User:Bob K/sandbox: Difference between revisions
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==Definitions== |
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Question: What is the difference between <math>x(t)\ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ X(f)</math> and <math>f(x)\ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ \hat{f}(\xi),</math> both found in Wikipedia? |
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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 ({{cite book |last1=Van Trees |first1=Harry L |title=Detection, Estimation, and Modulation Theory |volume=1 |date=1968 |publisher=John Wiley |location=New York |isbn=0-471-09517-6 |page=680}}) uses <math>s(t)\ (\text{or}\ s(x)) \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ S(j\omega).</math> |
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The Fourier transform is an ''analysis'' process, decomposing a complex-valued function <math>\textstyle f(x)</math> into its constituent frequencies. The inverse process is ''synthesis'', which recreates <math>\textstyle f(x)</math> from its transform. We can start with an analogy, the [[Fourier series]], which analyzes <math>\textstyle f(x)</math> on a bounded interval <math>\textstyle x \in [-P/2, P/2],</math> for some positive real number <math>P.</math> The constituent frequencies are a discrete set of ''harmonics'' at frequencies <math>\tfrac{n}{P}, n \in \mathbb Z,</math> whose amplitude and phase are given by the '''analysis formula:'''<blockquote><math>c_n = \tfrac{1}{P} \int_{-P/2}^{P/2} f(x) \, e^{-i 2\pi \frac{n}{P}x} \, dx.</math></blockquote>The actual '''Fourier series''' is the '''synthesis formula:'''<blockquote><math>f(x) = \sum_{n=-\infty}^\infty c_n\, e^{i 2\pi \tfrac{n}{P}x},\quad \textstyle x \in [-P/2, P/2].</math></blockquote>The analogy for a [[Lebesgue integration|Lebesgue integrable]] <math>\textstyle f(x)</math> can be obtained formally from the analysis formula by taking the limit as <math>P\to\infty</math>, while at the same time taking <math>n</math> so that <math>\tfrac{n}{P} \to \xi \in \mathbb R.</math><ref>{{Cite book |last1=Khare |first1=Kedar |title=Fourier Optics and Computational Imaging |last2=Butola |first2=Mansi |last3=Rajora |first3=Sunaina |publisher=Springer |year=2023 |isbn=978-3-031-18353-9 |edition=2nd |pages=13–14 |chapter=Chapter 2.3 Fourier Transform as a Limiting Case of Fourier Series |doi=10.1007/978-3-031-18353-9|s2cid=255676773 }}</ref> {{harv|Kaiser|1994|p=29}}, {{harv|Rahman|2011|p=11}}. There are several [[Fourier transform#Other conventions|common conventions]] for defining the Fourier transform. This article will primarily use the following form:{{Equation box 1 |
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|title =Fourier transform |
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|equation = {{NumBlk||<math>\widehat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{-i 2\pi \xi x}\,dx.</math>|{{EquationRef|Eq.1}}}} |
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My comments: |
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Evaluating {{EquationNote|Eq.1}} for all values of <math>\xi</math> produces the ''frequency-domain'' function.<ref group="note">Since we have assumed that <math>f\in L^1(\mathbb R)</math>, it can be shown that the function <math>\widehat f\in L^\infty\cap C(\mathbb R)</math> is bounded and [[uniformly continuous]] in the frequency domain, and moreover, by the [[Riemann–Lebesgue lemma]], it is zero at infinity.</ref> The complex number <math>\widehat{f}(\xi)</math>, in polar coordinates, conveys both [[amplitude]] and [[phase offset|phase]] of frequency <math>\xi.</math> The intuitive interpretation of {{EquationNote|Eq.1}} is that the effect of multiplying <math>f(x)</math> by <math>e^{-i 2\pi \xi x}</math> is to subtract <math>\xi</math> from every frequency component of function <math>f(x).</math><ref group="note">A possible source of confusion is the [[User:Bob K/sandbox#Frequency shifting|frequency-shifting property]]; i.e. the transform of function <math>f(x)e^{-i 2\pi \xi_0 x}</math> is <math>\widehat{f}(\xi+\xi_0).</math> The value of this function at <math>\xi=0</math> is <math>\widehat{f}(\xi_0),</math> meaning that a frequency <math>\xi_0</math> has been shifted to zero (also see [[Negative frequency#Simplifying the Fourier transform|Negative frequency]]).</ref> Any sinusoidal component that was at <math>\xi</math> ends up at frequency zero. The integral produces only its complex amplitude, because (at least formally) all the other components are oscillatory and integrate to zero over an infinite interval. If a frequency is not present, the transform has a value of 0 for that frequency. |
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*<math>x</math> is a universally recognized symbol for abscissa (independent variable)... the argument of a function. So I agree with the mathematicians on that count. |
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*But I agree with the EEs on reserving <math>f</math> for frequency (in ''cycles per unit of time or space''). |
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*I tried (and failed) to convince the mathematicians to substitute the less intimidating <math>\nu</math> instead of <math>\xi</math>. |
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*I reluctantly agree that the <math>\hat f</math> operator notation is more consistent in some applications; e.g. <math>\widehat{f\cdot g}</math> or <math>\widehat{f'},</math> instead of switching from cap letters to <math>\mathcal{F}\{f\cdot g\}</math> and <math>\mathcal{F}\{f'\},</math> but that seems like a minor consideration to me. |
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Bottom line: I like <math>s</math> (for "signal") and <math>S</math> (for "spectrum") instead of <math>x</math> and <math>X</math> or <math>f</math> and <math>\hat f</math>. I also prefer <math>f</math> instead of <math>j\omega.</math> |
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The corresponding synthesis formula is:{{Equation box 1 |
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|title = Inverse transform |
