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 (Van Trees, Harry L (1968). Detection, Estimation, and Modulation Theory. Vol. 1. New York: John Wiley. p. 680. ISBN 0-471-09517-6.) 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:

$s_{_{N}}[n]=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{N}}n}$

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:

${\begin{aligned}\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {m}{N}}n}\cdot s_{_{N}}[n]&=\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {m}{N}}n}\left[\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{N}}n}\right]\\&=\sum _{k=-\infty }^{\infty }S[k]\cdot \left[\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {m}{N}}n}\cdot e^{i2\pi {\frac {k}{N}}n}\right]\\&=\sum _{k=-\infty }^{\infty }S[k]\cdot \left[\sum _{n=0}^{N-1}e^{i2\pi {\tfrac {k-m}{N}}n}\right]\\&=\sum _{a=-\infty }^{\infty }N\cdot S[m+aN]\end{aligned}}$