Random Matrix Theory and Applications

This summary will briefly describe some recent results in random matrix theory and their applications. 1 Motivation 1.1 Multiple Antenna Gaussian Channels 1.1.1 The deterministic case Consider a gaussian channel with t transmitting and r receiving antennas. The received vector y ∈Cr will depend on the transmitted vector x ∈Ct by y = Hx+n where H is an rxt complex matrix of gains and n is a vector of zero-mean Gaussian noise with independent, equal variance real and imaginary parts, and x is a power-constrained input vector. To compute the capacity of this channel we use the singular value decomposition for H: we write H as H =UΣV T where U ∈ Crxr,V ∈ Ctxt are unitary while Σ is non-negative and diagonal. We now make a change of variables, x̃ =V T x, ỹ =UT y, ñ =UT n. Note that ñ has the same distribution as n, and because V is unitary out initial power constraint is unchanged. Thus, the original channel is equivalent to ỹ = Σx̃+ ñ which can be written in terms of the singular values as ỹi = λ 1/2 i x̃i + ñi for 1 ≤ i ≤ min{r, t} The capacity of the channel can now be found via water-filling (see Section 10.4 of [3]): set E[Re(x̃i)]+E[Im(x̃i)] = (μ−λ−1 i )+