A New Compact Delay, Doppler Stretch and Phase Estimation CRB with a Band-Limited Signal for Generic Remote Sensing Applications

Abstract

Since time-delay, Doppler effect and phase estimation are fundamental tasks in a plethora of engineering fields, tractable lower performance bounds for this problem are key tools of broad interest for a large variety of remote sensing applications. In the large sample regime and/or 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 contribution, we introduce firstly a new compact CRB expression for the joint time-delay and Doppler stretch estimation, considering a generic delayed and dilated band-limited signal. This generalizes known results for both wideband signals and the standard narrowband signal model where the Doppler effect on the band-limited baseband signal is not considered and amounts to a frequency shift. General compact closed-form CRB expressions for the amplitude and phase are also provided. 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. The new CRB expressions are validated in a positioning case study, both using synthetic and real data. These results show that the maximum likelihood estimator converges to the CRB at high signal-to-noise ratios, which confirms the exactness of the CRB. The CRB is further validated by comparing the ambiguity function and its 2nd order Taylor expansion where the perfect match also proves its exactness.

1. Introduction

Time-delay, Doppler stretch and phase estimation appear in a plethora of engineering fields such as navigation, radar, reflectometry, sonar or communications, to name a few [1,2,3,4,5,6,7,8,9,10,11,12], being the estimation of such parameters a key first stage of the receiver [7,10,11,12]. For any of these applications, when designing and assessing estimation techniques, it is of fundamental importance to know the ultimate achievable performance in the mean square error (MSE) sense, information which can be brought by lower bounds (LB) on the MSE. Even if several types of LBs exist [13], the family of Cramér–Rao bounds (CRB) [14,15] is the most popular, mainly due to its simplicity of calculation for various problems (see [6] (§8.4) and [13] (Part III)). In addition, the CRB gives an accurate estimation of the MSE of the maximum likelihood estimator (MLE) in the asymptotic region of operation under certain conditions, i.e., in the large sample regime and/or high signal-to-noise (SNR) regime of the Gaussian conditional signal model (CSM) [16,17]. Therefore, it is not surprising that several CRB expressions for the delay-Doppler estimation problem have been derived for the past decades, for finite [4,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32] or infinite [33] bandwidth signals, where the starting point is often either the Slepian–Bangs formulas [15] or general theoretical CRB expressions for Gaussian observation models [6,34,35,36].

Most of these CRB expressions [4,20,21,23,24,25,26,27,31,32] only address the standard narrowband signal model where the impact of the Doppler effect on the baseband signal is not taken into account and amounts to a frequency shift. However, in some applications [3,7,29], like wide/ultra-wide band (high range resolution, synthetic aperture or low interception probability) sonar or radar, or wide/ultra-wide band communications, the compression or stretch due to the range rate on the envelope of the received signal cannot be ignored. In radar, the standard narrowband signal model was first proposed by Woodward [1] for deterministic transmit waveforms and led to the conventional narrowband range-Doppler MLE and its associated narrowband radar ambiguity function. Following Woodward’s approach, various formulations for the wideband range-Doppler MLE and its associated ambiguity function were later investigated [2,3,7,37,38,39], and the significance of the difference between the two ambiguity functions has been shown to depend on the product of the signal duration, the signal bandwidth, and the target velocity [3]. In addition to the historical remote sensing systems as sonar and radar, in more recent Global Navigation Satellite Systems (GNSS), due to the very high velocity of the transmitter located on a satellite, it is also essential to incorporate the baseband signal dilatation due to the Doppler effect into the MLE formulation to reach the minimum achievable MSE, for instance in carrier phase-based precise positioning techniques [40]. In addition, this is also the case of GNSS-based reflectometry (GNSS-R) applications such as altimetry [12,41,42].

Surprisingly, although wideband ambiguity functions have been quite extensively studied for a while, CRB for the delay-Doppler estimation of wideband signals has received less attention [18,19,22,28,29,30] leading to lack of a general compact closed-form CRB expression. Indeed, [18,19] derive CRB expressions in active [18] and passive system [19] for a wideband signal with known amplitude and phase. Since, in many applications, it is unrealistic to assume a known complex amplitude, several CRB expressions for the joint estimation of delay-Doppler and complex amplitude [28,29] or phase-amplitude [22,30] have been proposed. However, all of these expressions have been derived under unnecessary restrictive assumptions on the signal, in particular if the signal is assumed to be band-limited, leading to unnecessary restrictive CRB expressions. Moreover, none of the existing CRBs [18,19,22,28,29,30] provide closed-form expressions for the amplitude and phase, which may be of great interest in applications like GNSS or GNSS-R where the phase estimation performance drives the performance of subsequent estimation techniques, such as GNSS carrier phase-based precise positioning [40] or phase-based GNSS-R [41,42]. Thus, the contributions of this article are:

• The first contribution is a general compact closed-form CRB expression for the delay-Doppler estimation of a generic band-limited signal which is only supposed to have a finite number on non-zero samples, thus encompassing all existing CRB expressions.
• A second contribution is the introduction of a general compact closed-form CRB expression for the amplitude and phase.
• The compact CRBs obtained are expressed in terms of the baseband signal samples, making it especially easy to use whatever the baseband signal considered. This allows to exploit such expressions in a plethora of remote sensing applications.
• The validity of the new CRB expressions is assessed in the context of GNSS, both using synthetic and real data.

Notice that these results encompass the preliminary results for the delay-Doppler estimation assuming a narrowband signal recently derived in [32]. In comparison with the literature, including [32], a certain number of new terms appear in the proposed CRB due to the Doppler effect on the baseband signal which can not be guessed from existing results. For instance, for a real signal, the estimation of both parameters is not decoupled any longer.

Notation

Italic indicates a scalar quantity, as in a; lower case boldface indicates a column vector quantity, as in $\mathbf{a}$; upper case boldface indicates a matrix quantity, as in $\mathbf{A}$. The matrix/vector transpose is indicated by a superscript ${(·)}^{\top }$ as in ${\mathbf{A}}^{\top }$, and the transpose conjugate ${(·)}^H$ as in ${\mathbf{A}}^H$.

2. Signal Model

We consider the transmission of a band-limited (bandwidth B) signal $s\left(t\right)$ over a carrier frequency $f_c$ (${\lambda }_c=c/f_c$) from a transmitter T, at position ${\mathbf{p}}_T\left(t\right)={\mathbf{p}}_T+{\mathbf{v}}_{T}t$, to a receiver R, at position ${\mathbf{p}}_R\left(t\right)={\mathbf{p}}_R+{\mathbf{v}}_{R}t$ (uniform linear motions). The band-limited signal can be expressed in time/frequency (to be exploited for the bound derivation) as

$$\begin{array}{cc}\hfill s\left(t\right)& =\sum _{n=N_1}^{N_2}s\left(\frac{n}{F_s}\right)\sin \mathrm{c}\left(\pi F_s\left(t-\frac{n}{F_s}\right)\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill s\left(f\right)& =\left(\frac{1}{F_s}\sum _{n=N_1}^{N_2}s\left(\frac{n}{F_s}\right)e^{-j2\pi n\frac{f}{F_s}}\right)1_{\left[-\frac{B}{2},\frac{B}{2}\right]}\left(f\right),\hfill \end{array}$$

