2D Fourier Transforms in Polar Coordinates - chapter

2D FOURIER TRANSFORMS IN POLAR COORDINATES Natalie Baddour Department of Mechanical Engineering, University of Ottawa, 161Louis Pasteur, Ottawa, Ontario, K1N 6N5, Canada Email: nbaddour@uottawa.ca

For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination of Hankel transforms and Fourier series -- even if the function does not possess circular symmetry. However, to be as useful as its Cartesian counterpart, a polar version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution etc. This paper derives the requisite polar version of the standard Fourier operations. In particular, convolution – two dimensional, circular and radial one dimensional – is discussed in detail.

It is shown that standard multiplication/convolution rules do apply as long as the correct definition of convolution is applied.

Short title: 2D Fourier transforms in polar coordinates keywords: 2D Fourier Transform, polar coordinates, convolution

1

1

INTRODUCTION

The Fourier transform needs no introduction and it would be an understatement to say that it has proved to be invaluable in many diverse disciplines such as engineering, mathematics, physics and chemistry. Its applications are numerous and include a wide range of topics such as communications, optics, astronomy, geology, image processing, signal processing and so forth. It is known that the Fourier transform can easily be extended to n-dimensions. The strength of the Fourier transform is that it is accompanied by a tool-set of operational properties that simplify the calculation of more complicated transforms through the use of these standard rules. Specifically, the standard Fourier tool-set consists of results for scaling, translation (spatial shift), multiplication and convolution, along with the basic transforms of the Dirac-delta function and complex exponential which are so essential to the derivation of the shift, multiplication and convolution results. This basic tool-set of operational rules is well known for the Fourier transform in single and multi-dimensions (Bracewell 1999; Howell 2000).

As is also known, the Fourier transform in 2D can be developed in terms of polar coordinates (Chirikjian & Kyatkin 2001) instead of the usual Cartesian coordinates, most usefully in the case where the function being transformed is naturally describable in polar coordinates. For example, this has seen application in the field of photoacoustics, (Y. Xu et al. 2002) and attempts have been made to translate ideas from the continuous domain to the discrete domain by developing numerical algorithms for such calculations (Averbuch et al. 2006). However, to the best of the author's knowledge, a complete interpretation of the standard Fourier operational tool-set in terms of polar coordinates is missing from the literature. Some results are known, such as the Dirac-delta function in both polar and spherical polar coordinates, but the results on shift, multiplication and in particular convolution, are incomplete.

This paper thus aims to develop the Fourier operational tool-set of Dirac-delta, exponential, spatial shift, multiplication and convolution for the 2 dimensional Fourier transform in polar coordinates. Of particular novelty is the treatment of the shift, multiplication and convolution theorems, which can also be adapted for the special cases of circularly symmetric functions that have no angular dependence. It is well known from the literature that 2D Fourier transforms for radially symmetric functions can be interpreted in terms of a (zeroth order) Hankel transform. It is also known that the Hankel transforms do not have a multiplication/convolution rule, a rule which has found so much use in the Cartesian version of the transform. In this paper, the multiplication/convolution rule is treated in detail for the curvilinear version of the transform and in particular it is shown here that the zeroth order Hankel transform does obey a multiplication/convolution rule once the proper interpretation of convolution is applied. This article carefully considers the definition of convolution and derives the correct interpretation of this in terms of the curvilinear coordinates so that the standard multiplication/convolution rule will once again be applicable.

The outline of the paper is as follows. For completeness, the Hankel transform and the interpretation of the 2D Fourier transform in terms of a Hankel transform and a Fourier series are introduced in sections 2 and 3.

Sections 4 and 5 treat the special functions of the Dirac-delta and complex exponential. Sections 6, 7 and 8 address the multiplication, spatial shift and convolution operations. In particular, the nature of the spatial shift and its role in the ensuing convolution theorem are discussed. Sections 9 and 10 discuss the spatial shift and convolution operators when dealing with the special case of radially (circularly) symmetric functions. Section 11 addresses the special case of a convolution of a radially symmetric function with one that is not, as this case has many important applications. Sections 12 and 13 discuss the special cases of 2

angular or radial convolution only – that is not a full 2D but a special 1D convolution as restricted to convolving over only one of the variables of the polar coordinates. In particular, it is shown that while the angular convolution yields a simple convolution relationship, the radial-only convolution does not. Section 14 derives the Parseval relationships. Sections 15 and 16 introduce some applications involving the Laplacian and the Helmholtz equation. Section 17 concludes the article. The operational toolset as derived is summarized in a table at the end of the paper.

2

HANKEL TRANSFORM

The nth order Hankel transform is defined by the integral (Piessens 2000) ∞

Fn ( ρ ) = H n ( f (r ) ) = ∫ f (r ) J n ( ρ r )rdr ,

(1)

0

where J n ( z ) is the nth order Bessel function and the overhat indicates a Hankel Transform as shown in equation (1). Here, n may be an arbitrary real or complex number. However, an integral transform needs to be invertible in order to be useful and this restricts the allowable values of n. If n is real and n > −1/ 2 , the transform is self-reciprocating and the inversion formula is given by ∞

f (r ) = ∫ Fn ( ρ ) J n ( ρ r ) ρ d ρ .

(2)

0

The inversion formula for the Hankel transform follows immediately from Hankel's repeated integral which ∞

states that under suitable boundary conditions and subject to the condition that

∫ f (r )

rdr is absolutely

0

convergent, then for n > −1/ 2 ∞

∞

0

0

1

∫ s ds ∫ f (r ) J n (sr ) J n (su ) rdr = 2 [ f (u +) + f (u −)] .

The most important cases correspond to n=0 or n=1. The Hankel transform exists only if the Dirichlet ∞

condition is satisfied, that is if the following integral exists:

∫r

1/ 2

f (r ) dr . The Hankel transform is

0

particularly useful for problems involving cylindrical symmetry. It is useful to note that the Bessel functions satisfy an orthogonality/closure relationship given by

3

(3)

∞

1

∫ J n (ux) J n (vx) xdx = u δ (u − v) .

(4)

0

3

CONNECTION BETWEEN THE 2D FOURIER TRANSFORM AND HANKEL

TRANSFORM The 2D Fourier transform of a function f ( x, y ) is defined similarly to its 1D counterpart: ∞

F (ω ) = F (ω x , ω y ) =

∞

∫ ∫

f ( x, y ) e

− j (ωx x +ω y y )

dx dy .

(5)

dω x dω y ,

(6)

−∞ −∞

The inverse Fourier transform is given by

f ( r ) = f ( x, y ) =

∞

1

∞

F (ω x , ω y )e ∫ ∫ 2 π ( ) −∞ −∞

jω ⋅r

2

(

)

where the shorthand notation of ω = ω x , ω y , r = ( x, y ) has been used.

Polar coordinates can be introduced as x = r cos θ , y = r sin θ and similarly in the spatial frequency domain as ω x = ρ cosψ ω y = ρ sinψ . It then follows that the two-dimensional Fourier transform can be written as ∞

F ( ρ ,ψ ) = ∫ 0

π

f (r ,θ ) e−ir ρ cos(ψ −θ ) rdrdθ .

∫

(7)

−π

Thus, in terms of polar coordinates, the Fourier transform operation transforms the spatial position radius and angle ( r , θ ) to the frequency radius and angle ( ρ ,ψ ) . The usual polar coordinates relationships

(

)

apply in each domain so that r 2 = x 2 + y 2 , θ = arctan ( y x ) and ρ 2 = ω x2 + ω y2 , ψ = arctan ω y ω x .

Using r to represent ( r , θ ) in physical polar coordinates and ω to denote the frequency vector ( ρ ,ψ ) in frequency polar coordinates, the following expansions are valid (Chirikjian & Kyatkin 2001)

eiω⋅r =

∞

∑ i n J n ( ρ r ) einθ e−inψ

n =−∞

4

(8)

∞

e−iω⋅r =

∑ i −n J n ( ρ r ) e−inθ einψ

(9)

n =−∞

These expansions can be used to convert the 2D Fourier transform into polar coordinates.

3.1 RADIALLY SYMMETRIC FUNCTIONS If it is assumed that f is radially symmetric, then it can be written as function of r only and can thus be taken out of the integration over the angular coordinate so that equation (7) becomes π

∞

F ( ρ ,ψ ) = ∫ rf (r )dr ∫ e−ir ρ cos(ψ −θ ) dθ .

0

(10)

−π

Using the integral definition of the zero order Bessel function

J 0 ( x) =

1 2π

π

−ix cos(ψ −θ ) dθ = ∫e

−π

1 2π

π

∫e

−ix cosα

dα

(11)

−π

equation (10) can be written as ∞

F ( ρ ) = F2 D { f (r )} = 2π ∫ f (r ) J 0 ( ρ r )rdr

(12)

0

which can be recognized as 2π times the Hankel transform of order zero. Thus the special case of the two-dimensional Fourier transform of a radially symmetric function is the same as the zeroth-order Hankel transform of that function:

F ( ρ ) = F2 D { f (r )} = 2π H 0 { f (r )} .

(13)

With reference to equation (13), f (r ) and F ( ρ ) are functions with radial symmetry in a 2D regime,

F2D {i} is an operator in a 2D regime, while H 0 {i} is an operator in a 1D regime

3.2 NON-RADIALLY SYMMETRIC FUNCTIONS When the function f (r , θ ) is not radially symmetric and is a function of both r and θ , the preceding result can be generalized. Since f (r , θ ) depends on the angle θ , it can be expanded into a Fourier series ∞

f ( r ) = f (r ,θ ) =

∑

n =−∞

5

f n (r ) e jnθ

(14)

where

f n (r ) =

1 2π

2π

∫

f (r , θ ) e− jnθ dθ .

(15)

0

This transform is well suited for functions that are separable in r and θ . This case is extensively treated in the widely-used text on Fourier Optics by Joseph Goodman (Goodman 2004).

Similarly, the 2D Fourier transform F ( ρ ,ψ ) can also be expanded into its own Fourier series so that

F (ω ) = F ( ρ ,ψ ) =

∞

∑ Fn ( ρ ) e jnψ

(16)

n =−∞

and

1 Fn ( ρ ) = 2π

2π

∫ F ( ρ ,ψ )e

− jnψ

dψ .

(17)

0

It is extremely important to note that Fn ( ρ ) is NOT the Fourier transform of f n (r ) . In fact, it is the relationship between f n (r ) and Fn ( ρ ) that we seek to define.

3.2.1 FORWARD TRANSFORM Expansions (9) and (14) are substituted into the definition of the forward Fourier transform to give ∞

F (ω ) =

∫

f (r )e− jω⋅r dr

−∞ ∞ 2π

0

=

∞

∫ m∑ =−∞

=∫

f m (r )e jmθ

0

∞

∑ 2π i

n =−∞

−n

inψ

e

∞

∑ i −n J n ( ρ r ) e−inθ einψ dθ rdr

(18)

n =−∞

∞

∫ fn (r ) J n ( ρ r ) rdr.

0

This follows since

2π

∫0

if n = 0 ⎧ 2π ⎩0 otherwise

einθ dθ = 2π δ n 0 = ⎨

6

(19)

where δ nm denotes the kronecker delta function. By comparing equations (16) and (18), we obtain the expression for Fn ( ρ ) , the Fourier coefficients of the Fourier domain expansion. Using equation (1), this can also be interpreted in terms of a Hankel transform as:

Fn ( ρ ) = 2π i

−n

∞

∫ f n (r ) J n ( ρ r ) rdr = 2π i

−n

H n { f n (r )} .

