The time shift property of the Fourier transform is defined as follows:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]x(n - n_o ) \Leftrightarrow e^{ - j\omega n_o } X(e^{j\omega } )[/itex]

I am confused by this notation....what does [itex]X(e^{j\omega } )[/itex] mean? I know that [itex]X(\omega)[/itex] is the value of the Fourier transform at a given angular frequency but I'm confused why it has been put in a complex exponent.

It is also sometimes written as:

[itex]h(x) = f(x - x_0)[/itex]

[itex]\hat{h}(\xi)= e^{-2\pi i x_0\xi }\hat{f}(\xi)[/itex]

This notation I think I understand...it's saying that, if I have the DFT of [itex]f(x)[/itex], then I can get the DFT for [itex]f(x - x_0)[/itex] by multiplying each point in the DFT by a scale factor that depends on the frequency. Specifically, from Euler's relation, I should scale the real/complex part by,

[itex]\cos(-2 \pi x_0 \xi ) + i \sin(2 \pi x_0 \xi )[/itex].

I tested this out by constructing a DFT that has a single spike, then taking the inverse FFT to reconstruct a time signal...and scaling the DFT spike and reconstructing again to see if I got a time shifted signal. I did not. When I thought about it more, I realized that this doesn't make sense, because some time shifts would result in multiplication by zero, which means that a shift followed by a negative shift could result in the signal being destroyed.

What have I got wrong?

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# Fourier time shift

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