2.14 The dtft as an “eigenbasis”

The dtft as an “eigenbasis”

We saw Parseval/Plancherel in the context of orthonormal basis expansions. This begs the question, do $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ just take signals and compute their representation in another basis?

Let's look at ${\mathcal{F}}^{-1}:L_2[-\pi ,\pi ]\to {\ell }_2\left(\mathbb{Z}\right)$ first:

Recall that $X\left(e^{j\omega }\right)$ is really just a function of $\omega$ , so if we replace $\omega$ with $t$ , we get

Does this seem familiar? If $X\left(t\right)$ is a periodic function defined on $[-\pi ,\pi ]$ , then ${\mathcal{F}}^{-1}\left(X\left(t\right)\right)$ is just computing (up to a reversal of the indicies) the continuous-time Fourier series of $X\left(t\right)$ !

We said before that the Fourier series is a representation in an orthobasis, the sequence of coefficients that we get are just the weights of thedifferent basis elements. Thus we have $\to x\left[n\right]=\mathcal{F}{\mathcal{F}}^{-1}\left(X\left(t\right)\right)$ and

What about $\mathcal{F}$ ? In this case we are taking an $x\in {\ell }_2\left(\mathbb{Z}\right)$ and mapping it to an $X\in L_2[-\pi ,\pi ]$ . $X$ represents an infinite set of numbers, and when we weight the functions $e^{j\omega n}$ by $X\left(\omega \right)$ and sum them all up, we get back the original signal

Unfortunately, $\left|\left|\frac{e^{j\omega n}}{\sqrt{2\pi }}\right|\right|=\infty$ ( $\ne 1$ ) so technically, we can't really think of this as a change of basis.

However, as a unitary transformation, $\mathcal{F}$ has everything we would ever want in a basis and more: We can represent any $x\in {\ell }_2\left(\mathbb{Z}\right)$ using ${\left\{e^{j\omega n}\right\}}_{\omega \in [-\pi ,\pi ]}$ , and since it is unitary, we have Parseval and Plancherel Theorems as well. On top of that, wealready showed that the set of vectors ${\left\{e^{j\omega n}\right\}}_{\omega \in [-\pi ,\pi ]}$ are eigenvectors of LSI systems – if this really were a basis, it would be called an eigenbasis .

Eigenbases are useful because once we represent a signal using an eigenbasis, to compute the output of a system we just need to know what itdoes to its eigenvectors (i.e., its eigenvalues). For an LSI system, $H\left(e^{j\omega }\right)$ represents a set of eigenvalues that provide a complete characterization of the system.