Variations on a theme by Schalkwijk and Kailath

This note describes coding and decoding strategies for discrete-time additive memoryless Gaussian-noise (DAMGN) channels with ideal feedback. It was shown by Shannon [14] in 1961 that feedback does not increase the capacity of memoryless channels, and was shown by Pinsker [10] in 1968 that fixed-length block codes on Gaussiannoise channels with feedback can not exceed the sphere packing bound if the energy per codeword is bounded independently of the noise realization. It is clear, however, that reliable communication can be simplified by the use of feedback, as illustrated by standard automatic repeat strategies at the data link control layer. There is a substantial literature (for example [11], [3], [9]) on using variable-length strategies to substantially improve the rate of exponential decay of error probability with expected coding constraint length. These strategies essentially use the feedback to coordinate postponement of the final decision when the noise would otherwise cause errors. Thus small error probabilities can be achieved through the use of occasional long delays, while keeping the expected delay small.

For DAMGN channels an additional mechanism for using feedback exists whereby the transmitter can transmit unusually large amplitude signals when it observes that the receiver is in danger of making a decoding error. The power (i.e., the expected squared amplitude) can be kept small because these large amplitude signals are rarely required. In 1966, Schalkwijk and Kailath [13] used this mechanism in a fixed-length block-coding scheme for infinite bandwidth Gaussian noise channels with ideal feedback. They demonstrated the surprising result that the resulting probability of decoding error decreases as a second order exponential1 in the code constraint length at all transmission rates less than capacity. Schalkwijk [12] extended this result to the finite bandwidth case, i.e., DAMGN channels. Later, Kramer [8] (for the infinite bandwidth case) and Zigangirov [15] (for the finite bandwidth case) showed that the above doubly exponential bounds could be replaced by kth order exponential bounds for any k > 2 in the limit of arbitrarily large block lengths. Later encoding schemes inspired by the Schalkwijk and Kailath approach have been developed for multi-user communication with DAMGN [16], [17], [18], [19], [20], secure communication with DAMGN [21] and point to point communication for Gaussian noise channels with memory [22].

The purpose of this paper is three-fold. First, the existing results for DAMGN channels with ideal feedback are made more transparent by expressing them in terms of a 1956 paper by Elias on transmitting a single signal from a Gaussian source via multiple uses of a DAMGN channel with feedback. Second, using an approach similar to that of Zigangirov in [15], we strengthen the results of [8] and [15], showing that error probability can be made to decrease with blocklength n at least with an exponential order anb for given coefficients a > 0 and b > 0. Third, a lower bound is derived. This lower bound decreases with an exponential order in n equal to an + b ′ (n) where a is the same as in the upper bound and b ′ (n) is a sublinear function 2 of the block length n.

Neither this paper nor the earlier results in [12], [13], [8], and [15] are intended to be practical. Indeed, these second and higher order exponents require unbounded amplitudes (see [10], [2], [9]). Also Kim et al [7] have recently shown that if the feedback is ideal except for additive Gaussian noise, then the error probability decreases only as a single exponential in block length, although the exponent increases with increasing signal-to-noise ratio in the feedback channel. Thus our purpose here is simply to provide increased understanding of the ideal conditions assumed.

We first review the Elias result [4] and use it to get an almost trivial derivation of the Schalkwijk and Kailath results. The derivation yields an exact expression for error probability, optimized over a class of algorithms including those in [12], [13]. The linear processing inherent in that class of algorithms is relaxed to obtain error probabilities that decrease with block length n at a rate much faster than an exponential order of 2. Finally a lower bound to the probability of decoding error is derived. This lower bound is first derived for the case of two codewords and is then generalized to arbitrary rates less than capacity.

Let X 1 , . . . , X n = X n 1 represent n > 1 successive inputs to a discrete-time additive memoryless Gaussian noise (DAMGN) channel with ideal feedback. That is, the channel outputs

where Z n 1 is an n-tuple of statistically independent Gaussian random variables, each with zero mean and variance σ 2 Z , denoted N (0, σ 2 Z ). The channel inputs are constrained to some given average power constraint S in the sense that the inputs must satisfy the second-moment constraint

Without loss of generality, we take σ 2

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