Adjusting radar transmitted waveform to its environment is one of the most important roles in cognitive radar; having the capability of updating transmitted waveforms in different applications is a key point. It has been shown in many studies that if the waveform is designed according to the target and clutter characteristics, the detection performance will improve significantly. The uncertainty of the target radar signatures decreases via maximizing MI and the probability of extended target detection is increases via maximizing SNR. In this paper, a waveform design approach based on maximizing both SNR and MI and with regard to target and clutter shape is presented. The detection performance for proposed waveform is compared with previous proposed waveforms. The present paper compares different scenarios of target and clutter and using the probability of detection as a cost function to investigate the advantages and disadvantages of each waveform in different scenarios which are mainly discussed in this text. The desired waveform for cognitive radar is selected based on simultaneously making compromises between SNR and MI, which plays an important role in cognitive radar systems and based on the assumption addressed in the text, the best waveform transmitted into the environment.
Cognitive radar is a radar system which selects its transmitted waveform to adapt to the radar environment by using feedback structure from the receiver to the transmitter. In these systems waveforms can be adaptively optimized based on preceding knowledge about the targets and environments; it leads to the improvement of the total performance of system [
The information-based approach in the presence of signal-dependent clutter is investigated in [
The contributions of this paper include an analysis of applying both the information-based and SNR-based approach to different deterministic extended targets and clutter scenarios considering the energy constraint. In this study, energy distribution for optimum waveform in clutter and noise scenario with the energy constraint have been addressed and we focus on the point that waveform should put a considerable amount of available energy on the frequency bands in which the target spectrum frequency components is considerable while the clutter is negligible. We derive various waveforms for each scenario and finally compare our results with linear frequency modulated (LFM) waveform to verify the performance of the proposed waveform.
This paper is organized as follows: first a radar system model is described in Section 2 and then the derivation of MI-based and SNR-based waveforms in the presence of Gaussian clutter and other scenarios are derived. In Section 3, using the obtained relations in Section 2, transmit waveform spectrum for two deterministic extended targets are illustrated and the energy allocation realized by obtaining waveforms are briefly discussed. Section 4 generally discusses the performance of obtaining waveforms and other predefined waveforms and a new closed-loop structure for detecting targets in different scenarios is proposed. Finally, in Section 5, we provide a general conclusion.
To analyze a radar system, all constituent blocks should be defined. We consider a signal model of a Gaussian extended target in the presence of clutter as shown in
Let x(t) be a finite-energy waveform with duration T. We assume x(t) is energy-limited and non-zero only in the time interval
c(t) is assumed to be sea clutter with known power spectral density (PSD) which can be approximated as follows:
Here fc is the peak locations of the Gaussian function; gc and
where ai and fi are respectively determined between a specific amplitude range and an appropriate interval due to clutter and noise specifications. The variance
In this part, the Energy Spectral Density (ESD) of MI-based is computed with the assumption of known target in different scenarios and compared with SNR-based waveform discussed in [
where
The desired waveform ESD is found by solving the constrained optimization problem:
With the energy constraint:
Now using the Lagrangian multiplier method, the waveform ESD will be obtained as follows:
We can equivalently maximize:
The first and the second derivatives of
Since the second derivative is negative for all εx(f),
where the parameter λ is found by solving the following equation:
In Equation (12), we see that the waveform ESD will be equal to zero for those frequencies satisfying:
Thus, we can consider λPn(f) as a threshold which depends on coefficient λ which means the waveform only puts energy into those frequencies where
The mutual information between the received signal and the target ensemble with the presence of clutter is:
And like previous step for deriving non-clutter case we have:
We can equivalently maximize:
Now the first and second derivatives of
where
And the parameter λ is found by solving the energy constraint equation:
In Equation (18), we see that the waveform ESD will put energy into those frequencies satisfying:
Here again the threshold λPn(f) is obtained by solving the energy constraint Equation (20) showing that the parameter λ is totally different for each energy constraint presented in this paper. A noteworthy point is that although the thresholds are absolutely similar, their values are different due to parameter λ. Consequently, the waveform energy allocation is different from the non-clutter case. It has been shown in simulation results that this waveform distributes most of its energy into frequency bands which the target PSD is high and the clutter PSD is low.