(2)

where $F_s\ge B$, and $N_1,N_2\in \mathbb{Z}$, $N_1\le N_2$. The complex analytic signal at the output of the receiver’s antenna can be written as

$$x_A\left(t\right)={\alpha }_A{s}_A\left(t\right)+n_A\left(t\right),$$

(3)

with $n_A\left(t\right)$ a zero-mean white complex Gaussian noise, and where the gain ${\alpha }_A$ depends on the transmitted signal power, the transmitter/receiver antenna gains and polarization vectors, and the radial distance between T and R. If this distance, ${\mathbf{p}}_{TR}\left(t\right)$, can be approximated by a first order distance-velocity model [2,3,7,39],

$$\begin{array}{cc}\hfill & ∥{\mathbf{p}}_{TR}\left(t\right)∥\triangleq ∥{\mathbf{p}}_R\left(t\right)-{\mathbf{p}}_T\left(t-\tau \left(t\right)\right)∥=c\tau \left(t\right)\simeq d+vt,\hfill \\ \hfill & \tau \left(t\right)\simeq \tau +bt,\tau =\frac{d}{c},b=\frac{v}{c},c\simeq 3·{10}^{8}m/s,\hfill \end{array}$$

(4)

so-called relative uniform radial movement, and characterized by a time-delay $\left(\tau \right)$ due to the propagation path and dilation $\left(1-b\right)$ induced by the Doppler effect. In this case [7,43],

$$s_A\left(t\right)=s\left(\left(1-b\right)\left(t-\tau \right)\right)e^{j2\pi f_c\left(1-b\right)t}{e}^{-j2\pi f_c\tau }.$$

(5)

Notice that the static signal model case in [44] is obtained with $b=0$ ($\tau \left(t\right)\simeq \tau$), and the standard narrow-band signal model [2,3,7,20,29,31,32] is obtained by approximating $s\left(\left(1-b\right)\left(t-\tau \right)\right)\simeq s\left(t-\tau \right)$. We can express the baseband output of the receiver’s Hilbert filter as

$$x\left(t\right)=\alpha s\left(t;\mathbf{\eta }\right)e^{-j{\omega }_{c}b\left(t-\tau \right)}+n\left(t\right),$$

(6)

where ${\omega }_c=2\pi f_c$, $s\left(t;\mathbf{\eta }\right)=s\left(\left(1-b\right)\left(t-\tau \right)\right)$, $\mathbf{\eta }={[\tau ,b]}^{\top }$ and the complex gain $\alpha ={\alpha }_A{e}^{-j2\pi f_c\left(1+b\right)\tau }$. If $F_s$ is the Hilbert filter bandwidth, then $n\left(t\right)$ is a complex white Gaussian noise within the bandwidth $F_s$ with unknown variance ${\sigma }_n^2$. In addition, we have the following Nyquist–Shannon sampling condition: $f\in \left[-\frac{F_s}{2},\frac{F_s}{2}\right]$ with $\frac{F_s}{2}\ge {\displaystyle \underset{b}{max}}\left\{\frac{B}{2}\left(1-b\right)\right\}$. Both propagation time-delay and Doppler effect dilation are made apparent in

$$\begin{array}{}\hfill \mathrm{(7)}& \begin{array}{cc}\hfill s\left(t;\mathbf{\eta }\right)& =s\left(\left(1-b\right)\left(t-\tau \right)\right)={\left.s\left(\left(1-b\right)u\right)\right|}_{t-\tau }\hfill \\ \hfill & \upharpoonleft \downharpoonright \hfill \end{array}\hfill \hfill \mathrm{(8)}& \begin{array}{cc}\hfill s\left(f;\mathbf{\eta }\right)& =\frac{1}{1-b}s\left(\frac{f}{1-b}\right)e^{-j2\pi f\tau }\hfill \end{array}\hfill \end{array}$$

The discrete vector signal model is build from $N^{\prime }=N_2^{\prime }-N_1^{\prime }+1$ ($N_1^{\prime }\ll N_1$, $N_2^{\prime }\gg N_2$) samples at $T_s=1/F_s$, meaning that the signal amplitude is assumed to be negligible outside the essential duration $\left[N_1^{\prime }{T}_s,N_2^{\prime }{T}_s\right]$,

$$\begin{array}{cc}\hfill \mathbf{x}& =\alpha \mathbf{a}\left(\mathbf{\eta }\right)+\mathbf{n},\hfill \\ \hfill \mathbf{x}& ={(x\left(N_1^{\prime }{T}_s\right),\dots ,x\left(N_2^{\prime }{T}_s\right))}^{\top },\hfill \\ \hfill \mathbf{n}& ={(n\left(N_1^{\prime }{T}_s\right),\dots ,n\left(N_2^{\prime }{T}_s\right))}^{\top },\hfill \\ \hfill \mathbf{s}\left(\mathbf{\eta }\right)& ={(s\left(N_1^{\prime }{T}_s;\mathbf{\eta }\right),\dots ,s\left(N_2^{\prime }{T}_s;\mathbf{\eta }\right))}^{\top },\hfill \\ \hfill \mathbf{a}\left(\mathbf{\eta }\right)& ={({\left(\mathbf{s}\left(\mathbf{\eta }\right)\right)}_1{e}^{-j{\omega }_{c}b\left(N_1^{\prime }{T}_s-\tau \right)},\dots ,{\left(\mathbf{s}\left(\mathbf{\eta }\right)\right)}_{N^{\prime }}{e}^{-j{\omega }_{c}b\left(N_2^{\prime }{T}_s-\tau \right)})}^{\top },\hfill \end{array}$$

(9)

where $\mathbf{n}\sim \mathcal{CN}\left(\mathbf{0},{\sigma }_n^2{\mathbf{I}}_{N^{\prime }}\right)$. Since the transmitter/receiver antenna gains and polarization vectors are in general unknown to a certain extent, $\alpha$ is assumed to be an unknown complex parameter [4,8,9,35,43]. Then, the unknown deterministic parameters [36] can be gathered in vector ${\underline{ϵ}}^{\top }=[{\sigma }_n^2,\alpha ,{\alpha }^{*},{\mathbf{\eta }}^{\top }]$, where ${\alpha }^{*}$ is the complex conjugate of $\alpha$. Note that the same signal model (9) can be obtained if instead of line-of-sight transmission, one considers transmission via diffraction (scatterer), reflexion (reflector) or combination of the three (multipaths) [8,43].

