Is a fourier transform a rotoation?

[RoyLB]

From my undergraduate textbook: Circuits, Signals, and Systems by Siebert, p 453

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Consider the two principal waveform representations schemes ...

$$x(t) = \int x(\tau)\delta(t - \tau)d\tau$$

$$x(t) = \int X(f)e^{j2\pi f t}df$$

If we consider the set of delayed impulses as determining one set of orthogonal vectors and the set of complex exponentials as determining another set, then $x(\tau) d\tau$ and $X(f) df$ are the components of $x(t)$ along the corresponding coordinates. The frequency-domain representations of $x(t)$ thus amounts to picking a coordinate system that is rotated from the time-domain coordinate system. And Parseval's Theorem

$$<br /> \int x^2 (t) dt = \int |X(f)|^2 df<br />$$
is just a statement of the fact that the length of a vector is independent of the coordinate system in which it is described
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​

Is this true? If so, then it should be possible to the find the axis of rotation, right? How does one go about that? Does the question make sense?

Thanks
Roy

 

[Hurkyl]

An "axis of rotation" only makes sense in a three-dimensional vector space. e.g. to make sense of it you could do the following:

Form the rotation matrix. Its eigenvalues will be one of the following:

• Three 1's
• One 1, two -1's
• One 1, one complex-conjugate pair of norm 1

The eigenspace associated to -1, or the space generated by the eigenspaces for the complex-conjugate pair describe a plane that is rotated by the rotation.

The eigenspace associated to 1 is a line that is fixed by the rotation.


The fact that the line uniquely determines a plane perpendicular to it is a unique feature of three-dimensional Euclidean space. That the list of possible sets of eigenvalues is so short is another unique feature of three-dimensional Euclidean space. (there simply aren't enough dimensions for the behavior of a rotation to be complex^*)


In finite dimensions, the space breaks up into a fixed space (which can be zero-dimensional in even dimension!), and planes that are rotated. (And some extra cases where the eigenvalues are repeated) I don't know if we can even say that in the infinite dimensional case you're considering.


*: used as an English word rather than a technical term[/size]

 

[RoyLB]

An "axis of rotation" only makes sense in a three-dimensional vector space.

Hurkyl,

Thanks for responding.

Perhaps my question was too literal. If this relation is true, is there some was to understand the transform as a rotation, besides the Parseval theorem being analogous to preserving a "length" in some infinite dimensional space? What if we restricted ourselves to a discrete time / discrete Fourier transform, to keep the dimensions finite?

- Roy

 

[Hurkyl]

FWIW, wikipedia's article lists the eigenvalues and one eigenbasis for the Fourier transform.

 

[RoyLB]

FWIW, wikipedia's article lists the eigenvalues and one eigenbasis for the Fourier transform.

Hurkyl,

Thanks for the link. I really should look more closely at wikipedia. According to

http://en.wikipedia.org/wiki/Fracti...pretation_of_the_Fractional_Fourier_Transform

and

http://en.wikipedia.org/wiki/Linear_canonical_transformation,

the Fourier transform is a rotation by 90 deg in the time-frequency domain. This may be related to the concept that position and momentum of Fourier transforms of one another according to quantum physics theory.

- Roy