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The DTFT as an “Eigenbasis”

Module by: Mark A. Davenport. E-mail the author

The DTFT as an “Eigenbasis”

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

Let's look at F-1:L2[-π,π]2(Z)F-1:L2[-π,π]2(Z) first:

F - 1 ( X ( e j w ) ) = 1 2 π - π π X ( e j ω ) e j ω n d ω . F - 1 ( X ( e j w ) ) = 1 2 π - π π X ( e j ω ) e j ω n d ω .
(1)

Recall that X(ejω)X(ejω) is really just a function of ωω, so if we replace ωω with tt, we get

F - 1 ( X ( t ) ) = 1 2 π - π π X ( t ) e j t n d t . F - 1 ( X ( t ) ) = 1 2 π - π π X ( t ) e j t n d t .
(2)

Does this seem familiar? If X(t)X(t) is a periodic function defined on [-π,π][-π,π], then F-1(X(t))F-1(X(t)) is just computing (up to a reversal of the indicies) the continuous-time Fourier series of X(t)X(t)!

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 the different basis elements. Thus we have x[n]=FF-1(X(t))x[n]=FF-1(X(t)) and

X ( t ) = n = - x [ n ] e - j t n 2 π . X ( t ) = n = - x [ n ] e - j t n 2 π .
(3)

What about FF? In this case we are taking an x2(Z)x2(Z) and mapping it to an XL2[-π,π]XL2[-π,π]. XX represents an infinite set of numbers, and when we weight the functions ejωnejωn by X(ω)X(ω) and sum them all up, we get back the original signal

x [ n ] = - π π X ( ω ) e j ω n 2 π d ω . x [ n ] = - π π X ( ω ) e j ω n 2 π d ω .
(4)

Unfortunately, ejωn2π=ejωn2π= (11) so technically, we can't really think of this as a change of basis.

However, as a unitary transformation, FF has everything we would ever want in a basis and more: We can represent any x2(Z)x2(Z) using {ejωn}ω[-π,π]{ejωn}ω[-π,π], and since it is unitary, we have Parseval and Plancherel Theorems as well. On top of that, we already showed that the set of vectors {ejωn}ω[-π,π]{ejωn}ω[-π,π] 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 it does to its eigenvectors (i.e., its eigenvalues). For an LSI system, H(ejω)H(ejω) represents a set of eigenvalues that provide a complete characterization of the system.

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