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Register a new userAn MGF-based Unified Framework to Determine the Joint Statistics of Partial Sums of Ordered Random Variables
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Added by
Sung Sik Nam
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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Order statistics find applications in various areas of communications and signal processing. In this paper, we introduce an unified analytical framework to determine the joint statistics of partial sums of ordered random variables (RVs). With the proposed approach, we can systematically derive the joint statistics of any partial sums of ordered statistics, in terms of the moment generating function (MGF) and the probability density function (PDF). Our MGF-based approach applies not only when all the K ordered RVs are involved but also when only the Ks (Ks < K) best RVs are considered. In addition, we present the closed-form expressions for the exponential RV special case. These results apply to the performance analysis of various wireless communication systems over fading channels.
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The joint statistics of partial sums of ordered random variables (RVs) are often needed for the accurate performance characterization of a wide variety of wireless communication systems. A unified analytical framework to determine the joint statistics of partial sums of ordered independent and identically distributed (i.i.d.) random variables was recently presented. However, the identical distribution assumption may not be valid in several real-world applications. With this motivation in mind, we consider in this paper the more general case in which the random variables are independent but not necessarily identically distributed (i.n.d.). More specifically, we extend the previous analysis and introduce a new more general unified analytical framework to determine the joint statistics of partial sums of ordered i.n.d. RVs. Our mathematical formalism is illustrated with an application on the exact performance analysis of the capture probability of generalized selection combining (GSC)-based RAKE receivers operating over frequency-selective fading channels with a non-uniform power delay profile. We also discussed a couple of other sample applications of the generic results presented in this work.
The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) $X_1 : Omega rightarrow {mathbb R}^{p_1}$ and $X_2 : Omega rightarrow {mathbb R}^{p_2}$, with respect to square-error distortion at the two decoders is re-examined using (1) Hotellings geometric approach of Gaussian RVs-the canonical variable form, and (2) van Puttens and van Schuppens parametrization of joint distributions ${bf P}_{X_1, X_2, W}$ by Gaussian RVs $W : Omega rightarrow {mathbb R}^n $ which make $(X_1,X_2)$ conditionally independent, and the weak stochastic realization of $(X_1, X_2)$. Item (2) is used to parametrize the lossy rate region of the Gray and Wyner source coding problem for joint decoding with mean-square error distortions ${bf E}big{||X_i-hat{X}_i||_{{mathbb R}^{p_i}}^2 big}leq Delta_i in [0,infty], i=1,2$, by the covariance matrix of RV $W$. From this then follows Wyners common information $C_W(X_1,X_2)$ (information definition) is achieved by $W$ with identity covariance matrix, while a formula for Wyners lossy common information (operational definition) is derived, given by $C_{WL}(X_1,X_2)=C_W(X_1,X_2) = frac{1}{2} sum_{j=1}^n ln left( frac{1+d_j}{1-d_j} right),$ for the distortion region $ 0leq Delta_1 leq sum_{j=1}^n(1-d_j)$, $0leq Delta_2 leq sum_{j=1}^n(1-d_j)$, and where $1 > d_1 geq d_2 geq ldots geq d_n>0$ in $(0,1)$ are {em the canonical correlation coefficients} computed from the canonical variable form of the tuple $(X_1, X_2)$. The methods are of fundamental importance to other problems of multi-user communication, where conditional independence is imposed as a constraint.
This paper introduces a new and ubiquitous framework for establishing achievability results in emph{network information theory} (NIT) problems. The framework uses random binning arguments and is based on a duality between channel and source coding problems. {Further,} the framework uses pmf approximation arguments instead of counting and typicality. This allows for proving coordination and emph{strong} secrecy problems where certain statistical conditions on the distribution of random variables need to be satisfied. These statistical conditions include independence between messages and eavesdroppers observations in secrecy problems and closeness to a certain distribution (usually, i.i.d. distribution) in coordination problems. One important feature of the framework is to enable one {to} add an eavesdropper and obtain a result on the secrecy rates for free. We make a case for generality of the framework by studying examples in the variety of settings containing channel coding, lossy source coding, joint source-channel coding, coordination, strong secrecy, feedback and relaying. In particular, by investigating the framework for the lossy source coding problem over broadcast channel, it is shown that the new framework provides a simple alternative scheme to emph{hybrid} coding scheme. Also, new results on secrecy rate region (under strong secrecy criterion) of wiretap broadcast channel and wiretap relay channel are derived. In a set of accompanied papers, we have shown the usefulness of the framework to establish achievability results for coordination problems including interactive channel simulation, coordination via relay and channel simulation via another channel.
In this paper we develop a finite blocklength version of the Output Statistics of Random Binning (OSRB) framework. The framework is shown to be optimal in the point-to-point case. New second order regions for broadcast channel and wiretap channel with strong secrecy criterion are derived.
A communication setup is considered where a transmitter wishes to convey a message to a receiver and simultaneously estimate the state of that receiver through a common waveform. The state is estimated at the transmitter by means of generalized feedback, i.e., a strictly causal channel output, and the known waveform. The scenario at hand is motivated by joint radar and communication, which aims to co-design radar sensing and communication over shared spectrum and hardware. For the case of memoryless single receiver channels with i.i.d. time-varying state sequences, we fully characterize the capacity-distortion tradeoff, defined as the largest achievable rate below which a message can be conveyed reliably while satisfying some distortion constraints on state sensing. We propose a numerical method to compute the optimal input that achieves the capacity-distortion tradeoff. Then, we address memoryless state-dependent broadcast channels (BCs). For physically degraded BCs with i.i.d. time-varying state sequences, we characterize the capacity-distortion tradeoff region as a rather straightforward extension of single receiver channels. For general BCs, we provide inner and outer bounds on the capacity-distortion region, as well as a sufficient condition when this capacity-distortion region is equal to the product of the capacity region and the set of achievable distortions. A number of illustrative examples demonstrates that the optimal co-design schemes outperform conventional schemes that split the resources between sensing and communication.
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