3. Maximum Likelihood and Ambiguity Function

Considering the signal model (9), the delay-Doppler MLE is defined as (Let $S=span\left(\mathbf{A}\right)$, with $\mathbf{A}$ a matrix, be the linear span of the set of its column vectors, $S^{\perp }$ the orthogonal complement of the subspace S, ${\mathbf{\Pi }}_{\mathbf{A}}=\mathbf{A}\left({\mathbf{A}}^H\mathbf{A}\right){\mathbf{A}}^H$ the orthogonal projection over S, and ${\mathbf{\Pi }}_{\mathbf{A}}^{\perp }=\mathbf{I}-{\mathbf{\Pi }}_{\mathbf{A}}$) [35]

$$\widehat{\mathbf{\eta }}=arg\underset{\mathbf{\eta }}{min}\left\{{\mathbf{x}}^H{\mathbf{\Pi }}_{\mathbf{a}\left(\mathbf{\eta }\right)}^{\perp }\mathbf{x}\right\}=arg\underset{\mathbf{\eta }}{max}\left\{\frac{{\left|\mathbf{a}{\left(\mathbf{\eta }\right)}^H\mathbf{x}\right|}^2}{\mathbf{a}{\left(\mathbf{\eta }\right)}^H\mathbf{a}\left(\mathbf{\eta }\right)}\right\}=arg\underset{\left(b,\tau \right)}{max}\left\{\frac{{\left|\underset{-\infty }{\overset{+\infty }{\int }}s{\left(t;\mathbf{\eta }\right)}^{*}{e}^{j{\omega }_{c}b\left(t-\tau \right)}x\left(t\right)dt\right|}^2}{\frac{1}{\left(1-b\right)}\underset{-\infty }{\overset{+\infty }{\int }}{\left|s\left(t\right)\right|}^{2}dt}\right\},$$

(10)

which is instrumental to validate a CRB expression because such estimator is known to be asymptotically efficient (e.g., in the high SNR regime) for the conditional signal model of interest [16,17]. Another interesting expression is the maximum SNR at the output of the MLE matched filter

$$\begin{array}{cc}\hfill & {\mathrm{SNR}}_{\mathrm{out}}=\frac{{\left|\underset{-\infty }{\overset{+\infty }{\int }}s{\left(t;\mathbf{\eta }\right)}^{*}\alpha s\left(t;\mathbf{\eta }\right)dt\right|}^2}{E\left[{\left|\underset{-\infty }{\overset{+\infty }{\int }}s{\left(t;\mathbf{\eta }\right)}^{*}n\left(t\right)dt\right|}^2\right]}=\frac{{\left|\alpha \right|}^2}{\left(\frac{{\sigma }_n^2}{F_s}\right)\left(1-b\right)}\underset{-\infty }{\overset{+\infty }{\int }}{\left|s\left(t\right)\right|}^{2}dt=\frac{{\left|\alpha \right|}^2\mathbb{E}}{\left(\frac{{\sigma }_{\mathbf{n}}^2}{F_s}\right)\left(1-b\right)},\hfill \end{array}$$

(11)

with $\mathbb{E}={\int }_{-\infty }^{+\infty }{\left|s\left(t\right)\right|}^{2}dt$ the energy of the signal. The corresponding ambiguity function is given by [2,3,7,43]

$$\begin{array}{cc}\hfill & \Xi \left({\mathbf{\eta }}^{\prime };\mathbf{\eta }\right)=\frac{1}{N^{\prime }}{\left(\mathbf{a}\left(\mathbf{\eta }\right)\alpha \right)}^H{\mathbf{\Pi }}_{\mathbf{a}\left({\mathbf{\eta }}^{\prime }\right)}\left(\mathbf{a}\left(\mathbf{\eta }\right)\alpha \right)=\frac{{\left|\alpha \right|}^2}{N^{\prime }}{∥\mathbf{a}\left(\mathbf{\eta }\right)∥}^2{\left|\frac{\mathbf{a}{\left(\mathbf{\eta }\right)}^H\mathbf{a}\left({\mathbf{\eta }}^{\prime }\right)}{∥\mathbf{a}\left(\mathbf{\eta }\right)∥∥\mathbf{a}\left({\mathbf{\eta }}^{\prime }\right)∥}\right|}^2,\hfill \end{array}$$

(12)

and if we define the following function

$$\mathbf{\Phi }\left(\mathbf{\eta }\right)={\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^T}}^H{\mathbf{\Pi }}_{\mathbf{a}\left(\mathbf{\eta }\right)}^{\perp }\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^T},$$

(13)

it can be approximated by its 2nd order Taylor expansion as [7] (Chapter 3.9.4)

$$\begin{array}{cc}\hfill \Xi \left(\mathbf{\eta }+d\mathbf{\eta };\mathbf{\eta }\right)& \simeq \frac{{\left|\alpha \right|}^2}{N^{\prime }}{∥\mathbf{a}\left(\mathbf{\eta }\right)∥}^2\left(1-\frac{1}{2}d{\mathbf{\eta }}^T\left(\frac{2\Re \left\{\mathbf{\Phi }\left(\mathbf{\eta }\right)\right\}}{{∥\mathbf{a}\left(\mathbf{\eta }\right)∥}^2}\right)d\mathbf{\eta }\right),\hfill \end{array}$$

(14)

where the second term which depends on $\Phi \left(\mathbf{\eta }\right)$ is directly related to the CRB (see (15)–(24)). Since the true and approximated ambiguity function must coincide around the maximum, this is also helpful to validate the CRB expression.

4. New Compact CRB for Delay, Doppler Stretch and Phase Estimation with a Band-limited Signal

The main goal of this contribution is to obtain a new compact analytic form of the CRB for the signal model (9), where both time-delay $\tau$ and time-delay drift b, also known as the Doppler effect leading to time compression or expansion, are to be estimated (i.e., $\tau \left(t\right)\approx \tau +bt$).

4.1. Background on CRB for the Single Source CSM

The CRB of $\mathbf{\eta }$ for a Gaussian conditional observation model (9) has been known for ages [34] and is given by [6,34,35,36]

$${\mathbf{CRB}}_{\mathbf{\eta }}=\frac{{\sigma }_n^2}{2{\left|\alpha \right|}^2}\Re {\left\{\mathbf{\Phi }\left(\mathbf{\eta }\right)\right\}}^{-1},\mathbf{\Phi }\left(\mathbf{\eta }\right)\triangleq \left(13\right).$$

(15)

Thus, there is no point in recomputing the global Fisher Information Matrix (FIM) [36]

$${\mathbf{F}}_{\underline{ϵ}}=E\left[\frac{\partial lnp\left(\mathbf{x};\underline{ϵ}\right)}{\partial \underline{ϵ}}{\left(\frac{\partial lnp\left(\mathbf{x};\underline{ϵ}\right)}{\partial \underline{ϵ}}\right)}^H\right]=-E\left[\frac{{\partial }^{2}lnp\left(\mathbf{x};\underline{ϵ}\right)}{\partial \underline{ϵ}\partial {\underline{ϵ}}^H}\right],$$

(16)

to extract ${\mathbf{CRB}}_{\mathbf{\eta }}$ (15) as proposed previously in [22,28,29,30] (even in the multiple sources case as in [29], since a multiple sources version of (15) exists as well [6,34,35,36]) which may lead to errors in its derivation. Interestingly enough, if we reparameterize the complex amplitude $\alpha$ as $\alpha =\rho e^{j\phi }$ where $\rho$ denotes the amplitude (modulus) and $\phi$ the phase, then (9) becomes

$$\mathbf{x}=\rho e^{j\phi }\mathbf{a}\left(\mathbf{\eta }\right)+\mathbf{n},$$

(17)

and the unknown deterministic parameter vector $\underline{ϵ}$ becomes ${ϵ}^{\top }=[{\sigma }_n^2,\rho ,\phi ,{\mathbf{\eta }}^{\top }]$. Under this equivalent parameterization of the conditional observation model, the compact ${\mathbf{CRB}}_{\mathbf{\eta }}$ (15) can be complemented in order to include amplitude and phase parameters [32], which leads to

$$\begin{array}{cc}\hfill & CR{B}_{\rho }=\frac{{\sigma }_n^2}{2{∥\mathbf{a}\left(\mathbf{\eta }\right)∥}^2}+{\rho }^2\frac{\mathrm{Re}\left\{{\mathbf{a}}^H\left(\mathbf{\eta }\right)\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^{\top }}\right\}{\mathbf{CRB}}_{\mathbf{\eta }}\mathrm{Re}{\left\{{\mathbf{a}}^H\left(\mathbf{\eta }\right)\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^{\top }}\right\}}^{\top }}{{∥\mathbf{a}\left(\mathbf{\eta }\right)∥}^4},\hfill \end{array}$$

