On Communication of Analog Data From a Bounded Source Space

Iii this paper we are concerned with communication of discrete-time analog data over a communication channel with a variety of fidelity criteria. The central assumption about the analog source is that its 3139 3144 T H E BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 19(59 outputs belong to a bounded set, typically the interval [--A/2, A/2]. We begin with a rough outline of our results, leaving the precise formulation and statement to Section I I . Proofs are found in Section I I I . Suppose that we have a data source which emits a sequence of symbols Xi , x2 , · · · e 9C (an arbitrary set) at a rate of ps per second. This sequence is fed into an "encoder" which assigns to each successive block of n source symbols, say x = (x 1 ) y ' ' ' ) ), a channel input of duration n/ps = T seconds. At the receiving end of the channel, the Tsecond output is transformed by a "decoder" into an n-sequence, say x = (4)x , x2, · · · , xn), which is delivered to the destination. The "distortion" between the source output sequence x and the received sequence x is defined as dM(x, x) = n~l d(xk, xk), where d(x, x) ^ 0 is an arbitrary function. The classical problem is that of a "memoryless" source, where successive source outputs are statistically independent with identical probability distribution. In this case it is meaningful to let the system performance criterion (fidelity criterion) be the statistical expectation of the distortion dM (x, x). A quantity of interest is d*(T), the smallest attainable value of the fidelity criterion when the coding delay is T seconds.