Discrete Fourier Transform: How does independent varialbe spacing change?

[4ierTrans4m3]

Hey guys,

I was imagining that I have a sine function: y = sin(x) where x represents a distance in meters for instance. Now let us say that I sample the function at x = 0,1,2,3...,10 (meters) producing a list of values: {sin(1), sin(2), sin(3),...,sin(10)} = {0.000, 0.841, 0.909, 0.141, -0.756, -0.958, -0.279, 0.656, 0.989, 0.412, -0.544}. Obviously I know that the "spacing" between each of these consecutive points would be 1 meter.

Now I take the DFT using Mathematica (using standard Fourier parameters), to get:

Fourier[{sin(1), sin(2), sin(3),...,sin(10)}] = {0.425 + 0.000*I, 0.570 - 0.270*I, -0.860 + 1.098*I, -0.034 + 0.218*I, 0.045 + 0.095*I, 0.066 + 0.028*I, 0.066 - 0.028*I,
0.045 - 0.095*I, -0.034 - 0.218*I, -0.860 - 1.098*I, 0.570 + 0.270*I}

Which is now a list of 11 complex numbers. It is my understanding that the independent variable will now have units of (1/meter)? Which would be a wavenumber? Let us just call this new independent variable k. Here is my real question. Imagine that now I plot the absolute value of this list of complex values on one axis vs k on the other axis. What would be the "spacing" (along the k axis) between each consecutive point in my transformed list?

Thanks for any help

 

[mathman]

I don't fully understand your question. However, the argument of a trig function must be a non-dimensional number - it can't be in meters, without an appropriate 1/meter factor.

 

[4ierTrans4m3]

Is not the Fourier transform of something in the time domain something in the frequency domain? I know that it does not make sense to put numbers with units into the argument of the sine function. But just imagine that the x variable "represents" a distance even though it is a pure number.

 

[olivermsun]

Before transforming, you had N = 11 samples with spacing dx = 1 m, with a total interval L = N dx = 11 m.

An FFT will return 11 Fourier coefficients corresponding to a spacing: dk = 1 / L = 1 / 11 [cycles/m].

However, the wavenumbers will be numbered (since this is an odd FFT) like this:
(-5, -4, ..., -1, 0, 1, 2, ..., 5)/11 [cycles/meter].