Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

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18 LINEAR SYSTEMS AND TRANSFORMS

When fy ¼ 0,

1ð

Uðfx; 0Þ ¼ ½h2ðxÞ h1ðxÞ&e j 2p fx xdx

1

¼ H2ðfxÞ H1ðfxÞ

Thus, Uðfx; 0Þ is the difference of the 1-D FT of the functions h2ðxÞ and h1ðxÞ.

2.6REAL FOURIER TRANSFORM

Sometimes it is more convenient to represent the Fourier transform with real sine and cosine basis functions. Then, it is referred to as the real Fourier transform (RFT). What was discussed as the Fourier transform before would then be the complex Fourier transform (CFT) [Ersoy, 1994]. For example, the analytic signal representation of nonmonochromatic wave fields can be more effectively derived using the RFT, as discussed in Section 9.4. In this and next sections, we will discuss the 1-D transforms only.

The RFT of a signal uðxÞ can be defined as

 

 

 

1

 

 

 

 

Uðf Þ ¼ 2wðf Þ

ð

uðtÞ cosð2p ft þ yð f ÞÞdx

ð2:6-1Þ

 

 

 

1

 

 

 

 

where

 

 

 

( 1=2 f

¼6 0

 

 

wðf Þ ¼

ð2:6-2Þ

 

 

 

 

1

f

0

 

 

 

 

 

 

 

¼

 

 

 

 

 

0

f

0

 

 

yðf Þ ¼ ( p=2 f < 0

ð2:6-3Þ

The inverse RFT is given by

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

uðtÞ ¼

ð

Uð f Þ cosð2p ft þ yð f ÞÞdf

ð2:6-4Þ

 

1

 

 

 

 

 

 

It is observed that cosð2pft þ yð f ÞÞ equals cosð2pftÞ for f ¼ 0 and sinð2pj f jtÞ for f < 0. This is a ‘‘trick’’ used to cover both the cosine and sine basis functions in a single integral. Negative frequencies are used for this purpose. It is interesting

REAL FOURIER TRANSFORM

 

 

 

 

 

 

 

 

 

 

 

 

19

 

1.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FunctionBasis

0.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

–0.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

–1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

–1.5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

–5

 

 

 

0

 

 

 

5

 

 

 

10

 

–10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Frequency

 

 

 

 

 

 

 

 

 

Figure 2.3.

p2 cos 2

ft

f

ÞÞ

for t

¼

0 25 and

10

<

f

<

10.

The basis function

 

ð

p

þ yð

 

:

 

 

 

 

p

to observe that 2 cosð2pft þ yð f ÞÞ is orthonormal with respect to f because Eq. (2.6-4) is true. Thus,

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

ð0

cosð2pft þ yð f ÞÞ cosð2pf t þ yð f ÞÞdf ¼ dðt tÞ

 

 

ð2:6-5Þ

where dð Þ is the Dirac-delta function.

 

ÞÞ

 

¼

 

:

 

10 < f < 10.

 

 

ð

2

p

ft

þ yð

f

is shown in Figure 2.3 for t

0

25 and

The basis function p2 cos

 

 

 

 

 

 

Equations (2.6-1) and (2.6-4) can also be written for f ¼ 0 as

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

U1ðf Þ ¼ 2wðf Þ

ð

uðtÞ cosð2p ftÞdt

 

 

ð2:6-6Þ

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

U0ðf Þ ¼ 2

ð

uðtÞ sinð2p ftÞdt

 

 

ð2:6-7Þ

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

uðtÞ ¼ ð

½U1ð f Þ cosð2p ftÞ þ U0ð f Þ sinð2p ftÞ&df

 

 

ð2:6-8Þ

0

20 LINEAR SYSTEMS AND TRANSFORMS

Thus, Uð f Þ equals U1ð f Þ for f ¼ 0 and X0ðj f jÞ for f < 0. X1ð f Þ and X0ð f Þ will be referred to as the cosine and sine parts, respectively. Equations (2.6-1), (2.6-6), and (2.6-7) are also referred to as the analysis equations. Equations (2.6-4) and (2.6-8) are the corresponding synthesis equations.

The relationship between the CFT and the RFT can be expressed for f 0 as

" Uc

 

Ucð0Þ ¼

Uð0Þ

j

#" U0

ð f Þ #f > 0

ð2:6-9Þ

 

ð

fÞ #

¼

2 "

1

Uc

 

f

 

1

 

1

j

U1

f

 

 

ð Þ

 

 

 

 

 

 

 

ð Þ

 

Equation (2.6-9) reflects the fact that U1ðf Þ and U0ðf Þ are even and odd functions, respectively.

