Multitone Waveform Design Toward Enhancing Conversion Efficiency for RF Electronics Detection

Nonlinear radar has attracted considerable attention in recent years due to its unique advantage in detecting nonlinear targets, such as radio frequency (RF) electronics, under strong clutter conditions. However, the low nonlinear RF-to-RF conversion efficiency (RRCE) severely limits its operational range and application scenarios. So far, the research community has mainly focused on improving the RRCE from the hardware aspects while ignoring the signal design. In this article, we study the waveform design problem that maximizes RRCE for detecting nonlinear targets. Assuming known channel state information and target characteristics, we develop a tractable block-oriented nonlinear model and formulate the waveform design as a constrained optimization problem. Then, two iterative algorithms based on successive relaxation are proposed to solve the nonconvex problem for different cases. Besides, the simulations and laboratory experiments demonstrate the effectiveness of the system model and the superiority of the proposed waveform design method over various baselines. More than 4-dB improvement in RRCE is observed for the given configurations in the experiment. The study provides a promising alternative for extending the maximum distance of nonlinear radar.


I. INTRODUCTION
Nonlinear target detection, also known as nonlinear radar, has gained significant attention in recent years [1], [2].It has been observed that targets containing nonlinear elements such as diodes and transistors exhibit the ability to reradiate nonlinear echoes when subjected to strong electromagnetic (EM) waves [3], often resulting in the generation of new frequencies, like harmonics and intermodulations.Common examples of electronic nonlinear targets include cell phones, radio frequency (RF) receivers, and diodeloaded antennas [4].In contrast to natural backgrounds that predominantly reflect in a linear manner, nonlinear radar leverages this property for target detection and recognition, offering a distinct advantage in finding RF electronics under clutter conditions.Consequently, nonlinear detection has proven to be valuable for various applications such as manmade electronics detection [5], [6], [7], [8], imaging [9], recognition [10], [11], [12], security checks [13], antiinterference communication [14], biological tracking [15], [16], [17], crack detection [18], and so on.
The generation of nonlinear echoes by the RF electronics can be considered as a nonlinear RF-to-RF conversion process.A major challenge faced by nonlinear radar is the low RF-to-RF conversion efficiency (RRCE) of that process, which severely limits the operating range and application scenarios.To improve the RRCE, the traditional approaches in the literature mainly focus on improving the hardware design of the radar systems, such as increasing the peak transmit power [19] or the receive sensitivity [20].Therefore, several or even hundreds of kilowatts of high power have been reported for harmonic radar [21], [22], [23].Trying to increase the power of a single transmitter has already reached its limits.Then, improving the nonlinear responses by new detection architectures, like the addition of phase-coherent auxiliary transmitters, has also recently been proposed [24].However, phase synchronization and beam alignment of auxiliary transmitters can be challenging under spatially distributed conditions.Another line of research focuses on collaborative applications such as insect tracking and maritime rescue, where special tags are used to improve RRCE.High-efficiency transponder designs have been studied extensively.And a variety of diode types [25] [16], topologies [17], and operating modes [26] are investigated to improve the harmonic conversion efficiency.However, the approaches are only effective for a limited number of cooperative scenarios.In general, most relevant research has been confined on circuit-or system-level designs.
Another complementary line of research is to explore the potential from signal-level aspect.Although various waveforms have been applied in nonlinear detection, most of the literature in this field utilizes the waveform as an alternative to obtain a high-resolution range profile or to improve isolation.For example, the pseudorandom code waveform is proposed to improve the range profile in [23] and to mitigate the interference from the self-leakage signal in [15].The frequency-modulated noise waveform is studied in [27] and [28] to improve the correlation properties between intermodulation responses and self-harmonic emission, thus increasing the isolation.Inspired by the bionic waveform, Leighton et al. [29] introduce the Gaussian modulated twininverted pulse to realize clutter suppression and nonlinear target identification.These studies demonstrate the potential of waveform diversity in nonlinear radar, but attempts for designing the efficient stimulus to maximize RRCE are still lacking.
The experience and knowledge in wireless power transmission (WPT) provide valuable insights for nonlinear radar because both tasks leverage the intrinsic nonlinearity of RF components (rectifier/RF electronics) to achieve frequency conversion.When a large EM wave impinges upon the targets, the nonlinear transfer characteristics induce nonlinear distortion [30], thereby physically generating dc and RF terms simultaneously.Therefore, RF-to-RF conversion efficiency and RF-to-dc conversion efficiency (RDCE) are actually two perspectives of one physical process.
For the WPT, the transmitted waveform has been shown to have a significant effect on the RDCE.Collado and Georgiadis [31] experimentally show that high peak-to-average ratio (PAPR) signals, such as orthogonal frequency-division multiplex and chaotic signals, can improve RF-to-dc efficiency.Besides, harmonically spaced multisine [32] and digitally modulated signals [33] have also been experimentally shown to be advantageous over tone signals.In addition, optimization-based waveform designs that consider channel information have also been proposed.Clerckx and Bayguzina [34] provide a systematic framework for channel-adaptive waveform design in a point-to-point WPT system, which sets a new benchmark for the WPT task.The work is extended to large-scale situations in [35] and low-complexity solution approximation in [36].The comprehensive and complete implementations for real-world experiments based on the framework are also reported in [37] and [38].Joint wireless information and power transmission has already become possible under a unified waveform design framework [39].Therefore, it is justifiable to believe that especially designed waveform may also have great impact on RRCE for the nonlinear radar.
However, an in-depth investigation of the new waveform design framework for maximizing RRCE is necessary.The nonlinear radar seeks to boost the reradiated nonlinear RF signal, and the stimulus propagates round-trip before being received.The WPT, however, attempts to boost the dc output and the stimulus propagates single-trip only.The purpose and workflow of these two tasks are inherently different.Therefore, the optimal stimulus for WPT is not necessarily optimal for the nonlinear radar.Among various nonlinear RF responses, the localized conjugate component (LCC) is a special intermodulation response that holds great potential in RF electronics detection [40] and could be taken as an example case.As shown in Fig. 1, a large signal is used to stimulate the electronic targets into a nonlinear state, while another small signal serves as the perturbation.When exposed to such mixed excitation, the RF electronics would generate the nonlinear LCC terms, while the clutter would reflect only linearly.In [40], the single tone is used as the large signal due to its simplicity for concept demonstration where the channel state information and target characteristics (CSITCs) are not fully exploited.Therefore, the new waveform design methodology for maximizing RRCE would be demonstrated by designing the large signal, which could propel the targets into a nonlinear state more efficiently.
The primary concern of this article is to establish a systematic waveform design methodology to maximize the RRCE for nonlinear radars.The work includes building the system model, formulating the optimization problems, proposing efficient solution algorithms, and designing validating experiments.We focus on the multicarrier waveforms due to their tractability in representation and potential in the field of joint communication and sensing scenarios.The main contribution of this work could be outlined as follows.These methods may also be useful for other cases of characteristic-adaptive waveform design.
The rest of this article is organized as follows.Section II presents the system model and formulates the waveform design problem.In Section III, two iterative solutions are developed to solve the nonconvex optimization problem.Section IV demonstrates the performance advantages of the proposed waveform design method through numerical simulation.Section V confirms the effectiveness of the model and methods through two types of detection experiments.Finally, Section VI concludes this article.
Notations: Throughout this article, the signal with and without ˜marker (e.g., x(t ) and x(t )) on the head stand for the complex envelope and the real bandpass RF signal, respectively.They are related by x(t ) = Re{x(t )e jω c t }, when the carrier frequency ω c is specified.The notations (•) T , (•) H , (•) * , and (•) represent the transpose, conjugate transpose, conjugate, and optimized solution, respectively.and stand for the dot product and convolution, respectively.

