Linear Circuits and Systems

Linear Circuits and Systems - 1 LINEAR CIRCUITS AND SIGNALS...

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Unformatted text preview: 1 LINEAR CIRCUITS AND SIGNALS Laplace Transform Department of Electrical and Computer Engineering University of Miami I. PRELIMINARIES Consider a LTI causal (LTIC) system h ( t ) H ( ) driven by the input u ( t ) = e st : u ( t ) = e st- LTIC h ( t ) H ( )- y ( t ) . (1) We know that y ( t ) = ( h * u )( t ) = Z h ( ) u ( t- ) d = Z h ( ) e s ( t- ) d = e st Z h ( ) e- s d. (2) The system being causal, we may change integration limits to [0 , + ) ; to fully capture an impulse response h ( t ) which may contain an impulsive signal at t = 0 , we will use the integration limits [0- , + ) . This leads us to u ( t ) = e st- LTIC h ( t ) H ( )- y ( t ) = H ( s ) u ( t ) = H ( s ) e st , (3) where H ( s ) = Z + t =0- h ( t ) e- st dt. (4) It is not necessary that s be restricted to be a real number. In general, it can be a complex number, and the integral in (4) converges only within a certain region in the complex plane; in other words, it is unlikely that it converges for all values in the complex plane. In fact, one can show that this region usually lies to the right of a vertical line in the complex plane. 2 II. LAPLACE TRANSFORM With the above discussion in place, we introduce Definition 1 (Laplace Transform (LT)): For a given time-domain signal h ( t ) , consider the integral H ( s ) = Z + t =0- h ( t ) e- st dt, where s = + j. The region in the complex plane where the above integral converges is referred to as the region of convergence (ROC); within the ROC, H ( s ) is referred to as the Laplace transform (LT) of h ( t ) . We denote this as H ( s ) = LT [ h ( t )] . The values of s for which the computed LT H ( s ) are referred to as the poles of H ( s ) ; the values of s for which H ( s ) = 0 are referred to as the zeros of H ( s ) . Remarks: In discussions related to the LT, the complex plane is usually referred to as the s-plane. When h ( t ) is the impulse response of an LTIC system, its LT is referred to as the systems transfer function (TF). Consider the special case of s = j , i.e., imaginary axis. Then, if the ROC contains the imaginary axis, we have H ( s ) s = j = Z + t =0- h ( t ) e- jt dt = H ( ) . (5) In other words, for a LTIC system, when the ROC contains the imaginary axis in the s- plane, the TF and FT of the system impulse response are identical. Recall that the FT of the impulse response is the system frequency response. Note the following: | H ( s ) | = Z + t =0- h ( t ) e- ( + j ) t dt Z + t =0- | h ( t ) e- t | dt. (6) Therefore, a sufficient condition for the convergence or existence of the LT is Z + t =0- | h ( t ) e- t | dt < , for some real , (7) i.e., absolute integrability of h ( t ) e- t . Compare this with the sufficient condition for the existence of the FT: Z + t =0- | h ( t ) | dt < BIBO stability. (8) 3 Note that (7) is less stringent than (8). Therefore, the LTNote that (7) is less stringent than (8)....
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Linear Circuits and Systems - 1 LINEAR CIRCUITS AND SIGNALS...

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