Wideband Waveform Optimization for Multiple Input Single Output Cognitive Radio with Practical Considerations

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1 The 1 Military Communications Conference - Unclassified Program - Waveforms Signal Processing Track Wideb Waveform Optimization for Multiple Input Single Output Cognitive Radio with Practical Considerations Zhen Hu Department of Electrical Computer Engineering Center for Manufacturing Research Tennessee Tech University Cookeville T, USA zhu1@tntech.edu an Guo Center for Manufacturing Research Tennessee Tech University Cookeville T, USA nguo@tntech.edu Robert Qiu Department of Electrical Computer Engineering Center for Manufacturing Research Tennessee Tech University Cookeville T, USA rqiu@tntech.edu Abstract This paper investigates the transmitted waveform optimization issues for wideb Multiple Input Single Output (MISO) cognitive radio. For cognitive radio, a spectral mask for the transmitted waveform is determined on spectrum sensing, arbitrary transmitted spectral shaping is required. Meanwhile, the interferences from primary radios should be canceled at the receiver of cognitive radio. The contribution of this paper is to optimize the MISO cognitive radio communication link by jointly considering the optimization objective, the spectral mask constraint at the transmitter the interference cancellation at the receiver. Meanwhile, reduction in transmitted peak power quantization is still very desirable, being concerned about implementation complexity power consumption. These motive us to consider various practical constraints for waveform optimization in the context of MISO cognitive radio. A number of solutions are provided in this paper in conjunction with numerical results showing the optimal wideb waveforms. Index Terms MISO, cognitive radio, wideb waveform, SDP I. ITRODUCTIO Waveform design or optimization is a key research issue in the current wireless communication system, the radar system the sensing or image system. Waveforms should be designed according to the different requirements objectives of performance. For example, waveform should be designed to carry more information to the receiver in terms of capacity. If the energy detector is employed at the receiver, the waveform should be optimized such that the energy of the signal in the integration window reaches the maximum [1] [] [3] [4]. For navigation geolocation, an ultra short waveform should be used to increase resolution. For multi-target identification, waveform should be designed so that the radar returns can bring more information back. In clutter dominant environment, maximizing the target energy minimizing the clutter energy should be considered. In the context of cognitive radio, waveform design or optimization gives us more flexibilities to design radio, which can coexist with other cognitive radios primary radios. From cognitive radio s point of view, spectral mask constraint at the transmitter the interference cancellation at the receiver should be seriously considered for waveform design or optimization, in addition to the traditional communication objectives constraints. Spectral mask constraint is imposed on the transmitted waveform such that cognitive radio has no interference to primary radio. At the same time the arbitrary notch filter is implemented at the receiver to cancel the interference from primary radio to cognitive radio. Multiple Input Single Output (MISO) system considered in this paper is one kind of multi-antenna system in which there are multiple antennas at the transmitter one antenna at the receiver. MISO system can explore the spatial diversity execute the beamforming to focus energy on the desired direction or point avoid interference to other radio systems. It is well known that waveform spatially diverse capabilities are made possible today due to the advent of lightweight digital programming waveform generator [5] or arbitrary waveform generator. This paper deals with wideb waveform optimization for MISO cognitive radio. The study on system performance, e.g. bit error rate (BER) or capacity, is out of scope of this paper. Semidefinite