Signal analysis

Probability theory and transform analysis techniques form the backbone of all communication theory.

A function can be represented approximately over a given interval vy a linear combination of members of an orthogonal set of functions.

s(t)≈∑_(n=-∞)^∞▒〖C_n g_n (t)〗

A orthogonal set of functions is a set with the property that a particular operation performed between any two distinct members of the set yields zero. For example, The dot product of any two vectors is zero.

You have learned that vectors are orthogonal if they are at right angles to each other.
This means that one vector has nothing in common with the other.
The projection of one vector onto another is zero.
A function can be considered an infinite-dimensional vector ( think of forming a sequence by sampling the function ), so the concepts from vector spaces have direct application to function spaces.

One such set of orthogonal functions is the set of harmonically related sines and cosines. That is, the functions
sin⁡〖2πf_0 t,〗 sin⁡〖4πf_0 t,〗 sin⁡〖6πf_0 t,…〗
cos⁡〖2πf_0 t,〗 cos⁡〖4πf_0 t,〗 cos⁡〖6πf_0 t,…〗
Form an orthogonal set for any choice of f_0. These functions are orthogonal over the interval between any starting point t_0. and t_0 + 1/f_0.