Physical Measurements from Discrete Fourier Transforms

In this activity, we examined the Fourier transforms of signals. The first signal used was a 1-D sinusoidal function. The sine function and its Fourier transform is shown below.

To get a Fourier transform of an image, we just need to use the 2-D form of the DFT. Since the magnitude of the Fourier transform contains most of the geometric information of the spatial domain image, it was displayed as the final image.

To accurately measure the FT of a fluorescent bulb, which flickers at around 120Hz, we need at least dt = 1/240 sec. This was obtained using the Nyquist formula, fmax = 1/(2*dt), where fmax is the maximum frequency the FT can detect without aliasing. So setting fmax = 120Hz, we get dt = 1/240 sec.

Effects of N and dt in the frequency domain.
The discrete frequency steps (df) in the frequency domain is given by df = (2*fmax)/N. So if N is increased, the frequency domain would be much more sensitive to small frequencies and it would be more accurate since there is a smaller frequency samples.
From the plots obtained(left: N = 256, right N = 1024), it can be seen that higher N results in a more accurate(narrower) value in the frequency domain. Although both peaks at 5, which is the frequency of the signal used, with N = 256, the plot range is 4.5-5.5 while with N = 1024, it is 4.9-5.1.
Combining the Nyquist theorem with the discrete frequency equation, we get df = 1/(dt*N). So as dt approaches 0, df approaches infinity. Therefore, I expect that decreasing dt would result in a less accurate(broader) frequency plot.
The plots obtained(left: dt = 2/256, right: dt = 2/64) shows that the expected results were correct.

For this activity, I give myself a grade of 10 since I performed all of the required activities.