Joint Delay-Doppler Estimation Performance in a Dual Source Context

Abstract

Evaluating the time-delay, Doppler effect and carrier phase of a received signal is a challenging estimation problem that was addressed in a large variety of remote sensing applications. This problem becomes more difficult and less understood when the signal is reflected off one or multiple surfaces and interferes with itself at the receiver stage. This phenomenon might deteriorate the overall system performance, as for the multipath effect in Global Navigation Satellite Systems (GNSS), and mitigation strategies must be accounted for. In other applications such as GNSS reflectometry (GNSS-R) it may be interesting to estimate the parameters of the reflected signal to deduce the geometry and the surface characteristics. In either case, a better understanding of this estimation problem is directly brought by the corresponding lower performance bounds. In the high signal-to-noise ratio regime of the Gaussian conditional signal model, the Cramér-Rao bound (CRB) provides an accurate lower bound in the mean square error sense. In this article, we derive a new compact CRB expression for the joint time-delay and Doppler estimation in a dual source context, considering a band-limited signal and its specular reflection. These compact CRBs are expressed in terms of the baseband signal samples, making them especially easy to use whatever the baseband signal considered, therefore being valid for a variety of remote sensors. This extends existing results in the single source context and opens the door to a plethora of usages to be discussed in the article. The proposed CRB expressions are validated in two representative navigation and radar examples.

1. Introduction

The joint delay-Doppler estimation problem has been addressed for years as it is a key step in a plethora of applications such as radar, sonar, communications, navigation or remote sensing [1,2,3,4,5,6,7]. For instance, this estimation is performed by a receiver in order to localize and track the position of sources such as a transmitting satellite, like in Global Navigation Satellite System (GNSS), or a radiated target from which the system receives the backscattered signal. Under nominal conditions the receiver collects only line-of-sight (LOS) signals. However, in challenging realistic scenarios like urban canyons, extended targets or in conventional reflectometry configurations, a series of non-line-of-sight (NLOS) echoes may be collected by the receiver. Those echoes, often referred to as multipath, are usually the consequence of good enough reflecting surfaces that are radiated by the sources of interest, which in turn reach the receiver on top of the LOS signals.

In some applications the user tries to mitigate the multipath effect because it tends to degrade the quality of the LOS signals of interest, directly affecting the overall system performance. This is the case in standard GNSS where numerous techniques were developed to get rid of the multipath interference (e.g., MEDLL [8], MMT [9], Vision Correlator [10]). Indeed, GNSS multipath remains one of the main sources of error in challenging urban environments, therefore being a key point to be addressed for safety-critical applications requiring a precise receiver position, such as in intelligent transportation systems. In other applications the user wants to collect and exploit such interference in order to extract features related to the secondary propagation medium or the reflecting surface properties. A typical example is GNSS reflectometry (GNSS-R) where the secondary signal is the main source of information. In GNSS-R, a first approach is to have two separated antennas: one pointing upward collecting the LOS signal and the other one pointing downward, collecting the reflected signal [11,12]. A second approach is to collect both signals with a single antenna and process the resulting superimposed signals [13,14]. Both GNSS multipath (i.e., considering one LOS signal and a main secondary path contribution) and the single antenna GNSS-R approach are typical scenarios of a dual source estimation problem, being the main problem of interest in this contribution.

When it comes to the estimation of deterministic parameters (i.e., such as time-delay and Doppler in this contribution), a powerful tool to determine the best statistical performance are Cramér-Rao Bounds (CRBs). Indeed, these bounds accurately describe the ultimate achievable minimum variance that an unbiased estimator can reach for a given problem. The CRBs for the delay and delay-Doppler estimation have been investigated in several contributions for different signal models, considering signals with a finite [1,15,16,17,18] or infinite [19] bandwidth. Regarding the contributions dealing with finite bandwidth signals it is common to resort to the narrowband signal assumption, that is, the impact of the Doppler effect on the baseband signal is neglected and only amounts to a time-varying frequency shift [20]. The derivation of such bounds are often based on the Slepian-Bangs formulas [21] or more general theoretical expressions for Gaussian observation models [22,23,24]. When considering a multiple source signal model the CRB derivation complexity increases significantly. Examples of those CRB expressions were derived for the case of extended targets [25] and in GNSS-R for altimetry purposes [26,27,28]. Despite these contributions, a dual source delay-Doppler estimation compact closed-form CRB expression for generic band-limited signals is not available in the literature, being an important missing point and a result of broad interest.

In either the single or multiple source context, the problem can be formulated as a Conditional Signal Model (CSM) [16,17,18,29,30,31,32,33]. Such model and the corresponding Conditional Maximum Likelihood Estimator (CMLE) lead to interesting properties: the CMLE performance is accurately described by the CRB in the asymptotic region of operation, i.e., it is efficient in the high signal-to-noise ratio (SNR) regime and/or the large sample regime [29,34]. This makes the CMLE an interesting tool to validate CRB expressions, as done for instance in [16,17,18,35]. Unfortunately, the main CMLEs drawback is their heavy computational load, which the practitioner usually wants to avoid, and therefore many solutions to reduce this load were proposed in the literature. A first approach consists in using sub-optimal techniques, where the multidimensional search is substituted with a simpler one-dimensional one, such as in the Capon [36] or MUSIC [37,38] methods. Another approach is to reduce the computational burden by splitting the multidimensional optimization search in a number of recursive searches. The Alternating Projection algorithm [39] and the relaxed version of the CLEAN algorithm [40] are two well-known examples of this approach.

