Digital Signal Processing Unit 2 Notes on discrete Fourier transform (DFT)

Digital Signal Processing Unit 2 Notes on  discrete Fourier transform (DFT)

                                             In mathematics, the discrete Fourier transform (DFT) is one of the specific forms of Fourier analysis. As such, it transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function (which is often a function in the time domain). But the DFT requires an input function that is discrete and whose non-zero values have a limited (finite) duration. Such inputs are often created by sampling a continuous function, like a person's voice. And unlike the discrete-time Fourier transform (DTFT), it only evaluates enough frequency components to reconstruct the finite segment that was analyzed. Its inverse transform cannot reproduce the entire time domain, unless the input happens to be periodic (forever). Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain discrete-time functions. The sinusoidal basis functions of the decomposition have the same properties.
Since the input function is a finite sequence of real or complex numbers, the DFT is ideal for processing information stored in computers. In particular, the DFT is widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, to solve partial differential equations, and to perform other operations such as convolutions. The DFT can be computed efficiently in practice using a fast Fourier transform (FFT) algorithm.
Since FFT algorithms are so commonly employed to compute the DFT, the two terms are often used interchangeably in colloquial settings, although there is a clear distinction: "DFT" refers to a mathematical transformation, regardless of how it is computed, while "FFT" refers to any one of several efficient algorithms for the DFT. This distinction is further blurred, however, by the synonym finite Fourier transform for the DFT, which apparently predates the term "fast Fourier transform" (Cooley et al., 1969) but has the same initialism.[edit] Definition
The sequence of N complex numbers x₀, ..., x_N−1 is transformed into the sequence of N complex numbers X₀, ..., X_N−1 by the DFT according to the formula:
where is a primitive N'th root of unity.
The transform is sometimes denoted by the symbol , as in or or .
The inverse discrete Fourier transform (IDFT) is given by
A simple description of these equations is that the complex numbers X_k represent the amplitude and phase of the different sinusoidal components of the input "signal" x_n. The DFT computes the X_k from the x_n, while the IDFT shows how to compute the x_n as a sum of sinusoidal components X_kexp(2πikn / N) / N with frequency k / N cycles per sample. By writing the equations in this form, we are making extensive use of Euler's formula to express sinusoids in terms of complex exponentials, which are much easier to manipulate. (In the same way, by writing X_k in polar form, we immediately obtain the sinusoid amplitude from | X_k | and the phase from the complex argument.) An important subtlety of this representation, aliasing, is discussed below.
Note that 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 X_k is the amplitude of a "positive frequency" 2πk / N. 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.

 Properties

 Completeness

The discrete Fourier transform is an invertible, linear transformation
with denoting the set of complex numbers. In other words, for any N > 0, an N-dimensional complex vector has a DFT and an IDFT which are in turn N-dimensional complex vectors.

 Orthogonality

The vectors form an orthogonal basis over the set of N-dimensional complex vectors:
where is the Kronecker delta. This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.

 The Plancherel theorem and Parseval's theorem

If X_k and Y_k are the DFTs of x_n and y_n respectively then the Plancherel theorem states:
where the star denotes complex conjugation. Parseval's theorem is a special case of the Plancherel theorem and states:
These theorems are also equivalent to the unitary condition below.

Periodicity

If the expression that defines the DFT is evaluated for all integers k instead of just for , then the resulting infinite sequence is a periodic extension of the DFT, periodic with period N.
The periodicity can be shown directly from the definition:
where we have used the fact that e ^{− 2πi} = 1. In the same way it can be shown that the IDFT formula leads to a periodic extension.

