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Solving the Fractional Fourier Integral Transform of $e^{j omega_{0} t}$

Mathematics Asked by The Dude on December 15, 2020

I want to solve the integral:

begin{align}
F_{alpha}(omega) = frac{1}{sqrt{2 pi}} cdot sqrt{1 – j text{cot}(alpha)} cdot e^{j frac{1}{2} text{cot}(alpha) omega^{2} } int_{-infty}^{infty} e^{-j text{csc}(alpha) omega t + j frac{1}{2} text{cot}(alpha)t^{2} } e^{ j omega_{0} t} dt
end{align}

where $j$ is the imaginary number. The problem is, the integral I get is not the answer I am expecting.


Worked Solution:

Combining the exponents gives us:
begin{align}
F_{alpha}(omega) = frac{1}{sqrt{2 pi}} cdot sqrt{1 – j text{cot}(alpha)} cdot e^{j frac{1}{2} text{cot}(alpha) omega^{2} } int_{-infty}^{infty} e^{- frac{1}{2} left[ -j text{cot}(alpha)t^{2} + 2j( omegatext{csc}(alpha) – omega_{0} ) t right]} dt
end{align}

Completing the square gives us:
begin{align}
F_{alpha}(omega) &= frac{1}{sqrt{2 pi}} cdot sqrt{1 – j text{cot}(alpha)} cdot e^{j frac{1}{2} text{cot}(alpha) omega^{2} } int_{-infty}^{infty} e^{-frac{1}{2} left[ -jtext{cot}(alpha) left( t + frac{omega_{0} – omega text{csc}(alpha)}{text{cot}(alpha)} right)^{2} + j frac{left( omega text{csc}(alpha) – omega_{0} right)^{2}}{text{cot}(alpha)} right] } dt \
F_{alpha}(omega) &= frac{1}{sqrt{2 pi}} cdot sqrt{1 – j text{cot}(alpha)} cdot e^{j frac{1}{2} text{cot}(alpha) omega^{2} } e^{-j frac{1}{2} frac{left( omega text{csc}(alpha) – omega_{0} right)^{2}}{text{cot}(alpha)}} int_{-infty}^{infty} e^{-frac{1}{2} left[ -jtext{cot}(alpha) left( t + frac{omega_{0} – omega text{csc}(alpha)}{text{cot}(alpha)} right)^{2} right] } dt end{align}

Now we let $z = t + frac{omega_{0} – omega text{csc}(alpha)}{text{cot}(alpha)} $, which means $dz = dt$, and the bounds don’t change. This means we have:
begin{align}
F_{alpha}(omega) &= frac{1}{sqrt{2 pi}} cdot sqrt{1 – j text{cot}(alpha)} cdot e^{j frac{1}{2} text{cot}(alpha) omega^{2} } e^{-j frac{1}{2} frac{left( omega text{csc}(alpha) – omega_{0} right)^{2}}{text{cot}(alpha)}} int_{-infty}^{infty} e^{j frac{1}{2} text{cot}(alpha) z^{2} } dz
end{align}

Well this is just a Gaussian integral, so we have:
begin{align}
F_{alpha}(omega) &= frac{1}{sqrt{2 pi}} cdot sqrt{1 – j text{cot}(alpha)} cdot e^{j frac{1}{2} text{cot}(alpha) omega^{2} } e^{-j frac{1}{2} frac{left( omega text{csc}(alpha) – omega_{0} right)^{2}}{text{cot}(alpha)}} sqrt{frac{2 pi}{-j text{cot}(alpha)}} \
F_{alpha}(omega) &= sqrt{ frac{ 1 – j text{cot}(alpha)}{-j text{cot}(alpha)} } cdot e^{j frac{1}{2} text{cot}(alpha) omega^{2} } e^{-j frac{1}{2} frac{left( omega text{csc}(alpha) – omega_{0} right)^{2}}{text{cot}(alpha)}}
end{align}

