2020/01/19 by Charalambos D. Charalambous, Charalambous, Charalambos D., Jan H. van Schuppen +1
Engineering · Computer Science · #Wireless Communication Security Techniques #Distributed Sensor Networks and Detection Algorithms
paper · pdf · doi:10.48550/arxiv.2001.06824
The Gray and Wyner lossy source coding for a simple network for sources that\ngenerate a tuple of jointly Gaussian random variables (RVs) X1 : \Ω\n\→ mathbb Rp1 and X2 : \Ω \→ mathbb\nRp2, with respect to square-error distortion at the two decoders is\nre-examined using (1) Hotelling's geometric approach of Gaussian RVs-the\ncanonical variable form, and (2) van Putten's and van Schuppen's\nparametrization of joint distributions bf PX1, X2, W by Gaussian RVs\nW : \Ω \→ mathbb Rn which make (X1,X2) conditionally\nindependent, and the weak stochastic realization of (X1, X2). Item (2) is\nused to parametrize the lossy rate region of the Gray and Wyner source coding\nproblem for joint decoding with mean-square error distortions bf\nE \||Xi-\Xi||_ mathbb Rpi2 \\≤ \Δi \∈\n[0,\∞], i=1,2, by the covariance matrix of RV W. From this then follows\nWyner's common information CW(X1,X2) (information definition) is achieved\nby W with identity covariance matrix, while a formula for Wyner's lossy\ncommon information (operational definition) is derived, given by\nCWL(X1,X2)=CW(X1,X2)\n = \(1)/(2) \∑j=1n\n \ln\n \(\n \(1+dj)/(1-dj)\n \), for the distortion region 0\≤ \Δ1 \≤\n\∑j=1n(1-dj), 0\≤ \Δ2 \≤ \∑j=1n(1-dj), and where 1 >\nd1 \≥ d2 \≥ \… \≥ dn>0 in (0,1) are em the canonical\ncorrelation coefficients computed from the canonical variable form of the\ntuple (X1, X2). The methods are of fundamental importance to other problems\nof multi-user communication, where conditional independence is imposed as a\nconstraint.\n