Adaptive OFDM waveform design for spatio-temporal-sparsity exploited STAP radar
2.5 Numerical results
In this section, we present the results of several numerical examples to illustrate the OFDM-STAP performance obtained by utilizing the spatio-temporal sparsity approach, and the achieved performance gain due to adaptive OFDM waveform design. First, we provide a description of the simulation setup and then discuss different numerical results.
Radar parameters—We considered a radar moving with a speed of vR
j j ¼100 m=s along the y-direction (see Figure 2.1). It used a linear array with M ¼10 sensor elements having interelement spacing d¼lc=2¼0:33 m, and a CPI havingN ¼8 temporal pulses to collect the spatio-temporal measurements.
The transmitted waveform was an L¼4 subcarrier OFDM signal with the fol- lowing parameters: carrier frequency fc¼450 MHz, bandwidth B¼5 MHz, subcarrier spacingDf ¼B=ðLþ1Þ ¼1 MHz, pulse widthTp ¼1=Df ¼1ms and PRI T ¼1:67 ms. This ensured an unity Doppler foldover factor (i.e., c¼2j jvRT=d¼1) for the clutter ridge on the spatio-temporal plane.
Interference parameters—We modelled the thermal noise covariance matrix ass2nILMNwiths2n¼1. The clutter responses were assumed to be equally spaced in azimuth angles withNc¼72 over the entire range gate. We simulated the clutter responses of eachkth patch from an independent complex Gaussian distribution CN0;RC;k
and scaledRC;k to satisfy the required clutter-to-noise ratio (CNR), defined as
CNR¼ PNc
k¼1trRC;k
LMNs2n (2.29)
For all the results presented in this section, we kept the CNR fixed at 40 dB.
In addition, to represent the non-idealistic real-world STAP scenario, we considered the effects of temporal decorrelation (or ICM) on our sparsity-based OFDM-STAP method. Various natural environmental variations, such as wind blowing over foliage, motion of ocean waves, induce pulse-to-pulse fluctuations in clutter reflectivity, which is commonly referred to as ICM [6, Chapter 4, 62]. In general, the temporal-only decorrelation can be modelled with a modified clutter covariance matrix as
RfC ¼RCblkdiagTD;01M;. . .;TD;L11M
(2.30) whereTD;l is anNN temporal correlation matrix at the lth subcarrier and 1M is anMM matrix having all the entries to be one. In this work, we constructed the temporal decorrelation for a water scenario with TD;l¼toeplitzðrD;lð0Þ;
rD;lð1Þ;. . .;rD;lðN1ÞÞ, where the temporal autocorrelation of the fluctuations had a Gaussian shape [2]:
rD;lðnÞ ¼exp 8p2s2vT2n2fl2 c2
(2.31)
withs2v representing the variance of the clutter spectral spread. In the simulations, we useds2v¼0:5.
Target parameters—The target was simulated at an azimuth direction ofyT¼0, and it was moving with a speed ofjvTj ¼20 m=s. The scattering coefficients of the target,zT ¼ ½zT;0;. . .;zT;L1T, were varied to construct two scenarios having dif- ferent target-energy distributions across different subchannels, i.e., Scenario I with zðIÞT ¼ ½1;2;3;1T and Scenario II withzðIIÞT ¼ ½1;5;1;0:5T. To create various signal- to-noise ratio conditions, these target scattering vectors were further scaled to satisfy
SNR¼kzTk22
Ls2n
(2.32) Simulation parameters—The spatio-temporal sparsity-based STAP problem was solved following the LASSO formulation after constructing the dictionary matrix withGa¼Gn¼144 grid points by equally partitioning the azimuth angley with an interval of 2:5. The LASSO estimator was implemented using CVX [64].
It was first operated separately on each secondary datas¼1;2;. . .;Ns, and then the recovered clutter responses were used to constructRbC. Finally, an estimate of the interference covariance matrix was obtained asRbI ¼RbCþhILMN withh¼10s2n. Performance metrics—To analyze the performance of the sparse-STAP tech- nique, we examined its characteristics with respect to the ROCs and SINR-loss measure. The evaluation of ROCs was essential as the STAP algorithms are mainly used in the detection problems. To compute the ROC curves, we used the rela- tionship betweenPFAandPDasPD¼Q ffiffiffiffiffiffiffiffiffiffiffi
pSINR
; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lnPFA
p
[1].
The SINR-loss performance metric characterizes the STAP performance rela- tive to what could be obtained in the absence of interference (clutter). In our case, if only thermal noise is present, i.e.,RI ¼ILMN, then the noise-only SINR value is equal toLMN. When clutter was present and we estimated the interference covar- iance matrix asRbI, the output SINR was computed as a function of target Doppler as
SINRðnÞ ¼ fTðnÞHRb1I fTðnÞ2
fTðnÞHRb1I RIRb1I fTðnÞ (2.33) where fTðnÞ ¼ ½ðfDðn0Þ fSða0;TÞÞT;. . .;ðfDðnL1Þ fSðaL1;TÞÞTT repre- sented a steering vector with fixed target angle and varying target Doppler. Then, the SINR-loss measure at the output of the proposed STAP filter was calculated as SINRlossðnÞ ¼SINRðnÞ=LMN.
