1. IntroductionThe forward and inverse Fourier transforms of two related functions f ( t ) and F ( ν ) can be defined in a symmetric form as [1] [2]:
F { f ( t ) } ( ν ) = F ( ν ) = ∫ − ∞ ∞ f ( t ) e − 2 π i ν t d t (1)
and:
F − 1 { F ( ν ) } ( t ) = f ( t ) = ∫ − ∞ ∞ F ( ν ) e 2 π i ν t d ν , (2)
where variables t and ν are the corresponding Fourier-transformed arguments in t-space and ν -space, respectively (time t vs. frequency ν , for example).
Fourier transform methods are widely used in many applications including signal processing [1] [2], spectroscopy [3] [4] and computational finance [5] [6] [7].
There are several efficient methods that have been reported for rational approximations in literature. For example, the rational approximations may be built on the basis of the Newman nodes [8], Chebyshev nodes [9], logarithmic nodes [10] and so on.
Recently we have reported a new method of rational approximation of the Fourier transform (1) as given by [11]:
F { f ( t ) } ( ν ) ≈ e 2 π i ν a ∑ m = 1 2 M − 1 A m ( σ + 2 π i ν ) + B m C m 2 + ( σ + 2 π i ν ) 2 , (3)
where M is an integer determining the number of summation terms 2 M − 1 , a is a shift constant, σ is a decay (damping) constant and:
A m = 1 2 M − 1 ∑ n = 0 N f ( n h − a ) e σ n h cos ( C m n h ) ,
B m = 1 2 M − 1 ∑ n = 0 N f ( n h − a ) e σ n h C m sin ( C m n h ) ,
C m = π ( 2 m − 1 ) 2 M h .
are expansion coefficients.
It has been noticed that approximation (3) is not purely rational and there was a question whether or not a rational function of the Fourier transform (1) in explicit form without any trigonometric multiplier of the kind:
e 2 π i ν a = cos ( 2 π ν a ) + i sin ( 2 π ν a ) , (4)
depending on the argument ν , can be obtained [12]. Theoretical analysis shows that this trigonometric multiplier originating from the shifting property of the Fourier transform can be indeed excluded. As a further development of our work [11], in this paper, we derive a rational function of the Fourier transform (1) that has no any trigonometric multiplier of the kind (4). Therefore, it can be used as an alternative to the Padé approximation. To the best of our knowledge, this method of rational approximation of the Fourier transform (1) for a non-periodic function f ( t ) has never been reported in scientific literature.
2. Derivation2.1. PreliminariesAssume that Re { f ( t ) } is even while Im { f ( t ) } is odd such that f : ℝ → ℂ , but Re { f } : ℝ → ℝ and Im { f } : ℝ → ℝ . Then it is not difficult to see that the Fourier transform (1) of the function f ( t ) can be expanded into two integral terms as follows:
F { f ( t ) } ( ν ) = F ( ν ) = 2 ∫ 0 ∞ Re { f ( t ) } cos ( 2 π ν t ) d t + 2 ∫ 0 ∞ Im { f ( t ) } sin ( 2 π ν t ) d t .
Assume also that the function f ( t ) behaves in such a way that for some positive numbers τ 1 and τ 2 the following integrals:
∫ τ 1 ∞ Re { f ( t ) } cos ( 2 π ν t ) d t ≈ 0
and:
∫ τ 2 ∞ Im { f ( t ) } sin ( 2 π ν t ) d t ≈ 0
are negligibly small and can be ignored in computation. Consequently, we can approximate the Fourier transform as given by:
F { f ( t ) } ( ν ) = F ( ν ) ≈ 2 ∫ 0 τ 1 Re { f ( t ) } cos ( 2 π ν t ) d t + 2 ∫ 0 τ 2 Im { f ( t ) } sin ( 2 π ν t ) d t . (5)
Further the values 2 τ 1 and 2 τ 2 will be regarded as widths (pulse widths) for the real and imaginary parts of the function f ( t ) , respectively.