1.1. Contributions

Keeping in mind the problem of interest, that is, the dual source delay-Doppler estimation with a generic band-limited signal, the main contributions of this article are as follows:

• A compact Fisher Information Matrix (FIM) for the general dual source narrowband CSM (i.e., where the Doppler effect on the band-limited baseband signal is not considered and amounts to a frequency shift) is provided in Section 3. This extends the work of [17,18] dealing with the single source narrowband CSM and completes the work of [28] in which the FIM is given for the specific case of a static, ground-based receiver, leading yo an identical Doppler effect for the two paths;
• The FIM is expressed in an easy to use form, based on the baseband signal samples;
• The theoretical results are supported and validated by the implementation of a dual source CMLE [1] and sub-optimal estimators such as a CLEAN RELAX Estimator (CRE) [40];
• The formal connection with existing literature [28] and possible uses of the new CRB are discussed;

1.2. Notation

The notation adopted is as follows: scalars are represented by italic lowercase characters as in b, vectors are represented by bold lowercase characters as in $\mathbf{b}$, and matrices are represented by bold uppercase characters as in $\mathbf{B}$. The transpose operation is indicated by the superscript ${}^T$, the conjugate transpose by the superscript ${}^H$, and the conjugate operation by the superscript ${}^{*}$. ${\mathbf{I}}_n$ represents the identity matrix of dimension n, $\mathrm{Re}\{·\}$ and $\mathrm{Im}\{·\}$ refer to the real part and the imaginary part.

1.3. Organization

Section 2 provides details on the context and the corresponding narrowband signal model. In Section 3 the CRB derivation is presented for the delay, Doppler, amplitude and phase estimation of a band-limited signal in the dual source context. Section 4 introduces the CMLE and a sub-optimal estimator. These two estimators are used to validate the new CRBs. Section 5 provides results for two representative examples. Section 6.1 relates the presented results to the literature and Section 6.2 gives an overview of how the current study can branch out into specific studies. Finally a conclusion is drawn in Section 7.

2. Signal Model

2.1. Single Antenna Receiver

In order to study the dual source problem we consider a receiver with a single antenna able to collect both a LOS signal (signal labelled 0) and a single NLOS reflection (signal labelled 1), which may be reflected for instance in a ground, sea or building surface (see Figure 1). Here we assume a purely specular reflection with no scattering effect as considered in [28,41]. Note that this model is also related to the multipath model used in the GNSS literature ((6.42) [42]).

We consider that a transmitter T and a receiver R have uniform linear motions such that the positions can be described as ${\mathbf{p}}_T\left(t\right)={\mathbf{p}}_T+{\mathbf{v}}_{T}t$ and ${\mathbf{p}}_R\left(t\right)={\mathbf{p}}_R+{\mathbf{v}}_{R}t$, where $\mathbf{p}$ and $\mathbf{v}$ are the position and velocity vectors, respectively and t is the time. Under such conditions, the distance between T and R can be approximated by a first order distance-velocity model [20]:

$$\parallel {\mathbf{p}}_{TR}\left(t\right)\parallel \triangleq \parallel {\mathbf{p}}_R\left(t\right)-{\mathbf{p}}_T(t-\tilde{\tau }\left(t\right))\parallel =c\tilde{\tau }\left(t\right)\approx d+vt,\mathrm{and}\tilde{\tau }\left(t\right)\approx \tau +bt,\tau =\frac{d}{c},b=\frac{v}{c},$$

(1)

where d is the absolute distance between T and R, v is the radial velocity between T and R, $\tau$ is the time-delay due to the propagation path and $(1-b)$ is the dilatation induced by the Doppler effect and c the speed of light in vacuum.

2.2. Narrowband Signal Model

A band-limited signal $s\left(t\right)$, with bandwidth B, transmitted over a carrier frequency $f_c$ (${\lambda }_c=c/f_c$) is considered in this study. This signal can be expressed as

$$s\left(t\right)=\sum _{n=N_1^{\prime }}^{N_2^{\prime }}s\left(\frac{n}{F_s}\right)\sin \mathrm{c}\left(\pi F_s\left(t-\frac{n}{F_s}\right)\right)\leftrightharpoons S\left(f\right)=\frac{1}{F_s}\sum _{n=N_1^{\prime }}^{N_2^{\prime }}s\left(\frac{n}{F_s}\right)e^{-j2\pi n\frac{f}{F_s}},-\frac{F_s}{2}\le f\le \frac{F_s}{2},$$

(2)

where sinc$(·)$ is the sine cardinal function, f is the frequency, $F_s\ge B$ is the sampling frequency, $N_1^{\prime },N_2^{\prime }\in {\mathbb{Z}}^2$, $N_1^{\prime }<N_2^{\prime }$ and ⇋ refers to the time-frequency pair that will be helpful for the bounds derivation. Using (1), the dual source complex analytic signal at the output of the receiver’s antenna can be expressed as

$$\begin{array}{cc}\hfill x_R\left(t\right)& =d_R(t;{\mathbf{\eta }}_0,{\rho }_0,{\varphi }_{R,0})+d_R(t;{\mathbf{\eta }}_1,{\rho }_1,{\varphi }_{R,1})+w_R\left(t\right),\hfill \\ \hfill d_R(t;{\mathbf{\eta }}_i,{\rho }_i,{\varphi }_{R,i})& ={\rho }_i{e}^{j{\varphi }_{R,i}}s\left((1-b_i)(t-{\tau }_i)\right)e^{j{\omega }_c(1-b_i)t}{e}^{-j{\omega }_c{\tau }_i},\hfill \end{array}$$

where $w_R\left(t\right)$ is a zero-mean white complex circular Gaussian noise, ${\omega }_c=2\pi f_c$, and for $i\in \{0,1\}$, ${\mathbf{\eta }}_i^T=[{\tau }_i,b_i]$, ${\rho }_i\in ]0,+\infty [$ and ${\varphi }_{R,i}$ the amplitude and phase of the complex coefficients induced by the propagation characteristics (fading, reflection, etc.), the polarization mismatches and the antenna gains.