II. PROBLEM FORMULATION
This section presents the system model for the RF electronics detection.The model establishes a relationship between the waveform parameters and the LCC responses generated by the target of interest.First, we introduce a block-oriented system model that effectively describes the steady-state nonlinear response when subjected to multicarrier excitation.Then, we employ the variational method to obtain the perturbed responses and the corresponding RRCE.Finally, we formulate the waveform design problems as constrained optimization problems tailored to specific tasks.The LCC in the fundamental frequency band is taken as an example due to its widespread usage.However, it is worth noting that the proposed methodology can be readily extended to other nonlinear responses in a similar manner.

A. Model for Steady-State Nonlinear Response
Consider a single-target detection scenario.In the frequency domain, a multitone signal offers a convenient representation for arbitrary waveforms.We assume that the transmitted signal consists of N tone parameters.The signal generation could be achieved either using a single transmitter with a switch-mode amplifier [41], [42] or multiple transmitters with spatial combining techniques [43].At this Fig. 2. Block-oriented model for the nonlinear detection system.The CSITCs are modeled by the cascaded linear dynamic blocks and nonlinear static blocks.
stage, we do not specify a concrete implementation scheme since the principle behind is the same.The total equivalent large signal at the transmitter, with a carrier frequency of ω L , is expressed as follows: A n e jω n t ( with A n = a n e jφ n , where a n and φ n refer to the amplitude and phase at the nth frequency, ω n .Without loss of generality, we assume that those frequencies are evenly spaced and central symmetry in the baseband where ω is the frequency interval.where A s and ω s are the complex amplitude and the frequency offset of the small signal, respectively.As shown in Fig. 2, the transmit signal propagates through the wireless multipath channel, characterized by the frequency-selected response, h c (t ), which could be further decomposed into range factor h u (r) and the normalized response hc (t ) where G tx and R tx are the gain and output impedance of transmitter's antenna, respectively.G a and R a are the gain and input impedance of target's antenna, respectively.λ and r are the wavelength of carrier frequency and range of the target, respectively.Then, the signal reaches the nonlinear target, which is modeled as a nonlinear dynamic system.In general, Volterra series could be applied for characterizing such a nonlinear transfer process.However, the number of coefficients for the full Volterra model grows too fast with the model order and the memory length.Therefore, a special class of pruned Volterra model, Wiener-Hammerstein model, is used here, which decomposes the whole nonlinear dynamic process into cascaded linear dynamic and nonlinear static blocks.h t (t ) and w t (t ) are the linear dynamic block before and after the nonlinear static block f NL (•), respectively.To simplify the notation, the linear blocks from the channel and the target could be merged.Then, the effective voltage signal ṽe (t ) right before the nonlinear static block could be derived as follows: x(t − τ 1 )h(τ 1 )dτ 1 (5) where h(t ) = h c (t ) h t (t ) contains the input CSITC.Then, the odd-order power series model is used to represent the static nonlinear block since the LCC in the fundamental frequency band would be studied here where ζ 2k+1 are coefficients of the series model and K denotes the maximum model order.Then, the receiving signal could be derived by substituting the expressions accordingly where w(t ) = w c (t ) w t (t ) contains the output CSITC.Similar to (4), wc (t ) could also be decomposed into range factor w d (r) and the normalized response wc (t ) where G rx and R rx are the gain and output impedance of receiver's antenna.It should be noted that h(t ) and w(t ) could be different when the frequency shift between the transmitting and receiving band is prominent.By assembling the above equations, the general system model between the parametric transmit signal and the nonlinear response is established.

