Generalized Fourier Series





Ai  (¹ 0 or ¥) if i = j

A set of functions {Fi(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.

Theorem: The Fourier Series

Given: {Fi(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]



Since {Fi(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 ith term of the sum

Solving for Ki



Q. E. D.

Examples of Complete, Orthogonal Sets



{ 1, sin(nw0t), cos(nw0t) }  ( 1 ³ n ³ ¥ )

[ -T/2, T/2] ( T = 2p/w0 )

{ exp(jnw0t) }  ( - ¥ ³ n ³ ¥ )

[ -T/2, T/2] ( T = 2p/w0 )

Pn(t) – The Legendre Polynomials
P0(t) = 1
P1(t)= t
P2(t)= ½( 3t2 – 1)
P3(t)= ½( 5t3 – 3t) …

[ -1, 1 ]

 - The 0th Order Bessel Function
rn is the nth root of J0(t) = 0

r0 = 2.41, r1 = 5.51, r2 = 8.64, r3 = 11.79, …

[0, a]


Example: The Sawtooth or Sweep function


f(t) = (A/T)*t

t in [0, T)










Use { 1, sin(nw0t), cos(nw0t) }  (w0 = 2*p/T)





First find a0

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 a0



             a0  =  A/2                    Note: This is the average or DC value of the waveform

Now find bn for n ³ 1


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 (x-y) 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




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



HW:  Show that the an are all zero for n ¹ 0

The Complex Exponential Series: Another form of the Sine-Cosine Series

Euler’s Formula


This also means that


The unit circle represents exp(x) as a unit vector of magnitude 1 and angle 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 Sine-Cosine 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