Bounds on Maximum Throughput for Digital Communications with Finite-Precision and Amplitude Constraints

The problem of finding the maximum achievable data rate over a linear time-invariant channel is considered under constraints different from those typically assumed. The limiting factor is taken to be the accuracy with which the receiver can measure the channel output. More precisely, we consider the following problem. Given a channel with known impulse response h(t), a transmitter with an output amplitude constraint, and a receiver that can distinguish between two signals only if they are separated in amplitude at some time f0 by at least some small positive constant d, what is the maximum number of messages, Nmax, that can be transmitted in a given time interval [0, T]? Lower bounds on Nmaxcan be easily computed by constructing a particular set of inputs to the channel. Our main result is an upper bound on Nmaxfor arbitrary h(t). The upper bound depends on the spread of h(t), which is the maximum range of values the channel output may take at some time f0 > 0 given that the output takes on a particular value a at time t = 0. For a particular h(t), computing the spread in discrete-time is equivalent to solving a linear program with bounded variables and one equality constraint. Solutions to linear programs in this class can be obtained very fast using, for example, a linear-time algorithm due to Witzgall. Numerical results are shown for different impulse responses, including two simulated telephone subscriber loop impulse responses. Assuming that the receiver resolution d is small, the upper bound is typically two to four times the lower bound for the cases examined.