Elsevier

Signal Processing

Volume 173, August 2020, 107604
Signal Processing

Performance bounds for two-channel delay-doppler estimation using unknown waveforms

https://doi.org/10.1016/j.sigpro.2020.107604Get rights and content

Abstract

In this paper, we derive Cramer-Rao Lower Bound (CRLB) expressions for two-channel radar target delay-Doppler estimation with unknown transmit waveforms. We have access to these unknown waveforms only via the noisy reference measurements. Further, we assume that the waveform source transmits periodic pulses as is common for most pulsed radar illuminators. Recent CRLB literature for radar has focussed on specific types of illuminators and derived the CRLB assuming these specific transmitted waveforms are known and incorporating them directly into the measurement model. Further, some papers while assuming the waveform itself to be unknown, make simplifying statistical assumptions on the waveforms or the waveform structure. Contrary to this, we compute a more general result that is not restricted to any specific waveform and we also include the noisy reference channel measurements in our derivation of the CRLB to obtain a more accurate performance analysis. The only assumption we make is on the periodicity of the transmit signal, which is true in practice. We demonstrate the CRLB using extensive numerical simulations both in the presence and absence of phase synchronization between the transmitter and receiver.

Introduction

Radar systems using illuminators of opportunity offer covert operation as well as low cost of deployment due to the lack of requiring expensive transmit equipment. Further, with growing limitations on the available bandwidth for exclusive use by radar systems, it becomes more important to efficiently use every informative illuminator to estimate the location and velocity of targets of interest irrespective of whether the transmit waveforms are known or unknown. As with any estimation algorithm, it is important to analyze the performance not merely to study the estimation accuracy but for system design like optimizing the bistatic geometry and transmitter selection since the performance is a function of the geometry. Cramer-Rao Lower Bound (CRLB) is important for analyzing the local estimation accuracy as it provides a lower bound on the error variance of unbiased estimators. Further, under certain conditions, the maximum likelihood estimators (MLE) asymptotically achieve the CRLB [1], [2], [3], [4]. Therefore, it is critical to analyze the CRLB for radar systems using illuminators of opportunity. This has motivated a lot of recent studies on this topic [5], [6], [7], [8], [9]. Further, CRLB has also been used for radar system design in terms of illuminator selection as well as receiver placement (geometry selection) [9], [10], [11].

While these papers consider a variety of illuminators, they ignore an important fact that the transmit waveforms are unknown and therefore directly plugging in the expressions for the illuminated waveforms of opportunity while computing the Fisher information matrix (FIM) does not provide accurate bounds for performance analysis. For example, [6], [7] consider third generation UMTS based wireless communications illuminators with known root-raised cosine waveforms instead of estimating the waveforms using the reference channel. Similarly, [8] and [9] derive the CRLB for L-band illuminators of opportunity and FM based illuminators but with a known transmit signal structure. While this kind of knowledge about the transmitted waveform may be available for some commercial illuminators, it is completely unknown in other instances. For example, the waveforms transmitted by a non-cooperative radar that is working independently are unknown. Similarly, even for commercial illuminators, the transmitted waveforms may be unknown when operating in certain environments.

Typically, in these scenarios, we have access to the unknown transmit waveform only via the noisy reference channel. Several recent papers have incorporated the noisy reference measurements into the signal model for passive radar detection problem [12], [13], [14], [15], [16]. In [13], [14], [15], multistatic/MIMO geometries were considered for passive radar. In multistatic/MIMO scenario, target detection is possible even without dedicated reference channels as one can use the signals at the other radar receivers as reference signals. However, for the bistatic case, we always need a dedicated reference channel for information about the unknown transmit waveform. Without a reference channel, the best that can be done is a simple energy detector. Please see Fig. 1 for the bistatic geometry involving reference and surveillance channels. In [12], we have proposed and analyzed the performance of a Constant False Alarm Rate (CFAR) detector for this two-channel problem along with experimental results in a controlled laboratory setting in [17]. However, this analysis with the two-channel signal model has been limited in the radar literature for the estimation problem as we have described above. This has motivated the two-channel radar CRLB study for unknown waveforms in the following papers [18], [19].

