**6. Numerical examples**

In this section, in order to evaluate the performance of Algorithms 1 and 2, we present some examples with a Gaussian signal *s* observed in the presence of Gaussian sensor noises.

The random signal *s* and observation noises *v*1, *v*2, … , *vL* are as follows:

$$H\_0: \ y\_j = \upsilon\_j; \qquad H\_1: \ y\_j = \mathfrak{s} + \upsilon\_j, \qquad for \, j = 1, \ldots, L,\tag{45}$$

where *v*1, *v*<sup>2</sup> … , *vL*, *s* are all mutually independent and

$$v\_j \sim \mathcal{N}(\mathbf{0}, \mathbf{0}.6), \ s \sim \mathcal{N}(\mathbf{1}, \mathbf{0}.4), \ for j = 1, \ldots, L.$$

Thus, given *H*<sup>0</sup> and *H*1, the two conditional probability density functions are

$$p(\boldsymbol{y}\_1, \boldsymbol{y}\_2, \dots, \boldsymbol{y}\_L | \boldsymbol{H}\_0) \sim N(\boldsymbol{\mu}\_0, \boldsymbol{\Sigma}\_0), \qquad p(\boldsymbol{y}\_1, \boldsymbol{y}\_2, \dots, \boldsymbol{y}\_L | \boldsymbol{H}\_1) \sim N(\boldsymbol{\mu}\_1, \boldsymbol{\Sigma}\_1),$$

where *μ*0, *μ*1, Σ0, Σ<sup>1</sup> are easily obtained from the relationship of *s*, *v*1, *v*2, … , *vL*.

Assume that each sensor is required to transmit a bit through a channel with probabilities of *Pceo <sup>j</sup>* <sup>¼</sup> *Pce*<sup>1</sup> *<sup>j</sup>* ¼ *p*, where *p* ¼ 0*:*05, 0*:*15, 0*:*3, for *j* ¼ 1, 2, … , *L*. In the cost function (2), let the cost coefficients *C*<sup>00</sup> ¼ *C*<sup>11</sup> ¼ 0 and *C*<sup>10</sup> ¼ *C*<sup>01</sup> ¼ 1. The receiver operating characteristics (ROC) curves are used to evaluate the performance of the algorithms. *Pf* and *Pd* denote the probability of false alarm and the probability of detection, respectively.
