) = x(t-4) + x(t)$$

Here the value of the output yˆ at any instant t is the sum of the values of the input x at t and at the instant 4 seconds earlier [at (t − 4)]. In this case, the output at any instant t does not depend on future values of the input, and the system is causal. The output of this system, which is yˆ(t), is identical to that in Eq. (1.26) or Fig. 1.30b except for a delay of 2 seconds. Thus, a noncausal system may be realized or satisfactorily approximated in real time by using a causal system with a delay.

A third reason for studying noncausal systems is that they provide an upper bound on the performance of causal systems. For example, if we wish to design a filter for separating a signal from noise, then the optimum filter is invariably a noncausal system. Although unrealizable, this

Noncausal systems are realizable with time delay!

noncausal system’s performance acts as the upper limit on what can be achieved and gives us a standard for evaluating the performance of causal filters.

At first glance, noncausal systems may seem to be inscrutable. Actually, there is nothing mysterious about these systems and their approximate realization through physical systems with delay. If we want to know what will happen one year from now, we have two choices: go to a prophet (an unrealizable person) who can give the answers instantly, or go to a wise man and allow him a delay of one year to give us the answer! If the wise man is truly wise, he may even be able, by studying trends, to shrewdly guess the future very closely with a delay of less than a year. Such is the case with noncausal systems—nothing more and nothing less.

DR ILL 1.15 A Noncausal System

Show that a system described by the following equation is noncausal:

$$y(t) = \int_{t-5}^{t+5} x(\tau) d\tau$$

Show that this system can be realized physically if we accept a delay of 5 seconds in the output.

1.7-5 Continuous-Time and Discrete-Time Systems

Signals defined or specified over a continuous range of time are continuous-time signals, denoted by symbols x(t), y(t), and so on. Systems whose inputs and outputs are continuous-time signals are continuous-time systems. On the other hand, signals defined only at discrete instants of time t0, t1, t2,…,tn,… are discrete-time signals, denoted by the symbols x(tn), y(tn), and so on, where n is some integer. Systems whose inputs and outputs are discrete-time signals are discrete-time systems. A digital computer is a familiar example of this type of system. In practice, discrete-time signals can arise from sampling continuous-time signals. For example, when the sampling is

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uniform, the discrete instants t0, t1, t2, … are uniformly spaced so that

$$t_{k+1} - t_k = T \qquad \text{for all } k$$

In such case, the discrete-time signals represented by the samples of continuous-time signals x(t), y(t), and so on can be expressed as x(nT), y(nT), and so on; for convenience, we further simplify this notation to x[n], y[n], …, where it is understood that x[n] = x(nT) and that n is some integer. A typical discrete-time signal is shown in Fig. 1.31. A discrete-time signal may also be viewed as a sequence of numbers …, x[−1], x[0], x[1], x[2], … Thus, a discrete-time system may be seen as processing a sequence of numbers x[n] and yielding as an output another sequence of numbers y[n].

Discrete-time signals arise naturally in situations that are inherently discrete time, such as population studies, amortization problems, national income models, and radar tracking. They may also arise as a result of sampling continuous-time signals in sampled data systems, digital filtering, and the like. Digital filtering is a particularly interesting application in which continuous-time signals are processed by using discrete-time systems, as shown in Fig. 1.32. A continuous-time signal x(t) is first sampled to convert it into a discrete-time signal x[n], which then is processed by the discrete-time system to yield a discrete-time output y[n]. A continuous-time signal y(t) is finally constructed from y[n]. In this manner, we can process a continuous-time signal with an appropriate discrete-time system such as a digital computer. Because discrete-time systems have several significant advantages over continuous-time systems, there is an accelerating trend toward processing continuous-time signals with discrete-time systems.

Figure 1.32 Processing continuous-time signals by discrete-time systems.

1.7-6 Analog and Digital Systems

Analog and digital signals are discussed in Sec. 1.3-2. A system whose input and output signals are analog is an analog system; a system whose input and output signals are digital is a digital system. A digital computer is an example of a digital (binary) system. Observe that a digital computer is a digital as well as a discrete-time system.

