Linear Circuits and Systems

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

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 time-domain 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 s-plane. • 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)....
View Full Document

## This note was uploaded on 10/28/2010 for the course EEN 307 taught by Professor Kamalpremeratne during the Spring '10 term at University of Miami.

### Page1 / 28

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online