(18)

$${\mathbf{CRB}}_{\theta }=\left[\begin{array}{cc}CR{B}_{\phi }& {\mathbf{CRB}}_{\mathbf{\eta },\phi }^{\top }\\ {\mathbf{CRB}}_{\mathbf{\eta },\phi }& {\mathbf{CRB}}_{\mathbf{\eta }}\end{array}\right],{\theta }^{\top }=\left(\phi ,{\mathbf{\eta }}^{\top }\right),$$

(19)

where ${\mathbf{CRB}}_{\mathbf{\eta }}$ is given by (15) and

$$\begin{array}{cc}\hfill & CR{B}_{\phi }=\frac{{\sigma }_n^2}{2{\rho }^2}\frac{1}{{∥\mathbf{a}\left(\mathbf{\eta }\right)∥}^2}+\frac{\mathrm{Im}\left\{{\mathbf{a}}^H\left(\mathbf{\eta }\right)\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^{\top }}\right\}{\mathbf{CRB}}_{\mathbf{\eta }}\mathrm{Im}{\left\{{\mathbf{a}}^H\left(\mathbf{\eta }\right)\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^{\top }}\right\}}^{\top }}{{∥\mathbf{a}\left(\mathbf{\eta }\right)∥}^4},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & {\mathbf{CRB}}_{\mathbf{\eta },\phi }=-{\mathbf{CRB}}_{\mathbf{\eta }}\frac{\mathrm{Im}{\left\{{\mathbf{a}}^H\left(\mathbf{\eta }\right)\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^{\top }}\right\}}^{\top }}{{∥\mathbf{a}\left(\mathbf{\eta }\right)∥}^2}.\hfill \end{array}$$

(21)

Last, since ${\mathrm{SNR}}_{\mathrm{out}}={\rho }^2/\left({\sigma }_{\mathbf{n}}^2\left(1-b\right)\right)\mathbb{E}{F}_s$ (11), it may be worth remembering that [6,35]

$$CR{B}_{{\sigma }_n^2}=\frac{1}{N^{\prime }}{\left({\sigma }_n^2\right)}^2,$$

(22)

which allows to compute $CR{B}_{{\mathrm{SNR}}_{\mathrm{out}}}$, of interest in some applications.

For the sake of completeness, under the reparametrized conditional observation model (17), the amplitude $\rho$ and phase $\phi$ MLEs are given by [35]

$$\widehat{\rho }=\left|\widehat{\alpha }\right|,\widehat{\phi }=arg\left\{\widehat{\alpha }\right\},\widehat{\alpha }=\frac{\mathbf{a}{\left(\mathbf{\eta }\right)}^H\mathbf{x}}{\mathbf{a}{\left(\mathbf{\eta }\right)}^H\mathbf{a}\left(\mathbf{\eta }\right)}=\frac{\underset{-\infty }{\overset{+\infty }{\int }}s{\left(t;\mathbf{\eta }\right)}^{*}{e}^{j{\omega }_{c}b\left(t-\tau \right)}x\left(t\right)dt}{\frac{1}{\left(1-b\right)}\underset{-\infty }{\overset{+\infty }{\int }}{\left|s\left(t\right)\right|}^{2}dt},$$

(23)

where $\widehat{\alpha }$ is the MLE of the complex amplitude $\alpha$.

4.2. A Preliminary Compact CRB for the Single Band-Limited Source CSM

Let $\beta =1-b$ and $s^{\left(1\right)}\left(t\right)=\frac{ds\left(t\right)}{dt}$. Thus, we look for the compact closed-form analytic expression,

$$\begin{array}{cc}\Re \left\{\mathbf{\Phi }\left(\mathbf{\eta }\right)\right\}\hfill & =\left[\begin{array}{cc}{\left(·\right)}_{1,1}& {\left(·\right)}_{1,2}\\ {\left(·\right)}_{1,2}& {\left(·\right)}_{2,2}\end{array}\right]\hfill \\ \hfill & =\Re \left\{{\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^{\top }}}^H\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^{\top }}\right\}-\Re \left\{\frac{1}{{∥\mathbf{a}\left(\mathbf{\eta }\right)∥}^2}{\left(\mathbf{a}{\left(\mathbf{\eta }\right)}^H\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^{\top }}\right)}^H\left(\mathbf{a}{\left(\mathbf{\eta }\right)}^H\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^{\top }}\right)\right\},\hfill \end{array}$$

(24)

which after tedious calculus (details in Appendix A) is given by

$$\begin{array}{c}{\left(·\right)}_{1,1}=F_s{\beta }^2\left(W_{3,3}-\frac{{\left|w_3\right|}^2}{w_1}\right),\hfill \\ {\left(·\right)}_{1,2}=F_s\beta \left(\begin{array}{c}{\omega }_c\Im \left\{w_4-\frac{w_3{w}_2}{w_1}\right\}-\frac{1}{w_1}\Re \left\{w_3{w}_4^{*}\right\}+W_{4,3}\end{array}\right),\hfill \\ {\left(·\right)}_{2,2}=F_s\left(\begin{array}{c}{\omega }_c^2\left(W_{2,2}-\frac{w_2^2}{w_1}\right)+2{\omega }_c\Im \left\{W_{4,2}-\frac{w_2{w}_4}{w_1}\right\}+W_{4,4}-\frac{{\left|w_4\right|}^2}{w_1}\end{array}\right),\hfill \end{array}$$

(25)

where

$${\mathbf{a}}^H\left(\mathbf{\eta }\right)\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^T}=F_s{\left(\begin{array}{c}j{\omega }_c\left(1-\beta \right)w_1-\beta w_3\hfill \\ -j{\omega }_c{w}_2-w_4\hfill \end{array}\right)}^{\top },$$

