What kind of spaces are useful in signals?

[Bipolarity]

Which spaces, as studied in branches of mathematics such as linear algebra and functional analysis, such as vector spaces, inner product spaces, normed linear spaces, metric spaces, Hilbert spaces, Banach spaces etc. are most useful/frequently encountered in signal processing?

My knowledge of mathematics is limited, but as I plan to converge on the field of signal processing very soon I would like to know which mathematics I place a bigger emphasis on as I continue my adventures in math.

Thankks!

BiP

 

[carlgrace]

Of the ones you mentioned vector spaces, inner product spaces and Hilbert spaces are the most commonly used. A lot of stuff is done in the frequency domain (Fourier for continuous time, Z for discrete time).

Unless you really get on the bleeding edge, though, you can typically learn enough of the math in the signal processing course itself.

 

[meBigGuy]

Actually, in the real word, you find yourself generally working in cubicle spaces.

 

[rbj]

i believe that Banach spaces are the same as normed linear spaces. and Hilbert spaces are inner-product spaces.

and all inner-product spaces are also normed linear spaces (the norm is the inner product of an element with itself and that quantity is square-rooted). and all normed linear spaces are simple metric spaces (the norm is the same as the distance metric to whatever the zero element is).

metric spaces and functional analysis are very good disciplines to have under your belt with DSP courses, particularly courses involving communications. but you also want a good course in probability, random variables, and random processes (sometimes called "stochastic processes"). but, unless you end up as an academic, you might never use the specific knowledge but you'll get a better feel for a "signal space" like those used in M-ary communication or QPSK.

and another math course you might want to take for signal processing is one in approximation theory. you might want to learn how the Remez exchange algorithm works. and you might want to learn about numerical methods, too. i presume you're solid with Calc, Diff Eq, and complex analysis (like line integrals and residue theory). how are you with matrices and determinants?

and, of course, you need to be solid with your transform analysis (Fourier, Laplace) and Linear System theory (sometimes called Signals and Systems). and you might want to learn about some analog signal processing, like about s-plane and Butterworth and Tchebyshev filters and the bilinear transform.

 

[micromass]

i believe that Banach spaces are the same as normed linear spaces. and Hilbert spaces are inner-product spaces.

No, that's false. Banach spaces and Hilbert spaces are complete. Normed linear spaces and inner-product spaces don't need to be.

 

[rbj]

No, that's false. Banach spaces and Hilbert spaces are complete. Normed linear spaces and inner-product spaces don't need to be.

okay, i didn't remember that distinction. i think for a signal processing engineer (or student), the difference is sort of esoteric.

but thanks for setting the record straight.