Fourier transform of exponential function

Question

It is a function $A(f,t)= e^{j 2 \pi (t/a-af)}$ I would like to take the inverse Fourier transform.

So:

$B(\tau,t)=\int A(f,t) e^{j2\pi f\tau}df=\delta(\tau-t_0)...???$

How to solve this integral? Or should I use $e^{j 2 \pi (t/a)}$ as constant?

Answer 1

Just recognize that, I think, $$ \mathcal{F}^{-1}_{f \to t'} \exp \left[ 2\pi j \left( \frac{t}{a}-af \right) \right] = e^{2\pi j t/a} \delta ( t'-a ) $$ if $t$ does not depend on $f$. Also, there are several different conventions on FT, you have to specify, but usually they differ up to a constant multiple. (See Wikipedia formula 102, 302)

Answer 2

Yes that's right and you should consider $t$ as constant and define a new, independent variable $\tau$ for time. The model in which the Fourier transform changes with some other variable ($t$ in this case) can be used for example to model a time-variant communication channel. In that case, even the channel is modeled stochastically with its FT dependent to time. In that case, the power spectral density is used as the FT of stochastic channel response and therefore varies with time, i.e. $$h(\tau;t)=\text{channel response varying over time}\\S(f;t)=\int E\{h(\tau+\tau_1;t)h(\tau_1;t)\}e^{-i2\pi f\tau_1}d\tau_1$$where $E\{\cdot\}$ stands for mathematical expectation and channel is assumed to be a stationary process.