Programming (SDP)-based iterative method are exploited to take care of wideb waveform design. Different designed waveforms are to be applied to different antennas. But these waveforms are optimized jointly in order to match the different spatial channels based on the objective of the system performance. SDP-based signal processing is becoming more more popular recently. It can be applied to control theory, statistics, circuit design, graph theory so on. The reasons for this is (1) more more practical problems can be formulated as SDP; () most interior-point methods for linear programming have been generalized to SDP [6]; (3) nowadays the computational capability is increased greatly SDP can be solved in real-time. Meanwhile, reduction in transmitted peak power quantization is still very desirable, being concerned about the implementation complexity power consumption. Thus, Peak-to-Average Power Ratio (PAPR) /1/$6. 1 IEEE 17

2 Tx FPGA Baseline Transmitter Arbitrary Waveform Generator D/A D/A Power Amplifier Power Amplifier h1(t) h(t) the RF front-ends at the transceivers such as power amplifier, LA arbitrary notch filter as well as antennas between the transmitter antenna n the receiver antenna. h n (t) is available at the transmitter [7] [8]. T h h n (t) dt E nh. denotes convolution operation. n(t) is AWG. x n (t) is the received noiseless bit- 1 waveform defined as Rx FPGA Decision Fig. 1. Tb L A Arbitrary otch Filter System architecture. binary waveform design are taken into account as the practical considerations in the context of MISO cognitive radio. The contribution of the proposed iterative methods is that the rank- 1 optimal solution can always be guaranteed. Making some of the constraints tighter will force the optimal solution to go toward rank-1 matrix. The rest of the paper is organized as follows. In Section II the system is described SDP-based iterative method is presented to solve the wideb waveform optimization issue. Practical wideb waveform design is discussed in Section III. umerical results are provided in Section IV, followed by some remarks given in Section V. II. WIDEBAD WAVEFORM OPTIMIZATIO USIG SDP-BASED ITERATIVE METHOD The system architecture considered in this paper is shown in Figure 1. We limit our discussion to a single pair of cognitive radios scenario. There are antennas at the transmitter one antenna at the receiver. On-off keying (OOK) modulation is used for transmission. Thus the transmitted signal at the transmitter antenna n is, s n (t) j d j p n (t jt b ) (1) where T b is the bit duration, p n (t) is the transmitted bit waveform defined over [,T p ] at the transmitter antenna n d j {, 1} is the j-th transmitted bit. Without loss of generality, the minimal propagation delay is assumed to be zero. The energy of transmitted waveforms is E p, Tp p n (t) dt E p () The received noise-polluted signal at the output of low noise amplifier (LA) is, r(t) h n (t) s n (t)+n(t) d j j x n (t jt b )+n (t) (3) where h n (t),t [,T h ] is the multipath impulse response that takes into account the effect of channel impulse response, x n (t) h n (t) p n (t) (4) We further assume that T b T h +T p def T x,i.e.noexistence of ISI. If the waveforms at different transmitter antennas are assumed to be synchronized, the k-th decision statistic is, r(kt b + t ) d j j x n (kt b + t jt b )+n (kt b + t ) d k x n (t )+n(kt b + t ) (5) In order to maximize the system performance, x n (t ) should be maximized. Thus the optimization problem can be formulated as follows to get the optimal waveforms p n (t), maximize x n (t ) Tp p n (t) dt E p t T b For simplicity in the following presentation, t is assumed to be zero, which will not degrade the optimum of the solution if such a solution exists. x (t) From Fourier transform, (6) x n (t) (7) x f n (f) hf n (f) pf n (f) (8) x f (f) h f n (f) p f n (f) (9) where x f n (f), h f n (f) p f n (f) are the frequency domain representations of x n (t), h n (t) p n (t) respectively. x f (f) is frequency domain representation of x(t). Thus, x () x n () x n () (1) x f n (f) df (11) 18

