Fourier Transform and properties




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TWF: Tutorial

Linear Canonical Transform

何宜靜 R00945009

Abstract

In this tutorial, I introduce some of the basic information about linear canonical transformation. LCT, a family of integral transform, is an important application tool in time frequency analysis. Because Fourier transform, Fractional Fourier Transform, and Fresnel Transform can all be represented as a form of LCT. Therefore, first, I introduce the definition of Fourier Transform and its properties, and then I give the derivation of LCT. Moreover, the motion of signals in time-frequency domain can be represented in matrix form such as horizontal shifting, vertical shifting, dilation, shearing, rotation, twisting and linearity properties. Finally, the application of LCT including filtering, Arnold's cat map, which can be an analysis tool in quantum chaos, and digital holography used for camera. I mainly summarize the previous student’s tutorial and add on some of the materials I read on the internet.

  1. Introduction

The linear canonical transform is a family of integral transform

(1.1)

,which has the form

(1.2)

Fourier transforms, Fractional Fourier Transform, and Fresnel Transform can also be classified as LCT, and the definition for these transforms are as follows:

  1. Fourier Transform(FT)
    (1.3)

  2. Fractional Fourier Transform(FRFT)
    (1.4)

,where

  1. Fresnel Transform
    (1.5)

    (1.6)

Fresnel Transform can be realized by two one-dimension LCT, one of them is horizontal shearing, and the other is vertical shearing.

Here, I present some of the definitions of Fourier Transform and its properties.

  1. Fourier Transform and properties

The definition of Fourier Transform is:

(2.1)

The inverse Fourier Transform is defined as:

(2.2)

From the properties of Fourier Transform
(2.3)

So

And (2.4)

So (2.5)

For convenient computation, we define 2 operators Q and P

Q{x(t)}=t*x(t), (2.6)

Operator P has a property of a rotation by pi/2 in a Q-P phase space, because of j=exp(j*pi/2).

There are four properties of P and Q.

Property (1)

F{P{x(t)}}=Q{F{x(t)}} in operator form FP=QF (2.7a)

Pf:



Property (2)

F{Q{x(t)}}=-P{F{x(t)}} in operator form FQ=-PF (2.7b)

Pf:





Property (3)

, i.e. they are self-adjoint operators (2.7c)

Pf:



Property (4)

[QP-PQ]=j*1 (2.7d)

Pf:

[QP-PQ]x(t)=QP{x(t)}-PQ{x(t)}=-jQ{}-P{t*x(t)}=-j*t*+j*t*=-j*t*+j(t*+x(t))=j*x(t)



3. Derivation of LCT

3.1 QP operators

We define a linear operator C which turns operators Q and P into linear combinations with each other:



(3.1)

, where the constants a,b,c,d are real. But there is a constraint:ad-bc=1.

Pf:

The commutator of [Q’,P’] defined as in (2.10) and using distributive law



(3.2)

So we define a matrix M and label the transformation C with this matrix

(3.3)

And means an act on an appropriate space and is linear. i.e.:



Due to (3.1), we define

So (3.4)

Similarly, transform of is

(3.5)

3.2 Integral transform and kernel

To realize the linear operator , we propose an integral transform with a kernel :

(3.6)

So (3.4) can be rewritten as below:

(3.7a)

Similarly,

(3.7b)

Where

So (3.6) can be rewritten as

(3.8a)

(3.8b)

to solve in, we assume the solution of the kind and find that A=ja/2b, B=-j/b, C=id/2b

(3.9)

, where

Finally, the integral transform (3.10)

This is the final form of Linear Canonical Form.

After arranging the consequence, we can have clear form

(3.11)

But what if b=0, we will discuss in next section.

3.2 Special case with b=0

To introduce the special case with b=0, we must know Gaussian function at first. Gaussian Bell function of width w:

(3.12)

then we use ad-bc=1 to replace d in d in (3.9)

so (3.13)

, where

and we must know

(3.14)

exist only when is bounded.

It means that or , I.e.

and if a=0, then Im(b)=0

the phase of Gaussian’s argument t is

so if

(3.15)

although is a complex argument, when |b|0 limit to a Dirac and shows that

(a) For any , the function either vanishes for or oscillates with infinite rapidity for

(b) The integral of the function over all real number is finite

(3.16)

, where is phase function defined below:

if (3.17)

if (3.18)

hence

(3.19)

substitute (3.19) into (3.13)

(3.20)

if near identity matrix, a is near to unity, c is near to zero, if we agree to let b approach zero from lower complex half-plane, including the real axis, then = (3.21)

this thus constitutes the identity for group of real linear canonical transform. This result determine our choice of phase of in (3.9)



and (3.21) also indicate the result when b0

(3.22)

so finally we have LCT formula with different condition or

thus

4. LCT application on time and frequency domain

Traditional Fourier Transform is one-dimensional form, so that there are only two possible movements including modulation and scaling.

Modulation:
Scaling:

There are various kinds of movements including horizontal shifting, vertical shifting, dilation, shearing and twisting for two-dimensional time-frequency analysis. LCT uses matrix to transform time and frequency signals from one shape into another, thus it can reduce sampling rate and other signal processing applications. So here I will introduce matrix representation of LCT for each kind of motion.

If

Then



  1. Horizontal shifting









  1. Vertical shifting






Ex.



  1. Dilation





Ex.



  1. Shearing




, we can get the argument b by trying some points.

Ex.







, and we can get c by trying some points.

Ex.



  1. Rotation

If we rotate the time-frequency signal clockwise by

We get , so when



Where a=0, b=1, c=-1, d=0



The following are the derivations of short time Fourier Transform, Gabor Transform, and Wigner Distribution Function applying FRFT



We can find out that Fourier Transform means rotate the time-frequency signal clockwise, so inverse Fourier Transform means rotate the T-F signal counterclockwise. Thus, we will get the original signal by Fourier Transform 4 times.




Here are some examples of rotating the signal by Gabor Transform and WDF.







Moreover, if a function is an eigenfunction of the Fourier Transform


Then its WDF and Gabor Transform will have the property of


A Gaussian function is an eigenfunction of FT, because the shape of a Gaussian function is a circle on time-frequency domain, and remaining a circle when we rotate it.

Ex.

untitled.jpg

  1. Twisting

As defined by the equation (1)



LCT can be expressed in matrix form with



It means that by transforming to , the shape of the signal can be twisted. Furthermore, if we know the shape of the twisted signal, the values of a,b,c,d can also be calculated.

Ex.



  1. Linearity

If we compare the transform matrix of horizontal and vertical shifting



And the other LCT can be expressed as



As we can see, the transform matrix of shifting are nonlinear, and the others are linear. To eliminate the non-linearity, we can add one dimension to matrix of shifting.

Ex.



While for rotation matrix



It might be more convenient to express the matrix in linear form.
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