(26)

and the terms $w_1$, $w_2$, $w_3$, $w_4$, $W_{2,2}$, $W_{3,3}$, $W_{4,2}$, $W_{4,3}$, $W_{4,4}$ are computed as

$$\begin{array}{cc}\hfill & w_1=\frac{1}{\beta }\underset{-\infty }{\overset{+\infty }{\int }}{\left|s\left(t\right)\right|}^{2}dt,w_2=\frac{1}{{\beta }^2}\underset{-\infty }{\overset{+\infty }{\int }}t{\left|s\left(t\right)\right|}^{2}dt,\hfill \\ \hfill & w_3=\frac{1}{\beta }\underset{-\infty }{\overset{+\infty }{\int }}{s}^{\left(1\right)}\left(t\right)s{\left(t\right)}^{*}dt,w_4=\frac{1}{{\beta }^2}\underset{-\infty }{\overset{+\infty }{\int }}t{s}^{\left(1\right)}\left(t\right)s{\left(t\right)}^{*}dt,\hfill \\ \hfill & W_{2,2}=\frac{1}{{\beta }^3}\underset{-\infty }{\overset{+\infty }{\int }}{t}^2{\left|s\left(t\right)\right|}^{2}dt,W_{3,3}=\frac{1}{\beta }\underset{-\infty }{\overset{+\infty }{\int }}{\left|s^{\left(1\right)}\left(t\right)\right|}^{2}dt,\hfill \\ \hfill & W_{4,2}=\frac{1}{{\beta }^3}\underset{-\infty }{\overset{+\infty }{\int }}{t}^2{s}^{\left(1\right)}\left(t\right)s{\left(t\right)}^{*}dt,W_{4,3}=\frac{1}{{\beta }^2}\underset{-\infty }{\overset{+\infty }{\int }}t{\left|s^{\left(1\right)}\left(t\right)\right|}^{2}dt,\hfill \\ \hfill & W_{4,4}=\frac{1}{{\beta }^3}\underset{-\infty }{\overset{+\infty }{\int }}{t}^2{\left|s^{\left(1\right)}\left(t\right)\right|}^{2}dt.\hfill \end{array}$$

(27)

4.3. Comparison with Existing Literature

In the existing literature [18,19,22,28,29,30], whenever the signal amplitude $\rho$ and phase $\phi$ are known, the observation model does not explicitly take into account a complex envelope modulated by a carrier frequency $f_c$ as in (6), (9), (17), but is simply [18] ((1.1), (1.8)), [19] ((2), (11)), [22] (1), [28] (4), [29] ((2), (5)), [30] (1),

$$x\left(t\right)=\alpha s\left(t;\mathbf{\eta }\right)+n\left(t\right),$$

(28)

where $s\left(t;\mathbf{\eta }\right)$ is a band-limited signal (since the noise $n\left(t\right)$ is always assumed to be band-limited, with a bandwidth being larger than (or equal to) the signal bandwidth). Thus (28) amounts to set $f_c=0$ in both observation models (6), (9), (17) and CRB terms (25), (26), leading to

$${\mathbf{CRB}}_{\mathbf{\eta }}=\frac{{\sigma }_n^2}{2{\rho }^2}{\mathbf{\Lambda }}^{-1},\begin{array}{c}{\Lambda }_{1,1}=F_s{\beta }^2\left(W_{3,3}-\frac{{\left|w_3\right|}^2}{w_1}\right),\hfill \\ {\Lambda }_{1,2}=F_s\beta \left(W_{4,3}-\frac{\Re \left\{w_3{w}_4^{*}\right\}}{w_1}\right),\hfill \\ {\Lambda }_{2,1}={\mathbf{\Lambda }}_{1,2},\hfill \\ {\Lambda }_{2,2}=F_s\left(W_{4,4}-\frac{{\left|w_4\right|}^2}{w_1}\right).\hfill \end{array}$$

(29)

If we consider the existing literature where the signal amplitude $\rho$ and phase $\phi$ are unknown [22,28,30], it takes into account that $\Re \left\{w_3\right\}=0$ (true for any $s\left(t\right)$, see next Section) but also that $\Re \left\{w_4\right\}=0$ which is an unnecessary restriction not satisfied for most of real-valued signals (see next Section), leading to [22] (14),[28] (10), [30] (Section IV) (with a few typos [22] ((12e–12g)), [30] (13)),

$${\mathbf{CRB}}_{\mathbf{\eta }}=\frac{{\sigma }_n^2}{2{\rho }^2}{\mathbf{\Lambda }}^{-1},\begin{array}{c}{\Lambda }_{1,1}=F_s{\beta }^2\left(W_{3,3}-\frac{\mathrm{Im}{\left\{w_3^{*}\right\}}^2}{w_1}\right),\hfill \\ {\Lambda }_{1,2}=F_s\beta \left(W_{4,3}-\frac{\mathrm{Im}\left\{w_3^{*}\right\}\mathrm{Im}\left\{w_4^{*}\right\}}{w_1}\right),\hfill \\ {\Lambda }_{2,1}={\mathbf{\Lambda }}_{1,2},\hfill \\ {\Lambda }_{2,2}=F_s\left(W_{4,4}-\frac{\mathrm{Im}{\left\{w_4^{*}\right\}}^2}{w_1}\right),\hfill \end{array}$$

(30)

which is (29) under $\Re \left\{w_3\right\}=\Re \left\{w_4\right\}=0$. Last, if we consider the existing literature where the signal amplitude $\rho$ is unknown but the phase $\phi$ is known [29], the single point target case detailed for a real-valued $\alpha$ reads,

$${\mathbf{CRB}}_{\mathbf{\eta }}=\frac{{\sigma }_n^2}{2{\rho }^2}{\mathbf{\Lambda }}^{-1},\begin{array}{c}{\Lambda }_{1,1}=F_s{\beta }^2{W}_{3,3},\hfill \\ {\Lambda }_{1,2}=F_s\beta W_{4,3},\hfill \\ {\Lambda }_{2,1}={\Lambda }_{1,2},\hfill \\ {\Lambda }_{2,2}=F_s\left(W_{4,4}-\frac{\Re {\left\{w_4\right\}}^2}{w_1}\right),\hfill \end{array}$$

(31)

where assumption ([18], (1.12)) is taken into account leading to ([18], (1.15)) $\Re \left\{w_4\right\}=-w_1/\left(2\beta \right)$ and $\Re {\left\{w_4\right\}}^2/w_1=w_1/\left(4{\beta }^2\right)$. Note that assumption ([18], (1.12)) is not satisfied by a great number of band-limited signals (for instance if $s\left(t\right)=s\left(0\right)$ sinc $\left(\pi F_{s}t\right)$). Additionally note that neither $CR{B}_{\rho }$ (18) nor ${\mathbf{CRB}}_{\theta }$ (19) have been introduced in the existing literature dealing with Doppler stretch. Thus, the proposed approach where the only assumption is to consider a band-limited signal having a finite number on non-zero samples (to ensure finiteness of all the integrals considered (27) as detailed in the next Section), and an explicit carrier frequency modulation, offer far more general (or new) CRB expressions than the existing ones (from which $w_2$, $W_{2,2}$, $W_{4,2}$ can not be guessed).