2.2. New Sampling MethodConsider a sampling Formula (see, for example, Equation (3) in [13] ):
f ( t ) = ∑ n = − N N f ( t n ) sinc ( π h ( t − t n ) ) + ε ( t ) , (6)
where:
sinc ( t ) = { sin t t , t ≠ 0 1, t = 0,
is the sinc function, t n is a set of sampling points, h is small adjustable parameter (step) and ε ( t ) is error term. François Viète discovered that the sinc function can be represented by cosine product1 [14] [15]:
sinc ( t ) = ∏ m = 1 ∞ cos ( t 2 m ) . (7)
In our earlier publications we introduced a product-to-sum identity [16]:
∏ m = 1 M cos ( t 2 m ) = 1 2 M − 1 ∑ m = 1 2 M − 1 cos ( 2 m − 1 2 M t ) (8)
and applied it for sampling [17] [18] as incomplete cosine expansion of the sinc function for efficient computation of the Voigt/complex error function. It is worth noting that this product-to-sum identity has also found some efficient applications in computational finance [6] [19] [20] involving numerical integration.
Comparing identities (7) and (8) immediately yields:
sinc ( t ) = l i m M → ∞ 1 2 M − 1 ∑ m = 1 2 M − 1 cos ( m − 1 / 2 2 M − 1 t ) .
Unlike Equation (7), this limit consists of sum of cosines instead of product of cosines. As a result, its application provides significant flexibilities in various numerical integrations [6] [17] [18] [19] [20].
Change of variable 2 M − 1 → M in the limit above leads to:
sinc ( t ) = l i m M → ∞ 1 M ∑ m = 1 M cos ( m − 1 / 2 M t ) .
Therefore, by truncating integer M and by making another change of variable t → π t / h we obtain:
sinc ( π h t ) ≈ 1 M ∑ m = 1 M cos ( π ( m − 1 / 2 ) M h t ) , − M h ≤ t ≤ M h . (9)
The right side of Equation (9) is periodic due to finite number of the summation terms. As a result, the approximation (9) is valid only within the interval t ∈ [ − M h , M h ] .
At equidistantly separated sampling grid-points such that t n = n h , the substitution of approximation (9) into sampling Formula (6) gives:
f ( t ) ≈ 1 M ∑ m = 1 M ∑ n = − N N f ( n h ) cos ( π ( m − 1 / 2 ) M h ( t − n h ) ) , − M h ≤ t ≤ M h . (10)
It is important that in sampling procedure the total number of the sampling grid-points 2 N + 1 as well as the step h should be properly chosen to insure that the widths 2 τ 1 and 2 τ 2 are entirely covered.
As we can see, the sampling Formula (10) is based on incomplete cosine expansion of the sinc function that was proposed in our previous works [17] [18] as a new approach for rapid and highly accurate computation of the Voigt/complex error function [21] [22] [23]. Computations we performed show that this method of sampling is particularly efficient in numerical integration.
2.3. Even FunctionSuppose that our objective is to approximate the sinc function sinc ( π ν ) . First we take the inverse Fourier transform (2) of the sinc function:
F − 1 { sinc ( π ν ) } ( t ) = ∫ − ∞ ∞ sinc ( π ν ) e − 2 π i ν t d ν = rect ( t ) ,
where:
rect ( t ) = { 1, if | t | < 1 / 2 1 / 2 , if | t | = 1 / 2 0, if | t | > 1 / 2 ,
is known as the rectangular function. This function is even since rect ( t ) = rect ( − t ) . The rectangular function rect ( t ) has two discontinuities at t = − 1 / 2 and t = 1 / 2 . Therefore, it is somehow problematic to perform sampling over this function. However, we can use the fact that:
rect ( t ) = l i m k → ∞ 1 ( 2 t ) 2 k + 1 . (11)
Thus, by taking a sufficiently large value for the integer k, say k = 35 , we can approximate the rectangular function (11) quite accurately as:
rect ( t ) ≈ f ( t ) = 1 ( 2 t ) 70 + 1 .