The inverse of Eq. (2.6-9) for f 0 is given by

U1ð0Þ ¼ Ucð0Þ

 

#" U

 

cð

fÞ

#f > 0

ð2:6-10Þ

" U1

ðf Þ #

¼

" j

 

j

 

U

f

 

1 1

 

U

f

 

 

 

0

ð Þ

 

 

 

 

 

cð Þ

 

 

Equations (2.6-9) and (2.6-10) are useful to convert from one representation to the other. When xðtÞ is real, U1ðf Þ and U0ðf Þare also real. Then, Eq. (2.6-9) shows that Ucðf Þ and Ucð f Þ are complex conjugates of each other.

2.7AMPLITUDE AND PHASE SPECTRA

Ucðf Þ can be written as

Ucðf Þ ¼ Uaðf Þ e jfðf Þ;

ð2:7-1Þ

where the amplitude (magnitude) spectrum Uað f Þ and the phase spectrum fð f Þ of the signal xðtÞ are defined as

 

 

 

 

1

hjU1

ð f Þj2 þ jU0ð f Þj2i

1=2

 

Uað f Þ ¼ jUcð f Þj ¼

 

ð2:7-2Þ

2

fð

 

Þ ¼

 

Real ½Uc½ð f Þ&

 

ð

Þ

 

f

 

tan 1

Imaginary

Ucð f Þ&

 

2:7-3

 

 

 

 

 

 

 

 

 

Uað f Þis an even function. With real signals, fð f Þ is an odd function and can be written as

fð f Þ ¼ tan 1½ U0ðf Þ=U1ð f Þ&

ð2:7-4Þ

HANKEL TRANSFORMS

21

Equation (1.2.2) for the inverse CFT can be written in terms of the amplitude and phase spectra as

 

 

1

 

 

uðtÞ ¼

ð

Uað f Þe j½2p ftþfð f Þ&df

ð2:7-5Þ

 

 

1

 

 

When uðtÞ is real, this equation reduces to

 

 

1

 

 

uðtÞ ¼ 2

ð0

Uað f Þ cos½2p ft þ fð f Þ&df

ð2:7-6Þ

because Ucð f Þ ¼ Uc ð f Þ, and the integrand in

1ð

j Uað f Þ sin½2p ft þ fð f Þ&df

1

is an odd function and integrates to zero.

Equation (2.7-6) will be used in representing nonmonochromatic wave fields in Section 9.4.

2.8HANKEL TRANSFORMS

Functions having radial symmetry are easier to handle in polar coordinates. This is often the case, for example, in optics where lenses, aperture stops, and so on are often circular in shape.

Let us first consider the Fourier transform in polar coordinates. The rectangular and polar coordinates are shown in Figure 2.4. The transformation to polar

Figure 2.4. The rectangular and polar coordinates.

22

 

 

 

 

LINEAR SYSTEMS AND TRANSFORMS

coordinates is given by

 

 

 

 

 

 

 

 

 

 

r ¼ ½x2 þ y2&1=2

 

 

 

 

 

 

y

 

 

 

 

y ¼ tan 1

 

 

 

 

 

 

 

 

 

x

ð

2:8-1

Þ

 

r ¼ ½fx2 þ fy2&1=2

 

 

f ¼ tan 1

 

fy

 

 

 

 

 

fx

 

 

 

The FT of uðx; yÞ is given by

ðð uðx; yÞe j2p ð fxxþfyyÞdxdy

 

 

 

Uðfx; fyÞ ¼

ð2:8-2Þ

 

1

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

f ðx; yÞ in polar coordinates is f ðr; yÞ. Fð fx; fyÞ in polar coordinates is Fð r; fÞ, given by

2p

1

 

 

 

 

 

 

 

Uðr; fÞ ¼ ð0

dy ð0

uðr; yÞe j2p rrðcos y cos fþsin y sin fÞrdr

ð2:8-3Þ

When uðx; yÞ is circularly symmetric, it can be written as

 

 

 

 

uðr; yÞ ¼ uðrÞ

ð2:8-4Þ

Then, Eq. (2.6-3) becomes

 

 

 

 

 

 

 

 

 

1

 

 

 

2p

 

 

Uðr; fÞ ¼ ð0

uðrÞrdr ð0

e j2prr cosðy fÞdy

ð2:8-5Þ

The Bessel function of the first kind of zero order is given by

 

 

 

 

 

 

2p

 

 

 

 

J0ðtÞ ¼

1

ð0

e jt cosðy fÞdy

ð2:8-6Þ

 

2p

It is observed that J0ðtÞ is the same for all values of f. Substituting this identity in Eq. (2.8-5) and incorporating an extra factor of 2p gives

1ð

Uðr; fÞ ¼ UðrÞ ¼ 2p uðrÞJ0ð2prrÞrdr

ð2:8-7Þ

0

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