4.4. A Versatile Compact CRB for Delay, Doppler Stretch and Phase Estimation with a Band-Limited Signal

Even if deriving the most general form of the CRBs (15), (18), (19), (25), (26) and (27) for a given observation model and a given class of signals is of theoretical importance, making it easy to apply for a large subset of the class, and ideally to the whole class, is also a challenging goal of great interest both from a practical and a theoretical point of view. Actually, such easy to use and versatile expressions can be introduced by exploiting the properties of a band-limited signal (1). Indeed, in that case, the terms $w_1$, $w_2$, …, $W_{4,3}$, $W_{4,4}$ (27) can also be expressed as (details in Appendix B)

$$\begin{array}{cc}\hfill & w_1=\frac{{\mathbf{s}}^H\mathbf{s}}{\beta F_s},w_2=\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\beta }^2{F}_s^2},w_3=\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{\beta },w_4=\frac{{\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{s}}{{\beta }^2{F}_s},W_{2,2}=\frac{{\mathbf{s}}^H{\mathbf{D}}^2\mathbf{s}}{{\beta }^3{F}_s^3},W_{3,3}=\frac{F_s{\mathbf{s}}^H\mathbf{Vs}}{\beta },\hfill \\ \hfill & W_{4,2}=\frac{1}{{\beta }^3{F}_s^2}\left({\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{Ds}-{\mathbf{s}}^H\mathbf{Ds}\right),W_{4,3}=\frac{1}{{\beta }^2}\left({\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}+{\mathbf{s}}^H\mathbf{VDs}\right),\hfill \\ \hfill & W_{4,4}=\frac{1}{{\beta }^3{F}_s}\left({\mathbf{s}}^H\mathbf{s}+{\mathbf{s}}^H\mathbf{DVDs}-2\Re \left\{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{Ds}\right\}\right),\hfill \end{array}$$

(32)

with $\mathbf{D}$, $\mathbf{\Lambda }$ and $\mathbf{V}$ defined as

$$\begin{array}{cc}\hfill & \mathbf{D}=\mathrm{diag}\left([N_1,N_1+1,\dots ,N_2-1,N_2]\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & {\left(\mathbf{V}\right)}_{n,n^{\prime }}=\left|\begin{array}{c}{n}^{\prime }\ne n:{\left(-1\right)}^{\left|n-n^{\prime }\right|}\frac{2}{{\left(n-n^{\prime }\right)}^2}\hfill \\ n^{\prime }=n:\frac{{\pi }^2}{3}\hfill \end{array}\right.,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & {\left(\mathbf{\Lambda }\right)}_{n,n^{\prime }}=\left|\begin{array}{c}{n}^{\prime }\ne n:\frac{{\left(-1\right)}^{\left|n-n^{\prime }\right|}}{\left(n-n^{\prime }\right)}\hfill \\ n^{\prime }=n:0\hfill \end{array}\right.,\hfill \end{array}$$

(35)

for $N_1\le n,n^{\prime }\le N_2$. Moreover, since $W_{3,3}>0$ if $\mathbf{s}\ne \mathbf{0}$, then $\mathbf{V}$ is a symmetric positive definite real-valued matrix $\left({\mathbf{V}}^T=\mathbf{V},\mathbf{V}>\mathbf{0}\right)$; and $\mathbf{\Lambda }$ is an anti-symmetric real-valued matrix $\left({\mathbf{\Lambda }}^T=-\mathbf{\Lambda }\right)$, which leads to $\Re \left\{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}\right\}=0$ and $\Re \left\{{\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{Ds}\right\}=0$. Finally, the terms in (25), (26) can be expressed as

$$\begin{array}{c}{\left(·\right)}_{1,1}=F_s\beta {\mathbf{s}}^H\mathbf{s}\left(\frac{{\mathbf{s}}^H\mathbf{Vs}}{{\mathbf{s}}^H\mathbf{s}}-{\left|\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\right|}^2\right)\hfill \\ {\left(·\right)}_{1,2}=\frac{{\mathbf{s}}^H\mathbf{s}}{\beta }\left(\begin{array}{c}\frac{{\omega }_c}{F_s}\Im \left\{\frac{{\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}-\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\right\}-\Re \left\{\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\right\}+\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}+\frac{{\mathbf{s}}^H\mathbf{VDs}}{{\mathbf{s}}^H\mathbf{s}}\hfill \end{array}\right)\hfill \\ {\left(·\right)}_{2,2}=\frac{{\mathbf{s}}^H\mathbf{s}}{F_s{\beta }^3}\left(\begin{array}{c}\frac{{\omega }_c^2}{F_s^2}\left(\frac{{\mathbf{s}}^H{\mathbf{D}}^2\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}-{\left(\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\right)}^2\right)+2+\frac{{\mathbf{s}}^H\mathbf{DVDs}}{{\mathbf{s}}^H\mathbf{s}}-{\left|1+\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\right|}^2\hfill \\ +2\frac{{\omega }_c}{F_s}\Im \left\{\frac{{\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}-\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\right\}-2\frac{{\omega }_c}{F_s}\Im \left\{\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\frac{{\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\right\}\hfill \end{array}\right)\hfill \end{array},$$

(36)

$$\mathbf{a}{\left(\mathbf{\eta }\right)}^H\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^T}=F_s{\mathbf{s}}^H\mathbf{s}{\left(\begin{array}{c}j\frac{{\omega }_c}{\beta F_s}\left(1-\beta \right)-\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}-j\frac{{\omega }_c}{{\beta }^2{F}_s^2}\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}-\frac{1}{{\beta }^2{F}_s}\frac{{\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\hfill \end{array}\right)}^{\top }.$$

(37)

From this result, it is straightforward to obtain compact closed-form CRB expressions for (15), (18), (19) (i.e., (24) is a $2\times 2$ matrix) which depend only on the baseband signal samples and known matrices $\mathbf{D}$, $\mathbf{V}$ and $\mathbf{\Lambda }$ (33)–(35). These appealing closed-form CRBs which depend only on the samples’ vector, can thus be used for any sensing application compliant with the CSM (6) in order to investigate its asymptotic theoretical performance, and possibly, to assess the threshold region by comparing simulated (or measured) MSE of MLEs with the CRBs values obtained.

In addition, if $s\left(t\right)$ is a real-valued signal, then $\Re \left\{{\mathbf{s}}^{\top }\mathbf{\Lambda }\mathbf{s}\right\}=\Re \left\{{\mathbf{s}}^{\top }\mathbf{D}\mathbf{\Lambda }\mathbf{Ds}\right\}=0$ and (36–37) simplify to

$$\begin{array}{c}{\left(·\right)}_{1,1}=F_s\beta {\mathbf{s}}^{\top }\mathbf{s}\frac{{\mathbf{s}}^{\top }\mathbf{Vs}}{{\mathbf{s}}^{\top }\mathbf{s}},\hfill \\ {\left(·\right)}_{1,2}=\frac{{\mathbf{s}}^{\top }\mathbf{s}}{\beta }\frac{{\mathbf{s}}^{\top }\mathbf{VDs}}{{\mathbf{s}}^{\top }\mathbf{s}},\hfill \\ {\left(·\right)}_{2,2}=\frac{{\mathbf{s}}^{\top }\mathbf{s}}{F_s{\beta }^3}\left(\begin{array}{c}\frac{{\omega }_c^2}{F_s^2}\left(\frac{{\mathbf{s}}^{\top }{\mathbf{D}}^2\mathbf{s}}{{\mathbf{s}}^{\top }\mathbf{s}}-{\left(\frac{{\mathbf{s}}^{\top }\mathbf{Ds}}{{\mathbf{s}}^{\top }\mathbf{s}}\right)}^2\right)+2+\frac{{\mathbf{s}}^{\top }\mathbf{DVDs}}{{\mathbf{s}}^{\top }\mathbf{s}}-{\left(1+\frac{{\mathbf{s}}^{\top }\mathbf{D}\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^{\top }\mathbf{s}}\right)}^2\hfill \end{array}\right),\hfill \end{array}$$

