Many things in life are periodic. A function f (defined now for all real numbers) is "periodic" if there is a positive number p (called a "period" for f) such that f(t+p) = f(t) for all t. In that case f(t+kp) = f(t) for any integer k. A constant function is periodic (for any positive p). If f is periodic with period p then it is also periodic with period kp for any positive integer k. For example, cos(omega t), sin(omega t), and e^{i omega t) are all periodic with period 2pi/omega (or any integer multiple of this number). In particular, for any positive integer n, cos(nt), sin(nt), and e^{int}, are periodic with period 2pi. For this reason it is convenient to choose time units so that our waves have period 2pi.
We may study a periodic function f by choosing some starting time a and viewing f between a and a+p. The choice of a is essentially arbitrary. There are two standard choices in the trigonometric example: a = 0, and a = -pi. We'll follow the book in adopting the second choice.
If f and g are periodic with the same period p, then so is any linear combination af + bg. Thus we have a very large family of functions which are periodic of period 2pi: expressions of the form
(a0)/2 + a1 cos(t) + b1 sin(t) + a2 cos(2t) + b2 sin(2t) + ...
= (a0)/2 + sum_{n=1}^infinity (an cos(nt) + bn sin(nt))
This is a "Fourier series." (There are good reasons for writing the constant term as (a0)/2.) All reasonable functions with period 2pi are well-approximated by such functions, which are in turn well-approximated by a finite number of terms in the sum. Given f(t), we can work out what the numbers an and bn have to be. Since sin and cos integrate to zero over a period,
pi a0 = 2pi(a0)/2 = integral_{-pi}^pi f(t) dt,
so a0 is twice the average value of f over a period. The coefficient am can be gotten by computing that
pi am = integral_{-pi}^pi cos(mt) f(t) dt
by using the trigonometric integrals
integral_{-pi}^pi cos(mt) cos(nt) dt = 0 if m is not n
integral_{-pi}^pi cos(mt) cos(mt) dt = pi (m = 1, 2, 3, ...)
integral_{-pi}^pi cos(mt) sin(nt) dt = 0.
The last of these is easy: cos is even, sin is odd, even.odd is odd, and the integral of an odd function over a symmetric interval is zero.
Using these and
integral_{-pi}^pi sin(mt) sin(nt) dt = 0 if m is not n
integral_{-pi}^pi sin(mt) sin(mt) dt = pi (m = 1, 2, 3, ...)
we find
pi bm = integral_{-pi}^pi sin(mt) f(t) .
This is another "transform": we're representing f by means of the sequence of numbers a0, a1, b1, a2, b2, .... These are the "Fourier coefficients" of f.
For example let
sq(t) = -1 for t between -pi and 0 = 1 for t between 0 and pi.
- the square wave. (We don't care about the values sq(0) or sq(pi) since these won't affect the values of the Fourier coefficients.) This is an odd function. Since cos is even, the integral defining an is zero. For bm we compute:
pi bm = integral_{-pi}^pi sin(mt) sq(t) dt
= 2 integral_0^pi sin(mt) dt (since sin(mt) is odd)
= 2 [-cos(mt)/m]_0^pi
so pi bm = 4/m if m is odd,
= 0 if m is even.
sq(t) has Fourier series
(4/pi) sin(t) + (4/3pi) sin(3t) + (4/5pi) sin(5t) + ....
You'll see this numerically for homework. I displayed the sum of the first one, two, three, six, and fifty terms in this series.
A "pure tone" is a pure sine or cosine wave: a1, b1 are the only nonzero Fourier coefficients. The contributions for higher m are "overtones."