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Kamran Kalbasi

Publications and source records attributed to Kamran Kalbasi.

5 recordsLinked to original sources
arXiv2020

On the relaxed maximum-likelihood blind MIMO channel estimation for orthogonal space-time block codes

This paper concerns the maximum-likelihood channel estimation for MIMO systems with orthogonal space-time block codes when the finite alphabet constraint of the signal constellation is relaxed. We study the channel coefficients estimation subspace generated by this method. We provide an algebraic characterisation of this subspace which turns the optimization problem into a purely algebraic one and more importantly, leads to several interesting analytical proofs. We prove that with probability one, the dimension of the estimation subspace for the channel coefficients is deterministic and it decreases by increasing the number of receive antennas up to a certain critical number of receive antennas, after which the dimension remains constant. In fact, we show that beyond this critical number of receive antennas, the estimation subspace for the channel coefficients is isometric to a fixed deterministic invariant space which can be easily computed for every specific OSTB code.

cs.IT↗
arXiv2019

On the probability distribution of the local times of diagonally operator-self-similar Gaussian fields with stationary increments

In this paper we study the local times of vector-valued Gaussian fields that are `diagonally operator-self-similar' and whose increments are stationary. Denoting the local time of such a Gaussian field around the spatial origin and over the temporal unit hypercube by $Z$, we show that there exists $λ\in(0,1)$ such that under some quite weak conditions, $\lim_{n\rightarrow +\infty}\frac{\sqrt[n]{\mathbb{E}(Z^n)}}{n^λ}$ and $\lim_{x\rightarrow +\infty}\frac{-\log \mathbb{P}(Z>x)}{x^{\frac{1}λ}}$ both exist and are strictly positive (possibly $+\infty$). Moreover, we show that if the underlying Gaussian field is `strongly locally nondeterministic', the above limits will be finite as well. These results are then applied to establish similar statements for the intersection local times of diagonally operator-self-similar Gaussian fields with stationary increments.

math.PR↗
arXiv2017

Feynman-Kac representation for the parabolic Anderson model driven by fractional noise

We consider the parabolic Anderson model driven by fractional noise: $$ \frac{\partial}{\partial t}u(t,x)= κ\boldsymbolΔ u(t,x)+ u(t,x)\frac{\partial}{\partial t}W(t,x) \qquad x\in\mathbb{Z}^d\;,\; t\geq 0\,, $$ where $κ>0$ is a diffusion constant, $\boldsymbolΔ$ is the discrete Laplacian defined by $\boldsymbolΔ f(x)= \frac{1}{2d}\sum_{|y-x|=1}\bigl(f(y)-f(x)\bigr)$, and $\{W(t,x)\;;\;t\geq0\}_{x \in \mathbb{Z}^d}$ is a family of independent fractional Brownian motions with Hurst parameter $H\in(0,1)$, indexed by $\mathbb{Z}^d$. We make sense of this equation via a Stratonovich integration obtained by approximating the fractional Brownian motions with a family of Gaussian processes possessing absolutely continuous sample paths. We prove that the Feynman-Kac representation \begin{equation} u(t,x)=\mathbb{E}^x\Bigl[u_o(X(t))\exp \int_0^t W\bigl(\mathrm{d}s, X(t-s)\bigr)\Bigr]\,, \end{equation} is a mild solution to this problem. Here $u_o(y)$ is the initial value at site $y\in\mathbb{Z}^d$, $\{X(t)\;;\;t\geq0\}$ is a simple random walk with jump rate $κ$, started at $x \in \mathbb{Z}^d$ and independent of the family $\{W(t,x)\;;\;t\geq0\}_{x\in\mathbb{Z}^d}$ and $\mathbb{E}^x$ is expectation with respect to this random walk. We give a unified argument that works for any Hurst parameter $H\in (0,1)$.

math.PR↗
arXiv2017

Anderson polymer in a fractional Brownian environment: asymptotic behavior of the partition function

We consider the Anderson polymer partition function $$ u(t):=\mathbb{E}^X\Bigl[e^{\int_0^t \mathrm{d}B^{X(s)}_s}\Bigr]\,, $$ where $\{B^{x}_t\,;\, t\geq0\}_{x\in\mathbb{Z}^d}$ is a family of independent fractional Brownian motions all with Hurst parameter $H\in(0,1)$, and $\{X(t)\}_{t\in \mathbb{R}^{\geq 0}}$ is a continuous-time simple symmetric random walk on $\mathbb{Z}^d$ with jump rate $κ$ and started from the origin. $\mathbb{E}^X$ is the expectation with respect to this random walk. We prove that when $H\leq 1/2$, the function $u(t)$ almost surely grows asymptotically like $e^{l t}$, where $l>0$ is a deterministic number. More precisely, we show that as $t$ approaches $+\infty$, the expression $\{\frac{1}{t}\log u(t)\}_{t\in \mathbb{R}^{>0}}$ converges both almost surely and in the $\mathcal{L}^1$ sense to some deterministic number $l>0$. For $H>1/2$, we first show that $\lim_{t\rightarrow \infty} \frac{1}{t}\log u(t)$ exists both almost surely and in the $\mathcal{L}^1$ sense, and equals a strictly positive deterministic number (possibly $+\infty$); hence almost surely $u(t)$ grows asymptotically at least like $e^{a t}$ for some deterministic constant $a>0$. On the other hand, we also show that almost surely and in the $\mathcal{L}^1$ sense, $\limsup_{t\rightarrow \infty} \frac{1}{t\sqrt{\log t}}\log u(t)$ is a deterministic finite real number (possibly zero), hence proving that almost surely $u(t)$ grows asymptotically at most like $e^{b t\sqrt{\log t}}$ for some deterministic positive constant $b$. Finally, for $H>1/2$ when $\mathbb{Z}^d$ is replaced by a circle endowed with a Hölder continuous covariance function, we show that $\limsup_{t\rightarrow \infty} \frac{1}{t}\log u(t)$ is a finite deterministic positive number, hence proving that almost surely $u(t)$ grows asymptotically at most like $e^{c t}$ for some deterministic positive constant $c$.

math.PR↗