The waveform ESD for SNR-based waveform for a known target with the presence of clutter is described in [
And the waveform ESD using the Lagrangian multiplier technique is [
where parameter
As mentioned in [
as we derived for MI-based waveforms and λPn(f) is considered as a threshold with different values for all waveforms. The last waveform discussed here is the waveform ESD for SNR-based waveform without the presence of clutter. In this case by putting
Using the Lagrange multiplier technique, we form the objective function
In summary, the transmit waveform x(t) that maximizes SNR is the Eigenfunction corresponding to the maximum eigenvalue of the kernel M(t) which is described by:
And finally the SNR has been just the product of this eigenvalue and the energy in the transmit waveform as follows:
Different methods are proposed to derive the optimum waveform for this scenario. One of them is the approach proposed in [
where the parameter λ is found by solving:
The ESD obtained using this method produces a waveform that totally follows the target spectrum. The energy of transmitted waveform is allocated to the target peaks in proportion to the amount of every peak; in other words, the proposed waveform allocates its energy in all of target spectrum frequency bands. This allocation is illustrated in
The waveforms discussed in this paper are: SNR-based waveform in clutter (CSNR-based) [
We cannot find a closed-form solution for the waveform derivations. In this section, we provide two numerical examples which can be obtained using the proposed waveforms. As it was mentioned in the previous section, target and clutter are supposed to have a Gaussian shape with the power spectrum
shown in
The target-to-noise ratio (TNR) is defined as the ratio of the area under target PSD to the area under noise PSD [
We repeat the experiment with target 2 and with the same clutter and noise PSD. For this target, we have TNR = 0.51 dB, CNR = 0.95 dB and TCR = −0.44 dB and corresponding waveforms for this target scenario are computed and illustrated in
As shown before, solving optimization problem requires knowledge of the target and clutter spectrum. It was shown in [
where
With a good approximation
And finally we have:
Using the above equation, since maximizing PD for arbitrary values of Pfa depends on the values of SINR for each waveform and by maximizing SINR, we can achieve higher values of PD. Previous research has shown that SNR-based waveform and LFM waveform are compared through this method and shown that the proposed method had better performance in terms of maximizing the probability of detection. Here, we obtained different waveforms with both SNR and MI values and now we want to select the desired waveform based on achieving maximizing probability of detection for each target and clutter scenarios regarding the point that the desired waveform should have a sufficient MI value in comparison with other waveforms. MI and SINR which are defined in (14), (21) respectively, are considered as objective functions. We computed their values for three different targets.
For each target the corresponding waveform according to target, clutter and noise spectrum is generated and utilized for computing SNR and MI. Now we suppose a specific scenario with known target, clutter and noise PSDs. We are interested in investigating the effects of target, clutter and noise separately. So, we consider the scenario 1 which includes a specific Gaussian extended target, Gaussian clutter and AWGN noise with known PSDs, then compute CNR, TCR and TNR values for this scenario. Afterwards, by changing clutter (target and noise are fixed), scenario 2 is obtained and similarly scenario 3 and scenario 4 are obtained by changing the target and noise, respectively. Target, Clutter and Noise PSD for each scenario are illustrated in
After defining scenarios, ESD of each waveform in accordance with each scenario is computed due to the obtained relations. Eventually for each scenario corresponding SNR and MI values for investigating the applied changes in target, clutter and noise are evaluated by Equation (15), (22) as summarized in
As indicated in
Value(dB) Scenario | CNR | TCR | TNR |
---|---|---|---|
Scenario 1 | 10.95 | −17.37 | −6.42 |
Scenario 2 | −6.62 | 0.2 | −6.42 |
Scenario 3 | 10.95 | −11.82 | −0.87 |
Scenario 4 | 20.95 | −17.37 | 3.58 |
Waveform Scenario | MI-based Waveform | CMI-based Waveform | CSNR-based Waveform | TSF Waveform | LFM Waveform |
---|---|---|---|---|---|
Scenario 1 | SNR = 17.16 MI = 16.82 | SNR = 17.94 MI = 17.45 | SNR = 18.09 MI = 17.1 | SNR = 15.66 MI = 15.55 | SNR = 8.25 MI = 8.24 |
Scenario 2 | SNR = 18.16 MI = 17.75 | SNR = 18.16 MI = 17.76 | SNR = 18.26 MI = 15.53 | SNR = 16.06 MI = 15.94 | SNR = 8.34 MI = 8.34 |
Scenario 3 | SNR = 19.93 MI = 19.47 | SNR = 22.5 MI = 21.51 | SNR = 22.87 MI = 20.67 | SNR = 19.76 MI = 19.52 | SNR = 13.59 MI = 13.56 |
Scenario 4 | SNR = 25.26 MI = 24.03 | SNR = 26.75 MI = 25.03 | SNR = 27.18 MI = 24.47 | SNR = 24.15 MI = 23.39 | SNR = 17.63 MI = 17.53 |
in one scenario and this is the CMI-based waveform that has greater MI amounts compared with other waveforms. As mentioned in [
In
It is obvious from
To evaluate the performance, we used receiver operating characteristic (ROC) curves corresponding to the obtained waveforms and LFM.
Now to better demonstrate the importance of waveform design in cognitive radar, we simulate different targets, clutters and noise realizations. To reach an aggregation, we compute SNR and MI values for each waveform by changing TCR, TNR and CNR values in a specific range. For example in
From MI consideration, as we expected, CMI-based and MI-based have better performance with a little difference compared with CSNR-based and TSF waveforms. Finally, we can conclude that indifferent values of TCR, all waveforms have 9 dB and 7 dB improvement in SNR and MI, respectively over LFM waveform.
The last case shows the SNR and MI value changes in dB for different CNRs. As indicated in
Therefore, a cognitive radar should be designed in such a way that firstly evaluates the TCR, TNR and CNR values and considering to the application and also waveform features which are mainly discussed in this paper, chooses the best waveform to transmit into the environment and achieve the maximum probable values of PD.
In
As indicated in this figure, the proposed cognitive radar structure includes an adaptive transmitter based on the feedback from the receiver and the interactions with a defined database. The radar transmitted its initial waveform into the environment. If the target is detected in the receiver, we have reached our goal; otherwise, cognitive radar loop is established using a feedback from the receiver to the transmitter and this procedure is done as long as the target of interest is identified. Surely, we can set the number of transmitting signals to a predetermined value to prevent the cognitive radar from being in an infinitive loop.
In this paper, we considered a deterministic Gaussian extended target with Gaussian clutter case for our experiments. The transmitted waveform optimization is done by maximizing the signal to noise ratio and mutual information between the target frequency responses of different targets and the received signal. This helps achieve a better performance in target recognition. We derived different MI-based, SNR-based and TSF waveforms and used an approximate analytical expression for the probability of detection for different waveforms and compared the performance of these waveforms based on this idea. The remarkable features for designing these waveforms for deterministic extended targets, are energy allocation in which waveforms should allocate their energy into frequency bands where target has the greatest PSD value and clutter has the smallest PSD value. This idea will cause the product of target and transmitted waveform got the maximum amount and then the performance of target detection in the presence of clutter will increase.
The derivations of MI-based waveforms for deterministic Gaussian extended targets and in the case of Gaussian clutter are new results. Moreover, TSF waveform in signal-dependent interference is a new result derived to show that depending on the application such as deterministic extended target, energy allocation is not done well. In fact for the scenarios outlined in the text, it is better to allocate most of transmitting waveform energy into frequency bands in which target has considerable peak amount and clutter amount is negligible instead of distributing energy into all target spectrum peaks. In this paper, different values of SNR and MI are obtained in different TCR, CNR and TNR amounts and it is proven that the obtained waveforms have various performances in different TCR, CNR and TNR and in cognitive radar. We propose a new closed-loop structure considering all features discussed in this paper and for each scenario of target, clutter and noise intelligently find the optimum solution. This work just uses deterministic Gaussian targets case in Gaussian clutter. Future work will attempt to investigate methods of waveforms designing for stochastic extended targets in Gaussian and non-Gaussian environments and suggest a general method for each clutter and target scenarios.
Vahid Karimi,Reza Mohseni,Yaser Norouzi,Mohammad Javad Dehghani, (2016) Waveform Design for Cognitive Radar with Deterministic Extended Targets in the Presence of Clutter. International Journal of Communications, Network and System Sciences,09,250-268. doi: 10.4236/ijcns.2016.96023