Under the narrowband signal hypothesis, i.e., the time-frequency product $B·T_c$ (with B the signal bandwidth and $T_c$ the coherent integration time) is smaller than the inverse Doppler term $c/v$, the Doppler effect on the band-limited baseband signal $s\left(t\right)$ is usually neglected so that: $s\left(\right(1-b\left)\right(t-\tau \left)\right)\approx s(t-\tau )$ (ch.9 [1]). Therefore, the baseband output of the receiver’s Hilbert filter with bandwidth $F_s$ containing a direct signal and a single specular reflection can be approximated by

$$\begin{array}{cc}\hfill x\left(t\right)& \triangleq x_R\left(t\right)e^{-j{\omega }_{c}t}=d(t;{\mathbf{\theta }}_0)+d(t;{\mathbf{\theta }}_1)+w\left(t\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill d(t;{\mathbf{\theta }}_i)& \triangleq {\rho }_i{e}^{j{\varphi }_i}s(t-{\tau }_i)e^{-j{\omega }_c{b}_i(t-{\tau }_i)},\hfill \end{array}$$

(4)

where for $i\in \{0,1\}$, ${\mathbf{\theta }}_i^T=[{\mathbf{\eta }}_i^T,{\rho }_i,{\varphi }_i]$, ${\varphi }_i={\varphi }_{R,i}-{\omega }_c(1+b_i){\tau }_i$. If we now consider the acquisition of $N=N_2-N_1+1(N_1\ll N_1^{\prime },N_2\gg N_2^{\prime })$ samples at a sampling rate $T_s=1/F_s$, the discrete signal model yields to the following dual source CSM,

$$\mathbf{x}=\mathbf{A}({\mathbf{\eta }}_0,{\mathbf{\eta }}_1)\mathbf{\alpha }+\mathbf{w},\mathbf{w}\sim \mathcal{CN}(0,{\sigma }_n^2{\mathbf{I}}_N),$$

(5)

with, for $n\in \left[N_1,N_2\right]$,

$$\begin{array}{cc}\hfill & {\mathbf{x}}^T=\left(\dots ,x\left(n{T}_s\right),\dots \right),\hfill \\ \hfill & \mathbf{A}({\mathbf{\eta }}_0,{\mathbf{\eta }}_1)=\left[{\mathbf{a}}_0,{\mathbf{a}}_1\right],\hfill \\ \hfill & {\mathbf{a}}_i^T=\left(\dots ,s(n{T}_s-{\tau }_i)e^{-j{\omega }_c{b}_i(n{T}_s-{\tau }_i)},\dots \right),\hfill \\ \hfill & {\mathbf{\alpha }}^T=\left({\rho }_0{e}^{j{\varphi }_0},{\rho }_1{e}^{j{\varphi }_1}\right),\hfill \\ \hfill & {\mathbf{w}}^T=\left(\dots ,w\left(n{T}_s\right),\dots \right).\hfill \end{array}$$

3. New Compact Dual Source CRB for Delay and Doppler Estimation with Band-Limited Signals

The main goal of this contribution is to derive a new compact analytic form of the delay-Doppler CRB for the dual source model in (5).

3.1. Problem Formulation

The parameters to be estimated are gathered in the following vector: ${\mathbf{ϵ}}^T=[{\sigma }_n^2,{\mathbf{\eta }}_0^T,{\rho }_0,{\varphi }_0,{\mathbf{\eta }}_1^T,{\rho }_1,{\varphi }_1]$. From (5), we can write $\mathbf{x}\sim \mathcal{CN}\left(\mathbf{A}({\mathbf{\eta }}_0,{\mathbf{\eta }}_1)\mathbf{\alpha },{\sigma }_n^2{\mathbf{I}}_N\right)$ and express the probability density function (pdf) as,

$$p(\mathbf{x},\mathbf{ϵ})=\frac{1}{{\left(\pi {\sigma }_n^2\right)}^N}{e}^{-\frac{1}{{\sigma }_n^2}{\parallel \mathbf{x}-\mathbf{A}({\mathbf{\eta }}_0,{\mathbf{\eta }}_1)\mathbf{\alpha }\parallel }^2}.$$

(6)

The corresponding CRB for the estimation of $\mathbf{ϵ}$ is defined as the inverse of the Fisher Information matrix (FIM) [43],

$${\mathbf{CRB}}_{\mathbf{ϵ}|\mathbf{ϵ}}={\mathbf{F}}_{\mathbf{ϵ}|\mathbf{ϵ}}^{-1}\left(\mathbf{ϵ}\right),{\mathbf{F}}_{\mathbf{ϵ}|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)=-E\left[\frac{{\partial }^{2}lnp(\mathbf{x},\mathbf{ϵ})}{\partial \mathbf{ϵ}\partial {\mathbf{ϵ}}^T}\right].$$

(7)

Note that for the Gaussian CSM of interest the Slepian-Bangs formula [21] provides the direct evaluation of the FIM terms,

$${\left({\mathbf{F}}_{\mathbf{ϵ}|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)\right)}_{k,l}\triangleq F_{{\mathbf{ϵ}}_k,{\mathbf{ϵ}}_l|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)=\frac{2}{{\sigma }_n^2}\mathrm{Re}\left\{{\left(\frac{\partial \mathbf{A}\mathbf{\alpha }}{\partial {\mathbf{ϵ}}_k}\right)}^H\left(\frac{\partial \mathbf{A}\mathbf{\alpha }}{\partial {\mathbf{ϵ}}_l}\right)\right\}+\frac{N}{{\sigma }_n^4}\frac{\partial {\sigma }_n^2}{\partial {\mathbf{ϵ}}_k}\frac{\partial {\sigma }_n^2}{\partial {\mathbf{ϵ}}_l},$$

(8)

where the noise power appears to be independent from the other parameters, and the FIM reduces to

$${\mathbf{F}}_{\mathbf{ϵ}|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)=\left[\begin{array}{ccc}\hfill F_{{\sigma }_n^2|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)& \mathbf{0}& \mathbf{0}\hfill \\ \mathbf{0}& {\mathbf{F}}_{{\mathbf{\theta }}_0|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)& {\mathbf{F}}_{{\mathbf{\theta }}_0,{\mathbf{\theta }}_1|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)\hfill \\ \mathbf{0}& {\mathbf{F}}_{{\mathbf{\theta }}_1,{\mathbf{\theta }}_0|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)& {\mathbf{F}}_{{\mathbf{\theta }}_1|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)\hfill \end{array}\right],$$

(9)

where:

• $F_{{\sigma }_n^2|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)=N/{\sigma }_n^4$.
• ${\mathbf{F}}_{{\mathbf{\theta }}_0|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)$ and ${\mathbf{F}}_{{\mathbf{\theta }}_1|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)$ correspond to the FIMs of the signals when they are totally decoupled. These matrices have been derived and studied in [16] without the Doppler frequency estimation, and in [17,18] for the general Gaussian CSM. The main results in [17,18] concerning the single source CSM FIM terms are summarized in Section 3.2.
• ${\mathbf{F}}_{{\mathbf{\theta }}_0,{\mathbf{\theta }}_1|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)$ = ${\mathbf{F}}_{{\mathbf{\theta }}_1,{\mathbf{\theta }}_0|\mathbf{ϵ}}{\left(\mathbf{ϵ}\right)}^T$ characterizes the interference between both signals. The derivation of such FIM terms is given in Section 3.3.

