高效自适应算法：随机丢失观测下的叠加指数信号参数估计
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"这篇研究论文提出了一种针对观测值随机丢失的叠加指数信号参数估计的高效自适应算法，适用于存在乘性与加性噪声的环境。该算法在保证估计精度的同时，具有较高的计算效率，旨在解决观测数据不完整时的信号处理问题。"
在信号处理领域，参数估计是关键任务之一，特别是在指数信号模型的应用中，如地震学、医学成像和通信系统等。叠加指数信号模型通常用于表示多个独立但相互叠加的指数过程，这些过程可能受到不同类型的噪声干扰。本文关注的是在这种模型中，当观测数据因随机因素而部分缺失时如何进行参数估计。
文章指出，传统的最小二乘估计器（Least Squares Estimators, LSEs）在面对缺失观测值时可能会失去有效性，因为它们通常假设所有观测数据都是完整的。为了解决这个问题，论文提出了一种新的自适应算法，该算法能够在数据缺失的情况下仍能提供无偏且一致的估计，即在样本数量趋于无穷大时，估计值能收敛到真实参数。
论文详述了新算法的设计原理，包括如何处理随机缺失的数据以及如何在乘性与加性噪声环境中保持估计的稳定性和准确性。算法的核心思想可能涉及递归更新和权重分配，以适应数据的缺失模式，并有效地估计信号参数。
此外，作者还可能对算法的性能进行了理论分析和数值模拟验证。理论分析部分可能涉及算法的渐近性质，如无偏性和一致性；数值模拟部分则可能展示了在不同数据丢失率和噪声水平下的估计精度，以证明算法的优越性。
关键词涵盖了关键概念，包括叠加指数信号、最小二乘估计、缺失观测、乘性与加性噪声，以及无偏且一致的估计器。这些关键词揭示了研究的焦点和方法论，表明了该算法在处理复杂噪声环境和不完整数据集时的能力。
这篇论文对于那些在实际应用中面临数据缺失问题的信号处理和数据分析专家来说具有重要的参考价值。通过提供一个计算效率高且适应性强的解决方案，该算法有望改善在不确定观测条件下的信号参数估计效果。
150 J. Bian et al. / Digital Signal Processing 48 (2016) 148–162
The rest of the paper is organized as follows. In Section 2, we
study the LSEs and obtain the asymptotic distribution of the pa-
rameters
of frequencies, amplitudes and phases for the considered
model. The initial estimation of the iterative algorithm is presented
in Section 3. The iterative algorithm is presented in Section 4. In
Section 5, we present the numerical experiments and finally the
conclusion is presented in Section 6. All the proofs are provided in
Appendices A–C.
2. Least squares estimators
It is known that the LSEs have the best convergence rate for
parameters estimation of superimposed exponential signals or har-
monic
signals in additive noise when the observations are com-
plete
[26]. However, the LSEs for the parameters of superimposed
exponential signals with randomly missing observations have not
been studied. It is necessary to study the LSEs for parameters of
model (1) first. The LSEs of the unknown parameters of model
(1) can be obtained by minimizing the residual sum of squares
namely,
Q (
ˆ
θ ) =
N
t=1
y(t) −
l
k=1
ˆ
μ
k
(1 − p)e
i(
ˆ
ω
k
t+
ˆ
ϕ
k
)
2
(2)
where
ˆ
θ = (
ˆ
θ
1
,
ˆ
θ
2
, ···,
ˆ
θ
l
) is the LSEs of θ = (θ
1
, θ
2
, ···, θ
l
). Here
ˆ
θ
k
= (
ˆ
ω
k
,
ˆ
ϕ
k
,
ˆ
μ
k
), θ
k
= (ω
k
, ϕ
k
, μ
k
) (k = 1, 2, ···, l). It is noted
that the second term in the square sum of Eq. (2) is actu-
ally
the mean of y(t). If we denote the block diagonal matrix
D as D = diag{D
1
, D
2
, ···, D
l
} where the block element D
j
=
diag{N
3/2
, N
1/2
, N
1/2
}, for j = 1, 2, ···, l, then the asymptotic dis-
tribution
of the LSEs of θ can be obtained as follows.
Theorem
1. Under the assumption of model (1), (
ˆ
θ −θ )D converges in
distribution to a 3l-variate normal distribution, with mean vector zero
and the covariance matrix which is a block diagonal matrix with the
structure = diag{
1
,
2
, ···,
l
}, where
j
=
⎡
⎢
⎣
σ
2
−σ
2
j
+p
k=j
μ
2
k
(1−p)μ
2
j
6 −3
−32
0
0
σ
2
+σ
2
j
+p
k=j
μ
2
k
2(1−p)
⎤
⎥
⎦
( j = 1, 2, ···, l) and σ
2
=σ
2
0
+
l
k
=1
σ
2
k
which is the total variance of
multiplicative and additive noise.
