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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 ( ω ) driven by the input u ( t ) = e st : u ( t ) = e st→ LTIC h ( t ) ↔ H ( ω )→ y ( t ) . (1) We know that y ( t ) = ( h * u )( t ) = Z τ h ( τ ) u ( t τ ) dτ = Z τ h ( τ ) e s ( t τ ) dτ = e st Z τ h ( τ ) e sτ dτ. (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 ( ω )→ y ( t ) = ˆ H ( s ) u ( t ) = ˆ H ( s ) e st , (3) where ˆ 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 timedomain signal h ( t ) , consider the integral ˆ H ( s ) = Z + ∞ t =0 h ( t ) e st dt, where s = σ + jω. The region in the complex plane where the above integral converges is referred to as the region of convergence (ROC); within the ROC, ˆ 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 ˆ H ( s ) → ∞ are referred to as the poles of ˆ H ( s ) ; the values of s for which ˆ H ( s ) = 0 are referred to as the zeros of ˆ H ( s ) . Remarks: • In discussions related to the LT, the complex plane is usually referred to as the splane. • When h ( t ) is the impulse response of an LTIC system, its LT is referred to as the system’s transfer function (TF). • Consider the special case of s = jω , i.e., imaginary axis. Then, if the ROC contains the imaginary axis, we have ˆ H ( s ) s = jω = Z + ∞ t =0 h ( t ) e jωt dt = H ( ω ) . (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:  ˆ H ( s )  = Z + ∞ t =0 h ( t ) e ( σ + jω ) t dt ≤ Z + ∞ t =0  h ( t ) e σt  dt. (6) Therefore, a sufficient condition for the convergence or existence of the LT is Z + ∞ t =0  h ( t ) e σt  dt < ∞ , for some real σ , (7) i.e., absolute integrability of h ( t ) e σt . 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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 Spring '10
 KAMALPREMERATNE
 Impulse response, Stability theory, e−st dt

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