3.2. Decoupled Fisher Information Matrix Terms

The two last diagonal blocks of the FIM (9) were derived for the single source CSM case in [17,18]. For the sake of completeness, the main results are recalled in the sequel:

$${\mathbf{F}}_{{\mathbf{\theta }}_i|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)=\frac{2F_s}{{\sigma }_n^2}\mathrm{Re}\left\{{\mathbf{Q}}_i\mathbf{W}{\mathbf{Q}}_i^H\right\},$$

(10)

where ${\mathbf{Q}}_i,i\in \{1,2\}$ is defined in (A9) and $\mathbf{W}$ is defined as:

$$\mathbf{W}=\left[\begin{array}{ccc}\hfill w_1& w_2^{*}& w_3^{*}\\ w_2& W_{2,2}& w_4^{*}\\ w_3& w_4& W_{3,3}\end{array}\right],$$

(11)

with the easy-to-use formulation w.r.t. the baseband signal samples:

$$\begin{array}{cc}\hfill & w_1=\frac{1}{F_s}{\mathbf{s}}^H\mathbf{s},w_2=\frac{1}{F_s^2}{\mathbf{s}}^H\mathbf{D}\mathbf{s},w_3={\mathbf{s}}^H{\mathbf{V}}^{\Delta ,1}\left(0\right)\mathbf{s},w_4=\frac{1}{F_s}{\mathbf{s}}^H\mathbf{D}{\mathbf{V}}^{\Delta ,1}\left(0\right)\mathbf{s},\hfill \\ \hfill & W_{2,2}=\frac{1}{F_s^3}{\mathbf{s}}^H{\mathbf{D}}^2\mathbf{s},W_{3,3}=F_s{\mathbf{s}}^H{\mathbf{V}}^{\Delta ,2}\left(0\right)\mathbf{s},\hfill \end{array}$$

(12)

where $\mathbf{s}$, the baseband sample vector, is defined in (A36), $\mathbf{D}$ in (A39), ${\mathbf{V}}^{\Delta ,1}$ in (A41) and ${\mathbf{V}}^{\Delta ,2}$ in (A42). The inversion of (10) proposed in [17,18] leads to a closed-form CRB expression for the delay, Doppler, phase and amplitude estimation, where a noteworthy feature is that the CRBs are expressed w.r.t. the baseband signal samples. Such CRBs were studied for different GNSS signals in [35,44].

3.3. Interference Fisher Information Matrix Terms

From the Slepian-Bangs formula the non-diagonal blocks of the FIM, referred to as interference Fisher information matrices, are expressed as follows:

$${\mathbf{F}}_{{\mathbf{\theta }}_1,{\mathbf{\theta }}_0|\mathbf{ϵ}}\left(\mathbf{ϵ}\right)=\frac{2F_s}{{\sigma }_n^2}\mathrm{Re}\left\{{\mathbf{Q}}_1{\mathbf{W}}^{\Delta }{\mathbf{Q}}_0^H\right\},$$

(13)

where ${\mathbf{Q}}_0$, ${\mathbf{Q}}_1$ are defined in (A9) and ${\mathbf{W}}^{\Delta }$ is defined as

$${\mathbf{W}}^{\Delta }=\left[\begin{array}{ccc}\hfill W_{1,1}^{\Delta }& W_{1,2}^{\Delta }& W_{1,3}^{\Delta }\hfill \\ \hfill W_{2,1}^{\Delta }& W_{2,2}^{\Delta }& W_{2,3}^{\Delta }\hfill \\ \hfill W_{3,1}^{\Delta }& W_{3,2}^{\Delta }& W_{3,3}^{\Delta }\hfill \end{array}\right]e^{j\Delta \psi }{e}^{-j{\omega }_c\Delta b{\tau }_0},$$

(14)

where $\Delta b$ and $\Delta \psi$ are defined in (A15) and with the different components of the matrix ${\mathbf{W}}^{\Delta }$ expressed w.r.t the baseband signal samples given by

$$\begin{array}{cc}\hfill W_{1,1}^{\Delta }& =\frac{1}{F_s}{\mathbf{s}}^H\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,0}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{s},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill W_{1,2}^{\Delta }& =\frac{1}{F_s^2}{\mathbf{s}}^H\mathbf{D}\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,0}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{s},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill W_{1,3}^{\Delta }& =-{\mathbf{s}}^H\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,1}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{s}+\frac{j{\omega }_c\Delta b}{F_s}{\mathbf{s}}^H\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,0}\mathbf{s},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill W_{2,1}^{\Delta }& =\frac{1}{F_s^2}{\mathbf{s}}^H\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,0}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{D}\mathbf{s},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill W_{2,2}^{\Delta }& =\frac{1}{F_s^3}{\mathbf{s}}^H\mathbf{D}\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,0}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{D}\mathbf{s},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill W_{2,3}^{\Delta }& =-\frac{1}{F_s}{\mathbf{s}}^H\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,1}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{D}\mathbf{s}+\frac{j{\omega }_c\Delta b}{F_s^2}{\mathbf{s}}^H\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,0}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{D}\mathbf{s},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill W_{3,1}^{\Delta }& ={\mathbf{s}}^H\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,1}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{s},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill W_{3,2}^{\Delta }& =\frac{1}{F_s}{\mathbf{s}}^H\mathbf{D}\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,1}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{s},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill W_{3,3}^{\Delta }& =F_s{\mathbf{s}}^H\mathbf{U}\left(\Delta b\right){\mathbf{V}}^{\Delta ,2}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{s}+j{\omega }_c\Delta b{\mathbf{s}}^H\mathbf{U}(\Delta b){\mathbf{V}}^{\Delta ,1}\left(\frac{\Delta \tau }{T_s}\right)\mathbf{s},\hfill \end{array}$$

(23)

where $\mathbf{s}$, the baseband sample vector, is defined in (A36), $\mathbf{D}$ in (A39), $\mathbf{U}$ in (A38), ${\mathbf{V}}^{\Delta ,0}$ in (A40), ${\mathbf{V}}^{\Delta ,1}$ in (A41) and ${\mathbf{V}}^{\Delta ,2}$ in (A42).

