Discrete Fourier Transform (general) - Definition

Definition

Let be any ring, let be an integer, and let be a principal nth root of unity, defined by:

  • for

The discrete Fourier transform maps an n-tuple of elements of to another n-tuple of elements of according to the following formula:

By convention, the tuple is said to be in the time domain and the index is called time. The tuple is said to be in the frequency domain and the index is called frequency. The tuple is also called the spectrum of . This terminology derives from the applications of Fourier transforms in signal processing.

If R is an integral domain (which includes fields), it is sufficient to choose as a primitive nth root of unity, which replaces the condition (1) by:

for

Proof: take with . Since, giving:

where the sum matches (1). Since is a primitive root of unity, . Since R is an integral domain, the sum must be zero. ∎

Another simple condition applies in the case where n is a power of two: (1) may be replaced by .

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