Fractional Fourier Transform

In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the ''n''-th power, where ''n'' need not be an integer — thus, it can transform a function to any ''intermediate'' domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition.

The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials.

However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups. Since then, there has been a surge of interest in extending Shannon's sampling theorem for signals which are band-limited in the Fractional Fourier domain.

A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber as essentially another name for a z-transform, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear chirp) and evaluating at a fractional set of frequency points (e.g. considering only a small portion of the spectrum). (Such transforms can be evaluated efficiently by Bluestein's FFT algorithm.) This terminology has fallen out of use in most of the technical literature, however, in preference to the FRFT. The remainder of this article describes the FRFT.

Introduction

The continuous Fourier transform $\mathcal$ of a function $f: \mathbb \mapsto \mathbb$ is a unitary operator of xv

4
$L^2$ that maps the function $f$ to its frequential version $\hat$ (all expressions are taken in the $L^2$ sense, rather than pointwise):

$\hat(\xi) = \int_^ f(x)\ e^\,\mathrmx$

and $f$ is determined by $\hat$ via the inverse transform $\mathcal^\, ,$

$f(x) = \int_^ \hat(\xi)\ e^\,\mathrm\xi\, .$

Let us study its ''n''-th iterated $\mathcal^$ defined by
\mathcal^ = \mathcal mathcal^[f and $\mathcal^ = (\mathcal^)^n$ when ''n'' is a non-negative integer, and \mathcal^ = f. Their sequence is finite since $\mathcal$ is a 4-periodic automorphism: for every function ƒ, \mathcal^4 = f.

More precisely, let us introduce the parity operator $\mathcal$ that inverts $x$, \mathcal colon x \mapsto f(-x). Then the following properties hold:
$\mathcal^0 = \mathrm, \qquad \mathcal^1 = \mathcal, \qquad \mathcal^2 = \mathcal, \qquad \mathcal^4 = \mathrm$
$\mathcal^3 = \mathcal^ = \mathcal \circ \mathcal = \mathcal \circ \mathcal.$

The FRFT provides a family of linear transforms that further extends this definition to handle non-integer powers $n = 2\alpha/\pi$ of the FT.

Definition

Note: some authors write the transform in terms of the "order " instead of the "angle ", in which case the is usually times . Although these two forms are equivalent, one must be careful about which definition the author uses.

For any real , the -angle fractional Fourier transform of a function ƒ is denoted by $\mathcal_\alpha (u)$ and defined by

Formally, this formula is only valid when the input function is in a sufficiently nice space (such as L1 or Schwartz space), and is defined via a density argument, in a way similar to that of the ordinary Fourier transform (see article), in the general case.

If is an integer multiple of π, then the cotangent and cosecant functions above diverge. However, this can be handled by taking the limit, and leads to a Dirac delta function in the integrand. More directly, since $\mathcal^2(f)=f(-t)~, ~~\mathcal_ ~ (f)$
must be simply or for an even or odd multiple of respectively.

For , this becomes precisely the definition of the continuous Fourier transform, and for it is the definition of the inverse continuous Fourier transform.

The FRFT argument is neither a spatial one nor a frequency . We will see why it can be interpreted as linear combination of both coordinates . When we want to distinguish the -angular fractional domain, we will let $x_a$ denote the argument of $\mathcal_\alpha$.

Remark: with the angular frequency ω convention instead of the frequency one, the FRFT formula is the Mehler kernel,
$\mathcal_\alpha(f)(\omega) = \sqrt e^ \int_^\infty e^ f(t)\, dt~.$

Properties

The -th order fractional Fourier transform operator, $\mathcal_\alpha$, has the properties:

Additivity

For any real angles ,
$\mathcal_ = \mathcal_\alpha \circ \mathcal_\beta = \mathcal_\beta \circ \mathcal_\alpha.$

Linearity

\mathcal_\alpha \left sum\nolimits_k b_kf_k(u) \right \sum\nolimits_k b_k\mathcal_\alpha \left _k(u) \right /math>

Integer Orders

If is an integer multiple of $\pi / 2$, then:
$\mathcal_\alpha = \mathcal_ = \mathcal^k = (\mathcal)^k$

Moreover, it has following relation

\begin
\mathcal^2 &= \mathcal && \mathcal (u)f(-u)\\
\mathcal^3 &= \mathcal^ = (\mathcal)^ \\
\mathcal^4 &= \mathcal^0 = \mathcal \\
\mathcal^i &= \mathcal^j && i \equiv j \mod 4
\end

Inverse

$(\mathcal_\alpha)^=\mathcal_$

Commutativity

$\mathcal_\mathcal_=\mathcal_\mathcal_$

Associativity

$\left (\mathcal_\mathcal_ \right )\mathcal_ = \mathcal_ \left (\mathcal_\mathcal_ \right )$

Unitarity

$\int f^*(u)g(u)du=\int f_\alpha^*(u)g_\alpha(u)du$

Time Reversal

$\mathcal_\alpha\mathcal=\mathcal\mathcal_\alpha$
\mathcal_\alpha (-u)f_\alpha(-u)

Transform of a shifted function

Define the shift and the phase shift operators as follows:

\begin
\mathcal(u_0) (u)&= f(u+u_0) \\
\mathcal(v_0) (u)&= e^f(u)
\end

Then
$\begin \mathcal_\alpha \mathcal(u_0) &= e^ \mathcal(u_0\sin\alpha) \mathcal(u_0\cos\alpha) \mathcal_\alpha, \end$

that is,

\begin
\mathcal_\alpha (u+u_0)&=e^ e^ f_\alpha (u+u_0 \cos\alpha)
\end

Transform of a scaled function

Define the scaling and chirp multiplication operators as follows:
\begin
M(M) (u)&= , M, ^ f \left (\tfrac \right) \\
Q(q) (u)&= e^ f(u)
\end

Then,
\begin
\mathcal_\alpha M(M) &= Q \left (-\cot \left (\frac\alpha \right ) \right)\times M \left (\frac \right )\mathcal_ \\ pt\mathcal_\alpha \left integral_transform