What the terms Basis functions and Orthogonal denote in case of signals?

Question

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 functions denote in case of signals?

Answer 1

If you think of a signal as a function of time $s(t)$ ($t$ could be a continuous variable or a discrete one $t_n$) that satisfies certain criteria, it can be thought of as a vector in some vector space of functions. It depends which vector space you want to use (e.g. $L^2$ or tempered distributions), but as most signals are finite and limited in space and/or time, it's usually safe to assume that they satisfy the criteria.

After that most rules from linear algebra can be used. Most importantly, a vector can be decomposed into a linear combination of basis vectors. An analog of standard basis vectors are shifted $\delta$ functions. If the basis vectors are orthonormal, the coefficients in the linear combination will be scalar products of the basis vector and the vector being decomposed. These coefficients are then known as Fourier coefficients.

So I suggest you revise your knowledge of linear algebra (mainly basis vectors, scalar product) and then try to look at functions as vectors and it will all fall into place.

Edit: to answer the "physical significance" part:

If you can attribute some meaning or behavior to basis functions in your system, then knowing how the signal is composed of them tells you how "similar" (but only in the linear sense of cross-correlation) it is to certain behaviors or patterns. A common example are the $\sin(mt)$ and $\cos(mt)$ basis functions used in Fourier analysis. In most physical systems they represent oscillation modes and if you can decompose a signal into these modes, you can understand it in terms of oscillation modes. That's pretty much what spectrograms do. In case of wavelets you can think of them as waveform packets that tell you not only what oscillation mode was present, but also where and/or when.

Edit: an intuitive way to look at functions as vectors:

If you think of a function as a mapping from one set to another (usually $\mathbb{R} \rightarrow \mathbb{R}$), simple array vectors are actually functions that map $\hat n = \{1, \dots, n\}$ onto $\mathbb C$ or $\mathbb R$. So you can imagine continuous functions as vectors that have so many elements that their length becomes infinite and the spacing between the element indices goes to zero, forming a "continuous" vector. This also means that in most formulas in linear algebra summations turn into integrals (they were always integrals, but the integration set was changed from $\hat n$ to $\mathbb R$). Of course, all this is much more complicated if done rigorously, because integrals as infinite sums can introduce infinite results.

It's a little simpler with finite discrete signals of length $N$ because then it's simple to look at them as array vectors of length $N$. However, any formulas derived for continuous signals might introduce edge defects.

Answer 2

The X and Y axis can be used to describe any point in an XY plane. Thus any non-zero vector in the X direction and one in the Y direction can be used in a linear combination to describe any point in that space. That linear combination is separable if the two basis vectors are orthogonal. Or you can turn or translate that XY pair to create other orthogonal basis vector pairs for that plane.

In frequency space, consider a 2 point DFT. the average and difference (DC + Nyquist) in the frequency domain can be used in a separable linear combination to describe any 2 samples in the time domain, thus can be used to form an orthogonal basis pair.