(38)

$$\mathbf{a}{\left(\mathbf{\eta }\right)}^H\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^T}=F_s{\mathbf{s}}^{\top }\mathbf{s}{\left(\begin{array}{c}j\frac{{\omega }_c}{\beta F_s}\left(1-\beta \right)-j\frac{{\omega }_c}{{\beta }^2{F}_s^2}\frac{{\mathbf{s}}^{\top }\mathbf{Ds}}{{\mathbf{s}}^{\top }\mathbf{s}}-\frac{1}{{\beta }^2{F}_s}\frac{{\mathbf{s}}^{\top }\mathbf{D}\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^{\top }\mathbf{s}}\hfill \end{array}\right)}^{\top }.$$

(39)

4.5. Standard Narrowband Signal Model

A simplified narrowband signal model is analyzed in [20,32], where the Doppler effect on the baseband signal $s\left(t\right)$ is not considered. In this case, $a\left(t;\mathbf{\eta }\right)$ is written as

$$s\left(\left(1-b\right)\left(t-\tau \right)\right)e^{-j{\omega }_{c}bt}\simeq s\left(t-\tau \right)e^{-j{\omega }_{c}bt}.$$

(40)

Under this hypothesis, (36) and (37) become [20,32]

$$\begin{array}{c}{\left(.\right)}_{1,1}=F_s{\mathbf{s}}^H\mathbf{s}\left(\frac{{\mathbf{s}}^H\mathbf{Vs}}{{\mathbf{s}}^H\mathbf{s}}-{\left|\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\right|}^2\right)\hfill \\ {\left(.\right)}_{1,2}=\frac{{\mathbf{s}}^H\mathbf{c}}{F_s}\left({\omega }_c\Im \left\{\frac{{\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}-\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\right\}\right)\hfill \\ {\left(.\right)}_{2,2}=\frac{{\mathbf{s}}^H\mathbf{s}}{F_s}\frac{{\omega }_c^2}{F_s^2}\left(\frac{{\mathbf{s}}^H{\mathbf{D}}^2\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}-{\left(\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\right)}^2\right)\hfill \end{array},$$

(41)

and [32]

$$\mathbf{a}{\left(\mathbf{\eta }\right)}^H\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^T}=F_s{\mathbf{s}}^H\mathbf{s}{\left(\begin{array}{c}j\frac{{\omega }_c}{\beta F_s}\left(1-\beta \right)-\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\hfill \\ -j\frac{{\omega }_c}{{\beta }^2{F}_s^2}\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\hfill \end{array}\right)}^{\top }.$$

(42)

Then, if $s\left(t\right)$ is a real signal, (41), (42), reduce to

$$\begin{array}{c}{\left(.\right)}_{1,1}=F_s{\mathbf{s}}^{\top }\mathbf{s}\frac{{\mathbf{s}}^{\top }\mathbf{Vs}}{{\mathbf{s}}^{\top }\mathbf{s}}\hfill \\ {\left(.\right)}_{1,2}=0\hfill \\ {\left(.\right)}_{2,2}=\frac{{\mathbf{s}}^{\top }\mathbf{s}}{F_s}\frac{{\omega }_c^2}{F_s^2}\left(\frac{{\mathbf{s}}^{\top }{\mathbf{D}}^2\mathbf{s}}{{\mathbf{s}}^{\top }\mathbf{s}}-{\left(\frac{{\mathbf{s}}^{\top }\mathbf{Ds}}{{\mathbf{s}}^{\top }\mathbf{s}}\right)}^2\right)\hfill \end{array},$$

(43)

and

$$\mathbf{a}{\left(\mathbf{\eta }\right)}^H\frac{\partial \mathbf{a}\left(\mathbf{\eta }\right)}{\partial {\mathbf{\eta }}^T}=j{F}_s{\mathbf{s}}^{\top }\mathbf{s}\frac{{\omega }_c}{\beta F_s}{\left(\begin{array}{c}1-\beta \hfill \\ -\frac{1}{\beta F_s}\frac{{\mathbf{s}}^{\top }\mathbf{Ds}}{{\mathbf{s}}^{\top }\mathbf{s}}\hfill \end{array}\right)}^{\top },$$

(44)

Notice that $\Re {\left\{\mathbf{\Phi }\left(\mathbf{\eta }\right)\right\}}_{2,2}$ has become invariant by time translation because it comes only from the term $e^{-j{\omega }_{c}b\left(t-\tau \right)}$.

4.6. Further Insights and Outlooks

The versatility of the proposed CRB expressions is a key feature that can be easily emphasized by the following (non exhaustive) list of examples of use. For instance, one important outcome offered by the new CRB expressions based on (36) and (37) is to allow a quantifiable assessment of the impact of a bandwidth increase on the CRB values, and also on the coupling between the delay and Doppler effect. Firstly, we can see that for real narrowband signals, the estimation of both parameters is decoupled because the cross terms are null (43), which is no longer the case when the bandwidth increases (38). Secondly, one can notice that an increase of bandwidth has mainly an impact on the Doppler estimation and a marginal impact on the delay estimation by coupling, since

$$\begin{array}{ccc}\hfill {\left(·\right)}_{1,1}^{WB}& =& F_s\beta {\mathbf{s}}^H\mathbf{s}\left(\frac{{\mathbf{s}}^H\mathbf{Vs}}{{\mathbf{s}}^H\mathbf{s}}-{\left|\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\right|}^2\right)={\left(·\right)}_{1,1}^{NB}\hfill \\ \hfill {\left(·\right)}_{2,2}^{WB}& \simeq & {\left(·\right)}_{2,2}^{NB}+\frac{{\mathbf{s}}^H\mathbf{s}}{F_s}\left(\begin{array}{c}2+\frac{{\mathbf{s}}^H\mathbf{DVDs}}{{\mathbf{s}}^H\mathbf{s}}-{\left|1+\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\right|}^2\hfill \\ 2\frac{{\omega }_c}{F_s}\Im \left\{\frac{{\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}-\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\right\}-2\frac{{\omega }_c}{F_s}\Im \left\{\frac{{\mathbf{s}}^H\mathbf{Ds}}{{\mathbf{s}}^H\mathbf{s}}\frac{{\mathbf{s}}^H\mathbf{D}\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\right\}\hfill \end{array}\right)\hfill \end{array}$$

where $WB$ and $NB$ designate the wideband and narrowband terms of $\Re \left\{\mathbf{\Phi }\left(\mathbf{\eta }\right)\right\}$.

On another note, from these expressions we recover the preliminary results in [44], which considered the time-delay estimation for a static system (no Doppler effect). Note that the CRBs (15), (18), (19) can be expressed as a function of the maximum SNR at the output of the MLE matched filter (11). In the static case, the CRB for the time-delay then reads

$$\begin{array}{cc}\hfill F_{\tau \mid \underline{ϵ}}\left(\underline{ϵ}\right)& ={\mathrm{SNR}}_{\mathrm{out}}\times 2F_s^2\left(\frac{{\mathbf{s}}^H\mathbf{Vs}}{{\mathbf{s}}^H\mathbf{s}}-{\left|\frac{{\mathbf{s}}^H\mathbf{\Lambda }\mathbf{s}}{{\mathbf{s}}^H\mathbf{s}}\right|}^2\right),\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill F_{\tau \mid \underline{ϵ}}\left(\underline{ϵ}\right)& ={\mathrm{SNR}}_{\mathrm{out}}\times 2F_s^2\left(\frac{{\mathbf{s}}^T\mathbf{Vs}}{{\mathbf{s}}^T\mathbf{s}}\right),\hfill \end{array}$$

(46)

for a complex $s\left(t\right)$ (45) and real $s\left(t\right)$ (46). Thus, this result can be exploited for optimal signal design, that is, to obtain a fixed-length band-limited signal ${\mathbf{s}}^b$ with a given fixed energy $\mathbb{E}$ (or equivalently ${\mathrm{SNR}}_{\mathrm{out}}$) which minimizes the CRB. We have shown in [44] that the optimal signal is related to the first eigenvector of $\mathbf{V}$, and the optimal CRB is given by

$$F_{\tau \mid \underline{ϵ}}\left(\underline{ϵ}\right)\le 2{\mathrm{SNR}}_{\mathrm{out}}{F}_s^2{D}_{1,1},$$

(47)

with $\pi \le D_{1,1}\le {\pi }^2$ the largest eigenvalue of $\mathbf{V}$.

