My notes on Fourier transform and Fast Fourier transform

The movement of a point in a unit circle with constant frequency \omega is given by cos(\omega)+ \iota sin(\omega). Fourier transform gives the energy of a signal at a certain frequency.

Helpful pages:

1. www.thefouriertransform.com

4. cygnus.ocean.dal.ca/public/klouden/ERTH4470/Intro2FTrev2012.pdf

5. http://blog.matthen.com/post/111638712111/how-a-series-of-waves-with-increasing-frequencies

6. http://physics.stackexchange.com/questions/202/fourier-transformation-in-nature-natural-physics

7. http://wiki.seg.org/wiki/Spatial_aliasing

8. https://www.youtube.com/watch?v=1JnayXHhjlg

Good parts:

http://physics.stackexchange.com/questions/202/fourier-transformation-in-nature-natural-physics

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.

http://www.thefouriertransform.com/transform/fourier.php

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

  1. http://www.robots.ox.ac.uk/~sjrob/Teaching/SP/l7.pdf

Screenshot 2017-04-17 01.20.57

Screenshot 2017-04-17 01.18.31

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 \Delta t=\frac{T_d}{N} while the sampling frequency or samples per seconds is given by f_s=\frac{1}{T_d} which also stands for bidirectional bandwidth. Therefore, \Delta f =\frac{f_s}{N}. Since the maximum frequency that we can display in the time domain is f_s, 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

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