User:Bob K/sandbox

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================================================= Question: What is the difference between  $$x(t)\ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ X(f)$$  and  $$f(x)\ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \hat{f}(\xi),$$  both found in Wikipedia?

Answer: Substantially nothing. Many/most EE texts use the 1st one to represent a Fourier transform pair. Mathematicians insist on the 2nd one. My favorite textbook author, Van Trees uses $$s(t)\ (\text{or}\ s(x)) \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ S(j\omega).$$

My comments:
 * $$x$$ is a universally recognized symbol for abscissa (independent variable)... the argument of a function.  So I agree with the mathematicians on that count.
 * But I agree with the EEs on reserving $$f$$  for frequency (in cycles per unit of time or space).
 * I tried (and failed) to convince the mathematicians to substitute the less intimidating $$\nu$$ instead of $$\xi$$.
 * I reluctantly agree that the $$\hat f$$ operator notation is more consistent in some applications; e.g. $$\widehat{f\cdot g}$$  or  $$\widehat{f'},$$  instead of switching from cap letters to  $$\mathcal{F}\{f\cdot g\}$$  and  $$\mathcal{F}\{f'\},$$  but that seems like a minor consideration to me.

Bottom line: I like $$s$$ (for "signal") and $$S$$ (for "spectrum") instead of $$x$$ and $$X$$ or $$f$$ and $$\hat f$$. I also prefer $$f$$ instead of $$j\omega.$$

Therefore: $$s(t)\ (\text{or}\ s(x)) \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ S(f).$$

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================================================= Failed derivation of S[m] for DFS

The expression:


 * $$s_{_N}[n] = \sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{N}n}$$

is N-periodic in n, without assuming S[k] is periodic. Following the example of the continuous time Fourier series, it seems that any coefficient S[m] can be computed from one period of $$s_{_N}$$ as follows:



\begin{align} \sum_{n=0}^{N-1} e^{ -i 2\pi \tfrac{m}{N}n }\cdot s_{_N}[n] &= \sum_{n=0}^{N-1} e^{-i 2\pi \tfrac{m}{N}n} \left[\sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{N}n} \right]\\ &= \sum_{k=-\infty}^\infty S[k]\cdot \left[ \sum_{n=0}^{N-1} e^{-i 2\pi \tfrac{m}{N}n}\cdot e^{i 2\pi \frac{k}{N}n}  \right]\\ &= \sum_{k=-\infty}^\infty S[k]\cdot \left[ \sum_{n=0}^{N-1} e^{i 2\pi \tfrac{k-m}{N}n} \right]\\ &= \sum_{a=-\infty}^\infty N\cdot S[m+aN] \end{align} $$