By Paul A. Lynn (auth.)
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Extra info for An Introduction to the Analysis and Processing of Signals
It should however be stressed that this elementary introduction to the Laplace transform does little to suggest the full value of the method. This will become clearer in chapter 7, where further aspects of the Laplace transform are discussed in the context of signal processing. Suffice it to say at this stage that the Laplace transform is a powerful mathematical tool which may be used to solve a great many problems other than those of signal analysis. In particular (and this mattel will also be discussed again in chapter 7) a problem stated as a set of differential equations may often be reduced to a set of much simpler algebraic equations if the Laplace transform is used.
Wave (a) is an even function, symmetrical about t = O. 1 that its fundamental component is (4/7t) cos Wit. Wave (b) is identical except that it is an odd function with a fundamental equal to (4/7t) sin Wit. This shift of time origin has therefore merely had the effect of converting a Fourier series containing only sine terms into one containing only cosine terms; the amplitude of a component at anyone frequency is, as we would expect, unaltered. The situation in (c) is however more complicated because the square wave is neither even nor odd, and must therefore be expected to include both sine and cosine terms in its Fourier series.
In such a diagram the real part of a complex variable is plotted along the abscissa, and the imaginary part along the ordinate. The poles and zeros of a function G(s) are in general complex values of s, so the Argand diagram gives a convenient method of displaying them, in which case it is widely referred to as the 's-plane' diagram. Suppose, for example, we have a time function [(1), the Laplace transform of which is G(s) = 4(s2 - 2s) 4(s)(s - 2) s2 + 2s + 10 (s + 1 + j3)(s + 1 - j3) Apart from the constant multiplier of 4, we may completely represent the function G(s) by drawing zeros at s = 0 and s = 2, and poles at s = -1 ± j3, in the complex plane.