3 If there is no spectral mask constraint, then according to the Cauchy Schwarz inequality, x () Ep h f n (f) pf n (f) df E nh (1) when p f n (f) is equal to the proportionable conjugate of h f n (f) for all f n, two equalities are obtained. The scale coefficient is, α E p (13) h f n (f) df In this case, p n (t) αh n ( t), which means the optimal waveform p n (t) is the corresponding proportionable time reversed multipath impulse response h n (t). If there is spectral mask constraint, then the following optimization problem will become more complicated, maximize x () Tp p n (t) dt E p p f n (f) c f n (f) (14) where c f n(f) represents the arbitrary spectral mask constraint at the transmitter antenna n. In order to solve the optimization problem 14, p n (t) h n (t) are uniformly sampled at yquist rate. Assume the sampling period is T s. T p /T s p p is assumed to be even, T h /T s h. p n (t) h n (t) are represented by p ni,i, 1,..., p h ni,i, 1,..., h respectively. Define, p n [p n p n1 p np ] T (15) h n [h nh h n(h 1) h n ] T (16) where T denotes transpose operation. If p h, then x n (t ) can be equivalent to h T n p n.define, Thus, p [p T 1 p T p T ] T (17) h [h T 1 h T h T ] T (18) h T n p n h T p (19) ( Maximization of h T p is the same as maximization of h T p ) as long as h T p is equal to or greater than zero. ( h T p ) ( h T p ) T ( h T p ) trace(hp) () where H hh T P pp T. P should be rank-1 positive semidefinite matrix. However, rank constraint is non-convex constraint, which will be omitted in the following optimization problems. Thus the optimization objective in the optimization problem (14) can be reformulated as, Meanwhile, (1) p p T p trace ( pp T ) trace(p) () the energy constraint in the optimization problem (14) can be reformulated as, trace (P) E p (3) For cognitive radio, there is a spectral mask constraint for the transmitted waveform. Based on the previous discussion, p n is assumed to be the transmitted waveform, F is the discrete-time Fourier transform operator, thus the frequency domain representation of p n is, p f n Fp n (4) where p f n is a complex value vector. If the i-th row of F is f i, then each entry in p f n can be represented by, ( p f n )i,1 f ip n,i1,,..., p +1 (5) where ( ) i,j means the entry in the matrix with the i-th row the j-th column. Define, F i fi H f i,i1,,..., p +1 (6) where H denotes conjugate transpose operation. Given the spectral [ mask constraint in ] terms of power T spectral density c n c n1 c n c n( p,so +1) ( p f ) n f i p n i,1 p T n f H i f i p n p T n F ip n c ni,i1,,..., p +1 (7) where is the modulus of the complex value. Define selection matrix S n R (p+1) (p+1), { 1, j i +(p +1)(n 1) (S n ) i,j (8), else So, p n S n p (9) 19

4 ( p f ) n i,1 p T n F ip n trace ( S T n F is n P ) (3) The optimization problem (14) can be reformulated as SDP based on (1), (3), (7) (3), trace (P) E p trace ( S T n F i S n P ) (31) c ni i 1,,..., p+1 n 1,,... If the optimal solution P to the optimization problem (31) is the rank-1 matrix, then the optimal waveforms can be obtained from the dominant eigen-vector of P.Otherwise, E p in the optimization problem (31) should be decreased to get the rank-1 optimal solution P to satisfy all the other constraints. An SDP-based iterative method (Algorithm I) is proposed to get the rank-1 optimal solution P : 1. Initialization of E p.. Solve the optimization problem (31) get the optimal solution P. 3. If the ratio of dominant eigen-value of P to trace (P ) is less than.99, thensete p to be trace (P ) gotostep. Otherwise, the method is terminated. The optimal waveforms can be obtained from the dominant eigen-vector of P, the dominant eigen-value of P Eq. (17). III. WAVEFORM DESIG WITH PRACTICAL COSIDERATIOS A. Peak-to-Average Power Ratio PAPR is one of major concerns in waveform design. Because of nonlinearity caused by nonlinear devices such as Digital-to-Analog Converter (DAC) Power Amplifier (PA), maximal transmitted power has to be backed up, resulting in inefficient utilization. PAPR in OFDM has been well studied. In this paper, PAPR is hled under a unified optimization framework. It is defined as, p n PAPR / (3) p n ( p +1) where p n max ( p n, p n1,, pnp ) (33) If the denominator of Eq. (3) is