[6.4 LTIC SYSTEM](#page-12-0) RESPONSE TO PERIODIC INPUTS

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6.4 LTIC SYSTEM RESPONSE TO PERIODIC INPUTS

A periodic signal can be expressed as a sum of everlasting exponentials (or sinusoids). We also know how to find the response of an LTIC system to an everlasting exponential. From this information, we can readily determine the response of an LTIC system to periodic inputs. A periodic signal x(t) with period T⁰ can be expressed as an exponential Fourier series

$$x(t) = \sum_{n = -\infty}^{\infty} D_n e^{jn\omega_0 t} \qquad \omega_0 = \frac{2\pi}{T_0}$$

638 CHAPTER 6 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES

In Sec. 4.8, we showed that the response of an LTIC system with transfer function H(s) to an everlasting exponential input ejω*^t* is an everlasting exponential H(jω)ejω*^t* . This input–output pair can be displayed as†

$$\underbrace{e^{j\omega t}}_{\text{input}} \Longrightarrow \underbrace{H(j\omega)e^{j\omega t}}_{\text{output}}$$

Therefore, from the linearity property,

$$\underbrace{\sum_{n=-\infty}^{\infty} D_n e^{jn\omega_0 t}}_{\text{input } x(t)} \Longrightarrow \underbrace{\sum_{n=-\infty}^{\infty} D_n H(jn\omega_0) e^{jn\omega_0 t}}_{\text{response } y(t)}$$

(6.29)

The response y(t) is obtained in the form of an exponential Fourier series and is therefore a periodic signal of the same period as that of the input.

We shall demonstrate the utility of these results by the following example.

EXAMPLE 6.12 Full-Wave Rectifier

A full-wave rectifier (Fig. 6.20a) is used to obtain a dc signal from a sinusoid sin t. The rectified signal x(t), depicted in Fig. 6.17, is applied to the input of a lowpass RC filter, which suppresses the time-varying component and yields a dc component with some residual ripple. Find the filter output y(t). Find also the dc output and the rms value of the ripple voltage.

First, we shall find the Fourier series for the rectified signal x(t), whose period is T⁰ = π. Consequently, ω⁰ = 2, and

$$x(t) = \sum_{n = -\infty}^{\infty} D_n e^{j2nt}$$

where

$$D_n = \frac{1}{\pi} \int_0^{\pi} \sin t e^{-j2nt} dt = \frac{2}{\pi (1 - 4n^2)}$$

(6.30)

Therefore,

$$x(t) = \sum_{n = -\infty}^{\infty} \frac{2}{\pi (1 - 4n^2)} e^{j2nt}$$

^† This result applies only to asymptotically stable systems. This is because when ^{s =} *^j*ω, the integral on the right-hand side of Eq. (2.39) does not converge for unstable systems. Moreover, for marginally stable systems also, that integral does not converge in the ordinary sense, and H(jω) cannot be obtained from H(s) by replacing s with jω.

Figure 6.20 (a) Full-wave rectifier with a lowpass filter and (b) its output.

Next, we find the transfer function of the RC filter in Fig. 6.20a. This filter is identical to the RC circuit in Ex. 1.17 (Fig. 1.35) for which the differential equation relating the output (capacitor voltage) to the input x(t) was found to be [Eq. (1.31)]:

$$(3D+1)y(t) = x(t)$$

The transfer function H(s) for this system is found from Eq. (2.41) as

$$H(s) = \frac{1}{3s+1}$$

and

$$H(j\omega) = \frac{1}{3j\omega + 1} \tag{6.31}$$

From Eq. (6.29), the filter output y(t) can be expressed as (with ω⁰ = 2)

$$y(t) = \sum_{n = -\infty}^{\infty} D_n H(jn\omega_0) e^{jn\omega_0 t} = \sum_{n = -\infty}^{\infty} D_n H(j2n) e^{j2nt}$$

Substituting Dn and H(j2n) from Eqs. (6.30) and (6.31) in the foregoing equation, we obtain

$$y(t) = \sum_{n = -\infty}^{\infty} \frac{2}{\pi (1 - 4n^2)(j6n + 1)} e^{j2nt}$$

640 CHAPTER 6 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES

Note that the output y(t) is also a periodic signal given by the exponential Fourier series on the right-hand side. The output is shown in Fig. 6.20b.

