Trigonometric Fourier Series

According to the Fourier theorem, any practical periodic function of frequencyω0 can be expressed as an infinite sum of sine or cosine functions. A periodic function satisfies:

f(t) = f(t +nT), where n is an integer and T is the period of the function

Suppose we have periodic signal f(t). The Fourier Theorem says that any practical periodic function can be expressed as a sum of sinusoidal functions of it's fundamental frequency wo.

• For a function to be represented by a Fourier series, it must meet the Dirichlet conditions:

  1. f(t) is single-valued everywhere (normal function).

  2. f(t) has a finite number of finite discontinuities each period (not too many breaks).

  3. f(t) has a finite number of maxima and minima in each period (not too jumpy).

  4. The integral ∫|𝑓(𝑡)|𝑑𝑡 < ∞ for any interval of one period (doesn't blow up)

Trigonometric Fourier Series

We need to basically solve a0, an, and bn

Example1:

Determine the Fourier series of the waveform shown below.

This is a periodic signal so f(t) = f(t +T) , where T = 2 in this case.

Use the formulas and wo = 2pi/T to solve a0, an, and bn

We have solved a0, an and bn so the solution is:

The more f(t) terms you add together , the closer the Fourier approximations gets to the original signal. When the sum goes to inf, its should exactly be the original signal. Below shows a plot of the first 11 terms of the Fourier series we solved.

Even and Odd signals

• A signal is Even if x(t) = x(-t)

• A signal is Odd if x(t) = -x(-t)

Note that cos is an even signal and sin is an odd signal

You can use this fact to quickly estimate some common functions.

• If f(t) is odd, then we expect zero pr little contribution from the cosine functions

• if f(t) is even then we expect zero or little contribution from the sine functions.