Fourier Analysis - Florida State University傅里叶分析-佛罗里达州立大学

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.
a period of 2p .
To determine the coefficients, we need to multiply the equation by either cos(mx) or sin(mx) and recognize that cos(mx) cos(nx) 1 [cos(n m )x cos(n m )x]
First developed by the French mathematician Joseph Fourier (1768-1830)
Fourier Series
Why using Fourier analysis? Analyze the phenomena from timebasis 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.
b m
1 p
p
f ( x ) sin(m x )dx, m 1, 2
p
In sum m ary, the E uler coefficients of a periodic function are:
a 0
1 2p
p
f (x)dx,
p
a n
1 p
p
f ( x ) cos(nx )dx, n 1, 2
n 1
To represent a function f(x) by the Fourier series, we need
to find the Fourier coefficients a0 , a1, b1, a2... and make sure that the sum of the series converges
p
b n
1 p
p
f ( x ) sin(nx )dx, n 1, 2
p
C hapter 10.2 in textbook (pp. 529-537)
a m
1 p
p
f(x)cos(m x)dx , m 1, 2,
p
E x :
a5
1 p
p
f(x)cos(5x)dx
p
Sim ilar practice can be used to determ in e the E uler coefficients
for the sine term s such that:
p
p
f(x)dx=
-p
p {a0
[an cos(nx ) bn sin(nx )]}dx
n 1
p
p
p f ( x )d x a 0 p d x a 0 ( 2p ),
a 0
1 2p
p
f (x)dx
p
A ll higher order terms vanish since they are all periodic with
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)
2 cos(m x) sin(nx) 1 [sin(n m ) x sin(n m ) x]
2
The new functions are still periodic with a period of 2p ,
except when n=m. Therefore, integrating these terms
with the real function f(x) as n .
Read pp. 534-536 in text for proof of convergence.
Euler Formulas
Represent a periodic function f(x) of period 2p by the Fourier
[an
p cos(nx)cos(mx)dx bn
sin(nx) cos(mx)dx]
p
n1
All cos(nx)cos(mx) terms All sin(nx)cos(mx) terms vanish Vanish except when n=m
cos(mx) cos(nx) 1 [cos(n m )x cos(n m )x] 2
1 [cos(2mx) cos(0)] 1 1 cos(2mx)
2
22
The latter term will vanish when integrating between -p and p .
T herefore,
p f(x )c os(m x )d x 1
p
2
p
p amdx p am
between -p to p will lead to the disppearance of all
terms except the cosine termsdic, vanishes
f(x)cos(mx)dx= -p
p a0 cos(mx)dx
p
p
series:
f ( x ) a0 [an cos(nx ) bn sin(nx )]
n 1
Determine the Euler coefficients by integrating both sides from
-p to p , taking advantage of the fact that a periodic function with a 2p period will vanish when integrated within this range.
A periodic function f(x) has a period of p is f(x)=f(x+p), 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
The Fourier series is the sum of the trigonometric terms:
a0 a1 cos( x ) b1 sin( x ) a2 cos(2 x ) b2 sin(2 x ) ... or
a0 [an cos(nx ) bn sin(nx )]