I found a decent free textbook that explains DSP fundamentals for
non-specialists. Here's the author's definition of a transform (e.g. Discrete Fourier Transform).

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Suppose you have a signal composed of 100 samples. If you devise some equation, algorithm, or procedure for changing these 100 samples into another 100 samples, you have yourself a transform. If you think it is useful enough, you have the perfect right to attach your last name to it and expound its merits to your colleagues. (This works best if you are an eminent 18th century French mathematician).
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