1
Parameters estimation using cuckoo search algorithm of harmonic signals in additive and multiplicative
noise
Haroun RAGUEB
Characterization and Instrumentation Division Welding and NDT Research Center
Cherraga, Algeria h.ragueb@csc.dz
Ali BADIDI BOUDA
Characterization and Instrumentation Division Welding and NDT research Center
Cherraga, Algeria a.badidi@csc.dz
Abstract—In this study we present a novel approach to estimate harmonics in multiplicative and additive noise using Cuckoo search algorithm. The estimation of parameters is formulated as a nonlinear optimization problem and simulations are carried out to assess the performance of the proposed algorithm.
Index Terms— Cuckoo search algorithm; Multiplicative noise;
Additive noise; Parameters estimation; Harmonics;
Metaheuristic.
I. INTRODUCTION
Additive noise models have been intensively studied due to the fact it contaminate a large class of physical mechanisms. However, multiplicative noise, or random amplitude modulation may be encountered in many practical systems [1], such as image processing systems [2,3], communication systems [4], and aerospace systems [5]. The problem of estimating harmonics in multiplicative noise is much more complicated than that of the additive case [6-8], unfortunately, most of the common estimating algorithm are unable to retrieve harmonics in the presence of multiplicative noise.
Cuckoo search is a metaheuristic search algorithm which has been proposed recently by Yang and Deb [9]. The algorithm is inspired by the reproduction strategy of cuckoos.
At the most basic level, cuckoos lay their eggs in the nests of other host birds which may be of different species. The host bird may discover that the eggs are not its own and either destroy the eggs or abandon the nest all to gather. This has resulted in the evolution of cuckoo eggs which mimic the eggs of local host birds [10]. To apply this as an optimization tool, Yang and Deb [9] used three idealized rules: (i) Each cuckoo lays one egg, which represents a set of solution co-ordinates at a time and dumps it in a random nest; (ii) A fraction of the nests containing the best eggs, or solution, will carry over to
the nest generation; (iii) The number of the nests is fixed and there is a probability that a host can discover an alien egg. If this happens, the host can either discard the egg or the nest and these results in building a new nest in a new location.
The CS algorithm has been applied successfully to variety of optimization problems [11-13]. In this paper, we propose the application of CS to harmonic estimation problems in multiplicative and additive noise.
II. PROBLEM STATEMENT
A signal with harmonics embedded in additive and multiplicative noise can be written as
1
( ) ( ) sin( ) ( )
N
n n n n
n
y t m t A ωt φ η t
=
=
∑
+ + (1)Where y(t) represents the distorted signal with additive noise η(t) and multiplicative noise mn(t). An, ωn and ϕn represent the amplitude, frequency and phase of the nth sinusoid respectively. The estimation of the amplitude An, frequencies ωn and phases ϕn in the presence of additive and multiplicative noise is an optimization problem. The problem is to minimize the objective functions based on error estimation. The estimated error is between the measured noisy harmonic signal and the estimated one, it is formulated as follow:
( ) ( ) ˆ( )
e t = y t −y t (2)
where
1
ˆ ˆ
ˆ( ) ˆ sin( )
N
n n n
n
y t µ ωt φ
=
=
∑
+ (3)With ˆ
µn is the estimated mean of the multiplicative noise, ˆ ωn
and φˆn are the estimated phase and frequency of the nth sinusoid. Let X denote the vector of all the unknown parameters as
[
1 2 1 2 1 2]
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ,ˆ , ...,ˆn, , , ..., n, , , ..., n
X = µ µ µ ω ω ω φ φ φ (4)
