In radar systems, many researchers focused on the waveform design to improve the performance. And many algorithms just only consider the energy of the transmitted waveform. However, in practice, only the energy constraint is insufficient to guarantee that the signal will satisfy common envelope requirements, such as constant modulus that is important to radar transmitter. In this paper, we propose a constant modulus waveform design method for rank-one target detection. Firstly the optimal waveform under the energy constraint is obtained. Then the optimal waveform owns the constant modulus property in time domain from its Fourier transform magnitude is obtained finally. Finally, simulation results show the effectiveness of the proposed algorithm

Bin WANG

Fengming XIN

Jinkuan WANG

Xin SONG

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Sensors & Transducers, Vol. 163 , Issue 1, January 2014, pp. 39-43  39  Sensors & Transducers© 2014 by IFSA Publishing, S. L. http://www.sensorsportal.com      Optimal Waveform Design with Constant Modulus Constraint for Rank-One Target Detection  1 Fengming XIN, 2 Bin WANG, 1 Jinkuan WANG, 2 Xin SONG 1 School of Information Science and Engineering, Northeastern University,  Shenyang 110004, China 2 Engineering Optimization and Smart Antenna Institute,  Northeastern University at Qinhuangdao, 066004, China 1 E-mail: wjk@mail.neuq.edu.cn, xfm_neu@126.com   Received: 14 October 2013   /Accepted: 9 January 2014   /Published: 31 January 2014   Abstract: In radar systems, many researchers focused on the waveform design to improve the performance. And many algorithms just only consider the energy of the transmitted waveform. However, in practice, only the energy constraint is insufficient to guarantee that the signal will satisfy common envelope requirements, such as constant modulus that is important to radar transmitter. In this paper, we propose a constant modulus waveform design method for rank-one target detection. Firstly the optimal waveform under the energy constraint is obtained. Then the optimal waveform owns the constant modulus property in time domain from its Fourier transform magnitude is obtained finally. Finally, simulation results show the effectiveness of the proposed algorithm. Copyright © 2014 IFSA Publishing, S. L.  Keywords: Waveform design, Cognitive radar, Relative entropy, Constant modulus constraint, Target detection.    1. Introduction  Cognitive radar is proposed as a new generation radar system by Haykin [1, 2], which is not only dependent on the adaptive signal processing in the radar receiver, but also through an adaptive radar transmitter in order to improve the performance. Thus the optimal waveform design problem is the key technology of the cognitive radar. Lots of research efforts have been focused on the radar waveform optimization. Bell proposed to maximize the mutual information (MI) between the received signal and the target impulse response (TIR) to optimize the transmitted waveform [3]. In [4], the authors extended the MI method to the multiple-input multiple-output (MIMO) radar and proposed another method that adopted the mean-square error (MSE) in estimating the TIR as the cost function and minimized MSE to optimize the waveform for target identification and classification. The proposed two methods have the same solution. Pillai et al. developed an eigensolution for optimal signal/filter pairs for target detection when the target and clutter can be seen as time invariant random processes [5, 6], which used the signal to clutter-plus-noise ratio (SCNR) as the cost function and maximized the SCNR to optimize the waveform. Goodman et al. published a series of papers about the waveform design based on the sequential hypothesis testing that controls when hard decision may be made with adequate confidence to design waveform. The detailed work can be found in [7, 8]. In [9], Kay derived the optimal NP detector firstly, which shows that the NP detection performance does not immediately lead to an obvious signal design criterion so that a divergence criterion is proposed for Article number P_1798Sensors & Transducers, Vol. 163 , Issue 1, January 2014, pp. 39-43  40 signal