# Fourier

From Colettapedia

## Vocab

- signal - function in time
- musical chord can be expressed in terms ofthe volume and frequencies of constituent notes

- magnitude - amount of a given frequency present in the original signal
- phase offset of the basic sinusoid

## General

- Some differential equations are easier to analyze in the frequency domain. After performing the desired operations, transformation of the result can be made back to the time domain.
- Differentiation in time domain corresponds to multiplication in the frequency domain.
- Convolution is also a multiplication operation

- Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa
- The Fourier transform of a Gaussian function is another Gaussian function.
- The fourier transform of a function f is denoted by f hat.

## Types

### NFFT

- Transforms a sequence of complex numbers
- into another sequence of complex numbers
- defined by
- for

#### Implementations

## Harmonics and Harmonic Analysis

- Nyquist sampling theorem - can only measure frequencies that are half the sampling rate. E.g., to detect a 10 kHz signal you have to sample at 20kHz or else you'll get aliasing/Moire pattern/beat phenomenon in wave interference.
- Harmonics - The Scientist and Engineer's Guide to Digital Signal Processing, chapter 11 - NICE FREE BOOK!
- Signal is periodic with frequency f
- The frequencies composing the signal are integer multiples of f, i.e., f, 2f, 3f, etc.
- The first harmonic is f, the second harmonic is 2f
- The first harmonic is given a special name - the fundamental frequency
- If the signal is symmetrical (Peak and trough of sine wave are mirror images of each other across horizontal axis) THE EVEN HARMONICS WILL HAVE A VALUE OF ZERO. In other words, ASYMMETRICAL distortion of signal produces fundamental plus BOTH even and odd harmonics.

## See Also

- Harmonic analysis - a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves.
- Convolution - a mathematical operation on two functions (f and g) that produces a third function expressing how the shape of one is modified by the other.
- Z-transform - Not to be confused with a z-score from statistics