Activity 5: Physical Measurements from Discrete fourier transform

For 1 dimension, the discrete fourier transform (DFT) of a set of N numbers xn is given by (taken from http://en.wikipedia.org/wiki/Discrete_Fourier_transform)

This function transforms the set of N Nnumbers xn usually in the time domain into a set of N numbers Xk in the frequency domain. The DFT of a set of numbers can be obtained using the fft(x) function of scilab.

Two dimensional DFT (i.e. for images) can be obtained using the following formula:
This function transforms a set of NxM numbers xn,m in the spatial domain to a set of numbers Xk,j in the frequency domain. Using scilab the function fft2(image) performs fourier transforms along the row and performs fft along the columns of the result.

Answers to questions:

a. The maximum sampling interval is give by
so if Fmax = 120Hz the maximum sampling interval will be 240Hz.

b. Increasing the number of samples would also result in a bigger axis since your minimum and maximum values will both increase this would mean that it can now detect more frequencies.
This is obvious in the following plots for N =100, 200 and 500 respectively,

c. Decreasing the sampling interval would result in an increase in Fmax as given by Nyquist Theorem this in turn would result in worse resolution than that for higher sampling interval since this would also increase df.
For the following plots, I changed T from 3, 1, and 0.5 respectively.


d. Changing N while keeping T constant would have the same result as b.


For this activity I will give myself a grade of 10 because I believe I have answered the questions adequately and I was able to verify my answers using Scilab.

Collaborators:
Eduardo David
Abraham Latimer Camba