Proof. 

refer to Appendix A. □

An interesting and reassuring fact is that when the difference between LOS and NLOS signals (i.e., signal 0 and signal 1) is set to zero: $\Delta \tau ={\tau }_1-{\tau }_0=0$, $\Delta b=b_1-b_0=0$, $\Delta \varphi ={\varphi }_1-{\varphi }_0=0$, ${\rho }_1={\rho }_0$, we have ${\mathbf{Q}}_0={\mathbf{Q}}_1$ and ${\mathbf{W}}^{\Delta }=\mathbf{W}$. This result could be expected but the FIM terms equations obtained confirm it. Indeed, if both signals are indistinguishable, they must have the same FIM and their interference FIM must coincide (total interference between both signals).

4. Maximum Likelihood and CLEAN RELAX Estimators

4.1. Dual Source Conditional Maximum Likelihood Estimator

From the dual source CSM (5) and its pdf (6), the estimator $\widehat{\mathbf{ϵ}}$ that maximizes the likelihood is such that $(\widehat{{\mathbf{\eta }}_0},\widehat{{\mathbf{\eta }}_1})$ maximizes the projection of the received signal onto the subspace defined by the mixing matrix $\mathbf{A}$ [22]. As a consequence, the estimated parameters ($\widehat{{\rho }_0},\widehat{{\rho }_1},\widehat{{\varphi }_0},\widehat{{\varphi }_1}$) directly depend on the delay/Doppler estimation. Finally, $\widehat{{\sigma }_n^2}$ is simply the orthogonal projection value:

$$\begin{array}{cc}\hfill \left(\widehat{{\mathbf{\eta }}_0},\widehat{{\mathbf{\eta }}_1}\right)& =arg\underset{{\mathbf{\eta }}_0,{\mathbf{\eta }}_1}{max}{∥{\mathbf{P}}_{\mathbf{A}({\mathbf{\eta }}_0,{\mathbf{\eta }}_1)}\mathbf{x}∥}^2\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \widehat{{\rho }_i}& =\left|{\left[{\left({\mathbf{A}}^H(\widehat{{\mathbf{\eta }}_0},\widehat{{\mathbf{\eta }}_1})\mathbf{A}(\widehat{{\mathbf{\eta }}_0},\widehat{{\mathbf{\eta }}_1})\right)}^{-1}{\mathbf{A}}^H(\widehat{{\mathbf{\eta }}_0},\widehat{{\mathbf{\eta }}_1})\mathbf{x}\right]}_i\right|\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill \widehat{{\varphi }_i}& =arg\left\{{\left[{\left({\mathbf{A}}^H(\widehat{{\mathbf{\eta }}_0},\widehat{{\mathbf{\eta }}_1})\mathbf{A}(\widehat{{\mathbf{\eta }}_0},\widehat{{\mathbf{\eta }}_1})\right)}^{-1}{\mathbf{A}}^H(\widehat{{\mathbf{\eta }}_0},\widehat{{\mathbf{\eta }}_1})\mathbf{x}\right]}_i\right\}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \widehat{{\sigma }_n^2}& =\frac{1}{N}{∥{\mathbf{P}}_{\mathbf{A}({\mathbf{\eta }}_0,{\mathbf{\eta }}_1)}^{\perp }\mathbf{x}∥}^2\hfill \end{array}$$

(27)

where the projectors are defined as ${\mathbf{P}}_{\mathbf{A}}=\mathbf{A}{\left({\mathbf{A}}^H\mathbf{A}\right)}^{-1}{\mathbf{A}}^H$, ${\mathbf{P}}_{\mathbf{A}}^{\perp }=\mathbf{I}-{\mathbf{P}}_{\mathbf{A}}$.

Even though this estimator is known to be computationally complex, it is also known to be asymptotically efficient (e.g., in the high-SNR regime [34]) for the dual source Gaussian CSM. This property makes it a powerful tool to validate the CRBs derived in Section 3.

4.2. Sub-Optimal Estimator: CLEAN-RELAX

In order to deal with the CMLE heavy computational load, alternative estimators can be used. In this paper, a relax version of the well-known CLEAN algorithm was implemented [40]. This estimator assumes that the two signals are decoupled. The method first estimates the parameters $(\widehat{{\tau }_0},\widehat{b_0},\widehat{{\rho }_0},\widehat{{\varphi }_0})$ of the strongest signal by using the cross-correlation function, it subtracts this estimated signal from the measured input, and then estimates the secondary signal parameters $(\widehat{{\tau }_1},\widehat{b_1},\widehat{{\rho }_1},\widehat{{\varphi }_1})$. These two estimation steps are iterated until the cost function, which is the resulting likelihood, is stabilized. The method used in this contribution only looks for two signals. In GNSS, the same algorithm is named Multipath Estimating Delay Lock Loop (MEDLL) and is typically used to mitigate multipath [8]. The CRE can be seen as a sub-optimal estimator because it works only under certain assumptions. Indeed, considering that the two signals can be decoupled stands only when $\Delta \tau$ is large enough compared to the sharpness of the cross-correlation function. This estimator is not expected to perform well when the two signals are very close in time. On the other hand, when the signals are clearly parted, the CRE asymptotically behaves as the CMLE. See Section 5 for simulated results on the CRE limit.

