1.
0 Ai
(¹
0 or ¥)
if i = j
A set of functions {F_{i}(t)}
is said to be orthogonal over an
interval [a,b] iff
2. A set of functions is said to be complete over an interval [a,b] if any waveform with a finite number of discontinuities on [a,b] can be expressed as a linear combination of members of that set.
Given: {F_{i}(t)} is a complete, orthogonal set on [a,b],
, and
f(t) has a finite number of
discontinuities on [a,b].
Then: over [a,b]
Where:
Since {F_{i}(t)} is a complete set on [a,b]
over [a,b]
Multiplying both sides of the equation by yields
Now integrating both sides over [a,b]
Reversing the order of integration and summation (linear operators) and moving the constants out of the integral leaves
But 0 for i ¹ j so we are left with only the i^{th} term of the sum
Solving for K_{i}
Q. E. D.
{F_{i}(t)} 
[a,b] 
{ 1, sin(nw_{0}t), cos(nw_{0}t) } ( 1 ³ n ³ ¥ ) 
[ T/2, T/2] ( T = 2p/w_{0} ) 
{ exp(jnw_{0}t) } (  ¥ ³ n ³ ¥ ) 
[ T/2, T/2] ( T = 2p/w_{0} ) 
P_{n}(t) – The Legendre Polynomials 
[ 1, 1 ] 
 The 0^{th}
Order Bessel Function r_{0} = 2.41, r_{1} = 5.51, r_{2} = 8.64, r_{3} = 11.79, …


f(t) = (A/T)*t
t in [0, T)
Use { 1, sin(nw_{0}t), cos(nw_{0}t) } (w_{0} = 2*p/T)
0 0
First find a_{0}
note: the second and third integrals are zero since the area under a sine or cosine over a integer number of cycles is always zero. They have equal areas above and below the axis.
Solving for a_{0}
a_{0} = A/2 Note: This is the average or
DC value of the waveform
Now
find b_{n} for n ³
1
again
Now multiply both sides by and integrate over the interval
The first integral is zero as before; the second is also zero since
HW: check it out
The third integral uses another trigonometric identity
The (x+y) terms again all yield zero and all of the (xy) terms are also zero except for the one where m=n. This yields
But cos(0) = 1 and the integral becomes T and we have
or
Integrating this requires the use of integration by parts
We want to get rid of the t in the integral so let u = t and dv be the rest. This yields
The integral is once again zero for the usual reason. The first term becomes
Simplifying further
or
HW: Show that the a_{n} are all zero for
n ¹ 0
Euler’s Formula
This also means that
The unit circle represents exp(x) as a unit vector of magnitude 1 and angle q
sin(x) cos(x) 1
q
The complex exponential is therefore closely related to the Phasor analysis used in AC circuits and
HW: find the complex exponential form for the Fourier Series of the sawtooth example function.
Note: the answer should be consistent with
Note: Both the SineCosine and Complex Exponential sets consist of members that are periodic functions. Each of their members has a frequency, which is a multiple of the fundamental frequency that has a period, T, equal to the length of the region of orthogonality. Each element is also periodic with period T and therefore their sum is also periodic.
Although we made no general statement in the theory about the behavior of the original function outside of the region of orthogonality, the Fourier series will only represent a periodic function. It, however, will represent a periodic function over all time.
Further note: at discontinuities, there will be some residual error