Figure 1 shows the function f ( t ) = 1 / ( ( 2 t ) 70 + 1 ) by blue curve. As we can see from this figure, the function very rapidly decreases at | t | > 1 / 2 with increasing t. Therefore, we can take τ 1 = 0.6 . Thus, the width of this function is 2 τ 1 = 1.2 .
Sampling of function f ( t ) = 1 / ( ( 2 t ) 70 + 1 ) in accordance with Equation (10) results in a periodic dependence. Consequently, due to periodicity on the right side of Equation (10) it cannot be utilized for rational approximation of the Fourier transform. However, this problem can be effectively resolved by sampling the function f ( t ) e σ t instead of f ( t ) itself. This leads to:
f ( t ) e σ t ≈ 1 M ∑ m = 1 M ∑ n = − N N f ( n h ) e σ n h cos ( π ( m − 1 / 2 ) M h ( t − n h ) ) , − M h ≤ t ≤ M h . (12)
Figure 2 shows the results of computation for even function f ( t ) = 1 / ( ( 2 t ) 70 + 1 ) by approximation (12) at M = 32 , N = 28 , h = 0.04 with σ = 0 (blue curve), σ = 0.25 (red curve) and σ = 0.75 (green curve). As we can see from this figure, all three curves are periodic as expected. However, if the constant σ is big enough, then slight rearrangement of Equation (12) in form:
f ( t ) ≈ e − σ t M ∑ m = 1 M ∑ n = − N N f ( n h ) e σ n h cos ( π ( m − 1 / 2 ) M h ( t − n h ) ) , (13)
can effectively eliminate this periodicity due to presence of the exponential decay multiplier e − σ t on the right side. This suppression effect can be seen from
Figure 3 illustrating the results of computation for the even function f ( t ) = 1 / ( ( 2 t ) 70 + 1 ) by approximation (13) at M = 32 , N = 28 with σ = 0 (blue curve), σ = 0.25 (red curve) and σ = 0.75 (green curve). As it is depicted by blue curve, at σ = 0 the function is periodic. However, as decay coefficient σ increases, the exponential multiplier e − σ t suppresses all the peaks (except the first peak at the origin) such that the resultant function tends to become solitary along the entire positive t-axis. This tendency can be observed by red and green curves at σ = 0.25 and σ = 0.75 , respectively. As a consequence, if the damping multiplier σ is big enough, say greater than unity, the approximated function becomes practically solitary as the original function f ( t ) = 1 / ( ( 2 t ) 70 + 1 ) itself.
Thus, substituting approximation (13) into Equation (5) and considering the fact that at sufficiently large σ the function becomes solitary along positive x-axis, the upper limit τ 1 of integration can be replaced by infinity as2:
F { f ( t ) } ( ν ) = F { Re { f ( t ) } } ( ν ) = F { 1 ( 2 t ) 70 + 1 } ( ν ) ≈ 2 ∫ 0 τ 1 [ e − σ t M ∑ m = 1 M ∑ n = − N N Re { f ( n h ) } e σ n h cos ( π ( m − 1 / 2 ) M h ( t − n h ) ) ] cos ( 2 π ν t ) d t ≈ 2 ∫ 0 ∞ [ e − σ t M ∑ m = 1 M ∑ n = − N N Re { f ( n h ) } e σ n h cos ( π ( m − 1 / 2 ) M h ( t − n h ) ) ] cos ( 2 π ν t ) d t .
This integral can be taken analytically in form of rational function now and after some trivial rearrangements that exclude double summation, it follows that
F { Re { f ( t ) } } ( ν ) ≈ ∑ m = 1 M α m + β m ν 2 κ m + λ m ν 2 + ν 4 , (14)
where the expansion coefficients are given by:
α m = 1 8 M π 4 ∑ n = − N N Re { f ( n h ) } e n h σ ( μ m 2 + σ 2 ) ( σ cos ( n h μ m ) + μ m sin ( n h μ m ) ) ,
β m = 1 2 M π 2 ∑ n = − N N Re { f ( n h ) } e n h σ ( σ cos ( n h μ m ) − μ m sin ( n h μ m ) ) ,
κ m = 1 16 π 4 ( μ m 2 + σ 2 ) 2 ,
λ m = 1 2 π 2 ( σ 2 − μ m 2 )
and:
μ m = π ( m − 1 / 2 ) M h .
