Download Understanding Fourier Analysis: Decomposing Periodic Functions into Sinusoidal Components and more Slides Advanced Physics in PDF only on Docsity!
Deconstructing periodic driving voltages (or any functions)
into their sinusoidal components:
It's easy to build a periodic functions by choosing coefficients
a
n
and b
n
or amplitudes A
n and phases φ n
to build periodic
functions of any sort via
What is less obvious is how to take a given periodic function
and find out what coefficients went into making it! With a little
experience, you can develop some intuition that makes it
possible to solve simple cases, but we need an analytical
method to do it for ANY periodic function. The technique is
called Fourier analysis and it closely allied with projecting
vectors onto their basis vectors to find components. It is
heavily used in signal and image processing, and you will
encounter related techniques is used in quantum mechanics
when you work with the Schrödinger equation.
f (^) ( t ) = a n
cos n! t + b
n
sin n! t
n " or^ An cos^ (^ n^! t^ +^ # n ) n "
We will need to know many integrals of this kind, and we ll
discuss the prefactors and the word projection later.
2 T sin(! t ) dt 0 T " =
2
! T cos(! t ) 0 T =
1
$ %cos(^2 $)^ #^ cos^ ( 0 ) & ' ( = 0 T 4 T 2 3 T 4 T A Time, t
A
f t ,g t
More important integrals …..
2 T sin 2 (! t ) dt = 2 T 1 2 " 1 2
cos(^2! t )
$ % & dt 0 T ' 0 T ' = 1 T dt 0 T ' = 1 T 4 T 2 3 T 4 T A Time, t
A
f t g t
More important integrals …..
2 T sin(! t )cos(! t ) dt 0 T " = 1 T sin( 2! t ) dt 0 T " = 0 T 4 T 2 3 T 4 T A Time, t
A
f t ,g t T 4 T 2 3 T 4 T A Time, t
A
f t g t
Integrate the product of two harmonic functions with frequencies that are integer multiples of the fundamental over one period of the longer period function and you get …… Zero (0) if the two harmonic functions have different frequencies. The functions are said to be orthogonal.
Black = red x blue
T 4 T 2 3 T 4 T
A Time, t
A
f t ,g t T 4 T 2 3 T 4 T A Time, t
A
f t g t
Integrate the product of two harmonic functions with frequencies that are integer multiples of the fundamental over one period of the longer period function and you get …… Not zero if the two harmonic functions have the same frequency.
Black = red x blue (red and blue are same)
T 4 T 2 3 T 4 T
A Time, t
A
f t ,g t T 4 T 2 3 T 4 T A Time, t
A
f t g t
We can write this more elegantly using the Kronecker delta. δ
pq
=1 if p = q ; = 0 if p ≠ q. 2 T sin (^) ( p! t ) 0 T " sin (^) ( q! t ) dt = 0 if p # q (integers) 1 if p = q (integers) $ % & '& 2 T cos (^) ( p! t ) 0 T " cos (^) ( q! t ) dt = 0 if p # q (integers) 1 if p = q (integers) $ % & '& 2 T sin (^) ( p! t ) 0 T " sin (^) ( q! t ) dt = # pq 2 T co s (^) ( p! t ) 0 T " cos (^) ( q! t ) dt = # pq
Why do we call this integral a projection? What is projecting onto
what? And what does it all mean? Up till now we have been doing
this integral with f ( t ) having the form of a sinusoidal function with
a frequency that in an integer multiple of ω.
f ( t ) g (^) ( t ) dt 0 T ! ! A i ! B = A 1 B 1
The projection of a vector A = (A
1
, A
2
, A
3
…) onto a vector B =
(B
1
, B
2
, B
3
…) is found by multiplying corresponding components
and adding them.
Can you see the similarity to thinking of two functions of a
variable t as long vectors f ( t ) = ( f ( t
1
), f ( t
2
), f ( t
3
) …) and g ( t ) = ( g
( t
1
), g ( t
2
), g ( t
3
) …) and multiplying and "adding" (integrating)?
Thus in some sense, the integral above is a projection of f ( t )
onto g ( t ). But it s not quite right yet ….
In projection language, we can say that the set of functions cos
( n ω t) and sin(n ω t); n integer; form and ORTHONORMAL SET
or ORTHONORMAL BASIS just like the unit vectors x(hat), y
(hat) and z(hat) ( sometimes also called i, j and k).
And just as you can write any vector that lives in 3-d space as a
sum of multiples of vectors x(hat), y(hat) and z(hat),
so, too, can you write any function that lives in "periodic space"
as a a sum of multiples of cos( n ω t) and sin(n ω t).
More powerful, because of the orthonormal nature of the basis
functions, you can project out the coefficient of any basis
function by projecting the function onto that basis vector.
Let's do an example.
Example of sine series: Take the red sawtooth function Multiply it by the sine term whose coefficient you want to find ……… (here choose n =1) Integrate over 1 period of the fundamental. Multiply by 2/ T. That number is the coefficient of the sin(1ω t ) term.
T 4 T 2 3 T 4 T
A Time, t
A
f t ,g t 0 T 4 T 2 3 T 4 T A Time, t
A
f t g t
Take the function (example of sine series)……… Multiply it by the sine term whose coefficient you want to find ……… (here choose n =10) Integrate over 1 period of the fundamental. Multiply by 2/ T. That number is the coefficient of the sin(10 t ) term.
T 4 T 2 3 T 4 T
A Time, t
A
f t ,g t 0 T 4 T 2 3 T 4 T A Time, t
A
f t g t
Are we sure there is no cosine contribution? Multiply it by the cosine term whose coefficient you want to find ……… (here choose n =1) Integrate over 1 period of the fundamental. Multiply by 2/ T. That number is the coefficient of the cos(1 t ) term. (ZERO)
T 4 T 2 3 T 4 T
A Time, t
A
f t ,g t 0 T 4 T 2 3 T 4 T A Time, t
A
f t g t
ODD functions of t have the property f ( t ) = f ( t ). Their Fourier representation must also be in terms of odd functions, namely sines. b n = 2 T f ( t )sin (^) ( n! t ) dt 0 T " Suppose we have an odd periodic function f ( t ) like our sawtooth wave and you have to find its Fourier series b
n
sin (^) ( n! t ) n = 1 ,2... " Then the unknown coefficients can be evaluated this way the function the harmonic Integrate over the period of the fundamental normalize properly Here s the coefficient of the sin(ωnt) term! Plot it on your spectrum!
a n = 2 T f ( t )cos (^) ( n! t ) dt 0 T " Suppose we have an even periodic function f ( t ) and you have to find its Fourier series a 0 2
- a n cos (^) ( n! t ) n = 1 , 2 ... " Then the unknown coefficients can be evaluated this way the function the harmonic Integrate over the period of the fundamental normalize properly Here s the coefficient of the cos(ωnt) term! Plot it on your spectrum! EVEN functions of t have the property f ( t ) = + f ( t ). Their Fourier representation must also be in terms of even functions, namely cosines.