omitted, reducing PAPR is equivalent to setting the upper bound for p n. The bound constraint p n b n can also be written as, b n p ni b n,i, 1,..., p (34) which can be further simplified as, (p ni ) (b n ),i, 1,..., p (35) Define selection vector s ni R 1 (p+1), { 1, j i +(p +1)(n 1) (s ni ) 1,j (36), else So, p ni s ni p (37) (p ni ) (s ni p) trace ( s T nis ni P ) (38) The optimization problem (14) or the optimization problem (31) together with PAPR consideration can be presented as SDP, trace (P) E p trace ( S T n F i S n P ) c ni trace ( s T ni s nip ) (39) (b n ) i 1,,..., p+1 n 1,,... Similarly, if the optimal solution P to the optimization problem (39) is the rank-1 matrix, then the optimal waveforms can be obtained from the dominant eigen-vector of P.Otherwise, E p in the optimization problem (39) should be decreased to get the rank-1 optimal solution P to satisfy all the other constraints. An SDP-based iterative method (Algorithm II) is proposed to get the rank-1 optimal solution P : 1. Initialization of E p.. Solve the optimization problem (39) get the optimal solution P. 3. If the ratio of dominant eigen-value of P to trace (P ) is less than.99, thensete p to be trace (P ) gotostep. Otherwise, the method is terminated. The optimal waveforms can be obtained from the dominant eigen-vector of P, the dominant eigen-value of P Eq. (17). B. Binary Waveform If the transmitted waveform is constrained to the binary waveform because of the hardware limitation or implementation simplicity, or equivalently if p ni E p, then The optimization problem (14) or the optimization problem (31) together with binary waveform design can be formulated as SDP, trace (P) E p trace ( S T n F is n P ) c ni trace ( s T ni s nip ) (4) E p i 1,,..., p+1 n 1,,... However, the constraints trace (P) E p trace ( s T ni s nip ) usually bring non-rank-1 E p 13

5 optimal solution P or invalid solution, i.e. no feasible region for the optimization problem because of the constraints, to the optimization problem (4). Thus the equality constraints are relaxed to the inequality constraints the optimization problem (4) is relaxed to, trace (P) E p trace ( S T n F is n P ) c ni trace ( s T ni s nip ) i 1,,..., p+1 n 1,,... E p (41) However, such relaxation forces us to verify the feasibility of the optimal solution P to the optimization problem (41). If the dominant eigen-value of P is the same as E p,which means trace ( s T ni s nip ) E is equal to p for all i n, then the optimal solution P is feasible the optimal binary waveforms can be obtained from the dominant eigen-vector of P the dominant eigen-value of P.Otherwise,E p in the optimization problem (41) should be decreased. An SDP-based iterative method (Algorithm III) is proposed to get the rank-1 optimal solution P : 1. Initialization of E p.. Solve the optimization problem (41) get the optimal solution P. 3. If the ratio of dominant eigen-value of P to E p is less than.9999, thensete p to be trace (P ) gotostep. Otherwise, the method is terminated. The optimal waveforms can be obtained from the dominant eigen-vector of P, the dominant eigen-value of P Eq. (17). IV. UMERICAL RESULTS The following setting has been used in generating numerical results: T s 1ns, T h 1ns, T p 1ns; E p is set to be 1 as the initial value; the number of transmitter antennas is equal to. Multipath impulse responses represented in the frequency domain between transmitter antennas receiver antenna are shown in Fig.. The nulling part from 9MHz to 39MHz in Fig. emulates the effect of arbitrary notch filter at the receiver, which means there is interference from primary radio in this kind of notched frequency b. The spectral mask constraints in the following simulations are established arbitrarily. Due to the page limitation, only results for channel 1 are shown to illuminate the performances of the proposed algorithms. All the SDPs presented in this paper are solved by the CVX tool [9] [1]. Fig. 3 shows the designed optimal waveform represented in frequency domain for channel 1 using SDP-based iterative method (Algorithm I). Obtained from the designed optimal waveforms, there are no powers allocated to