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|equation = {{NumBlk||<math>f(x) = \int_{-\infty}^{\infty} \widehat f(\xi)\ e^{i 2 \pi \xi x}\,d\xi,\quad \forall\ x \in \mathbb R.</math>|{{EquationRef|Eq.2}}}} |
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Therefore: <math>s(t)\ (\text{or}\ s(x)) \ \stackrel{\mathcal{F}}{\Longleftrightarrow}\ S(f).</math> |
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{{EquationNote|Eq.2}} is a representation of <math>f(x)</math> as a weighted summation of complex exponential functions. |
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This is also known as the [[Fourier inversion theorem]], and was first introduced in [[Joseph Fourier|Fourier's]] ''Analytical Theory of Heat''. |
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Failed derivation of S[m] for DFS |
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<ref>{{harvnb|Fourier|1822|p=525}}.</ref> |
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<ref>{{harvnb|Fourier|1878|p=408}}.</ref> |
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<ref>{{harv|Jordan|1883}} proves on pp. 216–226 the [[Fourier inversion theorem#Fourier integral theorem|Fourier integral theorem]] before studying Fourier series.</ref> |
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<ref>{{harvnb|Titchmarsh|1986|p=1}}.</ref> |
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The expression: |
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The functions <math>f</math> and <math>\widehat{f}</math> are referred to as a '''Fourier transform pair'''.<ref>{{harvnb|Rahman|2011|p=10}}.</ref> A common notation for designating transform pairs is:<ref>{{harvnb|Oppenheim|1999|p=58}}</ref> |
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:<math>f(x)\ \stackrel{\mathcal{F}}{\longleftrightarrow}\ \widehat f(\xi),</math> for example <math>\operatorname{rect}(x)\ \stackrel{\mathcal{F}}{\longleftrightarrow}\ \operatorname{sinc}(\xi).</math> |
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: |
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:<math>s_{_N}[n] = \sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{N}n}</math> |
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=== Physical conventions === |
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When the independent variable (<math>x</math>) represents ''time'' (often denoted by <math>t</math>), the transform variable (<math>\xi</math>) represents [[frequency]] (often denoted by <math>f</math>). For example, if time is measured in [[second]]s, then frequency is in [[hertz]]. The [[#Fourier transform on Euclidean space|Fourier transform on Euclidean space]] is treated separately, in which the variable <math>x</math> often represents position and <math>\xi</math> momentum. For other common conventions and notations, including using the [[angular frequency]] <math>\omega</math> instead of the [[frequency|ordinary frequency]] <math>\xi,</math> see [[#Other conventions|Other conventions]] and [[#Other notations|Other notations]] below. The [[#Fourier_transform_on_Euclidean space|Fourier transform on Euclidean space]] is treated separately, in which the variable <math>x</math> often represents position and <math>\xi</math> momentum. The conventions chosen in this article are those of [[harmonic analysis]], and are characterized as the unique conventions such that the Fourier transform is both [[Unitary operator|unitary]] on {{math|''L''<sup>2</sup>}} and an algebra homomorphism from {{math|''L''<sup>1</sup>}} to {{math|''L''<sup>∞</sup>}}, without renormalizing the Lebesgue measure.<ref>{{harvnb|Folland|1989}}.</ref> |
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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 <math>s_{_N}</math> as follows: |
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Many other characterizations of the Fourier transform exist. For example, one uses the [[Stone–von Neumann theorem]]: the Fourier transform is the unique unitary [[intertwiner]] for the symplectic and Euclidean Schrödinger representations of the [[Heisenberg group]]. |
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:<math> |
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<references /> |
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\begin{align} |
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\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]\\ |
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&= \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]\\ |
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&= \sum_{k=-\infty}^\infty S[k]\cdot \left[ \sum_{n=0}^{N-1} e^{i 2\pi \tfrac{k-m}{N}n} \right]\\ |
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&= \sum_{a=-\infty}^\infty N\cdot S[m+aN] |
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\end{align} |
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</math> |
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Revision as of 13:10, 16 May 2024
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Question: What is the difference between and 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
My comments:
- 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 for frequency (in cycles per unit of time or space).
- I tried (and failed) to convince the mathematicians to substitute the less intimidating instead of .
- I reluctantly agree that the operator notation is more consistent in some applications; e.g. or instead of switching from cap letters to and but that seems like a minor consideration to me.
Bottom line: I like (for "signal") and (for "spectrum") instead of and or and . I also prefer instead of
Therefore:
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Failed derivation of S[m] for DFS
The expression:
![{\displaystyle s_{_{N}}[n]=\sum _{k=-\infty }^{\infty }S[k]\cdot e^{i2\pi {\frac {k}{N}}n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e9cec45ca4a43f4ee48409199da7691010d2b3)
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 as follows:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e135a990886ccf3371b96a61ca63d2eba2956a57)