(20)

0

3.2.2 INVERSE TRANSFORM The corresponding 2D inverse Fourier transform is written as

f (r ) =

∞ 2π

1

∫ ∫ F (ω )e ( 2π ) 0 0

jω⋅r

2

dψ ρ d ρ .

(21)

Using equation (16) along with the expansion (8) yields

f (r ) =

∞

∞

1 ∑ 2π i n einθ ∫ Fn ( ρ ) J n ( ρ r ) ρ d ρ n =−∞ 0

(22)

so that

f n (r ) =

in 2π

∞

∫ Fn ( ρ ) J n ( ρ r ) ρ d ρ = 0

in H n {Fn ( ρ )} .

2π

(23)

Thus, it can be observed that the nth term in the Fourier series for the original function will Hankel transform into the nth term of the Fourier series of the Fourier transform function. However, it is an nth order Hankel transform for the nth term, so that all the terms are not equivalently transformed.

Furthermore, we recall that the general 2D Fourier transform for radially symmetric functions was shown to be equivalent to the zeroth order Hankel transform. Therefore the mapping from f n (r ) to Fn ( ρ ) , which is an nth order Hankel transform, is not a 2D Fourier transform.

3.2.3 DISCUSSION Most importantly, it can be seen that the operation of taking the 2D Fourier transform of a function is equivalent to 1) first finding its Fourier series expansion in the angular variable and 2) then finding the nth order Hankel transform (of the spatial radial variable to the spatial frequency radial variable) of the nth coefficient in the Fourier series and appropriately scaling the result. Clearly, for functions with cylindricaltype symmetries that are naturally described by cylindrical coordinates, the operation of taking a 3D Fourier transform will be equivalent to a) a regular 1D Fourier transform in the z coordinate, then 2) a Fourier series expansion in the angular variable and then 3) an nth order Hankel transform (of the radial 7

variable to the spatial radial variable) of the nth coefficient in the Fourier series. Since each of these operations involves integration over one variable only with the others being considered parameters vis-à-vis the integration, the order in which these operations are performed is interchangeable.

3.3 FOURIER PAIRS Thinking in terms of operators and paired functions is fairly common when dealing with Fourier transforms but less common when dealing with Fourier series. Fourier series are typically introduced as an equality f (θ ) =

∞

∑

n =−∞

f n e jnθ instead of as an operator. We thus propose here that a Fourier series should be

considered as a transform/operator in the same way that we consider a Fourier transform and use the

symbol FS { f (⋅)} to represent this transform. That is to say, we re-interpret equations (14) and (15) as forward and inverse Fourier Series Transforms so that we define the forward Fourier Series Transform as the operation of finding the Fourier series coefficients:

1 f n = FS { f (θ )} = 2π

2π

− jnθ ∫ f (θ ) e dθ

(24)

0

and the inverse transform is the one where the original function is returned from the coefficients by constructing the Fourier series itself:

f (θ ) = FS−1 { f n } =

∞

∑

n =−∞

Hence f n and f (θ ) are a Fourier (series) pair f (θ ) ⇔

f n e jnθ .

(25)

fn .

In working with Fourier transforms, we often refer to 'Fourier Pairs', meaning a pair of functions where one is the Fourier transform of the other. For full 2D Fourier transforms, we denote a Fourier pair as

f ( x, y ) ⇔ F (ωx , ω y ) which means that F (ω x , ω y ) = F2 D ( f ( x, y ) ) . Similarly, the

(

)

notation f ( r , θ ) ⇔ F ( ρ ,ψ ) implies that F ( ρ ,ψ ) = F2 D f ( r , θ ) . These functions are 'paired' via a Fourier transform, and this notation is captured with the ⇔ arrow. In terms of the Fourier series

(

expansion of the function and its transform, equation F ( ρ ,ψ ) = F2 D f ( r , θ ) ∞

∑

n =−∞

f n (r ) e

jnθ

⇔

) becomes

∞

∑ F ( ρ ) e jnψ

n =−∞

n

(26)

Since both sides of equation (26) contain an infinite series, where essentially θ is replaced with ψ and vice versa, it is actually far more convenient to denote the Fourier pair of (26) as

8

f n (r ) ⇔

Fn ( ρ )

(27)

where (i) the series is implied but dropped for shorthand and more importantly (ii) the reader must recall yet again that the relationship between the Fourier pair of (27) is not that of a Fourier transform as a naive interpretation of (27) would imply but rather is given by equations (20) and (23).

4

THE DIRAC-DELTA FUNCTION AND ITS TRANSFORM

The unit-mass Dirac-delta function in 2D polar coordinates is defined as

1 f ( r ) = δ ( r − r0 ) = δ ( r − r0 ) δ (θ − θ0 ) . r

(28)

To find the Fourier transform, the Fourier series expansion is required, followed by a Hankel transform, as previously discussed. Thus for the Dirac-delta function, the Fourier series expansions terms are

f n (r ) =

1 2π

2π

1

∫ r δ ( r − r0 ) δ (θ − θ0 ) e

−inθ

1

dθ =

2π r

0

δ ( r − r0 ) e

−inθ0

(29)

so that equation (28) can be written as ∞ 1 1 −inθ f ( r ) = δ ( r − r0 ) = δ ( r − r0 ) δ (θ − θ0 ) = ∑ δ ( r − r0 ) e 0 einθ r n =−∞ 2π r

(30)

Then the full transform is given by

F (ω ) =

∞

∑ Fn ( ρ )e

jnψ

=

n =−∞

=

∞

∑ 2π i

−n

inψ

e

n =−∞

∞

∫ f n (r ) J n ( ρ r ) rdr 0

δ ( r − r0 ) −inθ0 e J n ( ρ r ) rdr 2π r 0

∞

∑ 2π i −n einψ ∫

n =−∞

=

∞

∞

∑ i −n J n ( ρ r0 )e

−inθ0

(31)

einψ = e−iω⋅r0

n =−∞

where equation (31) is the 2D linear exponential phase function. The Fourier transform of the Dirac delta function is the exponential function, as would be expected from the results in Cartesian coordinates.

However, the more important result that we seek is that the coefficients of the Fourier transform are

Fn ( ρ ) = i − n J n ( ρ r0 )e 9

−inθ0

.

(32)

Hence the coefficients of the Fourier pair for the Dirac-delta function are given by

f n ( r ) = [δ ( r − r0 ) ]n =

1 2π r

δ ( r − r0 ) e

−inθ0

⇔ Fn ( ρ ) = i − n J n ( ρ r0 )e

−inθ0

= ⎡e ⎣

−iω⋅r0 ⎤

⎦n

. (33)

This Fourier pair is included in Table 1.

4.1 DIRAC-DELTA FUNCTION AT THE ORIGIN If r0 is at the origin, then it is multiply covered by the angular variable and the Dirac-delta function in 2D is given by

δ (r ) =

1 2π r

δ (r ) ,

(34)

where the difference in notation between δ ( r ) and δ ( r ) is emphasized. The notation δ ( r ) is used to represent the Dirac-delta function in the appropriate multi-dimensional coordinate system, whose actual form may vary. The notation δ ( r ) denotes the standard 1-D scalar version of the Dirac-delta function that is most familiar. Both equation (12) or equation (32) can be used to calculate the Fourier transform as they both yield the correct transform for the Dirac-delta function at the origin, which is 1. Because the function is radially symmetric, the series consists of only the zeroth order terms that is f n ( r ) = f ( r ) δ n 0 and

Fn ( ρ ) = F ( ρ ) δ n 0 . The Fourier pair is thus given by

fn ( r ) = or more compactly as

1 2π r

1 2π r

δ ( r ) δ n0

Fn ( ρ ) = i − n J n ( 0 ) = δ n 0

⇔

(35)

δ (r) ⇔ 1.

4.2 RING-DELTA FUNCTION An often-used function is the Ring delta function given by

f (r ) =

1 2π r

δ ( r − r0 ) .

(36)

The function given in equation (36) is non-zero only on the ring of radius r0 . The 2D Fourier transform of the Ring-delta is most easily found from equation (12) and is given by

10

∞

⎧ 1 ⎫ F ( ρ ) = 2π ∫ ⎨ δ ( r − r0 ) ⎬ J 0 ( ρ r )rdr = J 0 ( ρ r0 ) .

2π r ⎭ 0⎩

(37)

Because the function is radially symmetric, the full 2D transform is just a Hankel transform of order zero and the Fourier transform is also only radially symmetric. The series consists of only the zeroth order terms that is f n ( r ) = f ( r ) δ n 0 and Fn ( ρ ) = F ( ρ ) δ n 0 . So the full 2D Fourier transform pair is given by

1 2π r

5

δ ( r − r0 ) ⇔ J 0 ( ρ r0 )

(38)

THE COMPLEX EXPONENTIAL AND ITS TRANSFORM

From equation (8), the 2D complex exponential function can be written in polar coordinates as

iω0 ⋅r

f (r ) = e

=

∞

∑ i n J n ( ρ0 r ) e

−inψ 0

einθ

(39)

n =−∞

so that the Fourier coefficients can be directly seen from the previous equation or can be found from the formula by

1 f n (r ) = 2π

2π

∞

i m J m ( ρ 0 r )e ∫ m∑ =−∞

−imψ 0

eimθ e−inθ dθ

0

= i n J n ( ρ0 r ) e

−inψ 0

(40)

.

The full 2D Fourier transform is given by

F (ω ) =

∞

∑

2π i − n einψ

n =−∞

=

∫ f n (r ) J n ( ρ r ) rdr 0

∞

∑ 2π i − n einψ ∫ {i n J n ( ρ0 r ) e ∞

n =−∞

=

∞

∞

∑ 2π e

n =−∞

−inψ 0

0

−inψ 0

1

ρ

} J (ρ r ) rdr n

(41)

δ ( ρ − ρ0 ) einψ .

where the last line follows from the orthogonality of the Bessel functions (Arfken & Weber 2005). Thus the Fourier coefficients that we seek are given by

11

Fn ( ρ ) = 2π e

−inψ 0

1

ρ

δ ( ρ − ρ0 ) .

(42)

It is also noted that the closure of the complex exponentials (or equivalently finding the Fourier series of the 1D Dirac-delta function) gives

δ (ψ −ψ 0 ) =

∞

1 −inψ 0 inψ e e n =−∞ 2π

∑

(43)

so that equation (41) actually gives the traditional Fourier transform of the complex exponential as it should:

F (ω ) = ( 2π )

2

1

ρ

δ ( ρ − ρ0 ) δ (ψ −ψ 0 ) = ( 2π ) δ (ω − ω0 ) .

2

(44)

Equation (44) is the correct full 2D Fourier transform of the complex exponential, but the Fourier pair we seek are the coefficients of the function and coefficients of its transform which are given respectively by equations (40) and (42) as:

f n ( r ) = i n J n ( ρ0 r ) e

−inψ 0

⇔

Fn ( ρ ) = 2π e

−inψ 0

1

ρ

δ ( ρ − ρ0 )

(45)

This Fourier pair is also included in Table 1.

