The Laplace transform
Introduction
The Laplace transform possesses a distinct set of properties for analyzing signals LTI systems. Many of these properties parallel those of the Fourier Transform. For example, we shall see that continuous-time complex exponentials are eigenfunctions of LTI systems. As with complex sinusoids, one consequence of this property is that the convolution of time signals becomes multiplication of the associated Laplace transforms. Hence, the output of an LT system is obtained by multiplying the Laplace transform of the input by the Laplace transform of the impulse response, which is defined as the transfer function of the system. The transfer function generalizes the frequency response characterization of an LTI system’s input-output behavior and offers new insights into system characteristics. problems and
The Laplace transform comes in two varieties:
(1) unilateral, or one-sided, and
(2) bilateral, or two-sided.
The unilateral Laplace transform is a convenient tool for solving differential equations with initial conditions. The bilateral Laplace transform offers insight into the nature of system characteristics such as stability, causality, and frequency response, The primary role of the Laplace transform in engineering is the transient and stability analysis of causal LTI systems described by differential equations.
Eigenfunction property of e^st
Let us consider applying an input of the form x(t) = e^st to an LTI system with impulse response h(t). The system output will be given by:
y(t) = H{x(t)} = h(t) * x(t) ∫h(τ) x(t-τ) dτ.
We use x(t) = e^st to obtain:
y(t) = ∫h(τ) e^s(t-τ) dτ.
We define the transfer function:
H(s) = ∫h(τ) e^st dτ.
The action of the system on an input e” is multiplication by the transfer function H(s). Recall that an eigenfunction is a signal that passes through the system without being modified except for multiplication by a scalar. Hence, we identify e ^ (st) as an eigenfunction of the LTI system and as the corresponding eigenvalue.
Convergence
Our prior development indicates that the Laplace transform is the Fourier transform of x(t)e^-at. Therefore, a necessary condition for convergence of the Laplace transform is the absolute integrability of x(t) e^(- at).
The range for the Laplace transform is termed the region of convergence.
The Laplace transform exists for signals that do not have a Fourier transform. By limiting ourselves to a certain range of σ, we may ensure that is absolutely integrable, even though x(t) is not absolutely integrable by itself. For example, x(t) the Fourier transform of x(t)= e’ut does not exist, since x(t) is an increasing real exponential signal and is therefore not absolutely integrable. However, if σ>1, then x(t) e^-at = e^(1-σ)t u(t) is absolutely integrable, and so the Laplace transform, which is the Fourier transform of x(t) e^-at does exist.
The s-plane
It is convenient to represent the complex frequency s graphically in terms of a complex plane termed the s-plane. The horizontal axis represents the real part of s, that is the exponential damping factor, and the vertical axis represents the imaginary part of s, which is the sinusoidal frequency.
In the s-plane, σ = 0 corresponds to the imaginary axis. Therefore, we can say that the Fourier transform is given by the Laplace transform evaluated along the imaginary axis.
The jw-axis divides the s-plane in half. The region of the s-plane to the left of the jw-axis is termed the left half of the s-plane, while the region to the right of the jw-axis is termed the right half of the s-plane. The real part of s is negative in the left half of the s-plane and positive in the right half of the s-plane.
Poles and zeroes
The most commonly encountered polynomial encountered in engineering is the ratio of two polynomials in s, that is:
X(s) = bMs^M + bM-1s^M-1 + ….. + b0/ s^N + aN-1 S^N-1 + … + a1s + a0
It is useful to factor X(s) as a product of terms involving the roots of the denominator and the numerator polynomial.
The expression for the Laplace transform does not uniquely correspond to the signal x(t) if the ROC is not specified. That is, two signals may have identical Laplace transforms but different ROCs.
Reference