The movement of a point in a unit circle with constant frequency is given by . Fourier transform gives the energy of a signal at a certain frequency.
A piano acts as a fourier analyzer. Make a sound near the piano. some strings will vibrate more and others less. the more a string vibrates, the more of its fundamental frequency is in the sound.
Figure 3. The Box Function with T=10, and its Fourier Transform.
Figure 4. The Box Function with T=1, and its Fourier Transform.
“A fundamental lesson can be learned from Figures 3 and 4. From Figure 3, note that the wider square pulse produces a narrower, more constrained spectrum (the Fourier Transform). From Figure 4, observe that the thinner square pulse produces a wider spectrum (More amount of frequencies contributing effectively) than in Figure 3. This fact will hold in general: rapidly changing functions require more high frequency content (as in Figure 4). Functions that are moving more slowly in time will have less high frequency energy (as in Figure 3). ”
Discrete Fourier Transform
Fast Fourier Transform
Fast Fourier transform is the fast computation algorithm for Discrete Fourier Transform. Array of time-domain waveform samples (real valued) are transformed to array of frequency-domain spectrum samples (complex valued) using FFT. For the algorithm to work fast the length of the waveform sample must be a power of 2.
The sampling interval while the sampling frequency or samples per seconds is given by which also stands for bidirectional bandwidth. Therefore, . Since the maximum frequency that we can display in the time domain is , we typically display half of the frequency spectrum(lower half).
Spacial fourier transform
A very good resource from page 8: http://www.robots.ox.ac.uk/~az/lectures/ia/lect2.pdf