5.1 SPECIAL CASE As a special case of this, the Fourier transform of f (r ) = 1 can be computed by subsituting ω0 = 0 into the above formulas. This gives

F (ω ) = ( 2π )

2

1

ρ

δ ( ρ ) δ (ψ ) = ( 2π ) δ (ω ) 2

(46)

or alternatively, in series form the Fourier transform of f (r ) = 1 is given by

F (ω ) =

∞

∑

n =−∞

2π

ρ

The Fourier pair coefficients are given by

12

δ ( ρ ) einψ

.

(47)

fn ( r ) = i n J n ( 0) = δ n0

⇔

Fn ( ρ ) = 2π

1

ρ

δ (ρ)

(48)

where δ n 0 is the standard kronecker-delta, that is δ n 0 is 1 for n = 0 and 0 otherwise.

6

MULTIPLICATION ∞

We consider the product of two functions h(r ) = f (r ) g ( r ) where f (r ) =

∑

n =−∞

g (r ) =

∞

∑

n =−∞

g n (r )einθ , the Fourier series coefficients f n (r ) and g n (r ) are given by (15) and we seek to

find the equivalent coefficients hn (r ) of h(r ) = f ( r ) g (r ) =

∞

∑ hn (r )einθ . ∞

∑

n =−∞

g (r ) =

∑

n =−∞

This is accomplished by

n =−∞

finding the Fourier transform of h(r ) and using the expansions f (r ) = ∞

f n (r )einθ and

f n (r )einθ and

g n (r )einθ , along with equation (9) for the polar form of the 2D complex exponential:

H (ω ) =

∞

∫

f (r ) g (r ) e−iω⋅r dr

−∞ ∞ 2π

=∫ 0

∞

∞

∞

(49)

f n (r )einθ ∑ g m (r )eimθ ∑ i − k J k ( ρ r ) e−ikθ eikψ dθ rdr ∫ n∑ =−∞ m =−∞ k =−∞ 0

Performing the integration over the angular variable yields

H (ω ) =

∞

∑

2π i − k eikψ

k =−∞

=

∞

∞

f k −m (r ) g m (r ) J k ( ρ r ) rdr ∫ m∑ =−∞ 0

∞

(50)

∑ H k ( ρ )eikψ

k =−∞

where

H k ( ρ ) = 2π i

−k

∞

∞

f k −m (r ) g m (r ) J k ( ρ r ) rdr . ∫ m∑ =−∞ 0

However, it is known from equation (20) that 13

(51)

∞

H k ( ρ ) = 2π i − k ∫ hk (r ) J k ( ρ r ) rdr ,

(52)

0

hence it follows that

hk ( r ) =

∞

∑

m =−∞

f k −m (r ) g m (r ) ,

(53)

which is in fact the convolution of the Fourier series of f (r ) and g (r ) . In other words, the coefficients of the products of the two series is the convolution of the coefficients so that

( fg )k =

fk ∗ gk ,

(54)

with the convolution of two series defined as ∞

( f k ∗ g k ) (r ) ≡ ∑

m =−∞

f k −m (r ) g m (r ) .

(55)

The definition of the discrete convolution of two series given in equation (55) is the same as the standard definition given in the literature (Oppenheim & Schafer 1989). The Fourier pair is then given by

( fg )k =

∞

fk ∗ gk ⇔

2π i − k ∫ ( f k ∗ g k ) J k ( ρ r ) rdr

(56)

0

7

SPATIAL SHIFT

The correct expression for a Fourier series shifted in space is found by finding the inverse Fourier transform of the complex exponential-weighted transform. In other words, f ( r − r0 ) is found from

{

f ( r − r0 ) = F −1 e

−iω⋅r0

}

F (ω ) .

(57)

The reason for defining the shifted function according to equation (57) is that we have already found the expansion for the complex exponential as well as the rules for finding the product of two expansions. It is not sufficient to find any expression for the spatial shift but rather the expression that is sought is one that is (i) in the form of a Fourier series and (ii) in terms of the unshifted coefficients of the original function as this builds the rule for what to do to the coefficients if a shift is desired. Thus, by building on the previously-found results, the relevant spatial shift result can be found in the desired form.

14

Using the definition of the inverse Fourier transform given in equation (21), along with the expansions in equations (8), (9) and (16), then the desired quantity is given by

f (r − r0 ) =

∞ 2π

1

∞

∫ ∫ ∑ ( 2π ) 0 0 m=−∞ 2

i − m J m ( ρ r0 ) e

∞

∑ i k J k ( ρ r ) eikθ e−ikψ

−imθ0 imψ

e

∞

∑ Fn ( ρ )e jnψ i

n =−∞

(58)

dψ ρ d ρ

k =−∞

Performing the integration over ψ yields a non-zero value only if m + n − k = 0 , so that the preceding equation simplifies to

f (r − r0 ) =

1 2π

∞

∞

∑ ∑

i ne

−i ( k −n )θ0 ikθ

e

n =−∞ k =−∞

∞

∫ Fn ( ρ )J k −n ( ρ r0 ) J k ( ρ r ) ρ d ρ .

(59)

0

It can be observed from equation (59) that the kth Fourier coefficient of the shifted function is

[ f (r − r0 )]k (r ) =

1 2π

∞

∑

n =−∞

∞

i n e−i ( k −n)θ0 ∫ Fn ( ρ )J k −n ( ρ r0 ) J k ( ρ r ) ρ d ρ .

(60)

0

For reasons that will become apparent later, we may rewrite the indices in the previous equation as

1 [ f (r − r0 )]k (r ) = 2π

∞

∑

i

k − n −inθ0

e

n =−∞

∞

∫ Fk −n ( ρ )J n ( ρ r0 ) J k ( ρ r ) ρ d ρ

(61)

0

This gives the Fourier coefficients of the shifted function in r space in terms of the unshifted frequencyspace coefficients Fn ( ρ ) .

If the value of the shifted coefficients are desired in terms of the original unshifted coefficients f n (r) , where f (r ) =

∞

∑

n =−∞

f n (r )e jnθ , then the definition of the Fourier-space coefficients from equation (20) can

be used ∞

Fn ( ρ ) = 2π i − n ∫ f n (r ) J n ( ρ r ) rdr = 2π i − n H n { f n (r )} 0

15

(62)

and substituted into equation (60), along with a change in the order of integration so that

f (r − r0 ) =

∞

∑

k =−∞

∞

∞

eikθ

∑

n =−∞

e−i ( k −n)θ0 ∫ f n (u )Snk ( u, r , r0 ) udu

(63)

0

where S nk ( u , r , r0 ) is defined as a shift-type operator given by the integral of the triple-Bessel product

Snk

∞

( u, r , r0 ) = ∫ J n ( ρ u ) J k −n ( ρ r0 ) J k ( ρ r ) ρ d ρ .

(64)

0

Thus, the Fourier coefficients of the shifted function in terms of the unshifted function coefficients are given by

[ f (r − r0 )]k = e

−ikθ0

∞

inθ0

∑

e

n =−∞

∞

∫ fn (u )Sn ( u, r , r0 ) udu . k

(65)

0

The previous equation describes the shift operation in terms of Fourier polar coordinates. In other words,

equation (65) gives the rule on how to find the Fourier coefficients for the shifted function [ f (r − r0 )]k , if the original unshifted coefficients f n (r) are known. It can be seen from (65) that without going into the Fourier domain, this is given by multiplication of the original coefficients f n (r) by the shift operator

Snk ( u, r , r0 ) , integrating then summing over all possible values of n. Alternatively, the rule as given by equation (61) is to multiply the Fourier coefficients Fk − n ( ρ ) by i − n e

− inθ0

J n ( ρ r0 ) (the shift), sum over all

n and then transform back to spatial coordinates. What is interesting is that we need to multiply by all the possible i − n e

− inθ 0

J n ( ρ r0 ) (all values of n) and integrate to get a proper shift. Recalling that a shift in polar

coordinates really consists of a translation by r0 and then a rotation by θ 0 , we can interpret equation (61) as

[ f (r − r0 )]k (r ) =

1 2π

∞

∑

i k −n e

n =−∞ sum over all n

Hence i − n J n ( ρ r0 ) e

−inθ0

−inθ0

∞

∫ Fk −n ( ρ ) J n ( ρ r0 ) J k ( ρ r ) ρ d ρ

rotation 0 by θ0

(66)

translation by r0 transform back to spatial coordinates

is the kernel of the shift operation (translation + rotation) and in fact equation

(66) is a series convolution between i − n J n ( ρ r0 ) e

−inθ0

16

and Fk − n ( ρ ) , which will be shown below.

In the special case that r0 = 0 , then the definition of the shift operator along with the fact that J n (0) = δ n 0 implies that in this case the shift operator becomes ∞

1 Snk ( u, r , 0 ) = ∫ δ nk J n ( ρ u ) J n ( ρ r ) ρ d ρ = δ (u − r )δ nk u 0

(67)

so that equation (65) returns the correct unshifted value of f n ,as it should.

7.1 FOURIER DOMAIN COEFFICIENTS OF THE SHIFTED FUNCTION

The corresponding coefficients for the 2D Fourier transform can be found. If we define h(r ) = f ( r − r0 ) , then H (ω ) = e

−iω⋅r0

F (ω ) , and the Fourier coefficients H n ( ρ ) are sought. Thus H (ω ) is defined as a

product of the complex exponential and F (ω ) and it was previously shown that multiplication in the Fourier series domain implies convolution of the coefficients of the respective series, with the convolution operation defined in equation (53) . Since the coefficients for the complex exponential are given in equation (9) , it follows that −ikθ0 ⎤ H k ( ρ ) = ⎡⎢i − k J k ( ρ r0 ) e ⎥⎦ * Fk ( ρ ) ⎣

(68)

or more explicitly

Hk ( ρ ) =

∞

∑ i −n J n ( ρ r0 ) e

n =−∞

−inθ0

Fk −n ( ρ )

(69)

as per the definition of the convolution operation for two Fourier series coefficients. Converting the

coefficients H k ( ρ ) into the spatial domain to get hk ( r ) = [ f (r − r0 )]k (r ) will give us exactly equation (61), confirming the previous derivation.

Substituting n = k − m into equation (65) so that the indices more closely resemble equation (69), we can write the Fourier pair for the shift rule conceptually as −iω ⋅r0 −iω ⋅r0 ⎤ F (ω ) ⎤ = ⎡e *F ρ ⎡⎣ f ( r − r0 ) ⎤⎦ k ⇔ ⎡e ⎣ ⎦k ⎣ ⎦k k ( )

and more explicitly as

17

(70)

ik 2π

∫ {⎡⎣⎢i

∞

−k

0

=

∞

∑

e

n =−∞

−inθ0

∞

∫ 0

}

⎫ ⎧ ∞ −n −inθ * F ρ J ( ρ r ) ρ d ρ ( ) ⎪ k k ⎥⎦ i J n ( ρ r0 ) e 0 Fk −n ( ρ ) ∑ ⎪ ⎪ ⎪ (71) ⎬ ⇔ ⎨ n=−∞ ⎪ ⎪ = ⎡i − k J ( ρ r ) e−ikθ0 ⎤ * F ( ρ ) k f k −n (u )Sk −n ( u, r , r0 ) udu 0 k ⎪ ⎪⎩ ⎣⎢ ⎦⎥ k ⎭

J k ( ρ r0 ) e

−ikθ0 ⎤

where

Skk−n

∞

( u, r , r0 ) = ∫ J k −n ( ρ u ) J n ( ρ r0 ) J k ( ρ r ) ρ d ρ .

