2.6 INTUITIVE [INSIGHTS INTO](#page-8-0) SYSTEM BEHAVIOR

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2.6 INTUITIVE INSIGHTS INTO SYSTEM BEHAVIOR

This section attempts to provide an understanding of what determines system behavior. Because of its intuitive nature, the discussion is more or less qualitative. We shall now show that the most important attributes of a system are its characteristic roots or characteristic modes because they determine not only the zero-input response but also the entire behavior of the system.

2.6-1 Dependence of System Behavior on Characteristic Modes

Recall that the zero-input response of a system consists of the system’s characteristic modes. For a stable system, these characteristic modes decay exponentially and eventually vanish. This behavior may give the impression that these modes do not substantially affect system behavior in general and system response in particular. This impression is totally wrong! We shall now see that the system’s characteristic modes leave their imprint on every aspect of the system behavior. We may compare the system’s characteristic modes (or roots) to a seed that eventually dissolves in the

204 CHAPTER 2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS

ground; however, the plant that springs from it is totally determined by the seed. The imprint of the seed exists on every cell of the plant.

To understand this interesting phenomenon, recall that the characteristic modes of a system are very special to that system because it can sustain these signals without the application of an external input. In other words, the system offers a free ride and ready access to these signals. Now imagine what would happen if we actually drove the system with an input having the form of a characteristic mode! We would expect the system to respond strongly (this is, in fact, the resonance phenomenon discussed later in this section). If the input is not exactly a characteristic mode but is close to such a mode, we would still expect the system response to be strong. However, if the input is very different from any of the characteristic modes, we would expect the system to respond poorly. We shall now show that these intuitive deductions are indeed true.

Intuition can cut the math jungle instantly!

We have devised a measure of similarity of signals later (see in Ch. 6). Here we shall take a simpler approach. Let us restrict the system’s inputs to exponentials of the form e^{ζ t} , where ζ is generally a complex number. The similarity of two exponential signals e^{ζ t} and e^λt will then be measured by the closeness of ζ and λ. If the difference ζ − λ is small, the signals are similar; if ζ −λ is large, the signals are dissimilar.

Now consider a first-order system with a single characteristic mode e^λt and the input e^{ζ t} . The impulse response of this system is then given by Ae^λt , where the exact value of A is not important for this qualitative discussion. The system response y(t) is given by

$$y(t) = h(t) * x(t) = Ae^{\lambda t}u(t) * e^{\zeta t}u(t)$$

From the convolution table (Table 2.1), we obtain

$$y(t) = \frac{A}{\zeta - \lambda} [e^{\zeta t} - e^{\lambda t}] u(t)$$

\n(2.46)

Clearly, if the input e^{ζ t} is similar to eλ*^t* , ζ −λ is small and the system response is large. The closer the input x(t) to the characteristic mode, the stronger the system response. In contrast, if the input is very different from the natural mode, ζ − λ is large and the system responds poorly. This is precisely what we set out to prove.

We have proved the foregoing assertion for a single-mode (first-order) system. It can be generalized to an Nth-order system, which has N characteristic modes. The impulse response h(t) of such a system is a linear combination of its N modes. Therefore, if x(t) is similar to any one of the modes, the corresponding response will be high; if it is similar to none of the modes, the response will be small. Clearly, the characteristic modes are very influential in determining system response to a given input.

It would be tempting to conclude on the basis of Eq. (2.46) that if the input is identical to the characteristic mode, so that ζ = λ, then the response goes to infinity. Remember, however, that if ζ = λ, the numerator on the right-hand side of Eq. (2.46) also goes to zero. We shall study this interesting behavior (resonance phenomenon) later in this section.

We now show that mere inspection of the impulse response h(t) (which is composed of characteristic modes) reveals a great deal about the system behavior.

2.6-2 Response Time of a System: The System Time Constant

Like human beings, systems have a certain response time. In other words, when an input (stimulus) is applied to a system, a certain amount of time elapses before the system fully responds to that input. This time lag or response time is called the system time constant. As we shall see, a system’s time constant is equal to the width of its impulse response h(t).

