Estimation methods for elementary chirp model parameters Anjali Mittal Rhythm Grover Debasis Kundu and Amit Mitra

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Estimation methods for elementary chirp model

parameters

Anjali Mittal, Rhythm Grover, Debasis Kundu, and Amit Mitra

Abstract

In this paper, we propose some estimation techniques to estimate the elemen-

tary chirp model parameters, which are encountered in sonar, radar, acoustics, and

other areas. We derive asymptotic theoretical properties of least squares estimators

and approximate least squares estimators for the one-component elementary chirp

model. It is proved that the proposed estimators are strongly consistent and follow

the normal distribution asymptotically. We also suggest how to obtain proper initial

values for these methods. The problem of ﬁnding initial values is a diﬃcult problem

when the number of components in the model is large, or when the signal-to-noise

ratio is low, or when two frequency rates are close to each other. We propose

sequential procedures to estimate the multiple-component elementary chirp model

parameters. We prove that the theoretical properties of sequential least squares

estimators and sequential approximate least squares estimators coincide with those

of least squares estimators and approximate least squares estimators, respectively.

To evaluate the performance of the proposed estimators, numerical experiments are

performed. It is observed that the proposed sequential estimators perform well even

in situations where least squares estimators do not perform well. We illustrate the

performance of the proposed sequential algorithm on a bat data.

Index Terms- Chirp model, approximate least squares, least squares, sequential least

squares, frequency rate, consistency, asymptotic normality.

1 Introduction

We consider the following model:

y(t) =

k=1

keiβ0

kt2+(t) ; t= 1, . . . , N, (1)

where, A0

ks are the complex-valued non-zero amplitude parameters and i=√−1. The

β0

ks are the frequency rate parameters, which strictly lie between 0 and 2πand they

are distinct. Also, (t)0s are the complex-valued noise random variables present in the

observed signal y(t). Detailed assumptions on (t)0s are stated later (see assumption 1).

Further, pis the number of chirp components and is assumed to be known. For the ob-

served data y(1) , y (2) , . . . , y (N), the problem here is to estimate the unknown amplitude

parameters A0

ks and the unknown frequency rates β0

ks.

arXiv:2210.05162v1 [stat.ME] 11 Oct 2022

The model (1) is named as multiple-component elementary chirp model. Although the

model (1) is widely applicable, the literature on this model is rather limited (see, for

example, Mboup and Adali [16] and Casazza and Fickus [5] for few notable exceptions).

Here, estimation of the chirp rate i.e. the frequency rate is of utmost importance. Some

of the applications where chirp rate estimation is considered as of prime importance can

be found in sonar pulse detection [1], in micro-Doppler signal analysis [3], in acoustic

signal analysis [2], in focusing on the synthetic aperture radar images [6], and many more.

There are some estimation methods which mainly aim on the estimation of the instan-

taneous frequency rate (IFR), which is twice the chirp rate. Estimator based on cubic

phase function (CPF) [17] is one of them and other methods motivated by CPF such as

Nonparametric Chirp-Rate estimator based on CPF [7], viterbi algorithm [8], integrated

CPF (ICPF) [21] and product CPF (PCPF) [22], to name a few, have been discussed in

the literature.

In this paper, we propose some estimation methods to estimate the elementary chirp

model (1) parameters. We propose least squares estimators (LSEs), approximate least

squares estimators (ALSEs), sequential LSEs and sequential ALSEs, study their theo-

retical asymptotic properties and compare their numerical performances. Model (1) is a

non-linear regression model, as should be noted. In the literature, theoretical results on

the general non-linear regression model have been established by Jennrich [12] and Wu

[23]. It has been observed that the suﬃcient conditions of Jennrich [12] and Wu [23] are

not satisﬁed by model (1) for the LSEs to be consistent. Thus, one cannot apply the

results of Jennrich [12] and Wu [23] directly to establish the theoretical properties of the

LSEs.

