# What the terms "basis functions" and "orthogonal" denote in the case of signals?

I am a beginer. I have read that any given signal whether it is simple or complex one,can be represented as summation of orthogonal basis functions.

Here, what the terms Orthogonal and Basis function denote in case of signals?

It means the same things that for "usual" vectors.

• Two vectors $x$ and $y$ are orthogonal if $\langle x, y \rangle = 0$

With functions, the inner product is usually (for real valued functions)

$$\langle f, g \rangle = \int_{\Omega} f(x) g(x) dx$$

• A familly $(x_i)_{i \in I}$ of vectors from $E$ is a basis if
• every vector of $E$ can be written as a finite linear combinaison of some $x_i$
• the vectors $(x_i)_{i \in I}$ are lineary independants : if $\sum_{k=1}^n a_i x_{\sigma(i)} = 0$ then $a_1 = a_2 = \cdots = a_n = 0$

The hard part to understand is that functions are vectors. You can add them, multiply them by a scalar : every property of the "usual" vectors apply to the functions

Edit : your book may also be talking about Hilbertian basis, that are a little different from algebrical basis (what I explained). In particular, with Hilbertian basis, infinite linear combinaisons are allowed