5. Validation and Discussion

5.1. Methodology

To validate the new CRBs derived in Section 3, the asymptotic properties of the CMLE introduced in Section 4.1 are exploited. It is know that for a high SNR the statistical mean square error (MSE) of the delay and Doppler frequency CMLE are accurately described by the corresponding CRBs. Two representative signals are considered: (i) a Global Positioning System (GPS) L1 C/A signal [45], and (ii) a Linear Frequency Modulated (LFM) chirp signal classically defined as following:

$$s\left(t\right)=e^{j2\pi f\left(t\right)},f\left(t\right)=\frac{B}{2T}{\left(\frac{t}{T}-\frac{1}{2}\right)}^2,T=N{T}_s,$$

(28)

with B the chirp bandwidth equal to 8 MHz (which is the receiver bandwidth in this case) and N the number of samples. The received signal is built with two delayed and attenuated replicas with arbitrary values of Doppler frequencies. The parameters for the three different scenarios considered are given in

Table 1. Simulations scenarios for the CRB validation: (a) two signals totally parted ($\Delta \tau =2$ chips), (b) a secondary path very close to the main signal ($\Delta \tau =1/4$ chip), and (c) considered to showcase the difference between the CMLE and CRE ($\Delta \tau =1/8$ chip).

	Estimator	${\mathit{F}}_{\mathit{s}}$ (MHz)	$\mathbf{\Delta }\mathit{\tau }$ (L1 C/A Chips)	$\mathbf{\Delta }\mathit{\tau }$ (m)	${\mathit{\rho }}_{\mathbf{1}}\mathbf{/}{\mathit{\rho }}_{\mathbf{0}}$	$\mathbf{\Delta }\mathit{\varphi }$ ($°$)	${\mathit{Fd}}_{\mathbf{0}}\mathbf{/}{\mathit{Fd}}_{\mathbf{1}}$ (Hz)
(a)	CRE	8	2	600	0.5	15	20/50
(b)	CRE	8	1/4	75	0.5	15	20/50
(c1)	CRE	8	1/8	37.5	0.5	15	20/50
(c2)	CMLE	8	1/8	37.5	0.5	15	20/50

. Notice that the L1 C/A chips unit is $1/1023$ ms. In this table ${\rho }_1/{\rho }_0$ is the ratio between the reflected amplitude and the main one (also referred to as multipath-to-direct ratio in GNSS [42]) and $F_{di}=b_i{f}_c$ refers to the Doppler frequency in Hertz corresponding to the Doppler stretch $b_i$.

As discussed in Section 4, depending on the time-delay difference between the two replicas, $\Delta \tau$, one can apply a sub-optimal estimators such as the CRE [40], which may behave like a CMLE in the asymptotic region (high SNR). In this case it is possible to see the estimator threshold region in a reasonable computation time. However, when it comes to very small values of $\Delta \tau$, meaning less than $1/F_s$ chip delay, a sub-optimal algorithm like the CRE is in general not able to converge to the CRB. As a consequence, during the simulations, when the product $\Delta \tau F_s$ is smaller than or equal to 1, the CRE is no longer efficient. In such limit case, a direct implementation of the CMLE is preferred. Computational cost soars but at very high-SNR it is possible to limit the search area at $3\sigma$, where $\sigma$ is the expected variance of the estimated values, directly provided by the derived CRBs. Therefore the CMLE simulation is valid only at very high SNR, because in this asymptotic region the CMLE is Gaussian distributed and its variance is equal to the CRB.

Three scenarios are then considered: scenario (a) presents the simple case where signals are clearly parted. Scenarios (b) and (c) present the case when the secondary signal is within 1 chip, and therefore there exists a strong interference between both signals. For illustrative purposes we show the normalized cross-correlation function (CCF) in these three scenarios for both signals in Figure 2.

5.2. Results

5.2.1. Scenario (a): $\Delta \tau$ Larger than 1 Chip

In this scenario, we consider a relative delay of two L1 C/A chips, which corresponds to a path difference around 600 m. In this case, the two signals are clearly parted (see Figure 2 (top)) and the estimation of the main signal could almost be decoupled from the estimation of the secondary signal. Figure 3 (left hand side plots) shows the root mean square error (RMSE) for the time-delay estimation. One can see the threshold for the chirp signal at SNR_OUT = 20 dB (i.e., the SNR at the output of the matched filter) and for the GPS L1 C/A signal at SNR_OUT = 21 dB. For a SNR_OUT larger than this threshold the estimator reaches the CRB: (1) this validates the new CRB expression, and (2) provides the receiver conditions to be in a correct operation region. It is worth pointing out the impact that having two sources have on the time-delay estimation w.r.t. the single source case, where the threshold region for a GPS signal is at SNR_OUT = 15 dB (refer to [17,18]). Figure 3 (right hand side plots) show the RMSE for the Doppler frequency estimation. In this case, the performance for both direct and reflected signals is on the CRB in the range of SNR_OUT values considered. Notice that it does not make sense to consider lower SNR values because in such case the time-delay estimate is below the convergence threshold. In any case, this result validates again the CRB expression.

5.2.2. Scenario (b): $\Delta \tau$ Smaller Than 1 chip

In this scenario, we consider a relative delay of a quarter of a chip, which is about 75 m in path difference. In this case, there is a strong interference between the two signals (see Figure 2 (middle)). The results are shown in Figure 4 where again top plots are for the GPS L1 C/A signal, bottom plots for the chirp, left plots for the time-delay estimation and right plots for the Doppler frequency estimation. In this case we observe that even with such a strong interference, the time-delay estimation with a sub-optimal CRE is possible. We can notice that having a close secondary signal has an impact on the estimation threshold, which is now SNR_OUT = 22 dB for the chirp signal, and SNR_OUT = 27 dB for the GPS signal. As expected the impact is larger for the GPS signal because of its correlation function, as it is shown in Figure 2 (middle-left). This impact can also be acknowledged in the Doppler frequency estimation, where for the GPS signal threshold region appears.

5.2.3. Scenario (c): $\Delta \tau$ smaller than 1 chip

In this scenario, we consider a relative delay of an eigth of a L1 C/A chip, which accounts for a path delay of about 37.5m. In this case, there is such a strong interference between the two signals (see Figure 2 (bottom)) that the CRE cannot be efficient anymore. The CRE limit is clearly shown in Figure 5. For both signals, the time-delay estimation never reaches the optimal performances set by the CRB. Interestingly, the chirp signal seems to be more robust for the Doppler estimation, reaching the CRB for the direct signal but not for the reflected one.

