“Discrete Fourier Transform”的概念、定义、习题解题方法及翻译-数学辞典

单词

Discrete Fourier Transform

释义

Discrete Fourier Transform

The Fourier Transform is defined as

(1)

Now consider generalization to the case of a discrete function,

by letting

, where

, with

, ...,

. Choose the frequency step such that

(2)

with

, ..., 0, ...,

. There are

values of

, so there is one relationship between the frequencycomponents. Writing this out as per Press et al. (1989)

(3)

and

(4)

The inverse transform is

(5)

Note that

, 2, ..., so an alternate formulation is

(6)

where the Negative frequencies

have

, Positive frequencies

have

, with zero frequency

corresponds to both

and

. The discreteFourier transform can be computed using a Fast Fourier Transform.

The discrete Fourier transform is a special case of the z-Transform.

See also Fast Fourier Transform, Fourier Transform, Hartley Transform, Winograd Transform, z-Transform

References

Arfken, G. ``Discrete Orthogonality--Discrete Fourier Transform.'' §14.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 787-792, 1985.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Fourier Transform of Discretely Sampled Data.'' §12.1 in Numerical Recipes in C: The Art of Scientific Computing. Cambridge, England: Cambridge University Press, pp. 494-498, 1989.

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