# How to calculate the Fourier transform of Sinc[b (ω1 - ω2)]?

I would like to calculate the Fourier transform of Sinc[ b (ω1 - ω2)], but there are some problems as follows:

My target is

$$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\text{Sinc}(b (\text{\omega_1}-\text{\omega_2})) e^{-i \text{ \omega_1 t_1} } e^{-i \text{ \omega_2 t_2 } } d \text{\omega_1} d\text{\omega_2} =\frac{2 \pi ^2}{b} \Pi \left(\frac{\text{t_1}}{2 b}\right) \delta (\text{t_1}+\text{t_2})$$

The problem is as follow:

I use the following code

    Assuming[{b >0 }, FourierTransform[  Sinc[b (ω1 - ω2)], {ω1, ω2}, {t1, t2}, \!$$TraditionalForm\FourierParameters -> {1, \(-1$$}\)]]


After runing the above code for a long time, the out put is

    (I \[Pi] DiracDelta[t1+t2] (-Log[I b-I t1]+Log[-I b+I t1]+Log[-I (b+t1)]-Log[I (b+t1)]))/b


It can be further simplified to be 0, because

    Log[-I (b - t1)] + Log[-I (b + t1)] - Log[I (b - t1)] - Log[I (b + t1)]

= Log[I (b - t1)*I (b + t1)] - Log[I (b - t1)*I (b + t1)]

= Log[-b^2 + t1^2] - Log[-b^2 + t1^2]

= 0


This means $$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\text{Sinc}(b (\text{\omega_1}-\text{\omega_2})) e^{-i \text{ \omega_1 t_1} } e^{-i \text{ \omega_2 t_2 } } d \text{t_1} d\text{t_2} =0$$ ??

How to solve this porblem?

Clue 1: $$Sinc[ω1 - ω2]$$

    FourierTransform[ Sinc[ω1 - ω2], {ω1, ω2}, {t1,   t2}, \!$$TraditionalForm\FourierParameters -> {1, \(-1$$}\)]


The result is $$π^2 DiracDelta[t1 + t2] (Sign[1 - t1] + Sign[1 + t1])$$.

But in Mathematica, (Sign[1 - t1] + Sign[1 + t1]) does not equal to 2 UnitBox[t1/2], because FullSimplify[Sign[1 - t1] + Sign[1 + t1] - 2 UnitBox[t1/2]] = Piecewise[{{-1, t1 == -1 || t1 == 1}}, 0]

Clue 2: $$Sinc[b ω]$$

    FourierTransform[ Sinc[b ω], {ω}, {t}, \!$$TraditionalForm\FourierParameters -> {1, \(-1$$}\)]


The result is $$\frac{\pi}{2 b} (Sign[b - t] + Sign[b + t])$$.

Clue 3: $$Sinc[3 (ω1 - ω2)]$$

    Assuming[{a > b },  FourierTransform[Sinc[3 (ω1 - ω2)], {ω1, ω2}, {t1, t2}, \!$$TraditionalForm\FourierParameters -> {1, \(-1$$}\)]]


The result is 0.

Any comment or suggestion would be highly appreciated.

• Please include actual code (that can be copy-and-pasted) rather than just images thereof. – Daniel Lichtblau Apr 20 '19 at 14:40
• @Daniel Lichtblau, thank you for the comment. I have added the actual code and delete the image. – user14634 Apr 21 '19 at 1:00
• Actually, FullSimplify[Sign[1 - t1] + Sign[1 + t1] - 2 UnitBox[t1/2], Assumptions -> t2 > t1 > 1] simplifies to zero. Your clue #1 is incorrect. – bill s Apr 21 '19 at 3:44
• @ bill s , Thank you for the comment. But I think clue #1 is correct, because without any assumption, the result is not zero, i.e., FullSimplify[Sign[1 - t1] + Sign[1 + t1] - 2 UnitBox[t1/2]] simplifies to Piecewise[{{-1, t1 == -1 || t1 == 1}}, 0]. Generally speaking, t1 is a real number, and the assumption of t1>1 is not true. – user14634 Apr 21 '19 at 11:20
• I agree with @bills, but for a different reason: The rectangular function $\Pi(t)$ does not equal UnitBox[t]; rather, it is equal to 1/2 (Sign[1 - 2 t] + Sign[1 + 2 t]). Examine UnitBox[t/2] at t = ±1. – Michael E2 Apr 21 '19 at 12:19

By Fubini's theorem, we can relate multiple integrals to iterated integrals, i.e. $$\int f(x,y) dxdy=\int\left(\int f(x,y)dx\right)dy$$ as long as the integrand is sufficiently convergent. I am not sure if this is the case at hand, but I will assume it nonetheless.

In practise, this means that we can replace FourierTransform[f[a,b],{a,b},{x,y}] with FourierTransform[FourierTransform[f[a,b],a,x],b,y]. To simplify the calculations further, I will insert a FullSimplify between the FourierTransforms so that the calculation is faster and simpler.

