Discrete Fourier Transform - Definition

Definition

The sequence of N complex numbers x₀, ..., x_N−1 is transformed into another sequence of N complex numbers according to the DFT formula:

(Eq.1)

The transform is sometimes denoted by the symbol, as in or or .

Eq.1 can be interpreted or derived in various ways, for example:

• It completely describes the discrete-time Fourier transform (DTFT) of an N-periodic sequence, which comprises only discrete frequency components. (Discrete-time Fourier transform#Periodic data)
• It can also provide uniformly spaced samples of the continuous DTFT of a finite length sequence. (Sampling the DTFT)
• It is the discrete analogy of the formula for the coefficients of a Fourier series:

(Eq.2)

which is the inverse DFT (IDFT). Each is a complex number that encodes both amplitude and phase of a sinusoidal component of function .

The sinusoid's frequency is cycles per sample. Its amplitude and phase are:

where atan2 is the two-argument form of the arctan function. The normalization factor multiplying the DFT and IDFT (here 1 and 1/N) and the signs of the exponents are merely conventions, and differ in some treatments. The only requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that the product of their normalization factors be 1/N. A normalization of for both the DFT and IDFT makes the transforms unitary, which has some theoretical advantages. But it is often more practical in numerical computation to perform the scaling all at once as above (and a unit scaling can be convenient in other ways).

(The convention of a negative sign in the exponent is often convenient because it means that is the amplitude of a "positive frequency", . Equivalently, the DFT is often thought of as a matched filter: when looking for a frequency of +1, one correlates the incoming signal with a frequency of −1.)

In the following discussion the terms "sequence" and "vector" will be considered interchangeable.