Simultaneous communication and detection over compound channels

Abstract

t h x a , NY 14853.AbstractConsider a compound channel W with L component discrete memoryless channels (DMCs). We seek a channel code which serves two purposes: the code must achieve a communication rate Rover every DMC in W , and the decoder must accurately detect which of the DMCs in W is in effect. These two objectives are clearly conflicting: accurate detection of the channel is possible only if the code consists of relatively few codewords; such codes though convey little information. Suppose we measure detection performance using the hypothesis testing error exponent r. We ask the following question: what communication rates R and error exponents r are simultaneously achievable on W? For the case L = 2 and a Neyman-Pearson detection approach, we establish single-letter bounds and prove a strong converse. I. PROBLEM STATEMENTThe transmitter encodes one of M possible messages into one of M blocklength n codewords chosen from X", where X is the channel input alphabet. The codeword x is transmitted through a DMC with output alphabet Y . The DMC has a probability transition matrix given by either Wo(y1z) or Wl(ylz) under the null and alternative hypotheses, respectively. The received codeword y is processed to determine, as accurately a s possible, the transmitted message and the channel hypothesis. A (n, A, e, M , r ) simultaneous communication and detection (CD) code for the compound channel {WO, W I } consists oE 1. A codebook C = {xi E X n : 1 I i 5 M } and an encoder f : (1,. . . , M } --t Xn, such that f (2) 2 xi. set 2. A decoder q5 which maps received words to the message q5 : Y" + (1,. . . , M } . We choose q5 so that 3. A detection or hypothesis testing function 11 0-7803-5000-6/98/$10.00 0 1998 IEEE. Harish Viswanathan Lucent Technologies Bell Labs, HOH L272, 791 Holmdel Rd, Holmdel, NJ 07733. A rate-exponent pair ( R , r ) is said to be (A,€)-achievable if, for every 6 > 0 and all sufficiently large n, there exists a (n, A, E , 2"(R-6), r-6) CD code. A rate-exponent pair (R, r ) is achievable if it is (A, €)-achievable for every 0 < A, E < 1. Let R(A, e) and R be the set of all (A, €)-achievable and achievable rate-exponent pairs, respectively. Now define the exponentrate function A r(R) = sup r. ( R , r ) E a 11. MAIN RESULTS Define Theorem 1 r ( R ) 5 r a ( R ) . Now let P, Q be any distributions on X, and let 0 5 T I 1; for R 2 0, define + (1 -~)~(wollwilQ) + I T W , WO) -RI+, and rL(R) = rE[O.l],P,Q: max rl(.r,P,Q,R) A r z (~, p , Q , R l . Theorem 2 r(R) 2 rL(R). A R l r [ l ( P , W o ) h I ( P , W i )IExcept in special cases, our upper and lower bounds on r(R) do not agree. It is therefore important to prove a strong converse theorem and establish that R ( A , E ) is independent of (A, E ) . Theorem 3 (Strong Converse) LetA p = min W~( y l z ) . Z € X , 1 / E Y If p > 0, then f o r every 0 < A, E < 1, R(A, E ) = R. For proofs of these results, see [l]. REFERENCES[I] S. Jayaraman and H. Viswanathan. "Simultaneous Communication and Detection over Compound Channels"

The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel

“…Since entropy is maximized by a Gaussian distribution for any given covariance matrix (29) Choosing each and each to be Gaussian, we may combine (28) and (29) to yield (30) Therefore, the function in (22) is convex . Since the region over which we maximize is convex, any kind of hill-climbing technique can be used to perform the maximization.…”

“…If the probe is a pilot tone, then it will be constant. We do not address here the manner in which to allocate resources to the probe ( [30]). It is useful for us to determine in what way our probing should be performed so as to satisfy our constraint while reducing the variance of the error as much as possible.…”