Last, from a broader perspective, the proposed CRB expressions also pave the way for the definition of new cost-functions dedicated to waveform optimization exploring the trade-off between optimal accuracy and/or coupling when measuring time-delay and Doppler effect, and other operational constraints. For instance, they can easily be integrated to existing cost-functions used to build waveform sequences not only with good autocorrelation properties (low sidelobes level for a given energy) [45,46] or minimal peak-to-average power ratio [47], but also with optimal delay estimation accuracy, since they are fitted to any optimization method exploiting vector based cost functions [48,49].

5. Validation and Discussion

To assess the validity of the new compact closed-form CRB, a GNSS positioning example is considered [50]. In the case of GNSS, the signals broadcasted by different satellites use pseudo-orthogonal codes $s\left(t\right)$, which allow at the receiver to process different satellite signals using independent channels. Each channel estimates the time-delay and Doppler effect between the receiver and a given satellite, in order to construct a set of so-called pseudoranges and pseudorange rates, which in turn are used to solve a multilateration problem to find the receiver’s position, velocity and timing. If we omit the navigation data message (i.e., assuming data wipe-off or a pilot channel) a generic signal received from a GNSS satellite, at the output of the Hilbert filter, can be expressed as in (9) and (17).

5.1. Synthetic Signal

We first consider a simulated GNSS band-limited signal corresponding to a GPS L1 C/A PRN code of length 1023. As a result that GNSS codes are real, we compute the CRB using (38). The CRB and the corresponding MLE in (10), obtained from 1000 Monte Carlo (MC) runs with $\alpha =((1+j)/\sqrt{2})\sqrt{{\mathrm{SNR}}_{\mathrm{in}}}$, are shown in Figure 1 (top). Since (9) belongs to the class of conditional signal models [16], the MLE converges to the CRB at high SNR [17], which confirms the exactness of the CRB. This CRB is further validated by comparing the ambiguity function and its 2nd order Taylor expansion (14) in Figure 1 (bottom), where the perfect match also proves its exactness.

5.2. Real-Life GPS Data Experiment

The proposed CRB expression was also validated in a real-life GPS scenario. The signal acquisition was performed using a static L1 frequency patch antenna in open sky conditions, and recorded using an USRP X310 at $F_s=20$ MHz. To test standard front-end configurations, such a signal was converted to the equivalent $F_s=5$ MHz and $F_s=2.5$ MHz signals by averaging and decimating 4 and 8 samples, respectively. In a real-life data test it is not possible to perform MC runs, therefore the MSE was obtained via 100 different correlations, each one using a coherent integration time of 10 ms (i.e., $\mathbf{s}\left(\mathbf{\eta }\right)$ being 10 GPS L1 C/A PRN codes of length 1 ms), and separated by 20 ms. This leads to an overall observation interval of 1s during which the hypothesis of relative uniform radial movements (4) is regarded as valid [51]. For completeness, the scenario details on the satellites in view (PRN Id), their corresponding ${\mathrm{SNR}}_{\mathrm{out}}$ (dB), Doppler shift (Hz), elevation (${}^{°}$) and azimuth (${}^{°}$), are given in

Table 1. Real-life GPS scenario: satellites in view (PRN Id), ${\mathrm{SNR}}_{\mathrm{out}}$ ((dB)), Doppler shift (Hz), elevation (${}^{°}$) and azimuth (${}^{°}$). Data were collected on 22 April 2015, GPS week number 1841, 16:07-16:10 (UTC).

PRN Id	${\mathbf{SNR}}_{\mathbf{out}}$	Doppler	Elev	Azim
31	17.7	3170	19.3	−50.6
14	18.9	−490	44.8	−95.3
2	21.7	930	28.6	86.7
24	22.8	−2950	34.1	133
25	23.7	480	70.7	−41.7
29	25.6	2480	51.3	−162.7
12	27.1	−1780	53.9	56

. The time-delay and Doppler frequency shift of the GPS satellite signals were obtained with the MLE in (10), where a time-delay resolution of $0.0002$ chips ($0.06$ m) and a Doppler frequency shift resolution of $0.1$ Hz were considered. Firstly, to enforce the hypothesis of relative uniform radial movements (4), the dynamics of the signal was removed by fitting a $2\mathrm{nd}$ order polynomial. Secondly, notice that the ${\mathrm{SNR}}_{\mathrm{out}}$ taken into account is estimated, and thus it is a random variable whose variance decreases with the actual ${\mathrm{SNR}}_{\mathrm{out}}$ [35]. These results are shown in Figure 2 for the time-delay (top) and Doppler (bottom), which again validate the CRB expression at sufficiently high ${\mathrm{SNR}}_{\mathrm{out}}$. As highlighted by MC simulations in Figure 1, the SNR threshold for the delay estimation increases with $F_s$.

6. Conclusions

In this contribution, a new compact closed-form CRB expression for the delay-Doppler estimation of a generic band-limited signal has been derived, which may be exploited in a variety of remote sensing applications, i.e., navigation, radar, sonar, GNSS-R. Such a compact CRB only depends on the baseband signal samples, therefore it is especially easy to use. This generalizes known results on both wideband signals and narrowband signals where the impact of the Doppler effect into the baseband signal is not taken into account. In addition, general compact closed-form CRB expressions for the amplitude and phase have been introduced. The validity of the CRB has been assessed in a navigation example with both synthetic and measured signals. The comparison of the MLE obtained through Monte Carlo and the closed-form CRB allowed to proof the exactness of the proposed CRB. Indeed, as expected, the MLE converges to the CRB at high SNR (i.e., a known result for the Gaussian CSM). In addition, for the synthetic signal analysis, the perfect match of the ambiguity function and its 2nd order Taylor expansion also proves its exactness.

As outlooks, in addition to the performance limit characterization, these appealing closed-form CRBs, which depend only on the baseband samples’ vector, pave the way for the definition of new cost-functions dedicated to waveform optimization exploring the trade-off between optimal accuracy and/or coupling when measuring time-delay and Doppler effect, and other operational constraints. For instance, they can easily be integrated to existing cost-functions used to build waveform sequences not only with good autocorrelation properties (low sidelobes level for given energy) [45,46] or minimal peak-to-average power ratio [47], but also with optimal delay estimation accuracy, since they are fitted to any optimization method exploiting vector based cost functions [48,49].