SKEDSOFT

Properties of the Fourier Series:

Basic properties of Fourier Series which are given as follows:

Linearity:

F is a linear transformation and so superposition holds. In other words, assume that a and b are simple real numbers, that x and y are T-periodic functions, and that

Symmetry:

1. Even waveforms:

A waveform x is even if x (t) = x (-t)

If x is also a T-periodic function and F (x) = X, then

for all k.

2. Odd Waveforms:

A waveform x is odd if x (t) = - x (-t)

If x is also a T-periodic function and F (x) = X, then

3. Real Waveforms:

If x is a real value T-periodic function and F (x) = X then

X_k = X_-k * for all k. In other word, X_k equals the complex of X_-k. As a result part of X is even and the imaginary part is odd.

Differentiation:

Let x be a T-periodic functions and

for all t. If F (x) = X and F (y) = Y, then

for all k.

Integration:

Let x be a T-periodic functions and

for all t. If F (x) = X and F (y) = Y, then

For all k. In this case X₀ must be zero, otherwise the integral would not be a periodic function and its Fourier series would not exist.