Figure 4 shows the original sinc function sinc ( ν ) and its approximation (14) within the interval − 2 π ≤ ν ≤ 2 π at M = 32 , N = 28 , h = 0.04 and σ = 2.75 by black dashed and light blue curves, respectively. These two curves are not visually distinctive.
2.4. Odd FunctionConsider, as an example, the following function:
f ( t ) = i t ( 2 t ) 70 + 1 ≈ i t rect ( t ) .
We can see that the condition t r e c t ( t ) = − ( − t r e c t ( − t ) ) for odd function in its imaginary part is satisfied. The function Im { f ( t ) } = t / ( ( 2 t ) 70 + 1 ) is shown in Figure 1 by red curve. We can take τ 2 = 0.6 and the width is 2 τ 2 = 1.2 .
Using exactly same procedure as it has been described above and considering the fact that at sufficiently large σ the upper limit τ 2 of integration can be replaced by infinity, we can write3:
F { f ( t ) } ( ν ) = F { i Im { f ( t ) } } ( ν ) = F { i t ( 2 t ) 70 + 1 } ( ν ) ≈ 2 ∫ 0 τ 2 [ e − σ t M ∑ m = 1 M ∑ n = − N N Im { f ( n h ) } e σ n h cos ( π ( m − 1 / 2 ) M h ( t − n h ) ) ] sin ( 2 π ν t ) d t ≈ 2 ∫ 0 ∞ [ e − σ t M ∑ m = 1 M ∑ n = − N N Im { f ( n h ) } e σ n h cos ( π ( m − 1 / 2 ) M h ( t − n h ) ) ] sin ( 2 π ν t ) d t .
This leads to:
F { i Im { f ( t ) } } ( ν ) ≈ ∑ m = 1 M η m ν + θ m ν 3 κ m + λ m ν 2 + ν 4 , (15)
where the expansion coefficients are:
η m = 1 4 M π 3 ∑ n = − N N Im { f ( n h ) } e n h σ ( ( σ 2 − μ m 2 ) cos ( n h μ m ) + 2 σ μ m sin ( n h μ m ) )
and:
θ m = 1 M π ∑ n = − N N Im { f ( n h ) } e n h σ cos ( n h μ m ) .
The Fourier transform of the function i t rect ( t ) can be readily found analytically:
F { i t rect ( t ) } ( ν ) = ∫ − ∞ ∞ i t rect ( t ) e − 2 π i ν t d t = ∫ − 1 / 2 1 / 2 i t rect ( t ) e − 2 π i ν t d t = ∫ − 1 / 2 1 / 2 i t e − 2 π i ν t d t = sin ( π ν ) − π ν cos ( π ν ) 2 ( π ν ) 2 .
Gray curve in Figure 4 illustrates the Fourier transform of the function f ( t ) = i t / ( ( 2 t ) 70 + 1 ) obtained by using approximation (15) at M = 32 , N = 28 , h = 0.04 and σ = 3 . The original function:
F { i t rect ( t ) } ( ν ) = sin ( π ν ) − π ν cos ( π ν ) 2 ( π ν ) 2
is also shown for comparison by black dashed curve. These two curves in the interval − 2 π ≤ ν ≤ 2 π are also not distinctive visually.
3. AccuracyFigure 5 shows the absolute difference between original sinc function sinc ( π ν ) and its approximation (14) for input function f ( t ) = 1 / ( ( 2 t ) 70 + 1 ) calculated at M = 32 , N = 28 , h = 0.04 and σ = 2.7 . As we can see, the absolute difference within the interval − 2 π ≤ ν ≤ 2 π does not exceed 2.5 × 10−3. This accuracy is better than that of shown in our recent publication, where we used Equation (3) for the sinc function approximation (see Figure 6 in [11] ).