The shift theorem

Multiplying x_n by a linear phase for some integer m corresponds to a circular shift of the output X_k: X_k is replaced by X_{k − m}, where the subscript is interpreted modulo N (i.e., periodically). Similarly, a circular shift of the input x_n corresponds to multiplying the output X_k by a linear phase. Mathematically, if {x_n} represents the vector x then

if

then

and

Circular convolution theorem and cross-correlation theorem

The convolution theorem for the continuous and discrete time Fourier transforms indicates that a convolution of two infinite sequences can be obtained as the inverse transform of the product of the individual transforms. With sequences and transforms of length N, a circularity arises:
The quantity in parentheses is 0 for all values of m except those of the form n − l − pN, where p is any integer. At those values, it is 1. It can therefore be replaced by an infinite sum of Kronecker delta functions, and we continue accordingly. Note that we can also extend the limits of m to infinity, with the understanding that the x and y sequences are defined as 0 outside [0,N-1]:
which is the convolution of the sequence with a periodically extended sequence defined by:
Similarly, it can be shown that:
which is the cross-correlation of    and  

A direct evaluation of the convolution or correlation summation (above) requires O(N²) operations for an output sequence of length N. An indirect method, using transforms, can take advantage of the O(NlogN) efficiency of the fast Fourier transform (FFT) to achieve much better performance. Furthermore, convolutions can be used to efficiently compute DFTs via Rader's FFT algorithm and Bluestein's FFT algorithm.
Methods have also been developed to use circular convolution as part of an efficient process that achieves normal (non-circular) convolution with an or sequence potentially much longer than the practical transform size (N). Two such methods are called overlap-save and overlap-add^[1].

Trigonometric interpolation polynomial

The trigonometric interpolation polynomial

for N even ,

for N odd,

where the coefficients X_k /N are given by the DFT of x_n above, satisfies the interpolation property p(2πn / N) = x_n for .
For even N, notice that the Nyquist component is handled specially.
This interpolation is not unique: aliasing implies that one could add N to any of the complex-sinusoid frequencies (e.g. changing e ^{− it} to e^{i(N − 1)t} ) without changing the interpolation property, but giving different values in between the x_n points. The choice above, however, is typical because it has two useful properties. First, it consists of sinusoids whose frequencies have the smallest possible magnitudes, and therefore minimizes the mean-square slope of the interpolating function. Second, if the x_n are real numbers, then p(t) is real as well.
In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to N − 1 (instead of roughly − N / 2 to + N / 2 as above), similar to the inverse DFT formula. This interpolation does not minimize the slope, and is not generally real-valued for real x_n; its use is a common mistake.

 The unitary DFT

Another way of looking at the DFT is to note that in the above discussion, the DFT can be expressed as a Vandermonde matrix:
where
is a primitive Nth root of unity. The inverse transform is then given by the inverse of the above matrix:
With unitary normalization constants , the DFT becomes a unitary transformation, defined by a unitary matrix:
where det()  is the determinant function. The determinant is the product of the eigenvalues, which are always or as described below. In a real vector space, a unitary transformation can be thought of as simply a rigid rotation of the coordinate system, and all of the properties of a rigid rotation can be found in the unitary DFT.
The orthogonality of the DFT is now expressed as an orthonormality condition (which arises in many areas of mathematics as described in root of unity):
If is defined as the unitary DFT of the vector then
and the Plancherel theorem is expressed as:
If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a new coordinate system, then the above is just the statement that the dot product of two vectors is preserved under a unitary DFT transformation. For the special case , this implies that the length of a vector is preserved as well—this is just Parseval's theorem:

Expressing the inverse DFT in terms of the DFT

A useful property of the DFT is that the inverse DFT can be easily expressed in terms of the (forward) DFT, via several well-known "tricks". (For example, in computations, it is often convenient to only implement a fast Fourier transform corresponding to one transform direction and then to get the other transform direction from the first.)
First, we can compute the inverse DFT by reversing the inputs:
(As usual, the subscripts are interpreted modulo N; thus, for n = 0, we have x_{N − 0} = x₀.)
Second, one can also conjugate the inputs and outputs:
Third, a variant of this conjugation trick, which is sometimes preferable because it requires no modification of the data values, involves swapping real and imaginary parts (which can be done on a computer simply by modifying pointers). Define swap(x_n) as x_n with its real and imaginary parts swapped—that is, if x_n = a + bi then swap(x_n) is b + ai. Equivalently, swap(x_n) equals . Then
That is, the inverse transform is the same as the forward transform with the real and imaginary parts swapped for both input and output, up to a normalization (Duhamel et al., 1988).
The conjugation trick can also be used to define a new transform, closely related to the DFT, that is involutary—that is, which is its own inverse. In particular, is clearly its own inverse: . A closely related involutary transformation (by a factor of (1+i) /√2) is , since the (1 + i) factors in cancel the 2. For real inputs , the real part of is none other than the discrete Hartley transform, which is also involutary.

