ECE 3620: Laplace Transforms: Chapter 3: 3.1-3.4 Prof. K. Chandra ECE, UMASS Lowell September 21, 2016 1 Analysis of LTI Systems in the Frequency Domain Thus far we have understood the relationship between the input signal x(t), the system impulse response h(t) and the output y(t) through the convolution integral y(t) = x(τ)h(t τ)dτ (1) The independent variable has been t, representing time. Note that although we use time as a reference, this variable could also represent space. Signals and systems can be characterized with respect to their frequency content. This can lead to several advantages in their analysis. Consider for example the signal x(t) = cos(πt). If we wanted to represent this signal
digitally we would need to capture at least one time period of information and store that sequence of values. However, we can see that this signal is characterized by a single frequency ω = π. If we knew this frequency and the fact that the signal is a cosine function, the signal can be recreated anytime on any system with this one piece of frequency information. Often, the frequency information can lead to a more concise or compact representation of signals and systems. This is taken advantage of in the design of digital media. You have heard about standards such as MP3 audio. They all rely on using the frequency content in the audio signal and the perceptual features of the human ear to create codes that use the characteristic or relevant frequencies of the signal, leading to significantly reduced information that need to be stored. This has led to the compression formats and resulting high quality audio in CD s and DVD s. What follows is an introduction to the use of the complex frequency variable s to represent signals and systems. We effectively transform a signal x(t) in the time domain to a new function X(s) in the frequency domain. The same can be done to systems, by transforming h(t) to H(s). Before we do this, we introduce the concept of eigenfunctions, (eigen meaning characteristic in German ). These functions are fundamental to understanding the transformation from time to frequency domain. 2 Eigenfunctions of LTI Systems Consider the complex exponential function x(t) = e s 0t where s 0 = σ 0 + jω 0 is the complex frequency variable. This variable resides on the two-dimensional 2
complex frequency plane s = σ + jω, with σ and ω representing the real and imaginary components. The s-plane is a two dimensional region with horizontal axis being the real component < σ < and the vertical axis representing the imaginary component jω, < ω <. Let x(t) = e s0t be input to an LTI system with impulse response h(t). The output, per the convolution result is y(t) = = e s 0t x(t τ) h(τ)dτ = e s 0t H(s 0 ) = x(t)h(s 0 ) h(τ) e s 0τ dτ Note that in the result shown above, the output is equal to the input multiplied by a constant H(s 0 ). Any function which when input to a LTI system results in the output being the same function, only scaled by a constant is called as an eigenfunction. Therefore functions of the form e st are known as eigenfunctions of LTI systems. The complex constant is obtained by, H(s) = h(τ) e sτ dτ (2) evaluated for a specified frequency s = s 0. The function H(s) is known as the transfer function of the system. It is also known as the Laplace Transform (LT) of the system impulse response h(t). Notice that in Eqn. 2, the time varying function h(τ) is transformed to H(s) which is a function of the frequency variable s. 3
It is not guaranteed that H(s) will exist (i.e. 2 is integrable ) for all values of s in the complex frequency plane. To completely specify H(s), one has to also specify the region in the s plane for which the LT converges. This region is referred to as the Region of Convergence (ROC). We will determine how to define the ROC in the following examples. The eigenfunction property of complex exponentials makes it advantageous to represent input and output signals also in the s plane. In other words, complex exponential functions e st can be used as a basis for representing any signal x(t). In general any signal x(t) can be represented by a transformed signal X(s) denoted as its Laplace transform (LT) as follows, X(s) = x(t) e st dt (3) defined over an appropriate ROC. The identification of this region of convergence of X(s) in the s-plane is very important in the calculation of the LT. The region will be dependent on the function x(t). Consider the following examples. Let x(t) = e at u(t). Its LT is given by X(s) = = 0 e at e st dt e (a + s)t dτ = 1 [ e (a+σ)t e jωt] s + a 0 1 = iff (a + σ) > 0 s + a In step 3 above, we have used the condition for the limit to exist at t =, 4
recognizing that the function e jωt has a bounded amplitude of one for all t and the existence of the integral relies on the argument (a + σ) of the real exponential signal. This has to be positive for the upper limit to go to zero. The condition a + σ = a + Re[s] > 0 Re[s] > a is the region of convergence (ROC) of X(s) = 1 s+a, the LT of x(t) = e at u(t). Note that a can be positive or negative. Repeating the same problem for x(t) = e at u( t), and carrying out the same procedure as above, show that X(s) = 1 s+a with ROC, Re[s] < a. In summary the Laplace transform of any continuous time function x(t) X(s) = x(t) e st dt has an associated ROC. Failure to specify the ROC will result in an incomplete specification of X(s). When x(t) = x 1 (t) + x 2 (t) + x 3 (t), the Laplace transform X(s) = X 1 (s) + X 2 (s)+x 3 (s) with the ROC being the intersection of regions of convergence of X 1 (s), X 2 (s) and X 3 (s). In many cases the LT of x(t) is in the form of a ratio of two polynomials in s. That is, X(s) = N(s), a rational function. The roots of the numerator D(s) polynomial N(s) = 0 are denoted as the zeros of X(s), since at these values of s, X(s) is zero. The roots of the denominator, D(s) = 0 are the poles of the function X(s). At these values of s, X(s) goes to infinity. Note that the ROC should not contain any poles of X(s). In fact, the poles often define the boundaries of the ROC. In summary the following properties of the ROC are to be well understood: The ROC of X(s) consists of strips parallel to jω axis in s-plane. For rational LTs, ROC does not contain any poles 5
A finite duration signal x(t), that is absolutely integrable has an ROC that is the entire s-plane. A right sided infinite duration signal x(t) with a rational X(s) has a ROC that is to the right of the rightmost pole. A left sided infinite duration signal x(t) with a rational X(s) has a ROC that is to the left of the leftmost pole. A two sided signal x(t), has an ROC that is a strip in the s-plane. 6
Properties: Bilateral Laplace Transforms Given x(t) X(s) = x(t)e st dt and its ROC : R x(t t 0 ) e st 0 X(s), ROC : R e s 0t x(t) X(s s 0 ), ROC : R : Shifted by s 0 ; s 0 in general complex x(at) 1 a X ( s a) : R is scaled based on a Convolution: x 1 (t) x 2 (t) X 1 (s) X 2 (s) : ROC : R 1 R2 Differentiation : dx(t) dt X(s) has pole at s = 0 sx(s) : ROC : At least R or a larger ROC if tx(t) dx(s) ds ROC : R t x(τ) dτ 1 s X(s) : ROC : At least R Re[s] > 0 Initial Value Theorem: x(0+) = lim s [sx(s)] If x(t) = 0 no singularities at t = 0 t < 0 and Final Value Theorem: lim t x(t) = lim s 0 sx(s) If x(t) = 0 t < 0 and x(t) is finite as t Unilateral Laplace Transforms Given x(t) X(s) = 0 x(t)est dt with a ROC that is always in the right hand plane (RHP) dx(t) dt sx(s) x(0 ) 7