分數傅利葉轉換
维基百科，自由的百科全书
在數學的諧波分析領域中，分數傅利葉轉換(fractional Fourier transform, FRFT)是一種線性轉換，其將連續傅利葉轉換(簡稱「傅利葉轉換」)廣義化，可以想作傅利葉轉換的第n個冪，這個冪不一定要是整數。因此，其可將一個函數轉換到「介於」時間與頻率之間的範疇。它應用於濾波器設計、訊號分析，以及相位復原(phase retrieval)與模式識別(pattern recognition)等領域。
The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was given by Namias (1980), but it was not widely recognized until it was independently reinvented around 1993 by several groups of researchers (Almeida, 1994).
A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber (1991) as essentially another name for a z-transform, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear chirp) and evaluating at at fractional set of frequency points (e.g. considering only a small portion of the spectrum). (Such transforms can be evaluated efficiently by Bluestein's FFT algorithm.) This terminology has fallen out of use in most of the technical literature, however, in preference to the FRFT. The remainder of this article describes the FRFT.
See also the chirplet transform for a related generalization of the Fourier transform.
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[编辑] 定義
If the continuous Fourier transform of a function f(t) is denoted by 
, then 
, and in general 
; similarly, 
denotes the n-th power of the inverse transform 
of F(ω). The FRFT further extends this definition to handle non-integer powers n = 2α / π for any real α, denoted by 
and having the properties:
when n = 2α / π is an integer, and:
.
More specifically, is given by the equation:
Note that, for α = π / 2, this becomes precisely the definition of the continuous Fourier transform, and for α = − π / 2 it is the definition of the inverse continuous Fourier transform.
If α is an integer multiple of π, then the cotangent and cosecant functions above diverge. However, this can be handled by taking the limit, and leads to a 狄拉克δ函數 in the integrand. More easily, since 
, must be simply f(t) or f( − t) for α an even or odd multiple of π, respectively.
There also exist related fractional generalizations of similar transforms such as the 離散傅利葉轉換.
[编辑] 相關條目
其他的時間-頻率轉換：
[编辑] 外部連結
[编辑] 參考文獻
- V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," J. Inst. Appl. Math. 25, 241–265 (1980).
- Luís B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Sig. Processing 42 (11), 3084–3091 (1994).
- Soo-Chang Pei and Jian-Jiun Ding, "Relations between fractional operations and time-frequency distributions, and their applications," IEEE Trans. Sig. Processing 49 (8), 1638–1655 (2001).
- D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Review 33, 389-404 (1991). (Note that this article refers to the chirp-z transform variant, not the FRFT.)
- Haldun M. Ozaktas, Zeev Zalevsky and M. Alper Kutay. "The Fractional Fourier Transform with Applications in Optics and Signal Processing". John Wiley & Sons (2001). Series in Pure and Applied Optics.
