Abstract
We consider a real continuous-time bandlimited additive white Gaussian noise channel with 1-bit output quantization. On such a channel the information is carried by the temporal distances of the zero-crossings of the transmit signal. We derive an approximate lower bound on the capacity by lower-bounding the mutual information rate for input signals with exponentially distributed zero-crossing distances, sine-shaped transition waveform, and an average power constraint. The focus is on the behavior in the mid-to-high signal-to-noise ratio (SNR) regime above 10 dB. For hard bandlimited channels, the lower bound on the mutual information rate saturates with the SNR growing to infinity. For a given SNR the loss with respect to the unquantized additive white Gaussian noise channel solely depends on the ratio of channel bandwidth and the rate parameter of the exponential distribution. We complement those findings with an approximate upper bound on the mutual information rate for the specific signaling scheme. We show that both bounds are close in the SNR domain of approximately 10–20 dB.
Highlights
In digital communications, we typically assume that the analog-to-digital converter (ADC) at the receiver provides a sufficiently fine grained quantization of the magnitude of the received signal
Case a) represents tk = tk,transition interval (TI) = Tk′ = Tk + β2, i.e., the zeros-crossing of the kth pulse of x(t) in the TI. This corresponds to the zero-crossing-shift error analyzed to obtain the lower and the upper bound on the mutual information rate in Sect
In order to construct an upper bound on the mutual information rate, the intersymbol interference (ISI) is not considered and only n (Tk′ ) contributes to z(Tk′ )
Summary
We typically assume that the analog-to-digital converter (ADC) at the receiver provides a sufficiently fine grained quantization of the magnitude of the received signal. We derive approximate lower and upper bounds on the mutual information rate of the real and bandlimited continuous-time additive Gaussian noise channel with 1-bit output quantization under an average power constraint. The filtered noise process at time instants spaced by a temporal distance larger than β can assumed to be uncorrelated and the possibility of a noise event deleting two consecutive zero-crossings - and, an entire symbol - can be neglected This argument is supported by the simulation results presented in Appendix A. In order to analyze the achievable rate, we use the temporal separation of the two error events (shifts and insertions of zero-crossings) to separately evaluate their impact This separation is given as long as there is only one zero-crossing in each transition interval (TI) [Tk , Tk + β].
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