# A Decentralized Eigenvalue Computation Method for Spectrum Sensing Based on Average Consensus

• Jafar Mohammadi , Steffen Limmer and Sławomir Stańczak
From the journal Frequenz

## Abstract

This paper considers eigenvalue estimation for the decentralized inference problem for spectrum sensing. We propose a decentralized eigenvalue computation algorithm based on the power method, which is referred to as generalized power method GPM; it is capable of estimating the eigenvalues of a given covariance matrix under certain conditions. Furthermore, we have developed a decentralized implementation of GPM by splitting the iterative operations into local and global computation tasks. The global tasks require data exchange to be performed among the nodes. For this task, we apply an average consensus algorithm to efficiently perform the global computations. As a special case, we consider a structured graph that is a tree with clusters of nodes at its leaves. For an accelerated distributed implementation, we propose to use computation over multiple access channel (CoMAC) as a building block of the algorithm. Numerical simulations are provided to illustrate the performance of the two algorithms.

## Appendix

### Proof of convergence of ideal GPM

The proof follows from the same token given in [11], since for every eigenvalue λk we run the conventional power iteration. In order to satisfy the assumptions in the original proof by [11], we need the following lemma to hold:

Lemma 3:

The matrices Rk are all Hermitian, therefore diagonalizable.

Proof

It is straightforward if we write,

(22)Rk=Ri=1k1λiviviH.

Since R is Hermitian positive semi-definite, Rks are holding the same properties as well.

### Proof of Lemma 1

For GPM algorithm with qk1 as the initial input vector, and Theorem 8.2.1 in [11] we have:

dl2=|vlHvˆk1|2=(vlHRk1mqk1)2||Rk1mqk1||2

in which we used vˆk1=Rk1mqk1||Rk1mqk1||2. We further substitute

Rk1mqk1:=i=k1Kaiλimvi to achieve

dl2=vlHi=k1Kaiλimvi2i=k1Kai2λi2m
=al2λl2mi=k1Kai2λi2mblλlλk12m

where, the coefficients ai are defined as ai:=viHqk1 and bl:=al2ak12.

### Proof of Lemma 2

For GPM algorithm with qk1 is the initial input vector, and Theorem 8.2.1 in [11] we have:

λˆk1λk1=i=k1Kai2λi(2m+1)λk1i=k1Kai2λi(2m)
i=k1Kai2λi(2m+1)ak12λk1(2m+1)
=1+bk1′′λkλk1(2m+1).

where, ai:=viHqk1 and bk1′′:=i=kKai2ak12.

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