ALGORITHM EXTENSION OF CUBIC PHASE FUNCTION
FOR ESTIMATING QUADRATIC FM SIGNAL
Pu Wang, Jianyu Yang
School of Electronic Engineering
Univ. of Elec. Sci. and Tech. of China
610054 Chengdu, P. R. China
{pwang, jyyang }@uestc.edu.cn
Igor Djurovi´ c
Electrical Engineering Department
University of Montenegro
81000 Podgorica, Montenegro
igordj @cg.ac.yu
ABSTRACT
In this paper, an extended algorithm for parameter estimation
of quadratic FM signal is derived by exploring the time di-
versity in the cubic phase (CP) function. The performance of
the proposed algorithm is analyzed in terms of estimate bias
and variance, and compared with other methods. Although
the proposed algorithm employs a fourth-order nonlinearity
which results in higher threshold SNR, it provides a number
of advantages, such as low mean-square error (MSE) for the
estimates at high SNR and simply extension for multicompo-
nent signals. Extension to cubic FM signal is also discussed.
The theoretical analysis is verified by the simulation results.
Index Terms— Parameter estimation, FM signal, statisti-
cal signal processing.
1. INTRODUCTION
In the signal processing literature, considerable attention has
been paid to parameter estimation of the frequency-modulated
(FM) signal from noisy observations. The FM signal can be
found in a number of applications such as radar, sonar, geo-
physics, and biomedicine [1], [2]. This paper focuses on the
quadratic FM signal and also discusses the cubic FM signal.
The most accurate way for analyzing the quadratic FM
signal is the maximum likelihood (ML) estimation. It yields
optimal results but requires a three-dimensional maximiza-
tion, and thus it is computational exhausting. Moreover, if
the objective function is not convex, the maximization is easy
to converge to local maxima. To avoid the multidimensional
search, the suboptimal approaches such as the high-order am-
biguity function (HAF) [1], [3], integrated general ambiguity
function (IGAF) [4], and product HAF (PHAF) [5], are pro-
posed. Recently, a bilinear transform — the CP function was
presented in [2] and [6]. For a quadratic FM signal defined as
s(n)= Ae
jφ(n)
=Ae
j(a0+a1n+a2n
2
+a3n
3
)
,
−
(N − 1)
2
≤ n ≤
(N − 1)
2
, (1)
where A, φ(n), and {a
i
}
3
i=0
are the amplitude, phase and
phase coefficients respectively, and N is odd, the CP func-
tion is presented as
CP(n, Ω) =
+∞
0
s(n + τ )s(n − τ )e
-jΩτ
2
dτ. (2)
Substituting s(n) in (2) with (1), the result is
s(n + τ )s(n − τ )= A
2
e
j2[φ(n)+(a2+3a3n)τ
2
]
. (3)
From (2) and (3), the CP function will peak at 2(a
2
+
3a
3
n), which is the instantaneous frequency rate (IFR) of (1)
[2], [6]. Once the IFR is obtained, the parameters, a
2
and a
3
,
can be estimated by selecting two different time positions and
solving the resulting equations set. In this paper, we explore
the time diversity in the CP function by using two special time
positions, which are symmetric with respect to origin, i.e.,
one is n and another −n. Although this extension results in
fourth-order nonlinearities, it offers the following advantages:
• low mean-square error (MSE) at high SNR for estimat-
ing a
3
;
• simple extension to multicomponent signals.
This paper is organized as follows. In Section 2, we present
the algorithm for estimating the quadratic FM signal in two
steps. Section 3 derives the statistical results for the estimate
using the first-order permutation analysis. In Section 4, ex-
tension to multicomponent case is considered. The simula-
tion results are provided to validate the theoretical results in
Section 5. Section 6 focuses on further development. Finally,
conclusions are drawn in Section 7.
2. THE PROPOSED ALGORITHM
Motivated by the CP function, we further exploit the time di-
versity on the basis of the CP function, which forms a fourth-
order nonlinear estimator — the Generalized Cubic Phase (GCP)
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