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Bounds on erate for Linear, Time-Invariant, Mu1 t iinpu t / Mu1 tiou tpu t Channels

By Dan Hajela, Michael and L. Honig


Abstract--Suppose that a multiinput/multioutput channel is described by a time-invariant, linear operator H, which maps an input vector waveform U(.) to an output vector waveform y(.). The input U(.) is assumed to be bounded in energy (L, norm) on the time interval [O,T]. Let N,,,,,(T,e) denote the maximum number of inputs to H for which any pair of distinct outputs are separated by at least E in L2 norm. The limit of [log, N,,,,,(T, E)] / T as T + 3 ~ is known as ‘‘€-rate.” Here we extend the bounds on €-rate given by Root for single-input/ single-output channels to multiinput/multioutput channels. This extension uses a result due to Lerer on the eigenvalue distribution of a convolution operator with a matrix kernel (impulse response). Our results are used to assess the increase in data rate attainable by an additive noise process, the statistics of which are unknown, but which is bounded by ~ / in 2 L, norm. Two channel outputs are therefore distinguishable if they are separated in L, norm by E. Let N,,,,,(T,e) denote the maximum number of distinguishable channel outputs, subject to the constraint that the corresponding inputs are bounded in L, norm over the interval [0, TI, where T> 0. The limit of [log, N,,,,,(T, E)] / T as T CC has been called both ‘‘€-rate ” and ‘‘€-capacity, ” and can be used to estimate the maximum achievable data rate for the situation designing input signals which exploit the multidimensional nature of the just described. In this paper we will refer to this quantity channel, relative to treating each constituent channel in isolation. Numerical results based upon a simple model for two coupled twisted-pair wires are presented. as the ‘‘€-rate, ” since it has been pointed out to the authors that “E-capacity ” has other meanings in the contexts of both information theory and approximation theory. Upper and lower bounds on erate for linear time-I

Year: 2008
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