Timing[ FourierTransform[ FullSimplify[ FourierTransform[Sinc[a( x-y)],x,t,FourierParameters->{1,-1}] ,{t,y,a}\[Element]Reals ],y,s,FourierParameters->{1,-1} ] ]

yields

{0.039023,(\[Pi]^2 DiracDelta[s+t] (Sign[a-t]+Sign[a+t]))/a}

on my laptop.

• Thanks a lot for this simple and smart answer. This is what I need. – user14634 Apr 25 '19 at 22:34

Still not a full answer, but maybe pointing towards the solution.

PiecewiseExpand[
FourierTransform[
Sinc[ω1 - ω2],
{ω1, ω2}, {t1, t2},
FourierParameters -> {1, -1}
],
t1 ∈ Reals
] This does not work with Sinc[b (ω1 - ω2)], though and I don't know why. It should only be a matter of the transformation formula, but apparently, Mathematica is not clever enough to apply it automatically.

• Thanks a lot for this answer. I aggree with you that Mathematica is not clever enough to sovle this problem. – user14634 Apr 24 '19 at 14:17

Starting with the previous result as $$f$$ and using Simplify we can obtain

f = π/b I DiracDelta[t1 + t2]
(-Log[I b - I t1] + Log[-I b + I t1] + Log[-I (b + t1)] - Log[I (b + t1)]);
Simplify[f, Assumptions -> 0 < b < t1]
Simplify[f, Assumptions -> -b < t1 < b]
Simplify[f, Assumptions -> t1 < -b < 0]

(*
0
(2 π^2 DiracDelta[t1 + t2])/b
0
*)


However, we get another factor of 2 when we evaluate

FourierTransform[ Sinc[b (ω1 - ω2)], {ω1, ω2}, {t1, t2},
FourierParameters -> {1, -1}, Assumptions -> -b < t1 < b]

(*   (4 π^2 DiracDelta[t1 + t2])/b   *)

• Thanks a lot for this answer. – user14634 Apr 25 '19 at 22:31

The problem becomes simpler when you do a unitary variable substitution:

$$s = \frac{t_1+t_2}{\sqrt{2}} \qquad \sigma = \frac{t_1-t_2}{\sqrt{2}}\\ x = \frac{\omega_1+\omega_2}{\sqrt{2}} \qquad y = \frac{\omega_1-\omega_2}{\sqrt{2}}\\$$

The double Fourier integral then separates into two easy ones:

$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\text{sinc}[b(\omega_1-\omega_2)]e^{-i\omega_1t_1}e^{-i\omega_2t_2}d\omega_1d\omega_2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\text{sinc}(\sqrt{2}by)e^{-isx}e^{-ity}dx\,dy\\ = \left( \int_{-\infty}^{\infty} e^{-isx} dx\right) \left( \int_{-\infty}^{\infty} \text{sinc}(\sqrt{2}by)e^{-ity}dy\right)$$

These we can do with Mathematica:

Sqrt[2π]*InverseFourierTransform[1, x, s]
(* 2π*DiracDelta[s] *)

Sqrt[2π]*InverseFourierTransform[Sinc[Sqrt*b*y], y, t] // Simplify

(* π*(Sign[Sqrt*b-t] + Sign[Sqrt*b+t])/(2*Sqrt*b) *)


Hence the result is

$$2\pi\delta(s) \cdot \frac{\pi[\text{sign}(\sqrt{2}b-t)+\text{sign}(\sqrt{2}b+t)]}{2\sqrt{2}b}\\ = \frac{\pi^2}{\sqrt{2}b}\delta\left(\frac{t_1+t_2}{\sqrt{2}}\right)\left[\text{sign}\left(\sqrt{2}b-\frac{t_1-t_2}{\sqrt{2}}\right)+\text{sign}\left(\sqrt{2}b+\frac{t_1-t_2}{\sqrt{2}}\right)\right]\\ = \frac{\pi^2}{b}\delta(t_1+t_2)\left[\text{sign}\left(b-\frac{t_1-t_2}{2}\right)+\text{sign}\left(b+\frac{t_1-t_2}{2}\right)\right]\\ = \frac{\pi^2}{b}\delta(t_1+t_2)\left[\text{sign}(b-t_1)+\text{sign}(b+t_1)\right]$$

where the last step uses the $$\delta$$-function to constrain $$t_2=-t_1$$ and thus to simplify the $$\text{sign}$$-functions' arguments with $$\frac{t_1-t_2}{2}=t_1$$.

If you're not too picky about the boundaries where $$t_1=\pm b$$, then the second half can be written as twice the UnitBox function $$\Pi$$:

$$\ldots = \frac{2\pi^2}{b}\delta(t_1+t_2)\cdot \Pi\left(\frac{t_1}{2b}\right)$$