Stuck on inverse fourier transform pair

[DragonPetter]

I have been trying to solve the inverse Fourier transform:

$\int_{-\infty}^{\infty}\left[e^{-j2\pi ft_0}e^{j\theta}\right]e^{j2\pi ft}df$

I know that the Fourier transform pair says

$e^{-j2\pi ft_0}e^{j\theta} \leftrightarrow \delta(t-t_0)$

but the extra phase term $e^{j\theta}$ makes it so I can't use this pair. Can I just consider it a constant? If so, then it gives me a weird time based function of an imaginary number.

 

[pmsrw3]

Yes, $e^{j\theta}$ is independent of f (unless there's something you're not telling us), so you can just pull it out of the integral. I don't understand why you say it gives you "a weird time based function of an imaginary number." It gives you $e^{j\theta}\delta\left(t-t_0\right)$, which is just the delta function multiplied by a phase factor (assuming theta is real), with maybe a factor of $\frac{1}{\sqrt{2\pi}}$ or something like that.

 

[DragonPetter]

Yes, $e^{j\theta}$ is independent of f (unless there's something you're not telling us), so you can just pull it out of the integral. I don't understand why you say it gives you "a weird time based function of an imaginary number." It gives you $e^{j\theta}\delta\left(t-t_0\right)$, which is just the delta function multiplied by a phase factor (assuming theta is real), with maybe a factor of $\frac{1}{\sqrt{2\pi}}$ or something like that.

Yes, but this phase factor has an imaginary component for any angle that is not a whole multiple of 0 or pi. I know what this phase means for my application, but I don't understand how a time function can have an imaginary component. In Euler's formula, there are imaginary components, but they always cancel out to a real value, and in this case they don't.

 

[pmsrw3]

Yes, but this phase factor has an imaginary component for any angle that is not a whole multiple of 0 or pi. I know what this phase means for my application, but I don't understand how a time function can have an imaginary component. In Euler's formula, there are imaginary components, but they always cancel out to a real value, and in this case they don't.

Why shouldn't a function of time have an imaginary part? It all depends on the application. Sometimes you're calculating something that, because of the physics it represents, must be a real number, and in those cases the imaginary parts must indeed cancel out. But you say that you know what the phase means for your application, which (correct me if I misunderstand) means that for your purposes, complex numbers do have a physical meaning. This is not uncommon, actually.

 

[blue_raver22]

I would suggest checking the properties of the Fourier transform to see if there is a way to handle the extra phase term. It is possible that there is a property or formula that can help simplify the problem. If not, then considering the extra phase term as a constant could be a valid approach, but it may lead to a complex-valued function as you mentioned. In this case, it may be helpful to plot the function to gain a better understanding of its behavior. Additionally, consulting with a colleague or referencing a textbook or online resource may provide further insight and assistance in solving the inverse Fourier transform.