Inverse Fourier Transform of a function

[CantorSet]

Hi everyone, this is not a homework question but from my reading of a signals processing paper.

This paper says if f(t) is the inverse Fourier transform of a function

f(\lambda) = e^{-2i\pi\lambda d}

then we can "easily see" that f(t) will have a peak d.

Part of the issue here is my shaky of the Fourier transform, which up til this point, I understand as a frequency decomposition of a signal. That is, let f(x) be a signal function supported on [-a,a]. Then,

f(\lambda) = \int_{-a}^{a}f(x)e^{-2i\pi x\lambda}dx

is the Fourier transform with the property that |f(\lambda)| quantifies the "amount" of frequency \lambda in the original signal function f(x).

But returning to my original problem, if we take the inverse Fourier transform of

f(\lambda) = e^{-2i\pi\lambda d}, then we have

f(t) = \frac{1}{2\pi}\int e^{-2i\pi\lambda d} e^{2i\pi\lambda t} d \lambda = \frac{1}{2\pi}\int e^{2i\pi\lambda (t-d)} d \lambda. But I can't see how |f(t)| is maximized at t=d, as it becomes the integral of 1.

Am I missing something?

 

[HallsofIvy]

It might make more sense if you did not use "f" both for the function and its Fourier transform. And what are the limits of integration in the final integral?

 

[jackmell]

f(t) = \frac{1}{2\pi}\int e^{-2i\pi\lambda d} e^{2i\pi\lambda t} d \lambda = \frac{1}{2\pi}\int e^{2i\pi\lambda (t-d)} d \lambda. But I can't see how |f(t)| is maximized at t=d, as it becomes the integral of 1.

Am I missing something?

First evaluate the integral:

\int_{-a}^a e^{2\pi i \lambda(t-d)}d\lambda

and then take the limit as t goes to d.

 

[CantorSet]

First evaluate the integral:

\int_{-a}^a e^{2\pi i \lambda(t-d)}d\lambda

and then take the limit as t goes to d.

Oh, I see...

So we have

\int_{-a}^a e^{2\pi i \lambda(t-d)}d\lambda = \frac{Sin(2a\pi (d-t)}{\pi(d-t)}

So the function

\frac{Sin(2a\pi (d-t)}{\pi(d-t)}

achieves its max of \frac{2a}{\pi} when d = t.

Thanks for the help. By the way, was there an easy way to see this without having to integrate? The original function

f(\lambda) = e^{-2 i \pi \lambda d}

is a Fourier basis elements on L^2[-a,a].