design, also based on the relative entropy in signal input multiple output (SIMO) radar scenario. However, the Kay’s method just only considers the energy constraint in the transmitter side. However the constant modulus is also important. Since the radar systems commonly require constant modulus waveforms in order to utilize nonlinear power amplifiers, otherwise the target detection performance will be degraded. In this paper, we solve the optimal waveform design with constant modulus problem for target detection in rank-one target model. In next section, the signal model and problem formulation is given. The optimal waveform design considering the constant modulus constraint is solved in section 3. Simulation results are shown in section 4. The conclusion is summarized in section 5.   2. Signal Model and Problem Formulation  When the transmitted signal band B  is sufficiently large so that B  is comparable to / 2c d , where c  is the speed of the light and d  is the range span of the target, the target is not within a signal range cell. In this case, it should be modeled as extended target whose impulse response function can be expressed as [10]:  ,1( ) ( )Lt t l llh t a t   , (1)  where , , 1,2,t la l L   are the complex reflection coefficients of the target. The clutter impulse response function can be expressed as   ,1( ) ( )Mc c m mmh t a t   , (2)  where , , 1,2,c ma m M  are the complex reflection coefficients of the clutter. So the received signal in additive Gaussian noise condition can be expressed as  ( ) ( ) ( ) ( ) ( ) ( )t cx t s t h t s t h t n t     , (3)  where ( )s t  is the transmitted signal and ( )n t  is the complex additive Gaussian noise.   denotes convolution operator. According to [11], formula (3) can be expressed in frequency domain, that is   ( ) ( )t ct cdiag diag    X SH SH NH s H s N, (4)  where  /2 / 2 1( ), , ( )TN NS F S F s  , ( )diagS s   /2 1 /2/2 1 1 /2 1,1 ,exp( 2 ) exp( 2 )exp( 2 ) exp( 2 ), ,N N LtN N LTt t Mj F j Fj F j Fa a             H  , (5)   /2 1 /2/2 1 1 /2 1,1 ,exp( 2 ) exp( 2 )exp( 2 ) exp( 2 ), ,N N McN N MTc c Mj F j Fj F j Fa a             H  , (6)  tH  is a zero mean random Gaussian vector with covariance htR . When ( ) 1htRank R , the amplitude vector of the target scatters function a is fixed, the phase of the target scatters function je   may fluctuate randomly due to changes in the distance between the target and the radar that is the scatters of a rigid target exit no relative Doppler. Some papers named this kind of target after rank-one target. The covariance htR  can be decomposed Hht ht htR V V  by the low rank decomposition, the htV  is defined by the target signature vector. The clutter is also complex random Gaussian vector with zero mean and covariance hcR . And the hcR  is also can be decomposed as Hhc hc hcR V V .  When a target exists in radar environment, the detection procedure is given by decision on one of the two possible hypotheses as  01::ccHH   X SH NX SH SH N, (7)  Our aim is to design the optimal waveform to maximize the probability of detection. In next section, we will elaborate how to optimize the waveform design. This document is prepared according to our journal's manuscript instructions. You can use it as a template.   3. Waveform Design with Constant Modulus Constraint  According to model (5), the probability density function (PDF) under 0H  can be written as  10 21( | ) exp ( )det( )HcnNcnp H   X X R XR, (8)  The PDF under 1H  can be written as Sensors & Transducers, Vol. 163 , Issue 1, January 2014, pp. 39-43  411 211( | )det( )exp ( )Ns cnHs cnp H    XR RX R R X, (9)  where cnR  is the covariance matrix of c SH N , sR  is the covariance matrix of SH  and Hs s sR VV . Thus, the likelihood radio test can be expressed as  101 1( | )( ) ln( | )( )H cn s cnp Hlp Hc       XXXX R R R X, (10)  where   11det ( )lndets cncnc    R RR. The first item in the right of equal sign in (10) is a constant. According to the Woodbury’s identity, (10) can be expressed as  2111 1( )1HHcn ss cn sH Hs cn s s cn sl c   X R VV R VXV R V V R V, (11)  The relative entropy is defined as the exponent in probability of error in a hypothesis test between distribution 0p  and 1p [12]. There exists the probability of error in each hypothesis can be expressed as  0 1 1 00011101 10 10 0( ) ( )( | )( | )ln( | )( | )( | )ln( | )( | ) ( | )ln ln( | ) ( | )D D p p D p pp Hp H dp Hp Hp H dp Hp H p HE Ep H p H            XX XXXX XXX XX X , (12)  where ( )i jD p p  denotes the relative entropy in favor of iH  against jH . In radar systems, the target detection problem has two probabilities of error, false alarm probability faP  and miss probability mP ,  0 1exp ( )N D p p    and  1 0exp ( )N D p p   are equal to faP  and mP , respectively. It can be seen that maximizing the relative entropy could result in minimizing the faP  and mP . Assuming 1 1Hs cn sV R V  [9], Substitute (11) into (12), we have  12/2 1/22 2/2 12/2( )( ) ( )( ) ( )( ) ( ) ( )Hfa m s cn sNs nn N c n n nNn ht nn N n hc n n nD P PV FP F P FS F V FS F P F P F  V R V, (13) Note that 1m dP P  , where dP  is the target detection probability. We hope the target detection probability dP  as large as possible while the false alarm probability faP  as small as possible. Thus, the optimization criterion can be designed as    0 1 1 0max max ( ) ( )D D p p D p p   , (14)  Considering the total energy limit, the optimization model is  2 2/ 22/22/2/ 2( ) ( )max( ) ( ) ( ). . ( )BBhc nBsBS F V FD dFS F P F P Fs t S F dF E, (15)  After using Lagrange multiplier, the optimal waveform is  2 ( ) ( ) / ( )( ) max ,0( )n nhcV F P F P FS FP F     , (16)  The parameter   is found by solving  /2/2( ) ( )/ ( )max ,0( )B ht n nsBhcV F P F P FdF EP F     , (17)  The optimal waveform is the same as the result of [12]. However, the optimal waveform of equation (16) just only considers the energy constraint. In modern radar systems, the constant modulus waveforms are commonly required in practice. Since the nonlinear power amplifiers in radar systems require the input signal to be constant modulus. Thus the optimal waveform with the constant modulus constraint could be considered.  The method to the constant modulus is based on the technique in [13].the constant modulus signal is generated in time domain after given its Fourier transform magnitude. Let MC  denotes the set of function  ( )g t  that have the same Fourier transform magnitude  M   over the frequency interval  . A magnitude projection operator MP  defined can make the arbitrary function  f t  project to a “nearest neighbor”  MP f t  that belongs to MC . For arbitrary function  f t , its Fourier transform can be representation by      jF F e    . The magnitude projection of  x t  can be defined as         ',,jMM eP f tF     , (18) Sensors & Transducers, Vol. 163 , Issue 1, January 2014, pp. 39-43  42 Equation (18) guarantees that radar systems maintain the transmit signal to own the optimal waveform in frequency domain. Continue the constant modulus element. Let AC  denote the set of functions   g t  which have constant envelope level A , There exists a amplitude projection operator AP to assign an arbitrary function  f t  a" nearest neighbor”  AP f t  on AC  so that there is no element Ag C  to satisfy Ax g x P x   . Given an arbitrary function      j tf t a t e  , the projection procedure is        ,,j tA j tAe t TP f ta t e otherwise  , (19)  The magnitude and amplitude projection are combined according to   1( )k A M kf t P P f t  , (20)  The equation (20) has the error reduction property [13]:         1 1 1k k A M k k A M k kd f t PP f t f t PP f t d       , (21)  where  kf t  is the thk  projection iterated function. When the optimal waveform without considering the constant modulus ( )s t  that owns the magnitude    2M S f   is obtained, the optimal waveform  g t  with the constant modulus could be obtained by the equation (18), (19) and (20). The equation (21) guarantee the error reduce when the k  . After a number of iteration, the signal  s t  has the constant modulus envelope with the prescribed Fourier transform magnitude.   