Hybrid Cramer-Rao Bound On Carrier And Sampling Frequency Offset Estimation For OFDM Systems In Rayleigh Fading Channels

For carrier frequency offset (CFO) and sampling (clock) frequency offset (SFO) estimation, the hybrid Cramer-Rao bound (HCRB) is developed when the CFO, SFO, information-bearing symbols are deterministic (non-random) and channel coefficients are random. Both noise and channel coefficients are complex Gaussian. For the HCRB to be applicable, it is necessary for deterministic parameters to be identifiable (uniquely determined). Some necessary identifiability conditions of some deterministic parameters are found and presented. The HCRB is dependent on the initial time instant. The HCRB is compared with some existing methods via simulation. Our results demonstrate that this bound is not tight enough. Further effort is needed to develop a tighter bound.

Rapid growth of shear strain in weakened zones of the lithosphere

A weakened zone in the lithosphere plunging into the mantle can lead to an earthquake after the application of a shear stress only in the case if the effective viscosity of this zone is very low. At low viscosity, in the short time that elapses after the application of stress, significant displacements of the walls of the zone emerge causing high-amplitude seismic waves. The Andrade law describing the transient creep under constant stress applied at the initial time instant leads to very low effective viscosity a few first seconds after the initial time instant. The effective viscosity also decreases due to the temperature rise in the weakened zone caused by the dissipative release of heat. However, this process is not rapid enough to noticeably change the temperature and effective viscosity in a short time.

The explicit probability distribution of the sum of two telegraph processes

We consider two independent Goldstein–Kac telegraph processes X1(t) and X2(t) on the real line ℝ, both developing with constant speed c > 0, that, at the initial time instant t = 0, simultaneously start from the origin 0 ∈ ℝ and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate λ > 0. Closed-form expressions for the transition density φ(x, t) and the probability distribution function Φ(x, t) = Pr {S(t) < x}, x ∈ ℝ, t > 0, of the sum S(t) = X1(t) + X2(t) of these processes at arbitrary time instant t > 0, are obtained. It is also proved that the shifted time derivative g(x, t) = (∂/∂t + 2λ)φ(x, t) satisfies the Goldstein–Kac telegraph equation with doubled parameters 2c and 2λ. From this fact it follows that φ(x, t) solves a third-order hyperbolic partial differential equation, but is not its fundamental solution. The general case is also discussed.

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Consider two independent Goldstein-Kac telegraph processes X1(t) and X2(t) on the real line ℝ. The processes Xk(t), k = 1, 2, describe stochastic motions at finite constant velocities c1 > 0 and c2 > 0 that start at the initial time instant t = 0 from the origin of ℝ and are controlled by two independent homogeneous Poisson processes of rates λ1 > 0 and λ2 > 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X1(t) - X2(t)|, t > 0, between these processes at an arbitrary time instant t > 0. Some numerical results are also presented.

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Consider two independent Goldstein-Kac telegraph processesX1(t) andX2(t) on the real line ℝ. The processesXk(t),k= 1, 2, describe stochastic motions at finite constant velocitiesc1&gt; 0 andc2&gt; 0 that start at the initial time instantt= 0 from the origin of ℝ and are controlled by two independent homogeneous Poisson processes of rates λ1&gt; 0 and λ2&gt; 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X1(t) -X2(t)|,t&gt; 0, between these processes at an arbitrary time instantt&gt; 0. Some numerical results are also presented.