[4.13 SUMMARY](#page-11-0)

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4.13 SUMMARY

This chapter discusses analysis of LTIC (linear, time-invariant, continuous-time) systems by the Laplace transform, which transforms integro-differential equations of such systems into algebraic equations. Therefore solving these integro-differential equations reduces to solving algebraic equations. The Laplace transform method cannot be used for time-varying-parameter systems or for nonlinear systems in general.

The transfer function H(s) of an LTIC system is the Laplace transform of its impulse response. It may also be defined as a ratio of the Laplace transform of the output to the Laplace transform of the input when all initial conditions are zero (system in zero state). If X(s) is the Laplace transform of the input x(t) and Y(s) is the Laplace transform of the corresponding output y(t) (when all initial conditions are zero), then Y(s) = X(s)H(s). For an LTIC system described by an Nth-order differential equation Q(D)y(t) = P(D)x(t), the transfer function H(s) = P(s)/Q(s). Like the impulse response h(t), the transfer function H(s) is also an external description of the system.

Electrical circuit analysis can also be carried out by using a transformed circuit method, in which all signals (voltages and currents) are represented by their Laplace transforms, all elements by their impedances (or admittances), and initial conditions by their equivalent sources (initial condition generators). In this method, a network can be analyzed as if it were a resistive circuit.

Large systems can be depicted by suitably interconnected subsystems represented by blocks. Each subsystem, being a smaller system, can be readily analyzed and represented by its input–output relationship, such as its transfer function. Analysis of large systems can be carried out with the knowledge of input–output relationships of its subsystems and the nature of interconnection of various subsystems.

LTIC systems can be realized by scalar multipliers, adders, and integrators. A given transfer function can be synthesized in many different ways, such as canonic, cascade, and parallel. Moreover, every realization has a transpose, which also has the same transfer function. In practice, all the building blocks (scalar multipliers, adders, and integrators) can be obtained from operational amplifiers.

The system response to an everlasting exponential est is also an everlasting exponential H(s)est. Consequently, the system response to an everlasting exponential ejω*^t* is H(jω) ejω*^t* . Hence, H(jω) is the frequency response of the system. For a sinusoidal input of unit amplitude and having frequency ω, the system response is also a sinusoid of the same frequency (ω) with amplitude |H(jω)|, and its phase is shifted by H(jω) with respect to the input sinusoid. For this reason |H(jω)| is called the amplitude response (gain) and H(jω) is called the phase response of the system. Amplitude and phase response of a system indicate the filtering characteristics of the system. The general nature of the filtering characteristics of a system can be quickly determined from a knowledge of the location of poles and zeros of the system transfer function.

Most of the input signals and practical systems are causal. Consequently we are required most of the time to deal with causal signals. When all signals must be causal, the Laplace transform analysis is greatly simplified; the region of convergence of a signal becomes irrelevant to the analysis process. This special case of the Laplace transform (which is restricted to causal signals) is called the unilateral Laplace transform. Much of the chapter deals with this variety of Laplace transform. Section 4.11 discusses the general Laplace transform (the bilateral Laplace transform), which can handle causal and noncausal signals and systems. In the bilateral transform, the inverse transform of X(s) is not unique but depends on the region of convergence of X(s). Thus, the region of convergence plays a very crucial role in the bilateral Laplace transform.

468 CHAPTER 4 CONTINUOUS-TIME SYSTEM ANALYSIS

REFERENCES

• 1. Lathi, B. P. Signal Processing and Linear Systems, 1st ed. Oxford University Press, New York, 1998.
• 2. Doetsch, G. Introduction to the Theory and Applications of the Laplace Transformation with a Table of Laplace Transformations. Springer-Verlag, New York, 1974.
• 3. LePage, W. R. Complex Variables and the Laplace Transforms for Engineers. McGraw-Hill, New York, 1961.
• 4. Durant, Will, and Ariel Durant. The Age of Napoleon, Part XI in The Story of Civilization Series. Simon & Schuster, New York, 1975.
• 5. Bell, E. T. Men of Mathematics. Simon & Schuster, New York, 1937.
• 6. Nahin, P. J. “Oliver Heaviside: Genius and Curmudgeon.” IEEE Spectrum, vol. 20, pp. 63–69, July 1983.
• 7. Berkey, D. Calculus, 2nd ed. Saunders, Philadelphia, 1988.
• 8. Encyclopaedia Britannica. Micropaedia IV, 15th ed., p. 981, Chicago, 1982.
• 9. Churchill, R. V. Operational Mathematics, 2nd ed. McGraw-Hill, New York, 1958.
• 10. Truxal, J. G. The Age of Electronic Messages. McGraw-Hill, New York, 1990.
• 11. Van Valkenberg, M. Analog Filter Design. Oxford University Press, New York, 1982.

PROBLEMS

• 4.1-1 By direct integration [Eq. (4.1)] find the Laplace transforms and the region of convergence of the following functions:
  • (a) u(t)−u(t −1)
  • (b) te−^t u(t)
  • (c) t cos ω0t u(t)
  • (d) (e2*^t* ⁻2e−^t )u(t)
  • (e) cos ω1t cos ω2t u(t)
  • (f) cosh(at)u(t)
  • (g) sinh(at)u(t)
  • (h) ^e−2*^t* cos(5*^t* ⁺θ )u(t)
• 4.1-2 By direct integration [Eq. (4.1)] find the Laplace transforms and the region of convergence of the following functions:

(a)

$$e^{-2t}u(t-5) + \delta(t-1)$$

(b)

$$\pi e^{3t} u(t+5) - \delta(2t)$$

(c)

$$\sum_{k=0}^{\infty} \delta(t - kT), T > 0$$

• 4.1-3 By direct integration find the Laplace transforms of the signals shown in Fig. P4.1-3.
• 4.1-4 Find the inverse (unilateral) Laplace transforms of the following functions:

(a)

$$$$