Time domain basis

Question

I have some troubles with understanding time domain, not on the intuitive level, but in math terms.

For example I have a vector signal $$ x = [x_0,x_1,x_2,...,x_{N-1}]$$ I understand that generally when we describe our signal in the form of a vector, we just put our information in a convenient form for further manipulation.

i can change the basis to the frequency domain via DFT analysis, and then I can get back (to time domain) with DFT synthesis. DFT synthesis just multiply my complex amplitude on complex sinudoids with different weights $w$. And it's perfectly correct on physics level like we just sum some periodic functions with different amplitudes.

But when we are talking about change of basis, we have in mind:

$$\alpha_k = \sum_{k=0}^{N-1} \beta_n \langle w^{(k)},w^{(n)} \rangle$$

where: $w^{(k)}$ - the first basis ; $w^{(n)}$ - the second basis ; $\alpha_k$ - expansion coefficients in second basis; $\beta_n$ - expansion coefficients in first basis;

where:$$x_n = \sum_{n=0}^{N-1}\beta_nw_n$$ - this is our information vector in time domain; and basis for frequency domain is $$w^{(k)} = w_k[n] = e^{\frac{2\pi i}{N}kn}$$

and my question is: do I understand right that time domain basis is a canonical basis for N-dimensional linear space like that: $$\hat{e}^{(k)} = \delta[n-k]$$

Answer 1

Given a $N$-dimensional linear space, the Kronecker $\delta[\cdot]$ basis is the most natural basis for describing discrete sampled signals, with

\begin{align} \delta_0 & = [1,0,0,\ldots,0]\\ \delta_1 &= [0,1,0,\ldots,0]\\ \delta_{N-1} &= [0,0,0,\ldots,1] \end{align}

and hence can be called canonical. You can refer to What does 'canonical' mean? for global ideas about the canon, with one answer:

The etymology refers to the canon, as a rule or a body of rules, or axiomatic or universal standards. It exists in arts: sculpture, music, script writing, etc. The notion of canon law is also used in the domain of religion.

In mathematics, and engineering, a canonical form is, similarly, a preferred notation, or a unique and natural form, or representation, of an object, a formula. For instance, a canonical basis is a basis of a vector space (an algebraic structure in general) that refers to a precise context, like the standard basis defined by the Kronecker delta.

However, I dare to say that this is disconnected from a "discrete-time signal" concept. You could consider a space where complex sines would be discretized and sampled in a frequency domain, and the canonical basis would be exactly the same. In this case, the same basis would be canonical, in that it represents well "frequency Diracs".

Wikipedia entry on canonical basis somewhat agrees with that (first item):

In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:

• In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta.