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We study the Cramer-Rao bounds of parameter estimation and coherence performance for the next generation radar (NGR). In order to enhance the performance of NGR, the signal model of NGR with master-slave architecture based on a single pulse is extended to the case of pulse trains, in which multiple pulses are emitted from all sensors and then integrated spatially and temporally in a unique master sensor. For the MIMO mode of NGR where orthogonal waveforms are emitted, we derive the closed-form Cramer-Rao bound (CRB) for the estimates of generalized coherence parameters (GCPs), including the time delay differences, total phase differences and Doppler frequencies with respect to different sensors. For the coherent mode of NGR where the coherent waveforms are emitted after pre-compensation using the estimates of GCPs, we develop a performance bound of signal-to-noise ratio (SNR) gain for NGR based on the aforementioned CRBs, taking all the estimation errors into consideration. It is shown that greatly improved estimation accuracy and coherence performance can be obtained with pulse trains employed in NGR. Numerical examples demonstrate the validity of the theoretical results.

Large aperture high-power phased array radar has played an important role in long-range surveillance, tracking and discrimination, owing to its capability of obtaining high signal-to-noise ratio (SNR) echoes. Typical such radars include the USA's Ground Based Radar-Prototype (GBR-P) and the Sea-Based X-Band (SBX) radar [

In this paper, we consider NGR with a master-slave architecture, where all the radars transmit signals and only the master radar receives the echoes. It is known that the maximum echo power can be achieved only when we make all the transmitted signals arrive at the target at the same time and in-phase, namely, the coherence gain is obtained via coherent processing. However, the distributed architecture of NGR makes it difficult to coherently combine signals for two reasons. First, the range from a target to different radars may be different, leading to echoes with different propagation time delays and phases; second, each radar has an independent local oscillator with different transmit and receive (T/R) phases, which also adds phase shifts to echoes. Since both the T/R phases and the phase caused by propagation delay can influence the coherence gain, we add the two phases together and name the sum as total phase. In NGR, both the mismatches of time and phase can cause performance degradation. To overcome this shortage, an operation procedure of two steps has been proposed [

Another problem in NGR lies in the constraint of system size,

A thorough review of the existing literature on NGR reveals that the present signal models in NGR are all based on single pulse schemes [

In addition, it is worth pointing out that the parameter estimation of NGR should be distinguished from the parameter estimation of MIMO radar which has been studied in [

In this paper, we make the following contributions which also answer the questions at the end of paragraphs two and three. All the contributions below are useful and instructive for the system design and performance analysis of NGR:

The NGR signal model based on a single pulse is extended to the case of pulse trains for the first time, and the concept of spatial coherence is extended to joint space-time coherence for NGR. The extension to pulse trains benefits the detection and tracking of weak targets and helps control the system scale of NGR.

The original coherence parameters (CPs) of NGR are extended to the generalized coherence parameters (GCPs), with

The closed-form CRBs of the GCPs are derived based on the signal model in (a), and verified through simulations, thus providing a lower bound for the estimation accuracy of the GCPs and a criterion for the performance evaluation of different estimation algorithms.

The formula of coherence gain for NGR is derived and the performance bound is analyzed based on the CRBs in (c) with all types of estimation errors considered, thus providing an upper bound for the SNR gain performance of NGR.

The paper is organized as follows: in Section 2, we present the NGR signal model with pulse trains and specifically define the GCPs. In Section 3, we derive the CRB for parameter estimation. In Section 4, we present the analytical formula of coherence performance. Simulation results and discussions are shown in Section 5, and Section 6 concludes the paper.

The system model of NGR with master-slave architecture is illustrated in

The pulse signal transmitted by the _{k}_{c} is the carrier frequency, and
_{k}_{k}_{k}_{k}

As mentioned above, we consider the case of transmitting multiple pulses. Assume that _{k}_{,0} is the initial range from the target to the _{k} is the range-rate of the target with respect to the

Then, the propagation time of the _{1}_{k}_{,0} = (_{1,0} + _{k}_{,0})/_{1}_{k}_{1} + _{k} represent the propagation time of the first pulse and the range-rate of the

For simplicity, we assume that the target is non-scintillating,

The received signal in _{1},_{n}_{1},_{n}

By denoting
_{k}_{1}_{k}_{,0} − _{11,0}, _{k}_{1}_{k}_{n}_{k}_{1}_{k}_{,0}) approximately, and _{1}_{k}_{,0} with _{k}_{k}_{k}_{1} represents the

It is obvious that
^{(k-1)×1}, Δ^{(k-1)×1}, ^{(k-1)×1} with their expressions as:

Note that the

In this section, we derive the CRBs of the GCPs. The Fisher information matrix (FIM) is calculated first. Then the CRBs are obtained by inverting FIM [

We find it difficult to compute the FIM of the GCPs directly, thus an intermediate parameter vector _{1}, _{2}, ⋯, _{k}]^{T}^{K×1}. Next we compute the FIM of

As the noise components in each pulse is independent and identically distributed (i.i.d.) Gaussian, the log-likelihood function of the received signal given in _{1,0}(_{1,1}(_{1},_{N}_{-1}(^{T}

The computation of _{K}_{p}_{×}_{q}_{p}_{×}_{q}_{p}_{×}_{q}_{p}_{r}_{i}^{T}_{r}_{i}^{T}_{r}_{i}^{T}_{θ}_{θ}^{−1}(

The CRBM of

Next we apply the chain rule of CRB to compute the CRBM of Δ

Extracting the diagonal elements of _{Δ}_{τ}

For clarity, we rewrite matrix

Using the matrix inversion lemma [_{r}_{i}^{T}^{−1}^{−1} corresponds to the CRBM of Δ^{−1}^{−1} corresponds to the CRBM of [_{r}_{i}^{T}

From

Then, using the matrix inversion lemma once again, we have:
^{−1}^{−1} and (^{−1}^{−1} correspond to the CRBM of Δ

Using the following two equations:
_{in}

Finally, the CRBs of total phase differences and Doppler frequencies are expressed as:

Based on

All the CRBs are irrelevant to the number of radars

The CRB of the total phase differences in _{c}/^{2}. Obviously, the latter is much higher than the former under the assumption of narrowband signals, and performance analysis based merely on the mismatch of T/R phases would be inappropriate and more or less discouraging, as we will see in Section 5.3.

When the pulse number _{k}_{k}^{3}, respectively. Intuitively, for the estimation of Δ_{k}_{k}_{k}

In this section, we analyze the coherence performance of NGR with pulse trains. The signal model in coherent mode is presented first. Then the formula of coherence gain is provided. Finally, some remarks are given.

We assume that the GCPs are stable when NGR switches from MIMO mode to coherent mode, so that the estimates for GCPs in MIMO mode can be applied to adjust the phases and time delays on transmit. The estimates for GCPs are defined as:

We use
_{k}) represents the time delay adjustment,
^{−}^{j}^{φ̂}_{k}^{•n} compensates the phase caused by Doppler.

From

Since all the GCPs have been compensated on transmit, the

For simplicity, _{k}_{k}

It is obvious from _{k}_{k}^{2} = ^{2}/12 and _{p}

First we calculate the averaged power of the output signal. The signal in _{1} where a peak after integration is expected, and the noise-free sampled signal can be expressed as:

The computation of the averaged power in _{1}(0) = _{2}(0) = 1 and _{1}(∞) = _{2}(∞) = 0. When
_{1} and _{2} can be approximated by Taylor expansion of

However, when
_{2}. In detail, the curves of
_{1} = −(^{2}/6, _{2} = (^{4}/40 and

From

Based on

If
^{2}^{2} is the ideal SNR gain of a master-slave single-pulse NGR consisting of

If

If we replace

In this section, numerical results are presented and discussions are conducted to verify the CRBs of the GCPs and evaluate the coherence gain formula.

Since the CRBs are not impacted by the number of radars

The MSE and CRB of the time delay difference

Note that the _{B}_{Δ}_{τk}

The MSE and CRB of the total phase difference and the first Doppler frequency (we have two Doppler frequencies to estimate since

The upper bound of SNR gain in

As mentioned in the second remark of Section 3.4, the CRB of the total phase differences in _{c}, while the CRB of the T/R phase differences derived in [_{c}/^{2}. The CRBs of these types of phase errors and the SNR gain based on each of them are shown in _{c} = 1 GHz and the bandwidth

We have extended the NGR model based on a single pulse to the case of pulse trains, so that the coherence gain can be obtained from both spatial and temporal integration. Accordingly, the coherence parameters (CPs) in [

This work was supported in part by National Natural Science Foundation of China (No. 61171120, No. 61201379 and No. 61102142), the Key National Ministry Foundation of China (No. 9140A07020212JW0101), the Foundation of Tsinghua University (No. 20101081772), the Foundation of National Laboratory of Information Control Technology for Communication System of China, and the Foundation of National Information control Laboratory.

In this Appendix, we develop the FIM for the intermediate parameter vector

Applying the properties of orthogonal waveforms, we have the second derivatives of the GCPs as:

From

In this Appendix, we develop the averaged power of the output signal. From

The i.i.d. property of the estimation errors and the following equations will be applied hereinafter:
_{k})]=_{k})],

This part of the sum can be expressed as:

This part of the sum can be expressed as:

This part of the sum can be expressed as:

In addition, we have:

From

The master-slave architecture of NGR.

The logarithmic MSE and CRB of (

The logarithmic MSE and CRB of (

SNR gain

Comparisons between total phase differences and T/R phase differences: (