Eigenvalues and eigenvectors

The eigenvalues of the DFT matrix are simple and well-known, whereas the eigenvectors are complicated, not unique, and are the subject of ongoing research.
Consider the unitary form defined above for the DFT of length N, where . This matrix satisfies the equation:
This can be seen from the inverse properties above: operating twice gives the original data in reverse order, so operating four times gives back the original data and is thus the identity matrix. This means that the eigenvalues λ satisfy a characteristic equation:

λ⁴ = 1.

Therefore, the eigenvalues of are the fourth roots of unity: λ is +1, −1, +i, or −i.
Since there are only four distinct eigenvalues for this matrix, they have some multiplicity. The multiplicity gives the number of linearly independent eigenvectors corresponding to each eigenvalue. (Note that there are N independent eigenvectors; a unitary matrix is never defective.)
The problem of their multiplicity was solved by McClellan and Parks (1972), although it was later shown to have been equivalent to a problem solved by Gauss (Dickinson and Steiglitz, 1982). The multiplicity depends on the value of N modulo 4, and is given by the following table:

size N	λ = +1	λ = −1	λ = -i	λ = +i
4m	m + 1	m	m	m − 1
4m + 1	m + 1	m	m	m
4m + 2	m + 1	m + 1	m	m
4m + 3	m + 1	m + 1	m + 1	m
Multiplicities of the eigenvalues λ of the unitary DFT matrix U as a function of the transform size N (in terms of an integer m).

Unfortunately, no simple analytical formula for the eigenvectors is known. Moreover, the eigenvectors are not unique because any linear combination of eigenvectors for the same eigenvalue is also an eigenvector for that eigenvalue. Various researchers have proposed different choices of eigenvectors, selected to satisfy useful properties like orthogonality and to have "simple" forms (e.g., McClellan and Parks, 1972; Dickinson and Steiglitz, 1982; Grünbaum, 1982; Atakishiyev and Wolf, 1997; Candan et al., 2000; Hanna et al., 2004).
The choice of eigenvectors of the DFT matrix has become important in recent years in order to define a discrete analogue of the fractional Fourier transform—the DFT matrix can be taken to fractional powers by exponentiating the eigenvalues (e.g., Rubio and Santhanam, 2005). For the continuous Fourier transform, the natural orthogonal eigenfunctions are the Hermite functions, so various discrete analogues of these have been employed as the eigenvectors of the DFT, such as the Kravchuk polynomials (Atakishiyev and Wolf, 1997). The "best" choice of eigenvectors to define a fractional discrete Fourier transform remains an open question, however.

 The real-input DFT

If are real numbers, as they often are in practical applications, then the DFT obeys the symmetry:
where the star denotes complex conjugation and the subscripts are interpreted modulo N.
Therefore, the DFT output for real inputs is half redundant, and one obtains the complete information by only looking at roughly half of the outputs . In this case, the "DC" element X₀ is purely real, and for even N the "Nyquist" element X_{N / 2} is also real, so there are exactly N non-redundant real numbers in the first half + Nyquist element of the complex output X.
Using Euler's formula, the interpolating trigonometric polynomial can then be interpreted as a sum of sine and cosine functions.

 Generalized/shifted 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 (or GDFT), also called the shifted DFT or offset 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 (X_{k + N} = − X_k) 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.
Another interesting choice is a = b = − (N − 1) / 2, which is called the centered DFT (or CDFT). The centered DFT has the useful property that, when N is a multiple of four, all four of its eigenvalues (see above) have equal multiplicities (Rubio and Santhanam, 2005).
The discrete Fourier transform can be viewed as a special case of the z-transform, evaluated on the unit circle in the complex plane; more general z-transforms correspond to complex shifts a and b above.