- #1

- 44

- 0

## Main Question or Discussion Point

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

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

[itex]f(\lambda) = e^{-2i\pi\lambda d}[/itex]

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

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 [itex]f(x)[/itex] be a signal function supported on [itex][-a,a][/itex]. Then,

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

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

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

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

[itex]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 [/itex]. But I can't see how [itex]|f(t)|[/itex] is maximized at [itex]t=d[/itex], as it becomes the integral of 1.

Am I missing something?

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

[itex]f(\lambda) = e^{-2i\pi\lambda d}[/itex]

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

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 [itex]f(x)[/itex] be a signal function supported on [itex][-a,a][/itex]. Then,

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

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

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

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

[itex]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 [/itex]. But I can't see how [itex]|f(t)|[/itex] is maximized at [itex]t=d[/itex], as it becomes the integral of 1.

Am I missing something?