Generalized sampling: stability and performance analysis
Generalized sampling: stability and performance analysis
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DOI：
10.1109/78.650255
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发表时间：
1997-12
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通讯作者：
M. Unser;J. Zerubia
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作者：
M. Unser;J. Zerubia
广义抽样提供了一种从m个线性移位不变系统的响应样本中恢复未知输入函数f(X)/SPL epsiv//SPL Hscr/的一般机制。该系统可以被设计成执行f(X)到重建子空间V(/SPLPhi/)=span{/SPLPhi/(x-k)}/subk/SPLepsiv/Z/上的投影；例如，具有/SPLPhi/(X)=sinc(X)的带限信号族。这意味着，当输入信号包括在V(/SPL Phi/)中时，重建将是完美的：V(/SPL Phi/)：Papoulis(1977)广义采样理论的传统框架。否则，我们得到一个信号近似f(X)/spl epsiv/V(/spl phi/)，它与f(X)在产生相同测量的意义上是一致的。为了刻画算法的稳定性，我们证明了出现在广义采样重构公式中的对偶综合函数构成了V(/SPL Phi/)的Riesz基，并利用相应的Riesz界定义了系统的条件数。然后，我们使用这些结果来分析交错抽样和导数抽样的各种情况的稳定性。接下来，我们考虑性能问题，一旦我们将该方法的适用性扩展到任意输入函数，即当/SPL Hscr/远大于V(/SPL Phi/)并且重构不再精确时，该问题就变得相关。我们证明了广义抽样解本质上等价于最优最小误差逼近。然后，我们对分析滤子在L/SUB 2/中的情况进行了详细的分析，并显式地确定了所有相关的束缚常数。最后，我们用一个交错抽样的例子来说明这些不同的计算。
Generalized sampling provides a general mechanism for recovering an unknown input function f(x)/spl epsiv//spl Hscr/ from the samples of the responses of m linear shift-invariant systems sampled at 1/mth the reconstruction rate. The system can be designed to perform a projection of f(x) onto the reconstruction subspace V(/spl phi/)=span {/spl phi/(x-k)}/sub k/spl epsiv/Z/; for example, the family of bandlimited signals with /spl phi/(x)=sinc(x). This implies that the reconstruction will be perfect when the input signal is included in V(/spl phi/): the traditional framework of Papoulis' (1977) generalized sampling theory. Otherwise, one recovers a signal approximation f(x)/spl epsiv/V(/spl phi/) that is consistent with f(x) in the sense that it produces the same measurements. To characterize the stability of the algorithm, we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a Riesz basis of V(/spl phi/), and we use the corresponding Riesz bounds to define the condition number of the system. We then use these results to analyze the stability of various instances of interlaced and derivative sampling. Next, we consider the issue of performance, which becomes pertinent once we have extended the applicability of the method to arbitrary input functions, that is, when /spl Hscr/ is considerably larger than V(/spl phi/), and the reconstruction is no longer exact. We show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation. We then perform a detailed analysis for the case in which the analysis filters are in L/sub 2/ and determine all relevant bound constants explicitly. Finally, we use an interlaced sampling example to illustrate these various calculations.