While the authors of [18] do treat the waveform itself to be unknown, they make statistical assumptions on the transmit waveform. The waveform is considered to be Gaussian distributed with appropriate correlations between the elements. However, when the illuminator is truly non-cooperative in nature, we cannot make any such assumption and have to treat the waveform as a deterministic unknown vector in each pulse interval. In [19], while the authors assume that the waveform is deterministic unknown, they make some additional assumptions on the structure of the waveform. They assume that linear time shift of the transmit waveform can be approximated by a circular time shift of the waveform vector. However, this is not always true in practice. Additionally, they treat the entire transmit waveform in the long observation interval to be deterministic unknown, thereby making the unknown parameter set to be huge and practically impossible to invert the FIM.

In this paper, we consider the realistic scenario where the signal of opportunity is deterministic unknown and we have access to it only via the noisy reference measurements. We have earlier used this approach to demonstrate some preliminary results in [20]. The only assumption we make on the structure of the transmitted signal is that it is periodic as is the case in reality for any pulsed radar-like illuminator both cooperative and non-cooperative. This is true even for commercial illuminators employing amplitude-modulation where the waveform in each pulse is the same and only the complex amplitude riding on the waveform changes from pulse to pulse [12]. In fact, this assumption is essential to get accurate performance analysis because treating the entire transmit waveform in the long observation interval to be deterministic unknown (as in [19]) as opposed to treating the waveform shape to be deterministic unknown only within one pulse interval (and repeating from pulse to pulse) will make it practically impossible to invert the FIM due to the dimensions. We include these assumptions in our derivation of the CRLB to obtain a more practical performance analysis that is essential to gauge the estimation accuracy as well as for system design using non-cooperative pulsed illuminators. The authors of [21] derive the two-channel radar CRLB for unknown waveform shapes but make a very unrealistic assumption that the signals in both the channels have the same amplitudes. This is never the case because the electromagnetic medium in the both the channels is completely different. Additionally, the target induces an unknown complex attenuation in the surveillance channel.

The rest of the paper is organized as follows. In Section 2, we present the measurement model with the appropriate unknown parameters involving both the reference and surveillance channels, forming the unknown parameter set for estimation. We derive the expressions for all the entries of the FIM to compute the CRLB in Section 3, followed by numerical simulations and discussion in Section 4. Finally, we end the paper with concluding remarks and future work in Section 5.

Section snippets

Signal model

Any practical radar system with unknown waveforms has two receive channels, one dedicated to a direct reference signal from the illuminator of opportunity whereas the other obtains surveillance measurements from the target echo signal. It is important to include both the measurement streams to analyze the system performance because the overall target estimation problem has relevant information in both the channels. In this paper, we consider a bistatic two-channel radar system. The surveillance

Cramer-Rao lower bound

CRLB is important for analyzing the local estimation accuracy as it provides a lower bound on the error variance of unbiased estimators. Further, under certain conditions, the MLE asymptotically achieve the CRLB [1], [4]. Given the log likelihood function logl(y; μ), then for any unbiased estimate μ^(y) of μ,covμ(μ^(y))I(μ)1,where the entries of the FIMIi,j(μ)=E{2logl(y;μ)μiμj}.In this two-channel radar estimation problem, the matrix I is of dimensions (2M+4)×(2M+4).

To compute the CRLB,

Numerical examples

We consider two different choices for the unknown transmit waveform in our numerical simulations. As we have mentioned earlier, the main focus of this paper is the scenario when the illuminator is non-cooperative in nature i.e, the transmit waveform in each pulse interval is unknown. This illuminator can be a non-cooperative radar operating in our region of interest or it can be a commercial illuminator operating in a hostile environment where we do not have access to the transmit waveform. We

Concluding remarks

In this paper, we have derived the CRLB for target delay and Doppler estimation using the two-channel radar signal model. We have assumed that the transmit waveform within each pulse duration is unknown and we have access to it only via the noisy reference channel measurements. Therefore, the dimensions of the unknown parameter set is higher, thereby making the computation of the bound more complex. Our bounds are very realistic and contrary to most existing papers on this topic that derive the

CRediT authorship contribution statement

Sandeep Gogineni: Conceptualization, Formal analysis, Investigation, Software, Writing - original draft. Muralidhar Rangaswamy: Supervision, Writing - review & editing. Michael Wicks: Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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