1.7-7 Invertible and Noninvertible Systems

A system S performs certain operation(s) on input signal(s). If we can obtain the input x(t) back from the corresponding output y(t) by some operation, the system S is said to be invertible. When several different inputs result in the same output (as in a rectifier), it is impossible to obtain the input from the output, and the system is noninvertible. Therefore, for an invertible system, it is essential that every input have a unique output so that there is a one-to-one mapping between an input and the corresponding output. The system that achieves the inverse operation [of obtaining x(t) from y(t)] is the inverse system for S. For instance, if S is an ideal integrator, then its inverse system is an ideal differentiator. Consider a system S connected in tandem with its inverse Si, as shown in Fig. 1.33. The input x(t) to this tandem system results in signal y(t) at the output of S, and the signal y(t), which now acts as an input to Si, yields back the signal x(t) at the output of Si. Thus, Sⁱ undoes the operation of S on x(t), yielding back x(t). A system whose output is equal to the input (for all possible inputs) is an identity system. Cascading a system with its inverse system, as shown in Fig. 1.33, results in an identity system.

In contrast, a rectifier, specified by an equation y(t) = |x(t)|, is noninvertible because the rectification operation cannot be undone.

Inverse systems are very important in signal processing. In many applications, the signals are distorted during the processing, and it is necessary to undo the distortion. For instance, in transmission of data over a communication channel, the signals are distorted owing to non-ideal frequency response and finite bandwidth of a channel. It is necessary to restore the signal as closely as possible to its original shape. Such equalization is also used in audio systems and photographic systems.

inverse results in an identity system.

EXAMPLE 1.14 Assessing System Invertibility

Determine whether the following systems are invertible: (a) y(t) = x(−t), (b) y(t) = tx(t), and (c) ^y(t) ^{= d} dt x(t).

⁽a) Here, the output is a reflection of the input, which does not cause any loss to the input. The input can, in fact, be exactly recovered by simply reflecting the output [x(t) = y(−t)], which is to say that a reflecting system is its own inverse. Thus, y(t) = x(−t) is an invertible system.

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(b) In this case, one might be tempted to recover the input from the output as ^x(t) ^{= 1 t} y(t). This approach works almost everywhere, except at t = 0 where the input value x(0) cannot be recovered. Due to this single lost point, the system y(t) = tx(t) is not invertible.

(c) Differentiation eliminates any dc component. For example, the inputs x1(t) = 1 and x2(t) = 2 both produce the same output y(t) = 0. Given only y(t) = 0, it is impossible to know if the original input was x1(t) = 1, x2(t) = 2, or something else entirely. Since unique inputs do produce unique outputs, we know that ^y(t) ^{= d} dt x(t) is not an invertible system.

1.7-8 Stable and Unstable Systems

Systems can also be classified as stable or unstable systems. Stability can be internal or external. If every bounded input applied at the input terminal results in a bounded output, the system is said to be stable externally. External stability can be ascertained by measurements at the external terminals (input and output) of the system. This type of stability is also known as the stability in the BIBO (bounded-input/bounded-output) sense. The concept of internal stability is postponed to Ch. 2 because it requires some understanding of internal system behavior, introduced in that chapter.

EXAMPLE 1.15 Assessing System BIBO Stability

Determine whether the following systems are BIBO-stable: (a) y(t) = x²(t), (b) y(t) = tx(t), and (c) ^y(t) ^{= d} dt x(t).

(a) This system squares an input to produce the output. If the input is bounded, which is to say that |x(t)| ≤ Mx < ∞ for all t, then we see that

$$|y(t)| = |x^2(t)| = |x(t)|^2 \le M_x^2 < \infty$$

Since the output amplitude is guaranteed to be bounded for any bounded-amplitude input, the system y(t) = x²(t) is BIBO-stable.

(b) We can prove that y(t) = tx(t) is not BIBO-stable with a simple example. The bounded-amplitude input x(t) = u(t) produces the output y(t) = tu(t) whose amplitude grows to infinity as t → ∞. Thus, y(t) = tx(t) is a BIBO-unstable system.

(c) We can prove that ^y(t) ^{= d} dt x(t) is not BIBO-stable with an example. The bounded-amplitude input x(t) = u(t) produces the output y(t) = δ(t) whose amplitude is infinite at ^{t =} 0. Thus, ^y(t) ^{= d} dt x(t) is a BIBO-unstable system.

DR ILL 1.16 A Noninvertible BIBO-Stable System

Show that a system described by the equation y(t) = x²(t) is noninvertible but BIBO-stable.