An input δ(t) to a system is instantaneous (zero duration), but its response h(t) has a duration Th. Therefore, the system requires a time Th to respond fully to this input, and we are justified in viewing Th as the system’s response time or time constant. We arrive at the same conclusion via another argument. The output is a convolution of the input with h(t). If an input is a pulse of width Tx, then the output pulse width is Tx + Th according to the width property of convolution. This conclusion shows that the system requires Th seconds to respond fully to any input. The system time constant indicates how fast the system is. A system with a smaller time constant is a faster system that responds quickly to an input. A system with a relatively large time constant is a sluggish system that cannot respond well to rapidly varying signals.

Strictly speaking, the duration of the impulse response h(t) is ∞ because the characteristic modes approach zero asymptotically as t → ∞. However, beyond some value of t, h(t) becomes negligible. It is therefore necessary to use some suitable measure of the impulse response’s effective width.

There is no single satisfactory definition of effective signal duration (or width) applicable to every situation. For the situation depicted in Fig. 2.21, a reasonable definition of the duration h(t) would be Th, the width of the rectangular pulse hˆ(t). This rectangular pulse hˆ(t) has an area identical to that of h(t) and a height identical to that of h(t) at some suitable instant t = t0. In Fig. 2.21, t⁰ is chosen as the instant at which h(t) is maximum. According to this definition,†

$$T_h h(t_0) = \int_{-\infty}^{\infty} h(t) dt$$

^† This definition is satisfactory when h(t) is a single, mostly positive (or mostly negative) pulse. Such systems are lowpass systems. This definition should not be applied indiscriminately to all systems.

or

$$T_h = \frac{\int_{-\infty}^{\infty} h(t) dt}{h(t_0)}$$

\n(2.47)

Now if a system has a single mode

^h(t) ⁼ Ae^λt u(t)

with λ negative and real, then h(t) is maximum at t = 0 with value h(0) = A. Therefore, according to Eq. (2.47),

$$T_h = \frac{1}{A} \int_0^\infty A e^{\lambda t} dt = -\frac{1}{\lambda}$$

Thus, the time constant in this case is simply the (negative of the) reciprocal of the system’s characteristic root. For the multimode case, h(t) is a weighted sum of the system’s characteristic modes, and Th is a weighted average of the time constants associated with the N modes of the system.

2.6-3 Time Constant and Rise Time of a System

Rise time of a system, defined as the time required for the unit step response to rise from 10% to 90% of its steady-state value, is an indication of the speed of response.† The system time constant may also be viewed from a perspective of rise time. The unit step response y(t) of a system is the convolution of u(t) with h(t). Let the impulse response h(t) be a rectangular pulse of width Th, as shown in Fig. 2.22. This assumption simplifies the discussion, yet gives satisfactory results for qualitative discussion. The result of this convolution is illustrated in Fig. 2.22. Note that the output does not rise from zero to a final value instantaneously as the input rises; instead, the output takes Th seconds to accomplish this. Hence, the rise time Tr of the system is equal to the system time constant

$$T_r = T_h$$

This result and Fig. 2.22 show clearly that a system generally does not respond to an input instantaneously. Instead, it takes time Th for the system to respond fully.

^† Because of varying definitions of rise time, the reader may find different results in the literature. The qualitative and intuitive nature of this discussion should always be kept in mind.

Figure 2.22 Rise time of a system.

2.6-4 Time Constant and Filtering

A larger time constant implies a sluggish system because the system takes longer to respond fully to an input. Such a system cannot respond effectively to rapid variations in the input. In contrast, a smaller time constant indicates that a system is capable of responding to rapid variations in the input. Thus, there is a direct connection between a system’s time constant and its filtering properties.