4. Simulations Results  In this section, we will give the simulation to demonstrate the effectiveness of the proposed method. Suppose the frequency of the signal is normalized to be  0,1 . The length of the target impulse response is 31. The sampling frequency is 2. Fig. 1 shows the target spectral signature and the clutter power spectral density (PSD). The energy is constrained to be 50 (energy unit). The noise PSD  is 0.15. Fig. 2 shows the optimal waveform without constant modulus constraint and the Fig. 3 shows the optimal waveform with constant modulus constraint. The two figures show that the optimal waveform with/without constant modulus constraint allocates the transmitted energy among most of the spectral peaks of the target signature and the two waveforms are similar, which demonstrates the optimal waveform considering the constant modulus constraint still guarantees the detection performance. The magnitudes of the constant modulus waveform are smaller than the one of the non-constant modulus waveform. However, the optimal waveform with constant modulus allocates the transmitted energy into additional frequency bands. Fig. 4 is the complex constellation of constant modulus constrained waveform. It can be seen that the optimal waveform has the constant envelope in time domain after considering the constant modulus constraint, which ensures the modulus of the transmitted signal could not exceed the maximum input value of DAC in practice.   0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.020.040.060.080.10.120.140.160.180.2Normalised FrequencyPower  Target spectral signatureClutter PSD  Fig. 1. The target spectral signature and the clutter PSD.   0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10123456Normalized frequencyEnergy spectral density  Optimal waveform without constant modulus constraint  Fig. 2. The optimal waveform without constant  modulus constraint.   0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.050.10.150.20.250.30.35Normalised FrequencyEnergy spectral density  Optimal waveform with constant modulus constraint  Fig. 3. The optimal waveform with constant  modulus constraint. Sensors & Transducers, Vol. 163 , Issue 1, January 2014, pp. 39-43  43-8 -6 -4 -2 0 2 4 6 8x 10-3-6-4-20246x 10-3Re(fk)Im(fk)  Fig. 4. Complex Constellation of optimal waveform  with constant waveform.   5. Conclusions  This paper focuses on the waveform design for target detection considering the constant modulus constraint. Maximizing the relative entropy as the optimization criterion designs the optimal waveform with the transmitted energy constraint for the rank-one target. In addition, considering the constant modulus in practice, the constant modulus constraint is applied to design the optimal waveform. The energy spectral density of the optimal waveform without constant modulus is regarded as he prescribed Fourier transform magnitude. Then the constant modulus waveform could be obtained in time domain. It is similar to the optimal waveform that only considers the total energy constraints. The simulation results present the proposed  method validation.   Acknowledgment   This work was supported by the National Natural Science Foundation of China (No. 61004052 and No. 61104005), the Natural Science Foundation of Liaoning province under Grant (No. 201102057), the Natural Science Foundation of Hebei Province (No. F2012501044 and No. F2013501075) and the Doctoral Scientific Research Foundation of Liaoning Province (No. 20131030). References  [1]. S. Haykin, Cognitive radar: a way of the future, IEEE Signal Processing Magazine, Vol. 23, Issue 1, January 2006, pp. 30-40. [2]. S. Haykin, Cognition is the key to the next generation of radar systems, in Proceedings of the IEEE Signal Processing Education Workshop, 2009, pp. 463-467. [3]. M. R. Bell, Information theory and radar waveform design, IEEE Transactions on Information Theory, Vol. 39, Issue 5, September 1993, pp. 1578-1597. [4]. Y. Yang and R. S. 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Beyer, Construction of constant envelope signals with given Fourier transform magnitude, in Proceedings of the IEEE International Radar Conference, 2009, pp. 4-8.  ___________________      2014 Copyright ©, International Frequency Sensor Association (IFSA) Publishing, S. L. All rights reserved. (http://www.sensorsportal.com) 

Optimal Waveform Design with Constant Modulus Constraint for Rank-One Target Detection

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