### 9.6.4 A Reduced-Complexity Optimal Sensing Decision-Making Algorithm with Independent Channels

Although it may not always be justifiable, an assumption that may hold sometimes in practice is that of independent channel dynamics. In the extreme case of all channels having independent idle/busy state dynamics, it can be shown that one may derive a new sufficient-statistic vector for the subband selection decision problem that has only a dimension of
, instead of 2^{M} as in the general case assumed in the previous section. Indeed, when channels are independent, a sufficient statistic for decision-making is simply the collection of *M* beliefs of each channel’s state given the history, where belief is the posterior probability of state given the history. Since each channel state is binary, we only need to provide one belief value for each channel (either the belief on state being idle or busy). Thus, an *M*-length vector of each channel’s belief state is sufficient. This dramatically reduces the computational complexity from being exponential in the total number of channels *M* to only being linear in *M*. This independent channel model was used in [129] in the context of DSS cognitive radios as we will discuss later in Chapter 13.

Let us keep track of the belief of each channel state *k*, for *k* = 1,… ,*M*, being idle (i.e., state 1):

where (*i, j*) pair for a given *k* is given by (9.20) and ( ...

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