COMPLEXITY OF PCM FORMATTING

Abstract

How difficult is it to develop a pulse code modulation (PCM) stream data format? Specifically, given a size, in bits, and a set of parameter sample rates, how hard is it to find a mapping of the sample rates that fits into the frame size -- if one even exists? Using telemetry trees this paper will show that the number of possible mappings for a given set of parameters and sample rates grows exponentially in terms of the number of parameters. The problem can thus be stated in terms of finding a specific instance, or showing that no such instance exists, among an exponentially large number of potential mappings. Although not proof, this provides strong evidence that the PCM format design problem is NP-complete (meaning it is a member of nondeterministic polynomial space but not a member of deterministic polynomial space). That is, that the problem can not be solved in polynomial time and would take a computer years or centuries to solve relatively small instances of the problem. However, if the problem requirements are relaxed slightly, telemetry trees can be used to reduce the PCM formatting problem to linear time in terms of the number of parameters. This paper describes a technique that can provide an optimal and fully packed PCM format.