Now we will work on simplifiying this expression. We start with the radical:
begin{align}
F_{alpha}(omega) &= sqrt{ 1 + j text{tan}(alpha) } cdot e^{j frac{1}{2} text{cot}(alpha) omega^{2} } e^{-j frac{1}{2} frac{left( omega text{csc}(alpha) – omega_{0} right)^{2}}{text{cot}(alpha)}}
end{align}

Now we combine the exponentials:
begin{align}
F_{alpha}(omega) &= sqrt{ 1 + j text{tan}(alpha) } cdot e^{j frac{1}{2} left[ text{cot}(alpha) omega^{2} – frac{left( omega text{csc}(alpha) – omega_{0} right)^{2}}{text{cot}(alpha)} right]}
end{align}

Now lets expand the exponent:
begin{align}
text{cot}(alpha) omega^{2} – frac{left( omega text{csc}(alpha) – omega_{0} right)^{2}}{text{cot}(alpha)} & = text{cot}(alpha) omega^{2} – frac{1}{text{cot}(alpha)} cdot left( omega^{2} text{csc}^{2}(alpha) – 2 omega_{0} omega text{csc}(alpha) + omega_{0}^{2} right)\
&= frac{text{cot}^{2}(alpha) – text{csc}^{2}(alpha)}{text{cot}(alpha)} omega^{2} + 2 frac{text{csc}(alpha)}{text{cot}(alpha)} omega_{0} omega – frac{1}{text{cot}(alpha)} omega_{0}^{2}
end{align}

Since $text{cot}(alpha) = frac{1}{text{tan}(alpha)} = frac{text{cos}(alpha)}{text{sin}(alpha)}$, and $text{csc}(alpha) = frac{1}{text{sin}(alpha)}$, we have:
begin{align}
&= left( text{cot}(alpha) – text{tan}(alpha) right) omega^{2} + 2 text{sec}(alpha) omega_{0} omega – text{tan}(alpha) omega_{0}^{2} \
&= -(omega^{2} + omega_{0}^{2}) text{tan}(alpha) + 2 text{sec}(alpha) omega_{0} omega + text{cot}(alpha) omega^{2} \
end{align}

So we have:
begin{align}
boxed{ F_{alpha}(omega) = sqrt{ 1 + j text{tan}(alpha) } cdot e^{jfrac{1}{2} text{cot}(alpha) omega^{2}} e^{-j frac{1}{2} (omega^{2} + omega_{0}^{2}) text{tan}(alpha) + j text{sec}(alpha) omega_{0} omega} }
end{align}


The Problem:

According to the paper "The Fractional Fourier Transform and Time-Frequency Representations" by Luis B Almeida, the answer should be:

begin{align}
boxed{ F_{alpha}(omega) = sqrt{ 1 + j text{tan}(alpha) } e^{-j frac{1}{2} (omega^{2} + omega_{0}^{2}) text{tan}(alpha) + j text{sec}(alpha) omega_{0} omega} }
end{align}

In other words, my answer has an extra $e^{-jfrac{1}{2} text{cot}(alpha) omega^{2}}$ term.


Question:

What have I done wrong? Am I correct, or is the paper incorrect?


Edit 1: I have tried plugging it into Mathematica, but it is of no help, even after trying all the simplify commands.

The Mathematica Answer:

(0.707107 E^((0. + 0.5 I) w^2 Cot[a] - (0. + 0.5 I) (1. v - 1. wCsc[a])^2 Tan[a]) Sqrt[1 - I Cot[a]])/Sqrt[(0. - 0.5 I) Cot[a]]

which is different from everyone’s….


Edit 2: Here is another twist in the saga.

In the paper, it says that Almeida’s result holds if $alpha – frac{pi}{2} neq n cdot pi$ where $n in mathbb{Z}$

However, when $alpha = n cdot pi + frac{pi}{2} = frac{2n+1}{2} cdot pi$, then $text{cot}(alpha) = text{cot}left( frac{2n+1}{2} cdot pi right) = 0$, so that extra complex exponential term is 1, which makes the my results agree with Almeida’s.

What in the world??

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