
202 ◾ Gannavarpu Rajshekhar, Sai Siva Gorthi, and Pramod Rastogi
BOX6.2 MAXIMUM LIKELIHOOD ESTIMATION
Consider a quadratic phase signal as follows:
Γ(y) = exp[jϕ(y)] + η(y)
with
φ=α+α+α∀∈yyyy() [1,128]
01 2
2
The coefficients are given as [α
0
,α
1
,α
2
] = [0.9, 0.2, 0.004], and the noise
standard deviation σ
η
= 0.1. The phase is shown in Figure6.3a.
To estimate the quadratic coefficients, a maximum likelihood estimation
(MLE) technique is applied. Using Equation (6.26), the bivariate objective
function to be maximized is given as
∑
=Γ −+
=
Ux xyjxyxy
y
(, )()exp[( )]
12
1
128
2
20 40 60 80 100 120
y
100
80
60
40
20
0.1
0
x
2
x
1
–0.1
–2
0
2
20 40 60 80 100 120