Figure 6 shows the results for the dual source CMLE. One can observe a good fit between the MSE and the CRB. This result supports the fact that the CRE limitations can be overcome by the CMLE, and also validates again the CRB expression for the dual source estimation problem derived in this contribution.

6. Further Insights and Outlooks

6.1. Comparison with Existing Literature

As previously mentioned, a formulation of the CRBs in the dual source context was proposed in [28]. In this reference, authors tackled the study of the best achievable performance in the estimation of a receiver altitude using GNSS-R. Starting from the Slepian-Bangs formula, they derived a general FIM that is similar to the one presented in Section 3. The main difference between the two formulations is that in [28], the study is done under the assumption of a static, ground-based receiver. This assumption allows the authors to set the two Doppler shifts at the same value. One can find a way to connect the results presented in Section 3 with the one proposed in [28]. Indeed, setting an equality such as $b_0=b_1$, which constrains the current model, can be seen as a reparameterization of the vector of unknown parameters $\mathbf{ϵ}$ [31]. We therefore consider the following vector transform,

$$b_0=b_1\iff \mathbf{ϵ}=\mathbf{ϵ}\left(\mathbf{\xi }\right),$$

(29)

where

$$\begin{array}{cc}\hfill \mathbf{ϵ}& ={\left[\begin{array}{ccccccccc}\hfill {\sigma }_n^2& {\tau }_0& b_0\hfill & {\rho }_0& {\varphi }_0& {\tau }_1& b_1& {\rho }_1& {\varphi }_1\end{array}\right]}^T,\hfill \\ \hfill \mathbf{\xi }& ={\left[\begin{array}{cccccccc}\hfill {\sigma }_n^2& {\tau }_0& b_0\hfill & {\rho }_0& {\varphi }_0& {\tau }_1& {\rho }_1& {\varphi }_1\end{array}\right]}^T.\hfill \end{array}$$

From this, if we note the 9-by-9 unity matrix ${\mathbf{I}}_9=\left[{\mathbf{e}}_1\dots {\mathbf{e}}_{\mathbf{9}}\right]$ with ${\mathbf{e}}_i$ the column unit vector with 1 at the ith component and 0 elsewhere, the relation (29) can be explicitly written as follows,

$$\mathbf{ϵ}\left(\mathbf{\xi }\right)=\left[\begin{array}{cccccccc}\hfill {\mathbf{e}}_1& {\mathbf{e}}_2& {\mathbf{e}}_3+{\mathbf{e}}_7\hfill & {\mathbf{e}}_4& {\mathbf{e}}_5& {\mathbf{e}}_6& {\mathbf{e}}_8& {\mathbf{e}}_9\end{array}\right]\mathbf{\xi }=\frac{\partial \mathbf{ϵ}\left(\mathbf{\xi }\right)}{\partial {\mathbf{\xi }}^T}\mathbf{\xi }.$$

(30)

Now if we consider a reparameterization of the pdf $p(\mathbf{x};\mathbf{ϵ})$ for the unknown parameters $\mathbf{ϵ}$ where

$$\mathbf{ϵ}=\mathbf{ϵ}\left(\mathbf{\xi }\right),dim\left\{\mathbf{\xi }\right\}=8,dim\left\{\mathbf{ϵ}\right\}=9,$$

then $p(\mathbf{x};\mathbf{\xi })=p(\mathbf{x};\mathbf{ϵ}(\mathbf{\xi }\left)\right)$ and by use of the derivation chain rule identity,

$$\frac{\partial lnp(\mathbf{x};\mathbf{\xi })}{\partial {\mathbf{\xi }}^T}=\frac{\partial lnp(\mathbf{x};\mathbf{ϵ}(\mathbf{\xi }\left)\right)}{\partial {\mathbf{ϵ}}^T}\frac{\partial \mathbf{ϵ}\left(\mathbf{\xi }\right)}{\partial {\mathbf{\xi }}^T}\iff \frac{\partial lnp(\mathbf{x};\mathbf{\xi })}{\partial \mathbf{\xi }}={\left(\frac{\partial \mathbf{ϵ}\left(\mathbf{\xi }\right)}{\partial {\mathbf{\xi }}^T}\right)}^T\frac{\partial lnp(\mathbf{x};\mathbf{ϵ}(\mathbf{\xi }\left)\right)}{\partial \mathbf{ϵ}},$$

we easily obtain that [46]

$${\mathbf{F}}_{\mathbf{\xi }|\mathbf{\xi }}\left(\mathbf{\xi }\right)=E\left\{\frac{\partial lnp(\mathbf{x};\mathbf{\xi })}{\partial \mathbf{\xi }}{\left(\frac{\partial lnp(\mathbf{x};\mathbf{\xi })}{\partial \mathbf{\xi }}\right)}^T\right\}={\left(\frac{\partial \mathbf{ϵ}\left(\mathbf{\xi }\right)}{\partial {\mathbf{\xi }}^T}\right)}^T{\mathbf{F}}_{\mathbf{ϵ}|\mathbf{ϵ}}\left(\mathbf{ϵ}\left(\mathbf{\xi }\right)\right)\frac{\partial \mathbf{ϵ}\left(\mathbf{\xi }\right)}{\partial {\mathbf{\xi }}^T}.$$

(31)

Developing the right-hand side of (31), one can obtain the following FIM (refer to Appendix B for more details on the different matrices),

$${\mathbf{F}}_{\mathbf{\xi }|\mathbf{\xi }}\left(\mathbf{\xi }\right)=\left[\begin{array}{ccc}{F}_{{\sigma }_n^2|\mathbf{\xi }}\left(\mathbf{\xi }\right)& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathbf{F}}_{{\mathbf{\theta }}_0|\mathbf{\xi }}\left(\mathbf{\xi }\right)& {\mathbf{F}}_{{\mathbf{\theta }}_0,{\tau }_1,{\rho }_1,{\varphi }_1|\mathbf{\xi }}\left(\mathbf{\xi }\right)\\ \mathbf{0}& {\mathbf{F}}_{{\tau }_1,{\rho }_1,{\varphi }_1,{\mathbf{\theta }}_0|\mathbf{\xi }}\left(\mathbf{\xi }\right)& {\mathbf{F}}_{{\tau }_1,{\rho }_1,{\varphi }_1|\mathbf{\xi }}\left(\mathbf{\xi }\right)\end{array}\right].$$

(32)

It appears that from the CRB presented in this contribution, it is possible to deduce the specific scenario studied in [28]. However, in other scenarios, the Doppler shifts of the LOS and the NLOS signals may be different (e.g., if the signal is reflected from a moving object [47]) and a more general approach is needed, for which the result in [28] is not valid and the proposed CRB provides an answer.