(72)

0

7.2 RULE SUMMARY The shift rule is summarized mathematically in equation (71), and this rule was actually derived in two different ways. The procedure can be explained as follows: to find the coefficients of the shifted function in spatial coordinates [ f (r − r0 )]k (r ) the steps are i.

Transform the unshifted function to the Fourier domain to find F (ω ) , meaning find its coefficients

Fn ( ρ ) . ii.

−iω⋅r0 Multiply the Fourier transform F (ω ) by the complex exponential e . In coefficients, this −iω⋅r0 means convolve the coefficients Fn ( ρ ) with the coefficients of e or

⎡i − n J ( ρ r ) e−inθ0 ⎤ * F ( ρ ) . n 0 ⎣ ⎦ n iii.

Transform these new coefficients back to the regular spatial domain to get [ f (r − r0 )]k (r )

−iω⋅r0 and then to The rule summary is to transform to the Fourier domain, 'multiply' (convolve) by e transform back. The shift operator can be used to avoid the transformation to the Fourier domain, and remain in the spatial domain to perform all calculations.

7.3 THE SHIFT OPERATOR The shift operator can be evaluated in closed form using results derived in (Jackson & Maximon 1972). In (Jackson & Maximon 1972), closed form results are found for the triple-product integral given by ∞

∫ J n (k1ρ ) J n (k2 ρ ) J n (k3 ρ )ρ d ρ 1

2

3

0

18

(73)

Now, in (Jackson & Maximon 1972), it is stated that the integral in (73) non-zero only if n1 + n2 + n3 = 0 .

This last condition is a result of integrating over angular variables and the requirement of a nonzero result.

The condition n1 + n2 + n3 = 0 is in fact far too restrictive, primarily due to the fact that

J − n ( x) = ( −1) J n ( x) . In fact, our shift operator arises after an integration over the angular variables in n

equation (58) where only the relationship m + n − k = 0 gives a nonzero result. Therefore, the tripleproduct integral that we are interested in, namely the shift operator ∞

Snk ( u, r , r0 ) = ∫ J n ( ρ u ) J k −n ( ρ r0 ) J k ( ρ r ) ρ d ρ = ( −1)

∞

∫ J n ( ρu ) J k −n ( ρ r0 ) J −k ( ρ r ) ρ d ρ (74)

k

0

0

requires no additional restrictions on the indices since the requirement for a nonzero angular integral has already been taken into account. Alternatively, replacing J k ( x) = ( −1) J − k ( x) in (74) guarantees that the k

indices add up to zero.

Closed form results are given in (Jackson & Maximon 1972), to which the interested reader is referred.

However, since the ultimate purpose of the shift operator is to multiply with the original Fourier coefficients and then to integrate, presumably this is not a computationally efficient way to compute the shifted coefficients. It is probably easier to compute the convolution in the Fourier domain and then inverse transform to the spatial domain than to attempt a direct calculation and subsequent integration with the shift operator. In other words, it is probably easier to compute

ik 2π

∫ {⎡⎣⎢i

∞

−k

J k ( ρ r0 ) e

}

−ikθ0 ⎤

* F ( ρ ) Jk (ρr) ρd ρ ⎦⎥ k

0

(75)

instead of ∞

∑

n =−∞

8

e

−inθ0

∞

∫ fk −n (u )Sk −n ( u, r , r0 ) udu . k

(76)

0

FULL TWO DIMENSIONAL CONVOLUTION

The two-dimensional convolution of two functions is defined by ∞

h(r ) = f (r ) **g (r ) =

∫ g (r0 ) f (r − r0 )dr0 .

−∞

19

(77)

The double-star notation of ** is used to emphasize that this is a 2D convolution and to distinguish it from a 1D convolution.

Using the Fourier expansions for g and the shifted version of f given by equation (63), the previous equation becomes ∞ 2π

∞

∫ m∑ =−∞

h( r ) = ∫ 0

∞

imθ0

g m (r0 )e

∑

k =−∞

0

∞

eikθ

∑

e

−i ( k −n)θ0

n =−∞

∞

∫ fn (u )Sn ( u, r , r0 ) udu dθ0 r0 dr0 k

(78)

0

The preceding equation can be simplified by performing the integration over the angular variable θ 0 so that

m = k −n: ∞

ikθ

∑e

h(r ) = 2π

k =−∞

∞

⎛∞ ⎞ k ⎜⎜ ∫ f n (u )Sn ( u, r , r0 ) udu ⎟⎟ g k −n (r0 ) r0 dr0 ⎝0 ⎠

∞

∫ n∑ =−∞ 0

(79)

Equation (79) says that convolution over coefficients, n ∞

∞

0

n =−∞

( f **g )k = 2π ∫ ∑

⎛∞

⎞ k ⎜⎜ ∫ f n (u )Sn ( u, r , r0 ) udu ⎟⎟ g k −n (r0 ) r0 dr0 ⎝0 ⎠

(80)

kernel of linear 'shift' for f roughly = f n ( r − r0 ) convolution over r

which we can loosely interpret as a combination of (i) 'shift' for f, (ii) convolution over the coefficients, that is series convolution and then (iii) another convolution over the r dimension. This is a nice little insight but serves little to aid in Fourier calculations. To produce a computationally relevant result, we proceed by simplifying equation (80) by using the definition of S nk ( u , r , r0 ) and rearranging to give

h( r ) =

∞

∑

k =−∞

2π e

ikθ

∞

∫ 0

⎧ ⎪ ⎪ ∞ ⎪ ⎨∑ ⎪ n =−∞ ⎪ ⎪⎩

∞

∞

∫ g k −n (r0 ) J k −n ( ρ r0 )r0 dr0 ∫ 0

0

i

k −n

2π

Gk − n ( ρ )

⎫ ⎪ ⎪ ⎪ f n (u ) J n ( ρ u )udu ⎬ J k ( ρ r ) ρ d ρ . (81) ⎪ ⎪ n i Fn ( ρ ) ⎪⎭ 2π

With the simplifications in the brackets as shown in the previous equation, this becomes

20

h( r ) =

1 2π

∞

∑

k =−∞

∞

⎧

0

⎩ n =−∞

⎫

∞

i k eikθ ∫ ⎨ ∑ Gk −n ( ρ ) Fn ( ρ )⎬ J k ( ρ r ) ρ d ρ . ⎭

(82)

Clearly this can be written in the form

h( r ) =

1 2π

∞

∑

k =−∞

∞

i k eikθ ∫ H k ( ρ ) J k ( ρ r ) ρ d ρ = 0

∞

∑ hk (r ) eikθ

(83)

k =−∞

where the coefficients H k ( ρ ) are clearly the convolution of the two series given by

Hk (ρ ) =

∞

∑

n =−∞

Gk −n ( ρ ) Fn ( ρ ) = Gk ∗ Fk .

(84)

In the results of the last section, it was shown that the convolution of two sets of Fourier coefficients is

equivalent to multiplication of the functions so that Gk ∗ Fk = ( GF )k , hence it follows from equation (84) that

H (ω ) = G (ω ) F (ω ) ,

(85)

as would be expected from the standard results of Fourier theory and serves to confirm the accuracy of the development. However, the main result we seek is equation (82), which gives the values of hk (r ) in terms of g k (r ) and f k (r ) (or rather the Hankel transform of those) and essentially defines the convolution operation for functions given in Fourier series forms. This equation defines the convolution operation so that to find the convolution of two functions given in Fourier series form, one must first find the kth coefficient of the Fourier transform of each function, namely Gk ( ρ ) and Fk ( ρ ) , and then subsequently convolve the resulting series as per equation (84) to get H k ( ρ ) . The final step is then to inverse Hankel transform the result to finally obtain hk (r ) . It is important to note that convolving the two functions f (r ) and g (r ) is not equivalent to convolving their series – this in fact was shown previously to be equivalent to the multiplication of the functions themselves.

This preceding result can be conceptually summarized as

( g ** f )k

⇔ Gk ∗ Fk = ( GF )k

where

21

(86)

∞

( Gk * Fk ) ( ρ ) = ∑

n =−∞

Gk −n ( ρ ) Fn ( ρ )

(87)

8.1 MULTIPLICATION REVISITED Equations (56) and (55) gave us the multiplication rule – that is how to find the Fourier coefficients of the products of two functions. We can now show here that the Fourier transform of a product is indeed a 2D convolution of their full transforms, F (ω ) **G (ω ) . For h ( r ) = f ( r ) g ( r ) , we restate equation (56) and insert (55) to get ∞ ⎛ ∞ ⎞ H k ( ρ ) = 2π i − k ∫ ⎜ ∑ f k −m (r ) g m (r ) ⎟ J k ( ρ r ) rdr ⎠ 0 ⎝ m =−∞

(88)

Now each Fourier coefficient can be written in the form of

f n (r ) =

in 2π

∞

∫ Fn ( ρ ) J n ( ρ r ) ρ d ρ

(89)

0

so that equation (88) becomes ∞ ∞ ⎞ i k −m im ( ) ( ) ( ) ( ) H k ( ρ ) = 2π i ∫ ⎜ ∑ F u J ur udu G v J vr vdv ⎟⎟ J k ( ρ r ) rdr (90) k m k m m m − − ⎜ 2π ∫0 2π ∫0 0 ⎝ m =−∞ ⎠ −k

∞⎛

∞

Recognizing the presence of the shift operator after a change of the order of integration, the preceding equation becomes

1 Hk ( ρ ) = 2π

∞

∞

∞

∫ m∑ ∫ =−∞ 0

Fk −m (u ) Skm−m

0

∞

( u, ρ , v ) udu ∫ Gm (v) vdv

(91)

0

Comparing equation (91) with the form of the Fourier coefficients of a convolution as given in equation (80), we see that equation (91) says that

Hk ( ρ ) =

1

( 2π )

2

⎡⎣ F (ω ) **G (ω ) ⎤⎦ k

or simply

22

(92)

H (ω ) =

1

( 2π )

F (ω ) **G (ω )

2

(93)

This is, of course, the result we would should obtain and the mathematics show that this result does indeed hold.

