（School of Information Science and Engineering，Southeast University，Nanjing 210096，China）

Detection optim ization for resonance region radar w ith densemulti-carrier waveform

Chen Peng Wu Lenan

（School of Information Science and Engineering，Southeast University，Nanjing 210096，China）

Unlike the existing resonance region radar systems（RRRS）that transmit the orthogonal frequency division multiplexing（OFDM）multi-carrier waveform，the dense multi-carrier（DMC）radar waveform which has a narrower frequency interval than the traditional OFDM waveform is proposed．Therefore，in the same frequency bandwidth，the DMC waveform containsmore sub-carriers and providesmore frequency diversity．Additionally，to further improve detection performance，a novel optimal weight accumulation target detection（OWATD）method is proposed，where the echo electromagnetic waves at different frequenciesare accumulated w ith the optimalweight coefficients．Then，w ith the signal-tonoise ratio（SNR）of echo waveform approaching infinity，the asymptotic detection performance is analyzed，and the condition that the OWATDmethod w ith the DMC outperforms the matched filter w ith the OFDM is presented．Simulation results show that the DMC outperforms the OFDM in the target detection performance，and the OWATD method can further improve the detection performance of the traditional methodsw ith both the OFDM and DMC radar waveform．

constant false alarm rate（CFAR）；dense multicarrier waveform；detection optimization；resonance region radar system

In the study ofmodern anti-stealth radar systems［1］，the electromagnetic waves resonate w ith the target when the target feature size is comparable w ith the wavelength of the transm itted waveform［23］．Hence，this resonation can be utilized to significantly improve target detection performance in the resonance region radar systems

As w ideband waveforms can provide better range and velocity resolu_tion，they are w idely utilized in traditional radar systems［78］．Furthermore，the w ideband waveform can cover the target resonating frequency to improve the anti-stealth ability．Therefore，themulti-carrier phase-coded（MCPC）waveform［9-10］as a realization of the w ideband s_ignal is applied to the radar systems by Levanon et al．［7，1113］，and the MCPC waveform can provide larger bandwidth-time（BT）product than the traditional w ideband ones［8］．The orthogonal frequency divisionmultiplexing（OFDM）signal can take advantage of the spectral efficiency and can be generated conveniently，so it is w idely adopted to the conventional MCPC waveform［14］．In addition，the target detection performance can be improved by the OFDM waveform，where frequency diversity is attained by the target scattering coefficients（TSC）at different frequencies［15］．

At the target detection part of the conventional OFDM radar systems，the echo waveform directly passes through the corresponding matched filter to obtain the maximal signal-to-noise ratio（SNR）while maintaining the performance of the range，which is known as the pulse compress process．However，when we consider the frequency diversity provided by the TSC，the target detection performance can be further improved．

In this work，a novel MCPC waveform based on the dense multi-carrier（DMC）is proposed for the RRRS，where the frequency interval is narrower than that of the OFDM signalandmore sub-carriers can be provided w ithin the same bandw idth．Therefore，the performance of the target detection can be further improved by the frequency diversity．Moreover，a novel target detection method，namely optimal weight accumulation target detection（OWATD），is proposed by accumulating the echo electromagnetic waves w ith the optimally weight coefficients at different frequencies．Then，the asymptotic detection performance is illustrated as the SNR of the echo waveform tends to infinity，and the condition that the OWATD method w ith DMC outperforms thematched filter with OFDM is proposed．

