Fourier time shift

[junglebeast]

The time shift property of the Fourier transform is defined as follows:

x(n - n_o ) \Leftrightarrow e^{ - j\omega n_o } X(e^{j\omega } )

I am confused by this notation....what does X(e^{j\omega } ) mean? I know that X(\omega) 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:

h(x) = f(x - x_0)
\hat{h}(\xi)= e^{-2\pi i x_0\xi }\hat{f}(\xi)

This notation I think I understand...it's saying that, if I have the DFT of f(x), then I can get the DFT for f(x - x_0) 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,

\cos(-2 \pi x_0 \xi ) + i \sin(2 \pi x_0 \xi ).

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?

[Redbelly98]

The time shift property of the Fourier transform is defined as follows:

x(n - n_o ) \Leftrightarrow e^{ - j\omega n_o } X(e^{j\omega } )

I am confused by this notation....what does X(e^{j\omega } ) mean?
Looks like a typo to me, I think it should be simply X(ω). Here's another link, they do have F(ω) (different notation):
http://cnx.org/content/m10100/latest/

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?
Not sure what is going on with your test. You might try starting in the time domain, with both a spike at t=0 and also a time-shifted spike. Take the FFT of both and compare.

BTW, if you want to represent purely real time-domain signals, the frequency-domain should have the property

X(-ω) = X*(ω),

where * denotes the complex conjugate. So the only way to have a single spike in the frequency domain is when that spike is at ω=0.

[junglebeast]

Alright, I found out where my confusion was...everything I said in my above post was correct, except for the place where I said "I tried this and it didn't work," because the reason it didn't work is I was doing the complex multiplication pointwise, and complex multiplication actually involves some addition! Now it all makes sense.


X(-ω) = X*(ω),

where * denotes the complex conjugate. So the only way to have a single spike in the frequency domain is when that spike is at ω=0.


Oh I was only referring to a single spike in the positive frequencies...because obviously the negative frequency range is just a mirror of the positive data as you point out.


I'm still confused about the X(e^jw) notation though...I've seen it in many different places so I don't think it's just a typo..

[Redbelly98]

I'm still confused about the X(e^jw) notation though...I've seen it in many different places so I don't think it's just a typo..
That is bizarre. If X is the signal in the frequency domain, the argument must be a real number ... this is the Fourier Transform, not Laplace, after all. I am equally baffled.