Fourier Analysis

1
Fourier Analysis
Fourier Series A Fourier series is a
representation of a function using a series of
sinusoidal functions of different frequencies.
(Recall Taylor other power series expansions
in Calculus II) They are extremely useful to be
used to represent functions of phenomena that are
periodic in nature. Fourier Integral
Transform Similar to Fourier series but extend
its application to both periodic and non-periodic
functions and phenomena. First developed by the
French mathematician Joseph Fourier (1768-1830)
2
Fourier Series
Why using Fourier analysis? Analyze the
phenomena from time-basis to a frequency-basis
analysis. It is simpler to describe a periodic
function using Fourier description. Ex a
periodic time series can always be described by
using its frequency and amplitude. A periodic
function f(x) has a period of p is f(x)f(xp),
that is, the function repeats itself every
interval of length p Trigonometric series 1,
cos(x), sin(x), cos(2x), sin(2x), These
functions all have the period of 2p
3
(No Transcript)
4
Euler Formulas
5
(No Transcript)
6
Periodic, vanishes
All sin(nx)cos(mx) terms vanish
All cos(nx)cos(mx) terms Vanish except when nm
7
(No Transcript)
8
(No Transcript)