# Fourier

## 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

1. Transforms a sequence of $N$ complex numbers $x_0, \ldots, x_{N-1}$
2. into another sequence of complex numbers $X_0, \ldots, X_{N-1}$
3. defined by $X_k = \sum_{n=0}^{N-1} x_n e^{-2\pi i p_n \omega_k}$
4. for $\quad 0 \leq k \leq N-1,$

## 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.