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On the Schalkwijk-Kailath coding scheme with a peak energy constraint

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1 Author(s)

In a recent series of papers, [2]-[4] Schalkwijk and Kailath have proposed a block coding scheme for transmission over the additive white Gaussian noise channel with one-sided spectral density N_{0} using a noiseless delayless feedback link. The signals have bandwidth W (W \leq \infty ) and average power \bar{P} . They show how to communicate at rates R < C = W \log (1 + \bar{P}/N_{0}W) , the channel capacity, with error probability P_{e} = \exp {-e^{2(C-R)T+o(T)}} (where T is the coding delay), a "double exponential" decay. In their scheme the signal energy (in a T -second transmission) is a random variable with only its expectation constrained to be \bar{P}T . In this paper we consider the effect of imposing a peak energy constraint on the transmitter such that whenever the Schalkwijk-Kailath scheme requires energy exceeding a \bar{P}T (where a > 1 is a fixed parameter) transmission stops and an error is declared. We show that the error probability is degraded to a "single exponential" form P_{e} = e^{-E(a)T+o(T)} and find the exponent E(a) . In the case W = \infty , E(a) = (a - 1)^{2}/4a C . For finite W, E(a) is given by a more complicated expression.

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IEEE Transactions on Information Theory  (Volume:14 ,  Issue: 1 )