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Register a new userThe Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points
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Alex Dytso
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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This paper studies an $n$-dimensional additive Gaussian noise channel with a peak-power-constrained input. It is well known that, in this case, when $n=1$ the capacity-achieving input distribution is discrete with finitely many mass points, and when $n>1$ the capacity-achieving input distribution is supported on finitely many concentric shells. However, due to the previous proof technique, neither the exact number of mass points/shells of the optimal input distribution nor a bound on it was available. This paper provides an alternative proof of the finiteness of the number mass points/shells of the capacity-achieving input distribution and produces the first firm bounds on the number of mass points and shells, paving an alternative way for approaching many such problems. Roughly, the paper consists of three parts. The first part considers the case of $n=1$. The first result, in this part, shows that the number of mass points in the capacity-achieving input distribution is within a factor of two from the downward shifted capacity-achieving output probability density function (pdf). The second result, by showing a bound on the number of zeros of the downward shifted capacity-achieving output pdf, provides a first firm upper on the number of mass points. Specifically, it is shown that the number of mass points is given by $O(mathsf{A}^2)$ where $mathsf{A}$ is the constraint on the input amplitude. The second part generalizes the results of the first part to the case of $n>1$. In particular, for every dimension $n>1$, it is shown that the number of shells is given by $O(mathsf{A}^2)$ where $mathsf{A}$ is the constraint on the input amplitude. Finally, the third part provides bounds on the number of points for the case of $n=1$ with an additional power constraint.
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This work considers a Poisson noise channel with an amplitude constraint. It is well-known that the capacity-achieving input distribution for this channel is discrete with finitely many points. We sharpen this result by introducing upper and lower bounds on the number of mass points. Concretely, an upper bound of order $mathsf{A} log^2(mathsf{A})$ and a lower bound of order $sqrt{mathsf{A}}$ are established where $mathsf{A}$ is the constraint on the input amplitude. In addition, along the way, we show several other properties of the capacity and capacity-achieving distribution. For example, it is shown that the capacity is equal to $ - log P_{Y^star}(0)$ where $P_{Y^star}$ is the optimal output distribution. Moreover, an upper bound on the values of the probability masses of the capacity-achieving distribution and a lower bound on the probability of the largest mass point are established. Furthermore, on the per-symbol basis, a nonvanishing lower bound on the probability of error for detecting the capacity-achieving distribution is established under the maximum a posteriori rule.
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This paper studies the capacity of the peak-and-average-power-limited Gaussian channel when its output is quantized using a dithered, infinite-level, uniform quantizer of step size $Delta$. It is shown that the capacity of this channel tends to that of the unquantized Gaussian channel when $Delta$ tends to zero, and it tends to zero when $Delta$ tends to infinity. In the low signal-to-noise ratio (SNR) regime, it is shown that, when the peak-power constraint is absent, the low-SNR asymptotic capacity is equal to that of the unquantized channel irrespective of $Delta$. Furthermore, an expression for the low-SNR asymptotic capacity for finite peak-to-average-power ratios is given and evaluated in the low- and high-resolution limit. It is demonstrated that, in this case, the low-SNR asymptotic capacity converges to that of the unquantized channel when $Delta$ tends to zero, and it tends to zero when $Delta$ tends to infinity. Comparing these results with achievability results for (undithered) 1-bit quantization, it is observed that the dither reduces capacity in the low-precision limit, and it reduces the low-SNR asymptotic capacity unless the peak-to-average-power ratio is unbounded.
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