The output Fourier series coefficient corresponding to n = 0 is the dc component of the output, given by 2/π. The remaining terms in the Fourier series constitute the unwanted component called the ripple. We can determine the rms value of the ripple voltage by using Eq. (6.27) to find the power of the ripple component. The power of the ripple is the power of all the components except the dc (n = 0). Note that Dˆ ⁿ, the exponential Fourier coefficient for the output y(t), is

$$\hat{D}_n = \frac{2}{\pi (1 - 4n^2)(j6n + 1)}$$

Therefore, from Eq. (6.28), we have

$$P_{\text{right}} = 2 \sum_{n=1}^{\infty} |D_n|^2 = 2 \sum_{n=1}^{\infty} \left| \frac{2}{\pi (1 - 4n^2)(j6n + 1)} \right|^2 = \frac{8}{\pi^2} \sum_{n=1}^{\infty} \frac{1}{(1 - 4n^2)^2 (36n^2 + 1)}$$

Numerical computation of the right-hand side yields Pripple = 0.0025, and the ripple rms value ⁼ Pripple ⁼ 0.05. This shows that the rms ripple voltage is 5% of the amplitude of the input sinusoid.

WHY USE EXPONENTIALS?

The exponential Fourier series is just another way of representing trigonometric Fourier series (or vice versa). The two forms carry identical information—no more, no less. The reasons for preferring the exponential form have already been mentioned: this form is more compact, and the expression for deriving the exponential coefficients is also more compact than those in the trigonometric series. Furthermore, the LTIC system response to exponential signals is also simpler (more compact) than the system response to sinusoids. In addition, the exponential form proves to be much easier than the trigonometric form to manipulate mathematically and otherwise handle in the area of signals as well as systems. Moreover, exponential representation proves much more convenient for analysis of complex x(t). For these reasons, in our future discussion we shall use the exponential form exclusively.

A minor disadvantage of the exponential form is that it cannot be visualized as easily as sinusoids. For intuitive and qualitative understanding, the sinusoids have the edge over exponentials. Fortunately, this difficulty can be overcome readily because of the close connection between exponential and Fourier spectra. For the purpose of mathematical analysis, we shall continue to use exponential signals and spectra; but to understand the physical situation intuitively or qualitatively, we shall speak in terms of sinusoids and trigonometric spectra. Thus, although all mathematical manipulation will be in terms of exponential spectra, we shall now speak of exponential and sinusoids interchangeably when we discuss intuitive and qualitative insights in attempting to arrive at an understanding of physical situations. This is an important point; readers should make an extra effort to familiarize themselves with the two forms of spectra, their relationships, and their convertibility.

DUAL PERSONALITY OF A SIGNAL

The discussion so far shows that a periodic signal has a dual personality—the time domain and the frequency domain. It can be described by its waveform or by its Fourier spectra. The timeand frequency-domain descriptions provide complementary insights into a signal. For in-depth perspective, we need to understand both these identities. It is important to learn to think of a signal from both perspectives. In the next chapter, we shall see that aperiodic signals also have this dual personality. Moreover, we shall show that even LTI systems have this dual personality, which offers complementary insights into the system behavior.

LIMITATIONS OF THE FOURIER SERIES METHOD OF ANALYSIS

We have developed here a method of representing a periodic signal as a weighted sum of everlasting exponentials whose frequencies lie along the ω axis in the s plane. This representation (Fourier series) is valuable in many applications. However, as a tool for analyzing linear systems, it has serious limitations and consequently has limited utility for the following reasons:

• 1. The Fourier series can be used only for periodic inputs. All practical inputs are aperiodic (remember that a periodic signal starts at t = −∞).
• 2. The Fourier methods can be applied readily to BIBO-stable (or asymptotically stable) systems. It cannot handle unstable or even marginally stable systems.

The first limitation can be overcome by representing aperiodic signals in terms of everlasting exponentials. This representation can be achieved through the Fourier integral, which may be considered to be an extension of the Fourier series. We shall therefore use the Fourier series as a stepping-stone to the Fourier integral developed in the next chapter. The second limitation can be overcome by using exponentials est, where s is not restricted to the imaginary axis but is free to take on complex values. This generalization leads to the Laplace integral, discussed in Ch. 4 (the Laplace transform).