Discrete Fourier transform Guide, Meaning , Facts, Information and Description

In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal. It is sometimes denoted by the symbol . It can be computed quickly using a fast Fourier transform (FFT) algorithm.

Table of contents

1 Definition 1.1 Properties 1.2 Generalized DFT 2 Applications 2.3 Signal analysis 2.4 Data compression 2.5 Partial differential equations 2.6 Multiplication of large integers 3 References

Definition

The discrete Fourier transform is an invertible, linear transformation

with C denoting the complex numbers. Moreover, if it is scaled by (see below), the DFT is unitary ("energy" conserving) by Parseval's theorem.

The n complex numbers x₀, ..., x_n-1 are transformed into the n complex numbers f₀, ..., f_n-1 according to the formula:

where e is the base of the natural logarithm, i () is the imaginary unit, and π is Pi.

The inverse discrete Fourier transform (IDFT) is given by

Written in matrix form, the DFT is:

where

is a primitive nth root of unity.

Note that the normalization factor multiplying the sum (here unity) and the sign of the exponent are merely conventions, and differ in some treatments. All of the following discussion applies regardless of the convention, with at most minor adjustments. The only important thing is that the forward and inverse transforms 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, but it is often more practical in numerical computation to perform the scaling all at once.

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.)

Properties

The function

whose coefficients f_j are given by the DFT of x_k, above, is called the trigonometric interpolation polynomial of degree n-1. It is the unique function of this form that satisfies the property: p(2πk/n) = x_k for k = 0, ..., n-1.

If x₀, ..., x_n-1 are real numbers, as they often are in the applications, then f_j = f_n-j^*, where the star denotes complex conjugation. Hence, the full information in this case is already contained in the first half (roughly) of the sequence f₀, ..., f_n-1. In this case, the "DC" element f₀ is purely real, and for even n the "Nyquist" element f_n/2 is also real, so there are exactly n non-redundant real numbers in the first half + Nyquist element of the complex output f. Using Euler's formula, the interpolating trigonometric polynomial can then be interpreted as a sum of sine and cosine functions.

The cyclic convolution x*y of the two vectors x = (x_j) and y = (y_k) is the vector x*y with components

(where we continue y cyclically so that y_-j = y_n-j for j = 1, ..., n-1. The discrete Fourier transform turns cyclic convolutions into component-wise multiplication: F(x*y)_j = F(x)_j F(y)_j (j = 0, ..., n-1).

Generalized DFT

It is possible to shift the transform sampling in time and/or frequency domain by some real shifts a and b, respectively. This is sometimes known as a generalized DFT and has analogous properties to the ordinary DFT:

Most often, shifts of 1/2 (half a sample) are used. While the ordinary DFT corresponds to a periodic signal in both time and frequency domains, a=1/2 produces a signal that is anti-periodic in frequency domain (f_j+n = -f_j) and vice-versa for b=1/2. Thus, the specific case of a=b=1/2 is known as an odd-time odd-frequency discrete Fourier transform (or O² DFT). Such shifted transforms are most often used for symmetric data, to represent different boundary symmetries, and for real-symmetric data they correspond to different forms of the discrete cosine and sine transforms.

The discrete Fourier transform can be viewed as a special case of the z-transform, evaluated on the unit circle in the complex plane.

Applications

The DFT has seen wide usage across a large number of fields; we only sketch a few examples below (see also the references at the end). All applications of the DFT depend crucially on the availability of a fast algorithm to compute discrete Fourier transforms and their inverses, a fast Fourier transform.

Signal analysis

Suppose a signal x(t) is to be analyzed. Here, t stands for time, which varies over the interval [0,T], and, in the case of a sound signal, x(t) is the air pressure at time t.

The signal is sampled at n equidistant points to get the n real numbers x₀ = x(0), x₁ = x(h), x₂ = x(2h), ..., x_n-1 = x((n-1)h), where h = T/n and n is even.

Then the discrete Fourier transform f₀, ..., f_n-1 is computed and the numbers f_{n/2 + 1}, ..., f_n-1 are discarded (they are redundant for real signals).

Then f₀/n approximates the average value of the signal over the interval, and for every j = 1, ..., n/2, the argument (see complex number) arg(f_j) represents the phase and the modulus |f_j|/n represents one half of the amplitude of the component of the signal having frequency j/T (see wave).

The reason behind this interpretation is that the f_j are approximations to the coefficients occurring in the Fourier series expansion of x(t). In general, the problem of using the DFT of discrete samples to approximate the Fourier transform of an infinite, continuous signal is called spectral estimation, and involves many more details than are described here. (For example, one often wants to window the data in order to reduce the distortion caused by the periodic boundary conditions implicit in the DFT.)

For an example see frequency spectrum.

Data compression

The field of digital signal processing relies heavily on operations in the frequency domain (i.e. on the Fourier transform). For example, several lossy image and sound compression methods employ the discrete Fourier transform: the signal is cut into short segments, each is transformed, and then the Fourier coefficients of high frequencies, which are assumed to be unnoticeable, are discarded. The decompressor computes the inverse transform based on this reduced number of Fourier coefficients. (Compression applications often use a specialized form of the DFT, the discrete cosine transform.)

Partial differential equations

Discrete Fourier transforms, especially in multidimensions, are often used to solve partial differential equations. The reason is that it expands the signal in complex exponentials e^ikx, which are eigenfunctions of differentiation: d/dx e^ikx = ik e^ikx. Thus, in the Fourier representation, a linear differential equation with constant coefficients is transformed into an easily solvable algebraic equation. One then uses the inverse DFT to transform the result back into the ordinary spatial representation. Such an approach is called a spectral method.

Multiplication of large integers

The fastest known algorithms for the multiplication of large integers or polynomials are based on the discrete Fourier transform: the sequences of digits or coefficients are interpreted as vectors whose convolution needs to be computed; in order to do this, they are first Fourier transformed, then component-wise multiplied, and transformed back.

References

• E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, NJ, 1988).
• A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, 1999).
• S. W. Smith, The Scientist and Engineer's Guide to Digital Signal Processing, (California Technical Publishing, San Diego, 2nd edition, 1999)

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