9

SPECIAL CASE: SPATIAL SHIFT OF RADIALLY SYMMETRIC FUNCTIONS

We now turn our attention to the special case of radially symmetric functions, those that are only a function of r. For a radially symmetric function, we recall that the 2D forward and inverse transforms are essentially Hankel transform of order zero. So from equation (12) the forward transform is given by ∞

F ( ρ ) = F2 D { f (r )} = 2π ∫ f (r ) J 0 ( ρ r )rdr

(94)

0

and the inverse transform is given by

f ( r ) = f ( r ) = F2−D1 { F ( ρ )} =

1 2π

∞

∫ F (ρ )J0 (ρ r)ρ d ρ

(95)

0

To get the correct expression for the shift of a radially symmetric function, we define the shifted function as before from equation (57) as

{

f ( r − r0 ) = F −1 e =

−iω ⋅r0 ∞ 2π

1

∞

i − m J m ( ρ r0 ) e ∑ ∫ ∫ ( 2π ) 0 0 m=−∞ 2

∞

∑i

k =−∞

=

}

F (ρ )

1 2π

ikθ −ikψ

Jk (ρ r) e

e

e

F ( ρ )i (96)

dψ ρ d ρ

∞

∞

∑

k

−imθ0 imψ

k =−∞

eik (θ −θ0 ) ∫ F ( ρ )J k ( ρ r0 ) J k ( ρ r ) ρ d ρ 0

Thus the Fourier coefficients of the shifted radially symmetric function are given by

[ f (r − r0 )]k =

∞

1 −ikθ0 e ∫ F ( ρ )J k ( ρ r0 ) J k ( ρ r ) ρ d ρ 2π 0

23

(97)

Although f (r ) is radially symmetric and only has the n = 0 term in its Fourier series, a properly shifted

f (r − r0 ) is not radially symmetric and so needs the full complement of entries in its Fourier series. By a proper shift, we mean the shift to imply the linear translation in r and angular rotation in θ . It is equally important to note that this is the case even if the new "center" r0 to which the function f has been shifted is located on the radial axis so that θ 0 = 0 . Equation (96) is the proper full shift (translation and rotation) and demonstrates that a function of r becomes a function of r and θ once shifted away from the origin.

Hence, f (r − r0 ) is the shift so that the radially symmetric f (r ) is now centered at r0 . This is not the same as f (r − r0 ) , which is a radially symmetric function. To illustrate these points, consider the Gaussian "dome" of f (r ) = e

⎛ r2 ⎞ −⎜ ⎟ ⎜ 4 ⎟ ⎝ ⎠

, f (r − 4) = e

⎛ ( r − 4) 2 ⎞ −⎜ ⎜ 4 ⎟⎟ ⎝ ⎠

and f (r − r0 ) as f (r ) = e

⎛ r2 ⎞ −⎜ ⎟ ⎜ 4 ⎟ ⎝ ⎠

shifted so that

is it centered at r0 = ( r = 4, θ = 0 ) . Clearly, the first two of these will be radially symmetric while the third is not.

The interpretation of equation (97) is as follows. We want to go from the original radially symmetric

f ( r ) to the kth coefficient [ f (r − r0 )]k of the shifted function. A "shift" in 2D polar space is really a

combination of a linear shift in the radial variable and a rotation by the angular variable. The rotation by the angle θ 0 is handled by a multiplication by e

− ikθ0

, as is typical in polar coordinates. It now remains to

cope with the linear translation portion of the shift. The 'rule' here is to take the Fourier transform of

f ( r ) , that is F ( ρ ) , and multiply it by J k ( ρ r0 ) which takes care of the linear shift by r0 of the kth coefficient. The shifted result is now transformed back into the usual r space by an inverse Hankel transform of order k. In essence, linear shift for functions that start with radial symmetry is a relatively simple operation as the translation from r to r − r0 is equivalent to a multiplication by J k ( ρ r0 ) in Fourier −iω⋅r

0 in space, much like a shift in Cartesian regular space is the equivalent of a multiplication by e Cartesian Fourier space. When the function to be shifted does not start out with radial symmetry, this multiplication operation to obtain the linear shift in the radial variable becomes a more complicated series convolution operation, as was previously demonstrated in equation (69).

9.1 FOURIER TRANSFORM OF THE SHIFTED RADIALLY SYMMETRIC FUNCTION The corresponding coefficients of the 2D Fourier transform can also be found. If we define

h(r ) = f ( r − r0 ) , then from the definition of a Fourier expansions, we know that the form of the

expansion must be ∞

∞

1 k ikθ h ( r ) = f ( r − r0 ) = ∑ i e ∫ Hk (ρ )Jk (ρ r) ρ d ρ n =−∞ 2π 0 Comparison of equations (98) and (96) gives immediately that 24

(98)

H k ( ρ ) = i − k e −ikθ0 F ( ρ ) J k ( ρ r0 )

(99)

and thus for a radially symmetric function, the shift property is to multiply the radially symmetric Fourier transform by the kernel for e

−iω⋅r0

H (ω ) =

. In fact, equation (99) implies that ∞

∑ H (ρ)e θ ik

k

k =−∞

= F (ρ)

∞

∑i

− k − ikθ 0

e

k =−∞

J k ( ρ r0 ) e

ikθ

=F ( ρ ) e

−iω ⋅r0

(100)

which is exactly the result we should obtain, verifying the development.

As previously mentioned, it is intuitively obvious from a 2D polar-coordinate view of the world that a radially symmetric function is no longer radially symmetric once shifted away from the origin. Thus, since the function is no longer radially symmetric, the 2D Fourier transform is no longer equivalent to a zeroth order Hankel transform as was the case for the radially symmetric function. In fact, equation (100) clearly − iω ⋅r

0 shows that it is the e term that destroys the radial symmetry of the 2D Fourier transform. In −iω⋅r0 particular, e = e−ir0 ρ cos(ψ −θ0 ) is a function of magnitude 1 but with a non-radially symmetric phase. Continuing with this argument, if Hankel transforms are considered to be merely transforming a function of a variable r to another function of a variable ρ , without consideration of its role in the 2D

perspective, it would be reasonable to look for a Hankel analog of the shift and convolution theorems of standard Fourier theory. Clearly, a 1D shift/convolution rule for Hankel transforms does not exist and it is only in considering the matter from the 2D perspective that the reason one cannot exist becomes obvious.

It is interesting to note that since J k ( 0 ) = 0 for k ≠ 0 and J 0 ( 0 ) = 1 , equation (96) for r0 = 0 becomes

1 f (r ) = 2π

ik (θ −θ0 )

∞

∑e

k =−∞

∞

∫ 0

1 F ( ρ ) J k ( 0) J k ( ρ r ) ρ d ρ = 2π

∞

∫ F ( ρ ) J0 ( ρ r ) ρ d ρ

(101)

0

as it should, verifying the development given herein.

9.2 RULE SUMMARY To summarize the preceding discussion, if f ( r ) is a radially symmetric function, the shifted function

f ( r − r0 ) is not radially symmetric and its Fourier coefficients are given by

[ f (r − r0 )]k =

∞

1 −ikθ0 e ∫ F ( ρ )J k ( ρ r0 ) J k ( ρ r ) ρ d ρ .

2π 0

25

(102)

Writing h(r ) = f ( r − r0 ) , then the Fourier coefficients in Fourier space are given by

H k ( ρ ) = i − k e −ikθ0 F ( ρ ) J k ( ρ r0 )

(103)

which establishes the Fourier-pair of coefficients

[ f (r − r0 )]k =

∞

1 −ikθ0 e F ( ρ )J k ( ρ r0 ) J k ( ρ r ) ρ d ρ ⇔ i − k e−ikθ0 F ( ρ ) J k ( ρ r0 ) ∫ 2π 0

(104)

9.3 SHIFT OF A RADIALLY SYMMETRIC FUNCTION IN TERMS OF THE SHIFT OPERATOR As previously done for the case of the shift of general function, we may find the coefficients of the shifted function in terms of the original function, without reference to its transform. We use the definition of the forward transform and switch the order of integration, so that equation (96) becomes

f (r − r0 ) =

∞

∑

k =−∞

∞

∞

eik (θ −θ0 ) ∫ f (u ) ∫ J 0 ( ρ u ) J k ( ρ r0 ) J k ( ρ r ) ρ d ρ udu .

0

(105)

0

For a slightly more compact notation, equation (105)can be written using equation (64) as

f (r − r0 ) =

∞

ik (θ −θ0 )

∑e

k =−∞

∞

f (u ) S0k ( u , r , r0 ) udu .

∫

(106)

0

which gives the shifted coefficients in terms of the original unshifted coefficients:

[ f (r − r0 )]k = e

−ikθ0

∞

∫

f (u ) S0k ( u , r , r0 ) udu

(107)

0

10 SPECIAL CASE: 2D CONVOLUTION OF TWO RADIALLY SYMMETRIC FUNCTIONS Two radially symmetric functions are functions for which both Fourier series expansions only include the n = 0 term. The 2D convolution of the two radially symmetric functions is defined as

26

∞

h(r ) = f (r ) **g (r ) =

∫ g (r0 ) f (r − r0 )dr0

(108)

−∞

where it is emphasized that the integration is over all r0 , which includes all possible values of radial and

angular variables. In other words, equation (108) must be properly interpreted as a 2D convolution along with equation (106) as ∞ 2π

h(r ) = f (r ) **g ( r ) = ∫ 0

∫ 0

∞

g ( r0 )

∑

e

−ikθ0 ikθ

e

k =−∞

∞

∫

f (u ) S0k ( u , r , r0 ) udu dθ 0 r0 dr0

(109)

0

and NOT interpreted as ∞

∫ g (r0 ) f (r − r0 )dr0 ,

(110)

0

which would be a 1D convolution. Equation (109) can be simplified by performing the integration over the angular variable, which is nonzero only if k = 0 , so that ∞

∞

f (r ) **g (r ) = f (r ) **g (r ) = 2π ∫ g (r0 ) ∫ f (u ) S00 ( u , r , r0 )udu r0 dr0 0

(111)

0

with

S00

∞

( u, r , r0 ) = ∫ J 0 ( ρ u ) J 0 ( ρ r0 ) J 0 ( ρ r ) ρ d ρ

(112)

0

so that equation (111) is hardly recognizable as a convolution operation. The double * notation has been used to highlight the fact that a 2D convolution is being taken. In fact, the proper, correct definition of the convolution, equation (109) can be interpreted from equation (111) as ∞

f (r ) **g (r ) = f (r ) **g (r ) = ∫ g (r0 ) Φ (r − r0 ) r0 dr0

(113)

0

where ∞

∞

2π

0

0

0

Φ ( r − r0 ) = 2π ∫ f (u ) S00 ( u , r , r0 )udu = ∫ F ( ρ ) J 0 ( ρ r0 ) J 0 ( ρ r )ρ d ρ =

27

∫

f (r − r0 )dθ 0 . (114)

Thus, we observe that the definition of convolution that we are tempted to write, as given by equation (110) , is almost correct, what needed to be done was to shift the function f from r to r0 (thus destroying radial symmetry) and then to integrate the resulting shifted function over all angular variables. The unshifted function g is still radially symmetric and thus is not affected by the integration over the angular variable and the final result is the form given in equation (113).

From equation (83) and (84), it follows that

h(r ) = f (r ) **g (r ) =

1 2π

∞

∫

G0 ( ρ ) F0 ( ρ ) J 0 ( ρ r ) ρ d ρ

(115)

0

since the interpretation of equation (84) in this case is as

Hk (ρ ) =

∞

∑

n =−∞

⎧G ( ρ ) F0 ( ρ ) k = 0 Gk −n ( ρ ) Fn ( ρ ) = ⎨ 0 otherwise ⎩ 0

(116)

In the previous two equations, G0 ( ρ ) and F0 ( ρ ) are the zeroth order coefficients in the expansions for

F (ω ) and G (ω ) themselves. Equation (116) can also be obtained by observing that the integration over the angular variable in equation (109) forces k to be zero. Thus, the 2D convolution of two radially symmetric functions yields another radially symmetric function, as can be seen from equation (115).

Moreover, by using the proper definition of a 2D convolution instead of using the tempting definition of a 1D convolution, the well-known relationship between convolutions in one domain leading to multiplication in the other domain is preserved, namely that

f (r ) **g (r ) = f (r ) **g (r ) ⇔ F ( ρ )G ( ρ ) .