the notched frequency b for two transmitter antennas. The powers allocated to the notched frequency b can not bring any benefit to the performance. Meanwhile, all the spectral mask constraints are satisfied Channel Fig.. Multipath impulse responses represented in the frequency domain between transmitter antennas receiver antenna Channel Waveform SDP Spectral Mask Fig. 3. Designed waveform represented in frequency domain for channel 1. Fig. 4 shows designed optimal waveform represented in frequency domain for channel 1 with PAPR consideration. Fig. 5 shows designed optimal waveform in corresponding time domain for channel 1. In this case, b 1 is set to be.1. Because of the waveform shape constraints in the time domain, for the designed optimal waveforms, there are still some powers allocated to the notched frequency b at the receiver for two transmitter antennas. This amount of power can not be saved in order to keep the specific shapes of waveforms in the time domain. Meanwhile, all the spectral mask constraints PAPR constraints are satisfied Channel Waveform SDP Spectral Mask Fig. 4. Designed waveform with PAPR consideration represented in frequency domain for channel

6 Energy Gap Time (ns) The umber Of Iterations Fig. 5. Designed waveform with PAPR consideration in time domain for channel 1. Fig. 8. Energy gap. Fig. 6 shows designed optimal binary waveform represented in frequency domain for channel 1. Fig. 7 shows designed optimal binary waveform in corresponding time domain for channel 1. Fig. 8 shows the convergence of energy gaps between E p the dominant eigen-value of P with the number of iterations. When the dominant eigen-value of P approaches E p very well, the optimal solution to the optimization problem (4) is obtained Channel Waveform SDP Spectral Mask Fig. 6. Designed binary waveform represented in frequency domain for channel Time (ns) Fig. 7. Designed binary waveform in time domain for channel 1. V. COCLUSIO This paper deals with wideb waveform optimization for MISO cognitive radio. Wideb waveforms are designed according to the optimization objective with the considerations of spectral mask constraint at the transmitter the influence of arbitrary notch filter at the receiver. Meanwhile, PAPR binary waveform design are also taken into account as the practical considerations in the context of MISO cognitive radio. The method of this paper can be easily extended to the passb waveform design, where the individual oscillator for each antenna can be tied together to achieve coherency [5]. ACKOWLEDGMET This work is funded by ational Science Foundation through grants (ECCS-914), (ECCS-81658), (ECCS-615). The authors want to thank Santanu K. Das Robert Ulman for their useful discussions. REFERECES [1]. Guo, J. Q. Zhang, P. Zhang, Z. Hu, Y. Song, R. C. Qiu, UWB Real-Time Testbed with Waveform-Based Precoding, in IEEE Military Conference, (San Diego, USA), ovember 8. []. Guo, Z. Hu, A. S. Saini, R. C. Qiu, Waveform-level Precoding with Simple Energy Detector Receiver for Wideb Communication, in IEEE Southeastern Symposium on System Theory, (Tullahoma, USA), March 9. [3] Z. Hu,. Guo, R. C. Qiu, Wideb Waveform Optimization for Energy Detector Receiver with Practical Considerations, in IEEE International Conference on Ultra-Wideb, (Vancouver, Canada), September 9. [4] Z. Hu,. Guo, R. C. Qiu, Wideb Waveform Optimization with Energy Detector Receiver in Cognitive Radio, in IEEE Military Conference, (Boston, USA), October 9. [5] M. C. Wicks, History of Waveform Diversity Future Benefits to Military Systems. A Lecture Series on Waveform Diversity for Advanced Radar Systems, July 9. [6] L. Venberghe S. Boyd, Semidefinite programming, SIAM review, vol. 38, no. 1, pp , [7] D. Singh, Z. Hu, R. C. Qiu, UWB Channel Sounding Channel Characteristics in Rectangular Metal Cavity, in Proc of IEEE Southeastern Symposium, (Huntsville, USA), April 8. [8] R. C. Qiu, C. Zhou, J. Q. Zhang,. Guo, Channel Reciprocity Time-Reversed Propagation for Ultra-Wideb Communications, in IEEE AP-S International Symposium on Antennas Propagation, 7. [9] boyd/cvx/. [1] S. Boyd L. Venberghe, Convex optimization. Cambridge Univ Pr, 4. 13