You already know how to look at signals as a function of time - you can plot them, graph them, list them, make pie charts - there are dozens of ways to try and get at the underlying meaning of the signal. Fourier methods give a completely different approach - by using sums of sinusoids, signals can be represented in terms of their frequency content. This is ideal for identifying periodicities, or regularities in data. It is ideal for making predictions - after all, once you've recognized a pattern, you can extrapolate into the future.
Even better, linear systems pass sinusoids undisturbed (they have the same frequencies at the output as they did at the input), so you can use the idea of the Fourier decomposition as a tool for understanding systems. This leads to the idea of the "frequency response" which is the best way to think about filters, communication systems, and linear systems in general. Sure - it all seems a bit complicated at first - but hey, the good things in life are never easy!
Time-domain response and convolution; frequency-domain response using Fourier series, Fourier transform, Laplace transform; discrete Fourier series and transforms; sampling; z-transform; relationships between time and frequency, descriptions of discrete and continuous signals and systems.
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