Frame properties and conditions are determined that would minimize the error in signal reconstruction or estimation in the presence of noise and erasures. The special focus here is on stochastic models. These include estimating a random signal with zero mean and a general covariance matrix, minimizing the mean-squared error (MSE) when the frame coefficients are erased according to some a priori probability distribution in the presence of random noise, and also studying the use of stochastic frames in estimating a random signal. In estimating a random signal from noisy coefficients, when a frame coefficient is lost or erased, it is established that the MSE is minimized under certain geometric relationships between the frame vectors and the signal. When the coefficients are erased according to some a priori distribution, conditions are found for the norms of the frame vectors in terms of the probability distribution of the erasure so that the MSE is minimized. Results obtained here also show how using stochastic frames can lead to more flexibility in design and greater control on the MSE.

中文翻译：

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带有随机模型的帧，删除和信号估计

确定帧属性和条件，以在存在噪声和擦除的情况下将信号重建或估计中的错误最小化。这里的重点是随机模型。这些包括估计具有零均值和通用协方差矩阵的随机信号，在存在随机噪声的情况下根据某些先验概率分布擦除帧系数时，最小化均方误差（MSE），以及研究使用估计随机信号中的随机帧。在根据噪声系数估计随机信号时，当丢失或删除帧系数时，可以确定在帧矢量和信号之间的某些几何关系下，MSE最小。当根据一些先验分布擦除系数时，根据擦除的概率分布为帧向量的范数找到条件，从而最小化MSE。此处获得的结果还表明，使用随机框架可以如何在设计上带来更大的灵活性，并更好地控制MSE。