A high-frequency sinusoid varies rapidly with time. A system with a large time constant will not be able to respond well to this input. Therefore, such a system will suppress rapidly varying (high-frequency) sinusoids and other high-frequency signals, thereby acting as a lowpass filter (a filter allowing the transmission of low-frequency signals only). We shall now show that a system

Figure 2.23 Time constant and filtering.

with a time constant Th acts as a lowpass filter having a cutoff frequency of fc = 1/Th hertz, so that sinusoids with frequencies below fc Hz are transmitted reasonably well, while those with frequencies above fc Hz are suppressed.

To demonstrate this fact, let us determine the system response to a sinusoidal input x(t) by convolving this input with the effective impulse response h(t) in Fig. 2.23a. From Figs. 2.23b and 2.23c we see the process of convolution of h(t) with the sinusoidal inputs of two different frequencies. The sinusoid in Fig. 2.23b has a relatively high frequency, while the frequency of the sinusoid in Fig. 2.23c is low. Recall that the convolution of x(t) and h(t) is equal to the area under the product x(τ )h(t − τ ). This area is shown shaded in Figs. 2.23b and 2.23c for the two cases. For the high-frequency sinusoid, it is clear from Fig. 2.23b that the area under x(τ )h(t − τ ) is very small because its positive and negative areas nearly cancel each other out. In this case the output y(t) remains periodic but has a rather small amplitude. This happens when the period of the sinusoid is much smaller than the system time constant Th. In contrast, for the low-frequency sinusoid, the period of the sinusoid is larger than Th, rendering the partial cancellation of area under x(τ )h(t −τ ) less effective. Consequently, the output y(t) is much larger, as depicted in Fig. 2.23c.

Between these two possible extremes in system behavior, a transition point occurs when the period of the sinusoid is equal to the system time constant Th. The frequency at which this transition occurs is known as the cutoff frequency fc of the system. Because Th is the period of cutoff frequency fc,

$$f_c = \frac{1}{T_h}$$

The frequency fc is also known as the bandwidth of the system because the system transmits or passes sinusoidal components with frequencies below fc while attenuating components with frequencies above fc. Of course, the transition in system behavior is gradual. There is no dramatic change in system behavior at fc = 1/Th. Moreover, these results are based on an idealized (rectangular pulse) impulse response; in practice these results will vary somewhat, depending on the exact shape of h(t). Remember that the “feel” of general system behavior is more important than exact system response for this qualitative discussion.

Since the system time constant is equal to its rise time, we have

$$T_r = \frac{1}{f_c} \qquad \text{or} \qquad f_c = \frac{1}{T_r} \tag{2.48}$$

Thus, a system’s bandwidth is inversely proportional to its rise time. Although Eq. (2.48) was derived for an idealized (rectangular) impulse response, its implications are valid for lowpass LTIC systems, in general. For a general case, we can show that [1]

$$f_c = \frac{k}{T_r}$$

where the exact value of k depends on the nature of h(t). An experienced engineer often can estimate quickly the bandwidth of an unknown system by simply observing the system response to a step input on an oscilloscope.

2.6-5 Time Constant and Pulse Dispersion (Spreading)

In general, the transmission of a pulse through a system causes pulse dispersion (or spreading). Therefore, the output pulse is generally wider than the input pulse. This system behavior can have serious consequences in communication systems in which information is transmitted by pulse amplitudes. Dispersion (or spreading) causes interference or overlap with neighboring pulses, thereby distorting pulse amplitudes and introducing errors in the received information.

Earlier we saw that if an input x(t) is a pulse of width Tx, then Ty, the width of the output y(t), is

$$T_{y}=T_{x}+T_{h}$$

This result shows that an input pulse spreads out (disperses) as it passes through a system. Since Th is also the system’s time constant or rise time, the amount of spread in the pulse is equal to the time constant (or rise time) of the system.

2.6-6 Time Constant and Rate of Information Transmission

In pulse communications systems, which convey information through pulse amplitudes, the rate of information transmission is proportional to the rate of pulse transmission. We shall demonstrate that to avoid the destruction of information caused by dispersion of pulses during their transmission through the channel (transmission medium), the rate of information transmission should not exceed the bandwidth of the communications channel.