It is established that the least squares estimation method and the approximate least

squares estimation method provide the same optimal rates of convergence for the ampli-

tude parameters and the frequency rate parameters, that is OpN−1

2and OpN−5

2,

respectively. Here, OpN−δmeans NδOpN−δis bounded in probability. In all the

proposed estimation methods, we have to perform non-linear optimization for which we

need to employ a numerical method. To employ a numerical technique, a set of initial

values of the non-linear parameters is required. In the absence of good initial values (near

to the true parameter values), due to high non-linearity of the least squares surface, the

algorithm rather than converging to a global minimum, may converge to a local minimum,

see Rice and Rosenblatt [18]. Therefore, to choose good initial values, we use the con-

ventional grid search method. For the model (1), it is observed that the general-purpose

iterative procedures like Newton-Raphson, Gauss-Newton, or their diﬀerent versions, re-

quire a long time to converge to the LSEs even from a set of good initial values. Therefore,

we use downhill simplex method, to compute the LSEs and ALSEs, eﬃciently.

To obtain the initial values for the proposed methods in case of model (1), we have to

do a multi-dimensional grid search which is a numerically intense problem in itself. Fur-

ther, when the variance of the error random variable is high or when the two frequency

rates are close to each other, multi-dimensional grid search may result in the initial values

which are not close to the true parameter values. This may lead to incorrect parameter

estimates. To overcome this problem, we propose a sequential least squares estimation

method and a sequential approximate least squares estimation method. These sequential

methods lower the computational complexity by reducing the p-dimensional optimization

problem to p, 1-D optimization problems. We also establish the theoretical properties

of sequential LSEs and sequential ALSEs and ﬁnd that they have the same theoretical

properties as their respective LSEs and ALSEs.

Furthermore, we perform extensive simulation studies to evaluate the performance of the

proposed estimation methods for various sample sizes and error variances. We also obtain

frequency rate estimates using other standard estimation methodologies and assess the

eﬀectiveness of the proposed methods in comparison to these techniques. For the one-

component elementary chirp model, dechirping method [4] and CPF method [17] and for

the multiple-component elementary chirp model, dechirping method and PCPF method

[22] have been used for the comparative study. It is noted that the proposed estimators

perform quite satisfactorily. The mean squared errors (MSEs) of the proposed estimators

are close to their respective theoretical asymptotic variances. Another interesting obser-

vation that came out of these experiments is that the proposed sequential estimators are

able to resolve the frequency rates even when two frequency rates are close to each other

whereas LSEs are unable to do so at times. This motivates us to use the proposed sequen-

tial estimators for the implementation purposes due to their good theoretical properties

and also excellent simulation results in diﬀerent presented scenarios. We also show how

well the proposed sequential estimators work by ﬁtting the elementary chirp model to a

real-world data set.

The rest of the paper is structured as follows. In the next section, we present the statistical

properties of the LSEs and the ALSEs for the one-component elementary chirp model.

In section 3, we present theoretical properties of the sequential LSEs and the sequential

ALSEs for the multiple-component elementary chirp model (1). We provide simulation

results to validate the theoretical results of the proposed methods in section 4. Real data

analysis is presented in section 5. In section 6, the paper is concluded. All the necessary

proofs and results are presented in the appendices section of supplementary material.

2 One-Component Elementary Chirp Model

In this section, we consider the following one-component elementary chirp model :

y(t) = A0eiβ0t2+(t) ; t= 1, . . . , N. (2)

Here, we need to estimate the unknown amplitude parameter and the frequency rate

parameter under the following assumption on the error random variables (t)0s.

Assumption 1 (t)0s are i.i.d. complex-valued random variables with mean 0 and vari-

ance σ2

2for both real and imaginary parts. Also, fourth order moment of (t)exists. It is

assumed that real and imaginary parts of (t)are independent.

We denote by ARand AI, the real and the imaginary part of the A; respectively, and the

real and the imaginary part of the (t) are denoted as R(t) and I(t); respectively. We

will use the following notations: θ= (AR, AI, β), the parameter vector, θ0= (A0

R, A0

I, β0),

the true parameter vector, ˆ

θ=ˆ

AR,ˆ

AI,ˆ

β, the LSE of θ0and ˜

θ=˜

AR,˜

AI,˜

β, the

ALSE of θ0.

Under the above assumption on the noise, we present two estimation techniques: the least

squares estimation technique and the approximate least squares estimation technique, in

the following subsections. We also establish the statistical properties of these estimators.

2.1 Least Squares Estimators

Least squares estimation method is one of the most intutive choices to estimate the

unknown parameters of the model. Let us denote Θ1= [−M, M ]×[−M, M]×[0,2π] as

a parameter space. The assumption on the unknown parameters is mentioned below:

Assumption 2 Let θ0be an interior point of the parameter space Θ1, and |A0|>0.