2.5.1 Sparsity-based STAP performance
Figure 2.3 depicts the spatio-temporal spectrum of the estimated clutter response by applying the proposed sparse recovery algorithm on a single secondary range gate measurement (i.e.,Ns¼1). In the ideal scenario (without any temporal decorrela- tion), we could clearly observe the diagonal clutter ridge of the estimated spectrum along with extremely few non-zero off-diagonal elements. In the presence of tem- poral decorrelation, we still saw a prominent diagonal ridge characteristic, but with
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Figure 2.3 Estimated clutter spectra (at L¼4 subcarriers) of the sparsity-based STAP method in the (a) ideal scenario (no decorrelation) and (b) presence of temporal decorrelation [the colorbar is in linear scale]
a slight increase in the number and strength of the off-diagonal elements. We want to mention here that, because of the inherent randomness of the clutter returns, the estimated spectra would look different when obtained from a different set of sec- ondary measurements.
The SINR-loss performance of the sparsity-based STAP technique is demon- strated in Figure 2.4. In this setup, we employed a conventional (non-adaptive) OFDM signal with equal magnitude spectral coefficients (i.e., al¼1= ffiffiffi
pL
8l). In the ideal scenario of Figure 2.4(a), it is clearly evident that we achieved within 3 dB of optimum SINR-loss by using only Ns¼2 secondary measurements (to estimate the interference covariance matrix). Furthermore, use of aboutNs¼5 secondary data resulted into a near-optimum performance. In the presence of
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Figure 2.4 SINR-loss performance of the sparsity-based STAP method with fixed (non-adaptive) OFDM signalling in the (a) ideal scenario (no decorrelation) and (b) presence of temporal decorrelation
temporal correlation of Figure 2.4(b), the optimum SINR-loss performance showed a much wider mainbeam clutter notch and a smaller maximum value when com- pared with the ideal scenario without any decorrelation. As before, we could still attain the near-optimum SINR-loss performance by utilizing onlyNs ¼25 sec- ondary data. Therefore, irrespective of whether any temporal decorrelation was present or not, our sparsity-based STAP approach employed considerably lesser number of secondary data to produce a near-optimum SINR-loss performance.
The ROC curves in Figure 2.5 show the detection performances of the sparsity- based STAP approach for two targetszðTIÞandzðTIIÞ, both in the absence and in the presence of temporal decorrelation effect. The resulting ROCs, obtained by utiliz- ing the interference covariance matrix estimated via sparsity-based STAP
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Signal-to-noise ratio (in dB) Probability of detection (PD)
Ideal RI, Target 1 Estimated RI, Target 1 Ideal RI, Target 2 Estimated RI, Target 2
Ideal RI, Target 1 Estimated RI, Target 1 Ideal RI, Target 2 Estimated RI, Target 2 (a)
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Signal-to-noise ratio (in dB) Probability of detection (PD)
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Figure 2.5 Receiver operating characteristics of the sparsity-based STAP method with fixed (non-adaptive) OFDM signalling in the (a) ideal scenario (no decorrelation) and (b) presence of temporal decorrelation
approach, were compared with their counterparts that assume the complete knowledge of the interference covariance matrix. ROCs of the sparsity-based STAP method were approximately within 1:2 dB of the optimum performance limit for both the target scenarios.
2.5.2 Performance improvement due to adaptive waveform design
Figures 2.6 and 2.7, respectively, demonstrate the SINR-loss and ROC perfor- mances when we used the adaptively designed OFDM coefficientsaopt. The SINR- loss performances of Figure 2.6 show that, both in the absence or presence of the
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Figure 2.6 SINR-loss performance of the sparsity-based STAP method with adaptive OFDM signalling in the (a) ideal scenario (no decorrelation) and (b) presence of temporal decorrelation
temporal decorrelation, the change in the transmit signal parameters did not affect the SINR-loss performance in a significant way. In Figure 2.6(a), we observe that the SINR-loss of the sparsity-based STAP approach remained within1 dB of the optimum performance by using onlyNs¼5 secondary measurements, whereas when temporal decorrelation was present, we achieved within2 dB of the opti- mum performance by utilizing the same number of secondary data.
The improvement in detection performance due to the adaptive waveform design is clearly noticeable when we compare the ROC plots in Figure 2.7 with their equivalents in Figure 2.5. For example, in the ideal scenario with no
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Signal-to-noise ratio (in dB) Probability of detection (PD)
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Ideal RI, Target 1 Estimated RI, Target 1 Ideal RI, Target 2 Estimated RI, Target 2 (a)
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Figure 2.7 Receiver operating characteristics of the sparsity-based STAP method with adaptive OFDM signalling in the (a) ideal scenario (no decorrelation) and (b) presence of temporal decorrelation
decorrelation effect, we observed almost 5.5 dB improvement in detection perfor- mance forzðIÞT and approximately 6 dB forzðIIÞT whenPD ¼0:5. On the other hand, when temporal decorrelation was present, the detection performance was improved by approximately 3 dB for both the targets atPD ¼0:5.