This paper deals with an extension of Papoulis’ generalized sampling expansion (GSE) of bandlimited signals to a case where a white noise, filtered to the signal's band, is added before sampling. We look for the best sampling scheme that maximizes the capacity or minimizes the mean-squared error (MSE) of the sampled channel between the input signal and the $ M$ sampled outputs signals, e.g. channels. Due to the noise it is beneficial to sample at a total rate higher than Nyquist, yet there is no gain in sampling each signal above Nyquist; thus the total rate considered, normalized by Nyquist rate, is between 1 to $M$ . To avoid preference to any channel, the paper assumes that the channels are composed of all-pass linear time-invariant (LTI) systems. For the case where the total normalized rate is between $ M-1$ and $ M$ , we show that the best scheme samples $ M-1$ outputs at Nyquist rate and the last output at the remaining rate. Surprisingly, equal sampling is suboptimal in general, but when there is an integer relation between $ M$ and the total normalized rate, equal sampling achieves the optimal performance. We also make a conjecture on other rates that the best scheme either samples some channels at Nyquist and one another channel at the remaining rate, or sample the channels equally. This conjecture is supported by simulation for $M\leq 5$ . Finally, we discuss the relation between maximizing the capacity and minimizing the MSE.

中文翻译：

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带限噪声多输出通道的最佳采样

本文讨论了Papoulis对带宽受限信号的广义采样扩展（GSE）的扩展，以至于在采样之前添加了过滤到信号频带的白噪声的情况。我们寻求最佳的采样方案，以最大程度地提高输入信号与输入信号之间的采样通道的容量或最小化均方误差（MSE）。$ M $采样的输出信号，例如通道。由于噪声的原因，以高于Nyquist的总速率进行采样是有益的，但是对高于Nyquist的每个信号进行采样都没有任何好处。因此，考虑的总速率（通过奈奎斯特速率归一化）介于1到$ M $ 。为了避免偏爱任何通道，本文假设通道由全通线性时不变（LTI）系统组成。对于总归一化率介于$ M-1 $ 和 $ M $ ，我们展示了最佳方案样本 $ M-1 $以奈奎斯特速率输出，最后一个以剩余速率输出。令人惊讶的是，一般情况下，相等采样次优，但是当两者之间存在整数关系时$ M $以及总归一化速率，相等采样可实现最佳性能。我们还对其他速率进行了猜测，即最佳方案是在Nyquist采样某些通道，然后以剩余速率采样另一个通道，或者对这些通道进行均等采样。该猜想得到了以下方面的仿真支持：$ M \ leq 5 $ 。最后，我们讨论了最大化容量和最小化MSE之间的关系。