Proof. See
Appendix A. 2
It is immediate from Theorem 1 that the LSE of the frequency
ω
j
( j = 1, 2, ···, l) has the following asymptotic distribution
N
3/2
(
ˆ
ω
j
−ω
j
)
L
−→ N
0,
6(σ
2
−σ
2
j
+ p
k=j
μ
2
k
)
(1 − p)μ
2
j
(3)
where ‘N (·)’ denotes normal distribution and ‘L ’ means conver-
gence
in distribution.
2
3. Initial estimator
The initial estimator for the frequencies of model (1) can be
obtained by maximizing the Lomb Periodogram function I(ω) as
follows
2
Here convergence in distribution means that for the continuous point t of the
distribution function of
ˆ
ω
j
, i.e. F
N
(t), lim
N→∞
F
N
(t) = F (t) where F (t) is the limit
distribution function of
ˆ
ω
j
.
I(ω) =
1
N
N
t=1
x(t)s(t)e
−iωt
.
(4)
The estimators obtained by finding p local maxima of I(ω) achieve
the best possible rate and are asymptotically equivalent to the
LSEs when there is no missing observation and is known as ap-
proximate
least squares estimators (ALSEs) in the literature [27].
If the values of the missing observations are taken as zeros, the
frequency can also be obtained by searching the peaks of the pe-
riodogram
maximizers, which is termed as Lomb periodogram es-
timation
[7]. We take the Lomb periodogram maximizers as the
initial estimator of the iterative algorithm. It is noted that the
Lomb periodogram estimators obtained under the condition that
frequencies are Fourier frequencies provide estimators with con-
vergence
rate only O
p
(N
−1
) [25]. Here a frequency is a Fourier
frequency if it is of the form λ =
2πk
N
, for some integer 1 ≤k ≤ N.
4. Iterative algorithm
In this section, we will discuss the iterative procedure and the
asymptotic property of the estimators of frequencies in model (1)
by
the iterative procedure below. Given a consistent estimator
˜
ω
j
of model (1), we compute
ˆ
ω
j
for j = 1, 2, ···, l as follows
ˆ
ω
j
=
˜
ω
j
+
f
s
N
2
Im
A
N
( j)
B
N
( j)
(5)
where
A
N
( j) =
N
t=1
y(t)(t −
N
2
)e
−i
˜
ω
j
t
, B
N
( j) =
N
t=1
y(t)e
−i
˜
ω
j
t
, (6)
f
s
=
N
3
(1 − p)
N
t
=1
s(t)t(t −
N
2
)
(7)
and Im[·] denotes the operation of taking the imaginary part of
a complex number. We can start with any consistent estimator
˜
ω
j
and improve it step by step using Eq. (5). The performance of the
estimators of frequencies for each iteration and the corresponding
asymptotic distribution are presented in the following theorem.
Theorem 2. If for j = 1, 2, ···, l,
˜
ω
j
− ω
j
= O
p
(N
−1−δ
), where δ ∈
(
0,
1
2
], then
(a)
ˆ
ω
j
−ω
j
= O
p
(N
−1−2δ
), if δ
1
4
,
(b) N
3
2
(
ˆ
ω − ω)
L
−−→ N
l
(0, ), if δ>
1
4
where
= diag
6(σ
2
−σ
2
1
+ p
k=1
μ
2
k
)
μ
2
1
(1 − p)
,
6(σ
2
−σ
2
2
+ p
k=2
μ
2
k
)
μ
2
2
(1 − p)
,
···,
6(σ
2
−σ
2
l
+ p
k=l
μ
2
k
)
μ
2
l
(1 − p)
and σ
2
=
l
j
=1
σ
2
j
+ σ
2
0
,
ˆ
ω = (
ˆ
ω
1
,
ˆ
ω
2
, ···,
ˆ
ω
l
), ω = (ω
1
, ω
2
,
···,
ω
l
).
Proof. See
Appendix B. 2
Remark 1. It is noted that the iterative statistics in Eq. (5) are
different from those in [21–23] as the coefficient f
s
in Eq. (5)
is
not fixed and it varies with different missing distributions of
time points in each iteration while the iterative coefficient used in
[21–23] is fixed to a constant. Moreover, the iterative coefficient in
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