Man and History: Discrete Fourier Series

2022年3月16日星期三

Discrete Fourier Series

2022/02/12

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https://pixabay.com/zh/photos/sphinx-egypt-hieroglyphics-temple-1175828/

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◎ 說明：

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首先是名詞的確定。

關於傅立葉變換，除了連續時間以外，還有離散時間傅立葉變換與離散傅立葉變換兩種。但是離散傅立葉級數與離散時間傅立葉級數兩者指的是同一件事，兩者都有人用 [1] - [7]。

維基百科使用的是「Discrete Fourier Series」[1], [2]，但在其中文條目裡又使用離散時間傅立葉級數，容易令人混淆。

Oppenheim 的 Signals and Systems 裡面使用「Discrete-Time Fourier Series」[3], [4]。

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Fig. 1. Discrete-Time Fourier Series [3].

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◎ 參考資料：

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1. 主值區間上的取值

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「離散傅立葉級數（DFS）與連續傅立葉級數相比有很大的區別。最大的不同在於離散時間傅立葉級數的係數序列是周期的。」「離散傅立葉變換實際上是離散時間傅立葉級數在主值區間上的取值。」[2]。

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2. DTFS

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「Introduces the discrete-time Fourier Series (closely related to the DFT) and shows how to find the Fourier series coefficients of sampled cosine and sine waveforms.」[5]。

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3. DFS

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「Discrete Fourier Series (DFS) is used for representing the periodic discrete-time signals in the frequency domain. This video presents a brief introduction and intuition to DFS.」[6]。

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4. inverse DFT

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「In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. Fourier) discrete real sinusoids or discrete complex exponentials, combined by a weighted summation. A specific example is the inverse discrete Fourier transform (inverse DFT).」[1]。

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5. DFT

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「Because of periodicity, xN can be represented as a DFS with N unique coefficients that can be extracted by an N-length DFT.」[1]。

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6. DTFT

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「The coefficients are useful because they are samples of the discrete-time Fourier transform (DTFT) of the x[n] sequence:」[1]。

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References

[1] Discrete Fourier series - Wikipedia

https://en.wikipedia.org/wiki/Discrete_Fourier_series