paper a day : pseudodifferential operators for wireless communications

It should really be “paper every two weeks” or something — mostly it’s laziness in the blogging half of the reading a paper a day rather than the reading part. This one was pretty heavy going though.

Pseudodifferential operators and Banach algebras in mobile communications
Thomas Strohmer
Appl.Comp.Harm.Anal., to appear.

This paper is pretty heavy on the math but does a nice job of explaining the engineering concepts as well. There are two main thrusts here : modeling wirelss channels and OFDM pulse design. Multipath fading and Doppler shift in wireless communication channels can be mathematized via pseudodifferential operators, and designing OFDM pulses can be viewed as trying to make a certain matrix that represent the channel equalization sparse or concentrated on its diagonal. The tools Strohmer uses are pretty advanced (for me), but that’s because I’m not up on my harmonic analysis.

Wireless fading channels are typically modeled by a convolution with a time-varying filter. If the filter were time-invariant we could take Fourier transforms and write the channel as a multiplication by a transfer function. In the time-varying case we can make a similar representation in terms of a time-varying transfer function, which makes the channel law into a pseudodifferential operator whose Kohn-Nirenberg symbol is precisely the time-varying transfer function. Thus modeling constraints on the symbol can be translated into constraints on the operator.

The decay profile from multipath fading and the effect of Doppler shift provide constraints on the localization of the symbol in the time-frequency plane. The engineering constraints don’t give a nice characterization of the symbol per se, but we can embed the class of channels into a Banach algebra of operators with certein weight functions. We can also embed the symbol into a specific modulation space called a Sjöstrand class.

Turning to the equalization problem, OFDM pulses form a Gabor system, which is a kind of time-frequency basis for a function space. We would like to choose a basis so that recovering the data modulated on these pulses is easy. It turns out that the whole equalization operation can be written as a matrix that is related to the set of pulses, so the condition we want is for this matrix and its inverse to be nearly diagonal or sparse.

The big theorem (Theorem 4.1) in the paper essentially states that for pulses with good time-frequency localization, if the channel’s K-N symbol is invertible, then the inverse of the equalization matrix belongs to a certain algebra. This is the mathematical statement of the pulses being a “good system for communication.” This plus some more advanced relationship between different spaces gives a way of actually engineering a pulse design that can trade off the spectral efficiency versus inter-symbol and inter-channel interference.

All in all, this was a good paper to read but I don’t think I have the background to go and use these tools and techniques on anything because the math is pretty far above my head.