1 Model of RRRSwith Multi-Carrier Waveform

The model of the RRRS considered in this work is shown in Fig．1．As the extended binary phase shift keying（EBPSK）modulation can provide higher bandw idth and power efficiency w ith the tighter spectrum than the traditional binary phase shift keying（BPSK）modulation［19］，the MCPC waveform based on the EBPSK modulation is utilized as the transm itted waveform．Thematrix S≜（s1，s2，…，sN）∈RL×Ndenotes the collection of thetransm itted waveform，where sidenotes the sample vector of the i-th sub-carrier signal；L denotes the length of the signal vector；and N denotes the number of the sub-carriers．The waveform is controlled by the coding matrix A∈RN×M，where M denotes the code length of each subcarrier．When the target feature size is comparable w ith the wavelength of the transm itted waveform，the electromagnetic waves resonate w ith the target，and the echo signal r can bemodeled as［20］

whereα≜｛α1，α2，…，αN｝Tdenotes the TSC vector at different frequencies；n～N（0，）denotes the additive white Gaussian noise（AWGN）w ith themean and covariance being 0 and，respectively；I denotes an identity matrix；denotes the variance of noise．To facilitate the analysis，we assume that the TSC follows the independent Gaussian distribution w ith the same variance but different means，i．e．，α～N（m，），where m≜｛m1,m2,…, mN｝Tdenotes the mean of the TSC，anddenotes the variance of the TSC．A fter receiving the echo waveform r，the detection method based on the constant false alarm rate（CFAR）is adopted to detect the presence of the target．In the following section，the traditional detection method based on thematched filter is presented．

Fig.1 RRRSmodel w ith multi-carrier waveform

2 Target Detection Based on OMM F M ethod

The target detectionmethod based on thematched filter and OFDM waveform is named the OMMFmethod in this paper．W ith the know ledge of the TSC，the compressed waveform can be obtained after passing the matched filter，

where v is the target detection threshold．When the target is absent，only noise can be received．Then the compressed waveform from Eq．（2）is

As the noise follows the Gaussian distribution，the linear transformation of n，denoted asλ0，also follows the Gaussian distribution，which can be determ ined by the mean and variance parameters as

where Psdenotes the transm itted power andε｛·｝denotes the expectation operation．Hence，the probability of false alarm based on the detectionmethod（3）can be expressed as

From the probability of the false alarm PFA，the threshold of the CFAR detection can be set as

Therefore，the target detection process represented in formula（3）is

After obtaining the threshold of the CFAR detection，the probability of target detection can be obtained．W ith the presence of target，the compressed waveform from Eq．（2）is

As the parametersαand n follow the Gaussian distribution，the linear transformation also follows the Gaussian distribution，which can be determ ined by the mean and the variance，

Eq．（10）indicates that the variance ofλ1is increased by the TSC，so the performance of target detection is decreased．Finally，we can obtain the probability of target detection based on the OMMFmethod as

The expectation of the probability of target detection w ithdifferentmeans of the TSC is

3 Target Detection Based on the OWATD Method

3.1 The OWATD method

In this subsection，we propose a OWATD method，which can maxim ize the probability of the target detection w ith themean know ledge of the TSC．In thismethod，a vector x≜｛x1,x2,…,xN｝Tis utilized to control the allocation of the weight coefficients．Therefore，the target detection based on the CFAR can be expressed as

where v（x）is the detection threshold，and

The vector of the weight coefficients x is optim ized to maxim ize the probability of target detection．

First，we give the threshold of the CFAR detection w ith the vector of the weight coefficients．When the target is absent，the probability of the false alarm is

whereλ0（x，α）≜（Sx）Tn follows the Gaussian distribution．Themean and variance respectively are

Then，the probability of false alarm is

Therefore，the threshold of the CFAR target detection w ith the vector of theweight coefficients can be expressed as

Secondly，when the target is present，the probability of target detection is

whereλ1（x，α）≜（Sx）T（Sα＋n）．When the mean TSC is known，λ1（x，α）in（19）follows the Gaussian distribution，and themean and variance are

Then，the probability of target detection in（19）can be expressed as

Since Q（·）is a monotonically decreasing function，the optimalweight coefficientswhich canmaxim ize the probability of the target detection are equal to

