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Fourier Analysis and Spectral Representation of Signals

We have seen in the previous chapter that the action of an LTI system on a sinusoidal or complex exponential input signal can be represented effectively by the frequency response H(Ω) of the system. By superposition, it then becomes easy—again using the frequency response—to determine the action of an LTI system on a weighted linear combination of si­ nusoids or complex exponentials (as illustrated in Example 3 of the preceding chapter). The natural question now is how large a class of signals can be represented in this manner. The short answer to this question: most signals you are likely to be interested in! The tool for exposing the decomposition of a signal into a weighted sum of sinusoids or complex exponentials is Fourier analysis. We first discuss the Discrete-Time Fourier Transform (DTFT), which we have actually seen hints of already and which applies to the most general classes of signals. We then move to the Discrete-Time Fourier Series (DTFS), which constructs a similar representation for the special case of periodic signals, or for sig­ nals of finite duration. The DTFT development provides some useful background, context and intuition for the more special DTFS development, but may be skimmed over on an initial reading (i.e., understand the logical flow of the development, but don’t struggle too much with the mathematical details).