Discrete Fourier series: Difference between revisions
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=== Relation to Fourier series === |
=== Relation to Fourier series === |
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The exponential form of Fourier series is given by: |
The exponential form of Fourier series is given by''':''' |
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:<math>s(t) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i2\pi \frac{k}{P} t},</math> |
:<math>s(t) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i2\pi \frac{k}{P} t},</math> |
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which is periodic with an arbitrary period denoted by <math>P.</math> When continuous time <math>t</math> is replaced by discrete time <math>nT,</math> for integer values of <math>n</math> and time interval <math>T,</math> the series becomes: |
which is periodic with an arbitrary period denoted by <math>P.</math> When continuous time <math>t</math> is replaced by discrete time <math>nT,</math> for integer values of <math>n</math> and time interval <math>T,</math> the series becomes''':''' |
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:<math>s(nT) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{P}nT},\quad n \in \mathbb{Z}.</math> |
:<math>s(nT) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{P}nT},\quad n \in \mathbb{Z}.</math> |
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With <math>n</math> constrained to integer values, we normally constrain the ratio <math>P/T=N</math> to an integer value, resulting in an <math>N</math>-periodic function: |
With <math>n</math> constrained to integer values, we normally constrain the ratio <math>P/T=N</math> to an integer value, resulting in an <math>N</math>-periodic function''':''' |
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{{Equation box 1 |
{{Equation box 1 |
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|indent = |
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|title=Discrete Fourier series |
|title=Discrete Fourier series |
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|indent= |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA |
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|equation = {{NumBlk|:|<math> |
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s(nT) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{N}n} |
|equation = :<math>s_{_N}[n] \triangleq s(nT) = \sum_{k=-\infty}^\infty S[k]\cdot e^{i 2\pi \frac{k}{N}n}</math> |
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⚫ | |||
</math>|{{EquationRef|Eq.1}}}} |
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|background colour=#F5FFFA}} |
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which are harmonics of a fundamental digital frequency <math>1/N.</math> |
which are harmonics of a fundamental digital frequency <math>1/N.</math> The <math>N</math> subscript reminds us of its periodicity. And we note that some authors will refer to just the <math>S[k]</math> coefficients themselves as a discrete Fourier series.<ref name=Nuttall/>{{rp|p.85 (eq 15a)}} |
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Due to the <math>N</math>-periodicity of the <math>e^{i 2\pi \tfrac{k}{N} n}</math> kernel, the summation can be "folded" as follows''':''' |
Due to the <math>N</math>-periodicity of the <math>e^{i 2\pi \tfrac{k}{N} n}</math> kernel, the infinite summation can be "folded" as follows''':''' |
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:<math> |
:<math> |
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\begin{align} |
\begin{align} |
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s_{_N}[n] &= \sum_{m=-\infty}^{\infty}\left(\sum_{k=0}^{N-1}e^{i 2\pi \tfrac{k-mN}{N}n}\ S[k-mN]\right)\\ |
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&= \sum_{k=0}^{N-1}e^{i 2\pi \tfrac{k}{N}n} |
&= \sum_{k=0}^{N-1}e^{i 2\pi \tfrac{k}{N}n} |
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\underbrace{\left(\sum_{m=-\infty}^{\infty}S[k-mN]\right)}_{\triangleq S_N[k]}, |
\underbrace{\left(\sum_{m=-\infty}^{\infty}S[k-mN]\right)}_{\triangleq S_N[k]}, |
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\end{align} |
\end{align} |
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</math> |
</math> |
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which is proportional (by a factor of <math>N</math>) to the '''inverse DFT''' of one cycle of the periodic summation, <math>S_N.</math><ref name=Oppenheim/>{{rp|p.542 (eq 8.4)}} <ref name=Prandoni/>{{rp|p.77 (eq 4.24)}} |
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S_N. |
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</math> |
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==References== |
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== Further reading== |
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{{reflist|1|refs= |
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<ref name=Oppenheim> |
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{{Cite book |ref=Oppenheim |last=Oppenheim |first=Alan V. |authorlink=Alan V. Oppenheim |last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer |last3=Buck |first3=John R. |title=Discrete-time signal processing |year=1999 |publisher=Prentice Hall |location=Upper Saddle River, N.J. |isbn=0-13-754920-2 |edition=2nd |chapter=4.2, 8.4 |url-access=registration |url=https://archive.org/details/discretetimesign00alan |quote=samples of the Fourier transform of an aperiodic sequence x[n] can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of x[n]. ... The Fourier series coefficients can be interpreted as a sequence of finite length for k=0,...,(N-1), and zero otherwise, or as a periodic sequence defined for all k.}}</ref> |
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<ref name=Prandoni> |
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{{cite book |last1=Prandoni |first1=Paolo |last2=Vetterli |first2=Martin |title=Signal Processing for Communications |date=2008 |publisher=CRC Press |location=Boca Raton,FL |isbn=978-1-4200-7046-0 |edition=1 |url=https://www.sp4comm.org/docs/sp4comm.pdf |accessdate=4 October 2020 |pages=72,76 |quote=the DFS coefficients for the periodized signal are a discrete set of values for its DTFT |
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⚫ | |||
}}</ref> |
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<ref name=Nuttall> |
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{{cite journal |
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| doi =10.1109/TASSP.1981.1163506 |
| doi =10.1109/TASSP.1981.1163506 |
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| last =Nuttall |
| last =Nuttall |
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| date =Feb 1981 |
| date =Feb 1981 |
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| url =https://zenodo.org/record/1280930 |
| url =https://zenodo.org/record/1280930 |
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}}</ref> |
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}} |
}} |
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Revision as of 22:13, 15 May 2024
In digital signal processing, a Discrete Fourier series (DFS) a Fourier series whose sinusoidal components are functions of discrete time instead of continuous time. A specific example is the inverse discrete Fourier transform (inverse DFT).
Introduction
Relation to Fourier series
The exponential form of Fourier series is given by:
which is periodic with an arbitrary period denoted by When continuous time is replaced by discrete time for integer values of and time interval the series becomes:
With constrained to integer values, we normally constrain the ratio to an integer value, resulting in an -periodic function:
which are harmonics of a fundamental digital frequency The subscript reminds us of its periodicity. And we note that some authors will refer to just the coefficients themselves as a discrete Fourier series.[1]: p.85 (eq 15a)
Due to the -periodicity of the kernel, the infinite summation can be "folded" as follows:
which is proportional (by a factor of ) to the inverse DFT of one cycle of the periodic summation, [2]: p.542 (eq 8.4) [3]: p.77 (eq 4.24)
References
- ^ Nuttall, Albert H. (Feb 1981). "Some Windows with Very Good Sidelobe Behavior". IEEE Transactions on Acoustics, Speech, and Signal Processing. 29 (1): 84–91. doi:10.1109/TASSP.1981.1163506.
- ^
Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. (1999). "4.2, 8.4". Discrete-time signal processing (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. ISBN 0-13-754920-2.
samples of the Fourier transform of an aperiodic sequence x[n] can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of x[n]. ... The Fourier series coefficients can be interpreted as a sequence of finite length for k=0,...,(N-1), and zero otherwise, or as a periodic sequence defined for all k.
- ^
Prandoni, Paolo; Vetterli, Martin (2008). Signal Processing for Communications (PDF) (1 ed.). Boca Raton,FL: CRC Press. pp. 72, 76. ISBN 978-1-4200-7046-0. Retrieved 4 October 2020.
the DFS coefficients for the periodized signal are a discrete set of values for its DTFT