(117)

The key point to this relationship is the proper definition of the convolution as a 2D convolution.

11 SPECIAL CASE: CONVOLUTION OF A RADIALLY SYMMETRIC FUNCTION WITH A NON-SYMMETRIC FUNCTION We consider the special case of the convolution of two functions where only one of the functions is radially symmetric. This case holds particular importance for certain physical problems involving nonhomogeneous partial differential equations which can be solved via a convolution of the forcing function with the system's Green's functions. For a large class of problems, it turns out that the Green's function is radially symmetric while the forcing function is typically not. This will be discussed further in one of the later sections of this paper, for now we consider the mathematics of the convolution without consideration of any applications, even though this special case has many applications. So, to set up the problem we are seeking the two-dimensional convolution of two functions defined as 28

∞

h(r ) = f (r ) **g ( r ) = f ( r ) **g (r ) =

∫ g (r0 ) f (r − r0 )dr0 .

(118)

−∞

where we assume that only f (r ) = f (r ) is radially symmetric and the notation in equation (118) is to remind the reader that a full radial and angular shift is required prior to integration over all possible shifts.

Using the usual Fourier expansion for g and the shifted version of f given by equation (96), the previous equation becomes ∞ 2π

h( r ) = ∫ 0

∞

∫ m∑ =−∞

imθ0

g m (r0 )e

0

⎧⎪ 1 ⎨ ⎩⎪ 2π

∞

∑

ik (θ −θ0 )

e

k =−∞

⎫⎪ ( ) ( ) F ρ J ρ r J ρ r ρ d ρ ( ) ⎬ dθ0 r0 dr0 (119) 0 k k ∫ 0 ⎭⎪

∞

Integrating over θ 0 gives k = m for nonzero results so that

h( r ) =

∞

∞

∑

k =−∞

⎧⎪∞

⎫⎪

⎩⎪ 0

⎭⎪

eikθ ∫ g k (r0 ) ⎨ ∫ F ( ρ )J k ( ρ r0 ) J k ( ρ r ) ρ d ρ ⎬ r0 dr0 .

0

(120)

This can be written as

h( r ) =

∞

∑

k =−∞

∞

eikθ ∫ g k (r0 )φk ( r − r0 ) r0 dr0

(121)

0

where we use the definition

⎧∞ ⎪ ∫ F ( ρ )J k ( ρ r0 ) J k ( ρ r ) ρ d ρ ⎪ φk (r − r0 ) = ⎨ ∞0 ⎪ ⎪ ∫ F ( ρ )J k ( ρ r ) ρ d ρ ⎩0

r0 ≠ 0 (122)

r0 = 0

The definition of (122) with a specific version given for the case r0 = 0 is necessary as otherwise J k (0) = 0 when k ≠ 0 . The notation of φk (r − r0 ) was used in the definition in (122) instead of φk (r , r0 ) or

f k (r − r0 ) since this function essentially arises as a result of shifting f (r ) to f (r − r0 ) and then integrating the result over all possible angles θ 0 .

This preceding result bears some discussion as it is extremely important. First of all, similarity between the

φk (r − r0 ) of equation (122) and the Φ ( r − r0 ) of equation (114) is pointed out. Those two functions have 29

very similar definitions and both arise as result of shift in a radially symmetric function and then integrating over all possible values of the rotation shift, θ 0 . The principle difference between them is that Φ ( r − r0 ) involves only the 0th order term. This is because Φ ( r − r0 ) is the application of the shift rule as part of a convolution of a radially symmetric function with another radially symmetric function and thus no higher order terms are required to compute the full convolution between them. In contrast, φk (r − r0 ) appears as the application of the shift rule to the radially symmetric function which shall then be used in a convolution with a non-radially symmetric function. Since the function that we are convolving with is now not radially symmetric, it shall have higher order terms in its expansion and the function with radial symmetry must now produce those corresponding terms for a convolution to make sense. If all the functions that are involved have radial symmetry than all Fourier transforms become equivalent to zeroth order Hankel transforms and a kth order Hankel transform is never required. As soon as one of the functions loses radial symmetry, a kth order Hankel transform becomes necessary to define a full Fourier transform, even if one of the functions being convolved is radially symmetric.

Second, we are essentially defining a new function (or rather the Fourier coefficients of this new function) as the kth order inverse Hankel transform of F ( ρ )

∞

φk ( r ) = ∫ F ( ρ )J k ( ρ r ) ρ d ρ

(123)

0

and the only role of this new set of coefficients is to enable a radially symmetric function F ( ρ ) to convolve with a non-radially symmetric function. Really, the function's coefficients are as defined in equation (123) and then if a shift in the function is required, we use the Hankel shift rule of multiplying by J k ( ρ r0 ) before taking the inverse Hankel transform so that the shifted coefficients are ∞

φk ( r − r0 ) = ∫ F ( ρ )J k ( ρ r0 ) J k ( ρ r ) ρ d ρ

r0 ≠ 0

(124)

0

As previously mentioned, the set of functions are really defined as from equation (123) with the Hankel shift rule given as in equation (124) because r0 = 0 in equation (124) would only yield the zeroth order term since J k ( 0 ) = 0 for k ≠ 0 and we'd be back to using only the zeroth order term.

If appearing by itself, a kth order Hankel transform of a radially symmetric function is never required.

However, once implicated with an operation involving non-symmetric functions, it becomes necessary.

Interestingly enough, with this new, powerful version of the radially symmetric function, equation (120) can be interpreted as

30

h( r ) =

∞

∑

k =−∞

∞

eikθ ∫ g k (r0 )φk ( r − r0 ) r0 dr0 = 0

∞

∑ eikθ ( gk *1D φk ) ( r )

(125)

k =−∞

so that the coefficients of h(r ) are the simple 1D convolution of the g k and φk : ∞

hk ( r ) = ( g k *1D φk ) ( r ) = ∫ g k (r0 )φk ( r − r0 ) r0 dr0

(126)

0

It is emphasized that the relationship is the usual 1D convolution, hence the use of the notation *1D instead of writing g k * φk , which would actually imply a series convolution.

11.1 IN TERMS OF THE FOURIER TRANSFORMS The preceding results can be interpreted completely in the Fourier domain without resorting to the spatial domain. Following the procedure for the functions without radial symmetry and using the fact that radially symmetric functions only have a zeroth term in their expansions, we can proceed to write the 2D convolution of a radially symmetric function with one that has no radial symmetric from equation (82) as

h(r ) = f (r ) **g (r ) =

∞

∑

k =−∞

∞

i k ikθ e ∫ Gk ( ρ ) F ( ρ ) J k ( ρ r ) ρ d ρ 2π 0

(127)

Clearly this can be written in the form

h( r ) =

1 2π

∞

∑

k =−∞

∞

i k eikθ ∫ H k ( ρ ) J k ( ρ r ) ρ d ρ = 0

∞

∑ hk (r ) eikθ

(128)

k =−∞

where

H k ( ρ ) = Gk ( ρ ) F ( ρ ) .

(129)

Comparing this with equation (84) when the two functions were not radially symmetric, we see that the step of convolving the two series has been eliminated – or rather the convolution of two series when one of them belongs to a spherically symmetric function (and thus consists of only the first term) becomes a simple matter of multiplication.

As a point of interest, note the connection between equations (126) and (129):

hk ( r ) = ( g k *1D φk ) ( r )

⇔ 31

H k ( ρ ) = Gk ( ρ ) F ( ρ ) .

(130)

This is yet another example of the convolution/multiplication rule in effect and it holds because (i) a proper 2D convolution was defined and (ii) the φk ( r ) ('shift') version of inverse Hankel of F ( ρ ) was used instead of naïvely using f ( r ) .

12 CIRCULAR (ANGULAR) CONVOLUTION For two functions f ( r ) = f ( r , θ ) and g ( r ) = g ( r , θ ) the notion of a circular or angular convolution can be defined. This is not the 2D convolution as previously discussed but rather a convolution over the angular variable only, so that it may be defined as

f ( r ) ∗θ g ( r ) =

1 2π

2π

∫ f ( r ,θ0 ) g ( r ,θ − θ0 ) dθ0 .

(131)

0

Note the notation of ∗θ used to denote the angular convolution. A Fourier relationship can be defined for this operation of angular convolution:

h(r , θ ) = f ( r ) ∗θ g ( r ) = =

1 2π ∞

2π

∫ 0

∞ ⎛ ∞ jm(θ −θ0 ) ⎞ jnθ0 ⎞ ⎛ ( ) f r e ⎜ ∑ n ⎟ ⎜ ∑ g m (r ) e ⎟ dθ 0 ⎝ n =−∞ ⎠ ⎝ m=−∞ ⎠ ∞

∑ ∑

n =−∞ m =−∞

1 f n (r ) g m ( r ) e jmθ 2π

2π

∫e

jnθ0 − jmθ0

e

(132)

dθ 0

0

The value of the integral in the preceding equation is 2π δ mn , which results in the following simplification:

f ( r ) ∗θ g ( r ) =

∞

∑

m =−∞

f n (r ) g n (r ) e jnθ .

(133)

In other words, the nth coefficient of the angular convolution of two functions is simply the product of the

nth Fourier coefficient of each of the two functions. Mathematically, if h(r , θ ) = f ( r ) ∗θ g ( r ) , then

hn (r ) = f n (r ) g n (r ) . This is also the same result as obtained with Fourier series of (1D) periodic functions.

13 RADIAL CONVOLUTION Similar to the definition of angular or circular convolution, we define the notion of a radial convolution as

32

∞

h(r ) = f (r ) *r g (r ) = ∫ g (r0 , θ ) f (r − r0 , θ )r0 dr0 .

(134)

0

Such convolutions are less often seen than their 2D or angular counterparts, however they are included herein for the sake of completeness. Using equation (63) for the radially-only shifted function yields ∞ ⎛ ∞ h(r ,θ ) = ∫ ⎜ ∑ g m (r0 ) eimθ 0 ⎝ m =−∞

⎞ ⎛ ∞ ikθ ∞ ⎟ ⎜⎜ ∑ e ∑ n =−∞ ⎠ ⎝ k =−∞

∞

⎞

∫ fn (u )Sn ( u, r , r0 ) udu ⎟⎟ r0 dr0 . k

0

⎠

(135)

Using the result for the product of two series so that the resulting Fourier coefficients are convolved,

fk ∗ gk ≡

∞

∑

m =−∞

f m (r ) g k −m (r ) , gives

h( r , θ ) =

∞

ikθ

∞

∞

∞

∑ e ∫ ∑ gk −m (r0 ) ∑ ∫ fn (u )Snm ( u, r , r0 ) udu r0dr0 k =−∞ m =−∞ n =−∞ 0

=

∞

∞

ikθ

0

∞

∞

∞

∑ e ∫ ∑ gk −m (r0 ) ∑ m =−∞ n =−∞

k =−∞

0

in 2π

∞

(136)

∫ Fn ( ρ )J m−n ( ρ r0 ) J m ( ρ r ) ρ d ρ r0dr0 0

Let us define

Gkm−−mn

∞

( ρ ) = ∫ g k −m (r0 )J m−n ( ρ r0 ) r0 dr0

(137)

0

which is the (m-n)th order Hankel transform of the (k-m)th Fourier coefficient. With this definition, equation (136) becomes

h( r , θ ) =

∞

∞

∞

∑e ∑ ∑

k =−∞

ikθ

m =−∞ n =−∞

∞

in Gkm−−mn ( ρ ) Fn ( ρ )J m ( ρ r ) ρ d ρ . ∫ 2π 0

(138)

This is the closest we can come to a convolution theorem for the radial convolution. Without the shift and integration over the angular portion of the function, the result is awkward because the order of the Hankel transform of one of the functions does not always correspond to the order of the Fourier coefficient as seen by the definition in equation (137). The angular shift and integration portion of the full 2D convolution effectively washes out all these cross terms to yield the nice result of the full 2D convolution, as previously discussed.