The optimalweight coefficients x can be obtained by letting the derivation of f（x）be equal to 0，i．e．，

where B＝STS and

Letting x*＝m，we can obtain

The condition that g（x）＝0 in（25）is that the nonzero eigenvalues of B are equal．Fig．2 depicts the eigenvalues distribution of B，when the frequency interval is 0．08 of the orthogonal one．Therefore，the nonzero eigenvalues of B are approximately equal under the condition of the DMC waveform considered in this work．Besides，when sub-carriers are orthogonal，the nonzero eigenvalues are also equal．Therefore，x*＝m is the optimalweight coefficients．This conclusion is sim ilar to the maximum ratio combiningmethod in the diversity theory，but the optimization objective function considered here is the probability of target detection．Then the process of the target detection based on CFAR is

Fig.2 RRRSmodel w ith multi-carrier waveform

Finally，we can obtain the probability of target detection w ith the optimal vector of the weight coefficients，

The detection probability w ith themean of the TSC is

3.2 Perform ance com parison between the DMC and OFDM waveform

This section presents the performance comparison between the dense and orthogonal multi-carrier waveform w ith the same signal bandwidth with themean know ledge of the TSC．The number of the sub-carriers in the DMC is

where B is the signal bandw idth；Δf≜ζΔfodenotes the frequency interval of the DMC waveform；Δfodenotes the frequency interval of orthogonalmulti-carrier waveform；andζ∈（0，1）denotes the dense parameter．Therefore， the number of dense sub-carriersof the orthogonal one

The probability of the target detection for differentζ can be obtained from（21）that

As SNR tends to infinity，we can obtain the asymptotic probability of the target detection from（30）in the RRRS w ith the DMC waveform that

The expectation of the asymptotic probability of target detection is

In order to analyze the performance of the target detection，the approximation result of（32）is given．If the TSC at different frequencies are the same，i．e．，¯m＝m1＝…＝mN,then（31）can be w ritten as

The expectation of（33）is

We simplify（34）as follows．For arbitrary m，n，if the follow ing holds

then

Then Eq．（33）can be simplified as

Eq．（37）shows that reducing the sub-carrier interval，which increases the number of sub-carriers Nd，can improve the target detection performance．The expectation of Eq．（37）is

To analyze the performance of the target detection，the assumption that the TSC at different frequencies are the same in（34）and the expression of the target detection is further simplified by the condition in（35）．The performance effect of these assumptions w ill be given in the next section．

3.3 The unknown mean know ledge of TSC

When themean of the TSC is unknown，we can adopt the best estimation method，which is the least square（LS）estimation under the condition of thiswork，to estimate themean of TSC＾m＝m＋δ，whereδ～N（0，Σ）is the vector of estimation errors，andΣis the covariance matrix．

The likelihood function f（r；m）of themean of the TSC m follows the Gaussian distribution as

The covariancematrix of the estimation errorδcan be obtained by the Fisher information，and it is

Substituting（39）into（40），we can obtain the entry of J at the p-th row and the q-th column as

Then the covariancematrix is the inverse of the Fisher information．

Therefore，the estimated mean of TSC＾m is utilized during the process of the target detection method（26）proposed in this work．

4 Simulation Results

In this section we conduct the simulation of the RRRS w ith the DMC waveform，and the simulation parameters are given in Tab．1．As the practical radar data is difficult to obtain，the simulation data is utilized to realize the proposed RRRS w ith the DMC waveform and the OWATD method．The simulation and theoretical probability of the target detection w ith and w ithout the mean know ledge of the TSC are，respectively，shown in Fig．3 and Fig．4，where the OWATD method proposed in this work is adopted．As shown in Fig．3 and Fig．4，the simulation results are consistent w ith the theoretical ones，which illustrates the accuracy of theoretical results obtained in（28）and（45）．

Tab.1 Simulation parameters

Fig.3 Comparison between theoretical and simulation results using the OWATD method w ith mean know ledge of TSC

Fig．5 depicts the comparison between the probability of target detection of the DMC waveform using the OWATD method and that of the OMMF method，where themean of the TSC is known and the bandw idth is the same．It can be seen from Fig．5 that different frequency intervals of the sub-carriers have significant effects on the probability of target detection，which can be further improved by reducing the frequency interval at high SNR．

