# What the terms orthogonal & basis function denote in case of signals

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1. May 2, 2015

### ramdas

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

Here, what the terms orthogonal and basis functions denote in case of signals? Can anyone explain concept with an example?

Also,what are the physical implications of basis functions?

Last edited: May 2, 2015
2. May 2, 2015

3. May 2, 2015

### Staff: Mentor

One set of basis functions that is used a lot in Fourier series is the set $\{\sin(t), \sin(2t), \sin(3t), \dots, \sin(nt), \dots\}$. These functions are orthogonal on the interval $[0, \pi]$, which means that the inner product of any two distinct functions in this set is zero. In other words, $\int_0^{\pi} \sin(kt) \sin(mt)~dt = 0$, if $k \neq m$.

The term basis is linear algebra terminology that has to do with vector spaces (or function spaces, which are nearly the same as vector spaces). For a given space, a basis is a set of vectors (or functions) that are (1) linearly independent and (2) span the space.

For a simple example of these concepts, let's take R2, the plane. This space (it's a vector space) has a natural basis, {<1, 0>, <0, 1>}. Every vector in R2 can be written as a linear combination of the two vectors in the basis. For example, <3, 4> = 3<1, 0> + 4<0, 1>. In a similar way, a function that represents a signal can be written as a linear combination of the basis functions.