Since an input pulse spreads out by Th seconds, the consecutive pulses should be spaced Th seconds apart to avoid interference between pulses. Thus, the rate of pulse transmission should not exceed 1/Th pulses/second. But 1/Th = fc, the channel’s bandwidth, so that we can transmit pulses through a communications channel at a rate of fc pulses per second and still avoid significant interference between the pulses. The rate of information transmission is therefore proportional to the channel’s bandwidth (or to the reciprocal of its time constant).†

The discussion of Secs. 2.6-2, 2.6-3, 2.6-4, 2.6-5, and 2.6-6) shows that the system time constant determines much of a system’s behavior—its filtering characteristics, rise time, pulse dispersion, and so on. In turn, the time constant is determined by the system’s characteristic roots. Clearly the characteristic roots and their relative amounts in the impulse response h(t) determine the behavior of a system.

EXAMPLE 2.15 Intuitive Insights into Lowpass System Behavior

Find the time constant Th, rise time Tr, and cutoff frequency fc for a lowpass system that has impulse response h(t) = te−^t u(t). Determine the maximum rate that pulses of 1 second

^† Theoretically, a channel of bandwidth fc can transmit correctly up to 2fc pulse amplitudes per second [4]. Our derivation here, being very simple and qualitative, yields only half the theoretical limit. In practice it is not easy to attain the upper theoretical limit.

210 CHAPTER 2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS

duration can be transmitted through the system so that interference is essentially avoided between adjacent pulses at the system output.

The system impulse response h(t) = te−^t u(t), which looks similar to the impulse response of Fig. 2.21, has a peak value of e−¹ = 0.3679 at a time t⁰ = 1. According to Eq. (2.47) and using integration by parts, the system time constant is therefore

$$T_h = \frac{\int_0^\infty t e^{-t} dt}{e^{-1}} = e^1 \left( -t e^{-t} \Big|_0^\infty + \int_0^\infty e^{-t} dt \right) = e^1 \left( 0 - e^{-t} \Big|_0^\infty \right) = e^1(1) = 2.7183$$

Thus,

$$T_h = 2.7183$$

s, $T_r = T_h = 2.7183$ s, and $f_c = \frac{1}{T_h} = 0.3679$ Hz

Due to its lowpass nature, this system will spread an input pulse of 1 second to an output with width

Ty = Tx +Th = 1+2.7183 = 3.7183 s

To avoid interference between pulses at the output, the pulse transmission rate should be no more than the reciprocal of the output pulse width. That is,

maximum pulse transmission rate ^{= 1} 3.7183 = 0.2689 pulse/s

By narrowing the input pulses, the pulse transmission rate could increase up to fc = 0.3679 pulse/s.

2.6-7 The Resonance Phenomenon

Finally, we come to the fascinating phenomenon of resonance. As we have already mentioned several times, this phenomenon is observed when the input signal is identical or is very close to a characteristic mode of the system. For the sake of simplicity and clarity, we consider a first-order system having only a single mode, e^λt . Let the impulse response of this system be†

$$h(t) = Ae^{\lambda t}$$

and let the input be

$$x(t) = e^{(\lambda - \epsilon)t}$$

The system response y(t) is then given by

$$y(t) = Ae^{\lambda t} * e^{(\lambda - \epsilon)t}$$

^† For convenience, we omit multiplying x(t) and h(t) by u(t). Throughout this discussion, we assume that they are causal.

2.6 Intuitive Insights into System Behavior 211

From the convolution table we obtain

$$y(t) = \frac{A}{\epsilon} \left[ e^{\lambda t} - e^{(\lambda - \epsilon)t} \right] = A e^{\lambda t} \left( \frac{1 - e^{-\epsilon t}}{\epsilon} \right)$$

(2.49)

Now, as → 0, both the numerator and the denominator of the term in the parentheses approach zero. Applying L’Hôpital’s rule to this term yields

$$\lim_{\epsilon \to 0} y(t) = A t e^{\lambda t}$$

Clearly, the response does not go to infinity as → 0, but it acquires a factor t, which approaches ∞ as t → ∞. If λ has a negative real part (so that it lies in the LHP), eλ*^t* decays faster than t and y(t) → 0 as t → ∞. The resonance phenomenon in this case is present, but its manifestation is aborted by the signal’s own exponential decay.