33

14 PARSEVAL RELATIONSHIPS A Parseval relantionship is important as it deals with the 'power' of a signal or function in the spatial and frequency domains. As previously mentioned, it is known that we can write ∞

f ( r ) = f (r ,θ ) =

∑

n =−∞

f n (r )e jnθ .

(139)

It is noted that in polar coordinates if r = ( r , θ ) then − r = ( r , θ + π ) . Hence ∞

∑

f ( −r ) = f (r , θ + π ) =

n =−∞ ∞

=

∑

n =−∞

f n ( r )e

f n (r ) ( −1) e− jnθ =

jn(θ +π ) ∞

n

∑

n =−∞

(140)

f − n (r ) ( −1) e jnθ n

where the overbar indicates a complex conjugate. Thus if we define a function g ( r ) = f ( −r ) , its coefficients are given by g n ( r ) = f − n (r ) ( −1) . To derive the Parseval relation, we evaluate that n

convolution of two functions f ( r ) and g ( r ) with g ( r ) = f ( −r ) and where the convolution is evaluated

( f * g ) as given by equation (77) at r = 0 . n J − n ( x) = ( −1) J n ( x) and J n ( 0 ) = δ n 0 so that

at r = 0 implying r = 0, θ = 0 . In other words, we evaluate This is done by making use of equation (81), along with

∞ ⎧ ∞ ⎪ − = g ( r ) f ( r ) dr 2 π ∫ 0 0 0 ∫ ⎨⎪n∑ −∞ 0 ⎩ =−∞ ∞

∞ ⎪⎫ g ( r ) J ( ρ r ) r dr ∫ − n 0 −n 0 0 0 ∫ fn (u ) J n ( ρu )udu ⎬⎪ ρ d ρ 0 0 ⎭

∞

(141)

which with the given choice of function for g becomes ∞

∫

−∞

∞

f (r0 ) f (r0 )dr0 = 2π ∫ 0

∞

∞

∑ ∫

n =−∞

0

∞

f n (r0 ) J n ( ρ r0 )r0 dr0 ∫ f n (u ) J n ( ρ u )udu ρ d ρ .

(142)

0

i− n Fn ( ρ ) 2π

in Fn ( ρ ) 2π

The preceding equation furnishes the desired Parseval relationship as ∞

∫

−∞

2

f (r ) dr =

1 2π

∞

∞

∑ ∫ n =−∞ 0

34

2

Fn ( ρ ) ρ d ρ .

(143)

Following this same procedure and evaluating the convolution of any two well-behaved functions f ( r ) and g ( − r ) at 0 yields the generalized Parseval relationship ∞

∫

g (r ) f (r )dr =

−∞

1 2π

∞

∞

∑ ∫

n =−∞

Gn ( ρ )Fn ( ρ ) ρ d ρ .

(144)

0

Furthermore, from the result on multiplication and observing from equation (140) that the coefficients of

g ( r ) are g − n (r ) , we have that

f (r ) g (r ) =

∞

∞

∑ ∑

k =−∞ m =−∞

f m (r ) g −( k −m ) (r ) eikθ .

(145)

Integrating both sides of the previous equation over all space gives ∞

∫

∞

∞

∑∫

f ( r ) g ( r ) dr = 2π

m =−∞ 0

0

f m (r ) g m (r ) rdr

(146)

which gives another version of the Parseval relationship in equation (144) as ∞

∞

∑∫

n =−∞ 0

f n (r ) g n (r ) rdr =

∞

∞

n =−∞

0

1

( 2π )

2

∑ ∫

Fn ( ρ )Gn ( ρ )ρ d ρ .

(147)

Clearly, it then follows from the preceding equation that ∞

∞

∑∫ n =−∞

2

f n (r ) rdr =

0

∞

1

∞

∑ ∫ ( 2π ) n=−∞ 0 2

2

Fn ( ρ ) ρ d ρ .

(148)

This last equation is the second Parseval relationship.

15 THE LAPLACIAN One of the most powerful applications of Fourier transforms is for the solutions of partial differential equations. Indeed, the preceding developments can be applied to simplify any PDE involving the Laplacian. In polar coordinates, the 2D Laplacian takes the form

∇2 =

∂2 1 ∂ 1 ∂2 + + ∂r 2 r ∂r r 2 ∂θ 2 35

(149)

Consider a typical function written in standard 2D polar form:

f ( r ) = f (r ,θ ) =

∞

∑

n =−∞

f n (r ) e

jnθ ,

(150)

Taking the Laplacian of f ( r ) gives

∞ ⎛ 2 ⎛ ∂2 1 ∂ 1 ∂2 ⎞ ∞ d f n 1 df n n 2 f n jnθ ( ) e f r ∇2 f ( r ) = ⎜ 2 + + 2 = ∑ ⎜ dr 2 + r dr − r 2 2 ⎟ ∑ n r r ∂ r r θ ∂ ∂ n n =−∞ =−∞ ⎝ ⎝ ⎠

⎞ jnθ (151) ⎟e ⎠

Hence for a function written in the form of (150), the required form of the Laplacian is denoted with ∇ 2n where this operator is defined by

∇ n2 =

d 2 1 d n2 + − . dr 2 r dr r 2

{

(152)

}

To obtain the full 2D Fourier transform, we seek F2 D ∇ 2 f (r ) which will be given by the series

{

2

∞

} ∑ 2π i

F2 D ∇ f (r ) =

−n

inψ

e

∞⎛

∫ ⎜⎜

d 2 fn ( r )

0⎝

n =−∞

dr 2

2 1 df n ( r ) n f n ( r ) ⎞ + − ⎟ J n ( ρ r ) rdr r dr r 2 ⎟⎠

(153)

A simple application of integration by parts along with the definition of a Bessel function gives ∞

∞

2 2 ∫ ∇n fn ( r ) J n ( ρ r ) rdr = − ρ ∫ fn ( r ) J n ( ρ r ) rdr 0

(154)

0

so that equation (153) becomes

{

}

F2 D ∇ 2 f (r ) = − ρ 2 = −ρ

2

∞

∑

2π i − n einψ

n =−∞

∞

∫ fn ( r ) J n ( ρ r ) rdr 0

∞

inψ

∑ Fn ( ρ ) e

(155)

= − ρ F ( ρ ,ψ ) 2

n =−∞

Thus in polar coordinates, taking the full 2D Fourier transform of the Laplacian, that is to say a Fourier series followed by an nth order Hankel transform of the Laplacian yields the original function multiplied by

− ρ 2 . It is emphasized that an nth order Hankel transform is necessary, in keeping with the definition of 36

the full 2D Fourier transform and also to get rid of the

n2 fn ( r ) r2

term that results from taking the

∂2 derivative as part of the Laplacian. Thus, although the original Laplacian operator is not a radially ∂θ 2 ∂2 symmetric operator due to the presence of the derivative, its Fourier space equivalent is a ∂θ 2 multiplication by − ρ 2 , which is in fact a radially symmetric operation. This seemingly innocent observation has many potential applications and is the primary motivation behind the section on convolutions between a radially symmetric function and one that is not radially symmetric. In summary, the Fourier pair is given by

∇ n2 f n =

d 2 fn ( r ) dr 2

+

2 1 df n ( r ) n f n ( r ) − ⇔ − ρ 2 Fn ( ρ ) 2 r dr r

(156)

16 APPLICATION TO THE HELMHOLTZ EQUATION All wave fields governed by the wave equation (such as acoustic waves) lead to the Helmholtz equation once a temporal Fourier transform is used to transform from the time domain to the frequency domain:

∇ 2u ( r , ω ) + k 2u ( r , ω ) = − s ( r , ω ) Here, the wave-numbers is k 2 =

ω2 cs2

(157)

. The temporal (time-to-frequency) Fourier transform is assumed to

use the same sign convention as given in equations (5) and (6). Here, s ( r , ω ) is the temporal Fourier transform of the inhomogeneous time and space-dependent source term for the wave equation. The variable u ( r , ω ) represents a physical variable that is governed by the wave equation, for example acoustic pressure. Both s ( r , ω ) and u ( r , ω ) are functions of position, r , and (temporal) frequency, ω . The variable cs represents the speed of the wave, which for an acoustic wave would be the speed of sound and for an electromagnetic wave would be the speed of light. For wave-fields governed by the wave equation,

k 2 is a real (and positive) quantity.

Equation (157) can also be used to represent other physical phenomena governed by diffusive waves (Mandelis 2001). For example, the equation for a Diffuse Photon Density Wave (DPDW) which describes the photon density u ( r , t ) in a solid due to incident energy intensity s ( r , t ) (optical source function). The standard heat equation can also be cast into the form of (157) once a temporal Fourier transform is taken.

In these latter two cases, the wave-number is complex, indicating a damped or diffusive wave. For example, the temporal Fourier transform of the heat equation gives a complex wave-number k 2 = −

37

iω

α

where α is the thermal diffusivity of a material, u ( r , t ) describes the temperature in the material as a function of time and space and s ( r , t ) is a time and space dependent heat source.

Thus, the general Helmholtz form of equation (157) can be used to describe several different physical phenomena, from the propagation of light or acoustic waves to the heavily damped nature of photonic or thermal waves. The exact form of the wave-number in each case indicates the propagation characteristics of a wave with a real k 2 indicating a propagating wave and a complex k 2 indicating a damped wave.

Generally, the term 'Helmholtz equation' refers to equation (157) with a real wave-number and a complex

k 2 can be referred to as 'pseudo-Helmholtz' equation. However, this terminology is rather cumbersome and we prefer to refer to equation (157) as a Helmholtz equation, regardless if k 2 is real or complex. Most of the mathematics can be made to yield the same results for real or complex wave-number if a proper choice of sign convention for k is chosen. For example, e ± ikx can be propagating or damped waves and the sign convention for a complex k will determine if the wave remains bounded at positive or negative infinity.

The quantity of interest is usually the wave number itself, namely k , which is the square root of the given squared wave-number in the Helmholtz equation. Each k can be considered as the sum of a real and an imaginary part, so that k = kr + iki with k r denoting the real part of k and ki denoting the imaginary part.

A proper choice of sign convention for k means defining the complex k as the square root of the corresponding k 2 such that k has a (choice of) positive or negative imaginary part. This can be done so that the wave(s) gives a physically reasonable result (boundedness) over the domain of interest of the problem. Choices of sign convention can also serve to make the book-keeping simpler in the sense that by choice of sign convention, results can be written the same way whether for real or complex k . The choice of sign convention for k also determines which expressions are considered to be outwardly radiating. This is important in the sense that mathematically the Helmholtz equation will allow solutions that are inwardly and outwardly propagating waves (damped or propagating), whereas to choose the most physically meaningful solution only one of these will usually be the right choice. The Sommerfeld Radiation Condition can be used to choose a radiating solution.