Another key difference between the two contributions is that the results in the present contribution are expressed directly using the baseband signal samples, whereas in [28] they are expressed with the signal Power Spectral Density (PSD). Therefore, w.r.t. [28], the calculations in this article went a little further: exploiting the Fourier transform properties over the hermitian product it is possible to jump from the PSD back to the baseband signal samples, as it is done in Appendix A and in [16]. From this remark, it is possible to express the CRB for the estimation of the altitude h of a receiver with the baseband signal samples, in the case where the signal is real and the path difference is large enough, so that the interference terms can be neglected,

$${\mathrm{CRB}}_{h|\mathbf{ϵ}}=\frac{{\sigma }_n^2{c}^2\left(1+{\left({\rho }_1/{\rho }_0\right)}^2\right)}{8{\rho }_1^2{sin}^2\left(e\right)F_s{W}_{3,3}}$$

(33)

where c is the speed of light in vacuum and e is the elevation angle of the transmitter seen from the specular point (see Figure 1). $W_{3,3}$ expression is recalled in (12). Note that this expression covers a broader area since it was obtained without any assumption on the Doppler shifts.

6.2. Possible CRB Applications

The CRB formulation proposed in this contribution being easy to manipulate (i.e., as it is expressed in terms of the baseband signal samples), several use cases can be identified. Hereafter is a non-exhaustive list of examples that could lead to further studies, to support the discussion and show that the proposed CRBs are of broad interest.

• As a direct extension of the results presented in the previous Section 6.1, one could look for the different parameters that can be obtained from $\mathbf{ϵ}$, and calculate a new CRB dedicated to these specific parameters. In [28], the study applied to the GNSS-R altimetry problem, investigates the receiver altitude and the complex ratio between the reflected and the direct amplitudes. A generalization of this method can easily be outlined as follows. The first step is to translate either the constraints or the new variables into a transform similarly to (29), then the general expression (31) yields the corresponding FIM. If it is simple enough, a matrix inversion might end up to a closed-form expression of the CRB.
• Another application could be the assessment of the impact the secondary signal has on the main signal’s parameters estimation. From the CRB expression it is indeed quite simple to extract the CRB for the time-delay estimation while there is no reflection, and the same CRB when the NLOS signals interfere with the LOS one. Such study may bring a new tool to characterize the signal robustness to multipath, being a key issue for instance in safety-critical GNSS applications, an ultimately drive the future generation of signal design.
• Again in the GNSS context, such CRBs can also be used to have an optimal performance assessment of the Carrier-to-Noise Density Ratio ($C/N_0$) estimation under multipath conditions. Notice that the $C/N_0$ is an important parameter used in several GNSS applications.
• Regarding precise positioning techniques such as real time kinematics (RTK), needed in modern applications, these CRBs may bring a valuable information on the impact that multipath may have on the final position estimate. Indeed, it is known that RTK solutions are not able to fix the carrier phase ambiguities under harsh propagation conditions such as strong multipath. This would extend the results presented in [17,18,35] to more realistic conditions.
• In the dual antenna GNSS-R context, the dual source CRB proposed in this contribution can be exploited in order to characterize the impact that an imperfect isolation between antennas may have on the final GNSS-R product. This may account for the reflected signal leaking into the upward antenna, or the LOS signal contaminating the downward one.
• Finally, a compact CRB expression can be exploited to do optimal signal design as suggested in [16]. As an example, we can study the case of the estimation of the receiver altitude (33). It was shown that the optimal signal can be linked to the first eigenvector of the matrix ${\mathbf{V}}^{\Delta ,2}\left(0\right)$. The optimal CRB would then be given by:

  $${\mathrm{CRB}}_{h|\mathbf{ϵ}}\ge {\mathrm{CRB}}_{h|\mathbf{ϵ}}^{\mathrm{opt}.}=\frac{{\sigma }_n^2{c}^2\left(1+{\left({\rho }_1/{\rho }_0\right)}^2\right)}{8{\rho }_1^2{sin}^2\left(e\right)F_s{w}_1{D}_{1,1}},$$

  (34)

  where $\pi \le D_{1,1}\le {\pi }^2$ is the largest eigenvalue of ${\mathbf{V}}^{\Delta ,2}\left(0\right)$.

7. Conclusions

In this contribution, a new compact CRB for the joint delay, Doppler and complex amplitude estimation of a dual source narrowband CSM was derived. A handy closed-form expression of these CRBs that only depends on the baseband signal samples was also proposed. Such formulation is especially easy to use whatever the baseband signal considered, therefore being valid for a large variety of applications and of broad interest. The new formulation was validated using the asymptotic properties of the dual source CMLE and a sub-optimal algorithm, the CRE, that behaves well under certain assumptions. The different simulations show a satisfying fit between the theoretical bounds and the statistical output of the estimators applied to two representative navigation and radar signals, which validated the proposed CRBs. In addition, a brief comparison with the existing literature was provided, showing the consistency with CRBs previously derived for GNSS-R applications. A number of applications can be thought of based on these CRBs, which were also discussed for completeness. For instance, the new CRBs can be exploited for optimal signal design, multipath characterization, precise positioning methods assessment under challenging propagation conditions, or for the estimation of specific application-dependent parameters that are function of the signals parameters (delay, Doppler shift, complex amplitude).