This discussion shows thatresonance is a cumulative phenomenon, not instantaneous. It builds up linearly with t. † When the mode decays exponentially, the signal decays too fast for resonance to counteract the decay; as a result, the signal vanishes before resonance has a chance to build it up. However, if the mode were to decay at a rate less than 1/t, we should see the resonance phenomenon clearly. This specific condition would be possible if Re λ ≥ 0. For instance, when Re λ = 0 so that λ lies on the imaginary axis of the complex plane (λ = jω), the output becomes

$$y(t) = A t e^{j\omega t}$$

Here, the response does go to infinity linearly with t.

For a real system, if λ = jω is a root, λ^∗ = −jω must also be a root; the impulse response is of the form Aejω*^t* + Ae−jω*^t* = 2Acos ωt. The response of this system to input Acosωt is 2Acosωt ∗ cosωt. The reader can show that this convolution contains a term of the form At cos ωt. The resonance phenomenon is clearly visible. The system response to its characteristic mode increases linearly with time, eventually reaching ∞, as indicated in Fig. 2.24.

Recall that when λ = jω, the system is marginally stable. As we have indicated, the full effect of resonance cannot be seen for an asymptotically stable system; only in a marginally stable system does the resonance phenomenon boost the system’s response to infinity when the system’s input

Figure 2.24 Buildup of system response in resonance.

^† If the characteristic root in question repeats r times, resonance effect increases as t ^r−1. However, t ^r−1eλ*^t* ^→0 as t → ∞ for any value of r, provided Re λ < 0 (λ in the LHP).

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is a characteristic mode. But even in an asymptotically stable system, we see a manifestation of resonance if its characteristic roots are close to the imaginary axis so that Re λ is a small, negative value. We can show that when the characteristic roots of a system are σ ± jω0, then the system response to the input ej^ω0*^t* or the sinusoid cosω0t is very large for small σ. † The system response drops off rapidly as the input signal frequency moves away from ω0. This frequency-selective behavior can be studied more profitably after an understanding of frequency-domain analysis has been acquired. For this reason we postpone full discussion of this subject until Ch. 4.

IMPORTANCE OF THE RESONANCE PHENOMENON

The resonance phenomenon is very important because it allows us to design frequency-selective systems by choosing their characteristic roots properly. Lowpass, bandpass, highpass, and bandstop filters are all examples of frequency-selective networks. In mechanical systems, the inadvertent presence of resonance can cause signals of such tremendous magnitude that the system may fall apart. A musical note (periodic vibrations) of proper frequency can shatter glass if the frequency is matched to the characteristic root of the glass, which acts as a mechanical system. Similarly, a company of soldiers marching in step across a bridge amounts to applying a periodic force to the bridge. If the frequency of this input force happens to be nearer to a characteristic root of the bridge, the bridge may respond (vibrate) violently and collapse, even though it would have been strong enough to carry many soldiers marching out of step. A case in point is the Tacoma Narrows Bridge failure of 1940. This bridge was opened to traffic in July 1940. Within four months of opening (on November 7, 1940), it collapsed in a mild gale, not because of the wind’s brute force but because the frequencies of wind-generated vortices, which matched the natural frequencies (characteristic roots) of the bridge, caused resonance.

Because of the great damage that may occur, mechanical resonance is generally to be avoided, especially in structures or vibrating mechanisms. If an engine with periodic force (such as piston motion) is mounted on a platform, the platform with its mass and springs should be designed so that their characteristic roots are not close to the engine’s frequency of vibration. Proper design of this platform can not only avoid resonance, but also attenuate vibrations if the system roots are placed far away from the frequency of vibration.