16.1 THE HELMHOLTZ TRANSFER FUNCTION For a Helmholtz equation that is two dimensional in space, taking the Fourier Series transform of equation (157) implies writing u ( r , ω ) =

∞

∑ u ( r , ω )e θ

n =−∞

in

n

and similarly for s ( r , ω ) and ∇ 2 , so that

u ( r , ω ) ⇔ un ( r , ω ) , s ( r , ω ) ⇔ sn ( r , ω ) and ∇ 2 ⇔ ∇ 2n and equation (157) becomes

∇ 2nun (r , ω ) + k 2un (r , ω ) = − sn (r , ω )

38

(158)

Taking an nth order Hankel transform of equation (158) turns this into a full 2D Fourier transform so that

un ( r , ω ) ⇔ U n ( ρ , ω ) , sn ( r , ω ) ⇔ S n ( ρ , ω ) and ∇ 2n ⇔ − ρ 2 . This and then rearranging gives

U n ( ρ,ω ) =

1 S ( ρ , ω ) = G ( ρ , ω ) Sn ( ρ , ω ) ρ − k2 n 2

(159)

where G ( ρ , ω ) is given by

G ( ρ,ω ) =

1 ρ − k2 2

(160)

and shall be referred to as the Helmholtz transfer function. The notation G ( ρ , ω ) is used as a reminder that the transfer function is (i) only a function of the frequency radial variable ρ in space and (ii) a function of temporal frequency ω via the wave-number k . Solving the Helmholtz equation for a given source

S (ω , ω ) is a matter of inverting equation (159). Note that equation (159) has the exact same form as equation (129) which means that this simple expression is the frequency domain equivalent of

g ( r , ω ) **s ( r , ω ) , where g ( r , ω ) = F2−D1 {G ( ρ , ω )} is the Green's function for the Helmholtz equation.

In fact, it is this example which was the primary motivator behind the section on convolutions of radially symmetric functions with non-radially symmetric functions.

To invert equation (159) back into the spatial domain implies finding

in un ( r , ω ) = 2π

∞

∫ 0

Sn ( ρ , ω ) Jn ( ρr ) ρd ρ ρ2 − k2

(161)

Results involving integrals of the type in equation (161) are considered elsewhere (Baddour 2009). Often several solutions to (161) are mathematically possible, in particular due to its nature as an improper integral. To aid in selecting the most physically meaningful result out of several mathematical possibilities, the Sommerfeld radiation condition must be used.

16.2 GREEN'S FUNCTION COEFFICIENTS The Green's function is defined as the solution to the Helmholtz equation for a delta-function source at r = r0 for real or complex k :

∇ 2 g (r , r0 , ω ) + k 2 g (r , r0 , ω ) = −δ ( r − r0 )

39

(162)

where we use g (r , r0 , ω ) to denote the Green's function. Taking the 2D Fourier transform of (162) gives

( −ρ

2

+ k 2 ) G (ω , r0 , ω ) = −e

− iω ⋅r0

(163)

For r0 = 0 , the definition of G in equation (163) is the same as that in (160). We convert to the polarcoordinate form so that the complex exponential is given by (9) and the Green's function is written as ∞

∑ G ( ρ, r ,ω ) e ψ

(164)

1 ⎡ − iω ⋅r0 ⎤ 1 e i − n J n ( ρ r0 ) e −inθ0 = 2 2 ⎣ 2 ⎦ n ρ −k ρ −k

(165)

G (ω , r0 , ω ) =

in

n =−∞

n

0

so that equation (163) becomes

Gn ( ρ , r0 , ω ) =

2

Although we arrived at equation (165) directly from the classical definition of a Green's function as the response to a delta function, we see from equation (99) that these are in fact the Fourier coefficients of a shifted radially symmetric function Helmholtz transfer function G ( ρ , ω ) =

1 where multiplication ρ − k2 2

− iω ⋅r by the coefficients ⎡e 0 ⎤ will provide the shift.

⎣

⎦n

With the interpretation of these coefficients as those the shifted function we can write g n (r , r0 , ω ) as

g n (r − r0 , ω ) , interpreting the shift. The coefficients of the Green's function in spatial (polar) coordinates as

g n (r , r0 , ω ) = g n (r − r0 , ω ) = =

e −inθ0 2π

∞

∞

in Gn ( ρ , r0 , ω ) J n ( ρ r ) ρ d ρ 2π ∫0

1

∫ ρ 2 − k 2 J n ( ρ r0 ) J n ( ρ r ) ρ d ρ

(166)

0

Where the notation g n (r , r0 , ω ) = g n (r − r0 , ω ) has been used to indicate that what we've found is actually −1 a shifted version of F2D ⎡⎣G ( ρ , ω ) ⎤⎦ . Of course, the fact that any Green's function is a shift of the response

to a delta function at the original is well known, but the point here was that this same interpretation also followed from the rules of Fourier transforms in polar coordinates developed herein.

40

17 SUMMARY AND CONCLUSIONS In summary, this article has considered the polar-coordinate version of the standard 2D Fourier transform and derived the operational toolset required for standard Fourier operations. As previously noted, the polarcoordinate version of the 2D Fourier transform is most useful for functions which are naturally described in terms of polar coordinates. Additionally, Parseval relationships were also derived. The results are concisely collected in the table at the end of the article. Of particular interest are the results on convolution and spatial shift. Notably, standard convolution/multiplication rules do apply for 2D convolution and 1D circular convolution but not for 1D radial convolution.

41

Table 1: Summary of Fourier Transform Relationships in Polar Coordinates

f (r ) f (r , θ ) = ∞

∑

n =−∞

f n (r )e jnθ

f (r , θ ) = ∞

∑

n =−∞

in 2π

f n (r )e jnθ

r

f (r , θ )e

dθ

2π i

1 2π

0

i n J n ( ρ0 r ) e

in 2π

−inψ 0

2π

∫ F ( ρ ,ψ )e

∑

1

ρ

Fn ( ρ )e jnψ

F ( ρ ,ψ ) =

dψ

∞

∑

n =−∞

−inψ 0

δ ( ρ − ρ0 )

⎡i − n J n ( ρ r0 )e−inθ0 ⎤ * Fn ( ρ ) = ⎣ ⎦

0

e

−imθ0

∞

∫ fn−m (u )Sn−m ( u, r , r0 ) udu n

0

∞

∑

i

−m

m =−∞

J m ( ρ r0 ) e

f n (r ) = hn (r ) * g n (r ) ∞

∑

=

m =−∞

h(r )**g (r )

− jnψ

0

2π e

∞

n =−∞

i − n J n ( ρ r0 )e−inθ0

∫ Fn ( ρ ) J n ( ρ r ) ρ d ρ =

m =−∞

h( r ) g ( r )

∫ f n (r ) J n ( ρ r ) rdr

Fn ( ρ )e jnψ

e−iω⋅r0

2 ( 2π ) δ (ω − ω0 )

∞

∞

∑

F ( ρ ,ψ ) =

∞ 0

δ ( r − r0 ) −inθ0 e 2π r

eiω0 ⋅r

−n

0

∫ Fn ( ρ ) J n ( ρ r ) ρ d ρ

δ (θ − θ 0 )

f (r − r0 )

∫

− jnθ

∞

δ ( r − r0 ) = δ ( r − r0 )

2π

1 2π

F (ω )

Fn ( ρ )

f n (r )

in 2π

∞

∫

2π i

hn −m (r ) g m (r )

−n

∞

∫

−imθ0

Fn −m ( ρ )

f n (r ) J n ( ρ r ) rdr

0

e−iω⋅r0 F (ω )

G (ω ) **H (ω )

( 2π )2

Fn ( ρ ) = H n ( ρ ) * Gn ( ρ ) Fn ( ρ ) J n ( ρ r ) ρ d ρ

=

0

∞

∑

m =−∞

42

Gn −m ( ρ ) H m ( ρ )

G (ω ) H (ω )

h ( r ) ∗θ g ( r )

∞

f n ( r ) = hn ( r ) g n ( r )

2π i − n ∫ f n (r ) J n ( ρ r ) rdr 0

f ( r − r0 )

(f

e−inθ0

symmetric )

2π

∞

∫ F ( ρ )J n ( ρ r0 ) J n ( ρ r ) ρ d ρ

i − n e −inθ0 F ( ρ ) J n ( ρ r0 )

∞

∑

n =−∞

Fn ( ρ )e jnψ

e−iω⋅r0 F ( ρ )

0

∞

f n (r ) = ∫ g k (r0 )φk ( r − r0 ) r0 dr0 0

g (r ) **h(r )

φk ( r − r0 ) =

( h symmetric )

2π i

−n

∞

∫ f n (r ) J n ( ρ r ) rdr 0

∞

∫ H ( ρ )J k ( ρ r0 ) J k ( ρ r ) ρ d ρ

F ( ρ ,ψ ) = ∞

∑

n =−∞

Fn ( ρ )e jnψ

0

g (r ) **h(r )

in 2π

( h symmetric )

∞

∫ Fn ( ρ ) J n ( ρ r ) ρ d ρ

Fn ( ρ ) = Gn ( ρ ) H ( ρ )

0

∞

∑ Fn ( ρ )e jnψ

n =−∞

∇ 2 f ( r ,θ ) f =

∞

∑

n =−∞

f n ( r )e

jnθ

∇ n2 f n

d 2 f n 1 df n n 2 = + − fn dr 2 r dr r 2

− ρ 2 Fn ( ρ )

43

− ρ 2 F ( ρ ,ψ )

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Averbuch, A. et al., 2006. Fast and accurate Polar Fourier transform. Applied Computational Harmonic Analysis, 21, 145--167.

Baddour, N., 2009. Multidimensional Wave Field Signal Theory: Fundamental Integrals with Applications. submitted to Journal of the Franklin Institute.

Bracewell, R., 1999. The Fourier Transform and its Applications, Englewood Cliffs, New Jersey: McGrawHill.

Chirikjian, G. & Kyatkin, A., 2001. Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups, New York: Academic Press.

Goodman, J., 2004. Introduction to Fourier Optics, Roberts & Company Publishers.

Howell, K., 2000. Fourier Transforms. In The Transforms and Applications Handbook. Boca Raton: CRC Press LLC, pp. 2.1 - 2.159.

Jackson, A.D. & Maximon, L.C., 1972. Integrals of Products of Bessel Functions. SIAM Journal on Mathematical Analysis, 3(3), 446-460.

Mandelis, A., 2001. Diffusion-Wave Fields, Mathematical Methods and Green Functions, New York: Springer.

Oppenheim, A. & Schafer, R., 1989. Discrete-time Signal Processing, Englewood Cliffs, New Jersey: Prentice-Hall.

Piessens, R., 2000. The Hankel Transform. In The Transforms and Applications Handbook. Boca Raton: CRC Press, pp. 9.1-9.30. Available at: internal-pdf://ch09_HankelTransform0350394163/ch09_HankelTransform.pdf.

Xu, Y., Xu, M. & Wang, L.V., 2002. Exact frequency-domain reconstruction for thermoacoustic tomography - II: Cylindrical geometry. IEEE Transactions on Medical Imaging, 21(7), 829-833.

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