Figure 6 shows the absolute difference between original function given by ( sin ( π ν ) − π ν cos ( π ν ) ) / ( 2 ( π ν ) 2 ) and its approximation (15) for input function f ( t ) = i t / ( ( 2 t ) 70 + 1 ) calculated at M = 32 , N = 28 , h = 0.04 and σ = 3 . We can see that the absolute difference within the interval − 2 π ≤ ν ≤ 2 π does not exceed 6 × 10−4.
It should be noted that with more well-behaved functions we can obtain considerably higher accuracies. For example, suppose that we need to obtain the Fourier transform of the function f ( t ) = π e − ( π t ) 2 + i [ π 3 / 2 t e − ( π t ) 2 ] by using approximations (14) and (15). Analytically, its Fourier transform is:
F { π e − ( π t ) 2 + i [ π 3 / 2 t e − ( π t ) 2 ] } ( ν ) = e − ν 2 + ν e − ν 2 ,
where e − ν 2 is the Fourier transform of π e − ( π t ) 2 while ν e − ν 2 is the Fourier transform of i π 3 / 2 t e − ( π t ) 2 .
Blue curve in Figure 7 corresponds to the absolute difference between function e − ν 2 and its approximation (14) for input function π e − ( π t ) 2 at M = 16 , N = 23 , h = 0.119 and σ = 6.9 . Red curve in Figure 7 corresponds to the absolute difference between function ν e − ν 2 and its approximation (15) for input function i π 3 / 2 t e − ( π t ) 2 at M = 16 , N = 23 , h = 0.119 and σ = 5.9 . We can see that with only 16 summation terms the absolute differences do not exceed 3 × 10−10 and 9 × 10−10. These results demonstrate that the rational approximations (14) and (15) can be highly accurate in the Fourier transform of well-behaved functions.
In our recent work [24] we applied alternative method of sampling by using incomplete cosine expansion of the Gaussian function of kind h e − ( t / c ) 2 / ( c π ) , where c and h are the fitting parameters. We have shown that this method of sampling can also be used to obtain high-accuracy computation of the Voigt/complex error function. In our future work will apply this method of sampling as an alternative that may reduce the absolute difference for rational approximations of the piecewise functions with discontinuities.
4. Alternative RepresentationFor a function f ( t ) = Re { f ( t ) } + i Im { f ( t ) } , where its real part Re { f ( t ) } is even and its imaginary part Im { f ( t ) } is odd, we can write:
F { f ( t ) } ( ν ) ≈ ∑ m = 1 M α m + β m ν 2 κ m + λ m ν 2 + ν 4 + ∑ m = 1 M η m ν + θ m ν 3 κ m + λ m ν 2 + ν 4
or:
F { f ( t ) } ( ν ) ≈ ∑ m = 1 M α m + η m ν + β m ν 2 + θ m ν 3 κ m + λ m ν 2 + ν 4 .
Using a Computer Algebra System (CAS) supporting symbolic programming it is not difficult to find coefficients p k and q k to represent this approximation as:
F { f ( t ) } ( ν ) ≈ P ( ν ) Q ( ν ) , (16)
where:
P ( ν ) = p 0 + p 1 ν + p 2 ν 2 + ⋯ + p 4 M − 2 ν 4 M − 2 + p 4 M − 1 ν 4 M − 1
and:
Q ( ν ) = q 0 + q 1 ν 2 + q 2 ν 4 + ⋯ + q 2 M − 1 ν 4 M − 2 + q 2 M ν 4 M ,
are polynomials of the orders 4 M − 1 and 4 M , respectively.
Padé approximation is one of the efficient methods to represent a function in form of ratio of two polynomials. Our preliminary numerical results show that the proposed new method of rational approximation may significantly extend the range [ ν min , ν max ] in coverage [12] than the conventional Padé approximation.