Fig.4 Comparison between theoretical and simulation results using the OWATD method w ithoutmean know ledge of TSC

Fig.5 Comparison between the OWATD method w ith DMC waveform and the OMMFmethod w ithmean know ledge of TSC

The comparison of the probability of target detection between the proposed OWATD method and the traditional OMMFmethod is shown in Fig．6，where themean of the TSC is known and the bandw idth are the same．As shown in Fig．6，when we consider the effect of the frequency interval of sub-carriers，the probability of target detection is improved by enlarging the frequency interval at low SNR．However，the probability of target detection is reduced by enlarging the frequency interval at high SNR．Therefore，the DMC waveform has the advantages of the detection performance at a high SNR．

Fig．7 depicts the asymptotic probability of target detection when SNR tends to infinity．In this paper，in order to analyze the performance of the target detection，we assume that the TSC at different frequencies are the same in（34），and further simplify the expression of the target detection w ith the condition in（35）．The performance effect of these assumptions is highlighted in Fig．7，and is compared w ith that w ithout approximation assumption．These assum ptions achieve a relatively small probability of the target detection，but have the same tendency as the frequency interval．Furthermore，the performance of the target detection w ith the DMC waveform via theOWATD method is better than that via the OMMFmethod for all frequency intervals of the sub-carriers，and the asymptotic probability of the target detection w ithout the mean know ledge of the TSC is lower than that using the OMMFmethod w ith the increase of the frequency interval．Therefore，for the asymptotic consideration，the DMC waveform via the OWATD method outperforms that via the traditional OMMFmethod．

Fig.6 Comparison between the OWATD method w ith DMC waveform and the OMMF method w ithoutmean know ledge of TSC

Fig.7 Asymptotic probability of the target detection

5 Conclusion

The problem of the target detection in the RRRS is considered，and the DMC waveform is proposed as the transm itted waveform．By exploiting the frequency diversity，the performance of target detection w ith the DMC waveform outperforms the traditional OFDM signal．Furthermore，we propose the OWATD method to further improve the target detection performance．As the SNR of the echo signal tends to infinity，the condition that the asymptotic detection performance of the OWATD method using the DMC waveform outperforms the matched filter using the OFDM signal has also been illustrated．Finally，simulation results demonstrate the efficiency of the OWATD method w ith the DMC waveform in the RRRS．In the futurework，the interference of clutterw ill be considered．

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基于密集多载波波形的谐振区雷达检测优化

陈 鹏 吴乐南

（东南大学信息科学与工程学院，南京210096）

区别于现有发射正交频分复用（OFDM）多载波波形的谐振区雷达系统（RRRS），提出了拥有比传统OFDM波形更窄频率间隔的密集多载波（DMC）雷达波形．在相同带宽内，DMC波形包含了更多的子载波，从而能提供更高的频率分集．为了进一步提高检测性能，提出了一种新的最优权重累积目标检测（OWATD）方法．该方法采用最佳权重系数来累积不同频率的电磁回波，分析了当回波信噪比（SNR）趋于无穷大时的极限检测性能，并给出了采用DMC的OWATD方法优于采用OFDM的匹配滤波方法的条件．仿真结果表明，DMC的目标检测性能优于OFDM，而且OWATD方法可以进一步提高采用DMC和OFDM波形的传统方法的检测性能．

恒虚警概率；密集多载波波形；检测优化；谐振区雷达系统

TN957．51

10．3969／j．issn．1003－7985．2015．03．002

2015-02-22．

Biographies：Chen Peng（1989—），male，graduate；Wu Lenan（corresponding author），male，doctor，professor，wuln＠seu．edu．cn．

s：The National Natural Science Foundation of China（No．61271204），the National Key Technology R＆D Program during the 12th Five-Year Plan Period（No．2012BAH15B00）．

：Chen Peng，Wu Lenan．Detection optim ization for resonance region radar w ith densemulti-carrier waveform［J］．Journal of Southeast University（English Edition），2015，31（3）：304- 310．

10．3969／j．issn．1003－7985．2015．03．002