Wireless Distributed Computing and Cognitive Sensing defines high-dimensional data processing in the context of wireless distributed computing and cognitive sensing. This book presents the challenges that are unique to this area such as synchronization caused by the high mobility of the nodes. The author will discuss the integration of software defined radio implementation and testbed development. The book will also bridge new research results and contextual reviews. Also the author provides an examination of large cognitive radio network; hardware testbed; distributed sensing; and distributed computing.

e−λt EeλSn = e−λt EeλXi . i Next, Taylor’s expansion and the mean zero and boundedness hypotheses can be used to show that, for every i, eλXi eλ 2 σ2 , where σ 2 var Xi , 0 λ 1. This results in p e−λt+λ 2 n var Xi . i=1 The optimal choice of the parameter λ ∼ min τ /2σ 2 , 1 implies Chernoff’s inequality p 2 max e−t /σ 2 , e−t/2 . 32 1 Mathematical Foundation 1.4.11 Extensions of Expectation to Matrix-Valued Random Variables If X is a matrix (or vector) with random

of A is called the spectrum of A, denoted spec(A), where each eigenvalue appears a number of times equal to its multiplicity. When f (t) is a polynomial or rational function with scalar coefficients and a scalar argument, t, it is natural to define f (A) by substituting A for t, replacing division by matrix inversion, and replacing 1 by the identity matrix [1, 20]. For example, 34 1 Mathematical Foundation f (t) = 1 + t2 −1 2 ⇒ f (A) = (I − A) (I + A) 1−t if 1 ∈ / spec (A). If f (t) has a

ψ(t) − ψ(0) t for all t. Taking the limit t → 0, we obtain log Tr eA+B Tr [eA ] Tr BeA . Tr [eA ] (1.66) Frequently, this consequence of Theorem 1.4.9 is referred to as the PeierlsBogoliubov Inequality. Not only are both of the functions H → log Tr eH and ρ → −S (ρ) are both convex, they are Legendre Transforms of one another. Here ρ is a density matrix. See [39] for a full mathematical treatment of the Legendre Transform. Example 1.4.10 (A Novel Use of Peierls-Bogoliubov Inequality for

of this fact is given in Sect. 2.13. 1.5 Decoupling from Dependance to Independence Decoupling is a technique of replacing quadratic forms of random variables by bilinear forms. The monograph [56] gives a systematic study of decoupling and its applications. A simple decoupling inequality is given by Vershynin [57]. Both the result and its proof are well known but his short proof is not easy to find in the literature. In a more general form, for multilinear forms, this inequality can be found in

must be recalled behind the mind. The non-commutative property is fundamental in studying random matrices. By using the eigenvalues and their variation property, it is very convenient to think of random matrices as scalarvalued random variables, in which we convert the two dimensional problem into one-dimensional problem—much more convenient to handle. 2.15 Linear Filtering Through Sums of Random Matrices The linearity of the expectation and the trace is so basic. We must always bear this mind.