The LSEs of the parameters of the model (2) are obtained by minimizing the following

residual sum of squares, say:

Q(θ) =

t=1 y(t)−Aeiβt2

,(3)

with respect to Aand βsimultaneously, where A=AR+iAI. In matrix notation, Q(θ)

can be expressed as follows;

Q(θ) = [Y−Z(β)A]H[Y−Z(β)A],(4)

where YN×1=y(1) , . . . , y (N)>and Z(β) = eiβ , . . . , eiβN2>.

From (4), note that Acan be separated from β, as it is a linear parameter. Therefore,

by using separable regression technique of Richards [19], for ﬁxed β, the LSE of Acan be

determined as

A(β) = hZ(β)HZ(β)i−1

Z(β)HY.(5)

By replacing Aby ˆ

A(β) in (4), we get

R(β) = Qˆ

A(β), β=YH[I−PZ]Y,(6)

where

PZ=Z(β)hZ(β)HZ(β)i−1

Z(β)H,

is the projection matrix on the column space spanned by the matrix Z(β). Thus, we can

obtain the LSE ˆ

βof β0by minimizing R(β) with respect to β. Then the LSE of βis used

to estimate the LSE of A, by substituting ˆ

βin (5).

The strong consistency and asymptotic normality of the LSEs are shown by the following

results.

Theorem 1 If assumptions 1 and 2 are satisﬁed, then ˆ

θis a strongly consistent estimator

of θ0, i.e.,

θa.s.

−−→ θ0as N→ ∞.

Proof See subsection A.1.

Theorem 2 If assumptions 1 and 2 hold true, then

(ˆ

θ−θ0)D−1d

−→ N3(0, σ2Σ−1)as N→ ∞,

where D=diag 1

√N,1

N2√Nand

Σ−1=





2+5A02

8|A0|2−5A0

RA0

8|A0|2

15A0

8|A0|2

−5A0

RA0

8|A0|21

2+5A02

8|A0|2−15A0

8|A0|2

15A0

8|A0|2−15A0

8|A0|245

8|A0|2







Proof See subsection A.1.

Although LSEs have the desired theoretical asymptotic properties, obtaining the least

squares estimators in practice is computationally quite challenging. For example, even

for a sinusoidal model, it has been studied that the least squares surface has a number

of local minima around the true parameter value (see, Rice and Rosenblatt [18], for more

details) and due to this reason most of the iterative methods converge to a local mini-

mum. Therefore, any iterative procedure requires a good set of initial values (close to the

true parameter value) for its convergence to the global minimum. We encounter a similar

problem for the elementary chirp model as well. Therefore, computing the LSEs for the

model (2) is also a numerically diﬃcult problem.

Periodogram estimators are one of the most prominent approaches for determining the

initial values of the sinusoidal model’s frequencies. Maximizing the following periodogram

function [20] provides these estimators:

I0(ω) = 1

N

t=1

y(t)e−iωt

,(7)

over the Fourier frequencies πk

N, k = 1, . . . , N −1. Now, we deﬁne a periodogram-type

function [11] analogous to the periodogram function which has the following mathematical

form:

I(β) = 1

N

t=1

y(t)e−iβt2

.(8)

Analogous to the periodogram estimator, periodogram-type estimator is obtained by max-

imizing (8) over the grid of the type 2πk

N2, k = 1, . . . , N2−1, which provides the estimator

of β0with the rate of convergence OP(N−2). This can be used as the initial value for the

frequency rate parameter.

It has been established in the literature that if I0(ω) is maximized over the continuous

range [0, π], then the obtained estimator possesses the same asymptotic properties as the

corresponding LSE and hence known as ALSE [20]. In the next subsection, we discuss

ALSEs for the elementary chirp model.

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EstimationmethodsforelementarychirpmodelparametersAnjaliMittal,RhythmGrover,DebasisKundu,andAmitMitraAbstractInthispaper,weproposesomeestimationtechniquestoestimatetheelemen-tarychirpmodelparameters,whichareencounteredinsonar,radar,acoustics,andotherareas.Wederiveasymptotictheoreticalpropertiesoflea...

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Estimation methods for elementary chirp model parameters Anjali Mittal Rhythm Grover Debasis Kundu and Amit Mitra

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