B. Model for Perturbed Nonlinear Response
The aforementioned part derives the general nonlinear response under large excitation, while, in this part, the perturbed response induced by the additional small signal, x s (t ), would be further derived using the variational method.First, we introduce the auxiliary parameter and the function ( ) where F (•) is the nonlinear operator that maps the transmit signal envelope to the receiving envelope.Mathematically, F (•) could also be regarded as the other function ( ) with scalar and real variable .Then, the first-order variational approximation could be written as follows: The nonlinear operator F (•) here is nonholomorphic since it explicitly contains the envelope magnitude, |x L (t )| and |x s (t )|, thus involving the corresponding conjugate terms.By applying the chain rule and Wirtinger calculus [44], the perturbed response could be obtained as shown by (11) shown at the bottom of this page.
As we can see, the conjugate term x * s (t ) exists only when k ≥ 1 (k = 0 corresponds to the linear response).Therefore, it is one of the indicators and fingerprints exclusive for the nonlinear target.
The general expression in (11) is complicated.However, when we analyze the maximum detection range, it is reasonable to ignore the high-order nonlinear terms (k ≥ 2) since the effective input voltage |ṽ e (t )| at the target port is usually a small term in that case.Then, the receiving LCC signal can be approximated by only keeping the first term as follows: ) So far, the relation between the LCC and the parametric large signal has been established.It should be noted that the input and output linear dynamic blocks h(t ) and w(t ) could be different when the target contains some nonreciprocal components, like the circulator.

C. Objectives and Constraints for Optimization
The RRCE of LCC is determined by the large-signal parameters when the perturbed signal remains relatively small [40].Therefore, the objective of waveform design is to maximize the RRCE under the fixed power of the large signal.Under the multitone framework, the RRCE could be derived from ( 12) by applying the Parseval theorem (detailed derivation is shown in Appendix A) (14) Suppose that the large and small signals are provided by different transmitters.Therefore, the power constraint by the large signal could be given as where A = [A 1 , . . ., A n ] T and P max is the maximum power of the transmitter.Then, the waveform design leads to the following optimization problem: P 0 is a nonconvex optimization problem with 2N variables, for which it is generally difficult to find the global optimum.Therefore, we propose two approximate solutions for the different application cases.First, we simplify the original objective using its upper bound by triangle inequality and rewrite the original constraint to the dimensionless equivalent form by the scaling property.According to the triangle inequality, we have max{N,p+N} n=max{1,p+1} where the equality holds when ∠c n = nφ 0 , i = 1, . . ., N, where φ 0 is an arbitrary phase offset.Therefore, optimal phases for A n could be chosen as where φ n stands for the optimal phase condition for the nth tone.φ 0 is set to be 0 without loss of generality.Then, the number of variables for optimization is reduced to N. Besides, every term in the RRCE is quartic, while that in the constraint is quadratic.Therefore, different values for the constraint actually lead to a scaled version of the same optimal solution.For example, if A as the optimal solution in ( 16) under the constraint of A 2 ≤ 1, then it is easy to prove that √ αA is the optimal solution for A 2 ≤ α.
Second, the original problem could be reformulated into a more compact form by introducing the shifted backward identity matrix (BIM), which is defined to have the element one on the cross diagonal, while all other elements are equal to zero.Then, the element of p-shifted N-dimensional BIM, J p [i, j], is defined as follows: where δ[n] is the discrete Dirac function, which has value 1 at the origin and 0 at all other points.Therefore, J 0 corresponds to the BIM and the other J p has the unit cross diagonal shifted upward (p < 0) or downward (p > 0).Then, under the optimal phase condition (18), RRCE could be reformulated as follows: where Then, the equivalent optimal problem could be written as follows: where is the matrix denotation for the input CSITC.Then, the new simplified optimization problem P 1 with reduced searching space is obtained, which could actually be regarded as the basic waveform design problem.Specific requirements for different applications could be fulfilled by adding additional constraints.

III. PROPOSED SOLUTIONS FOR WAVEFORM DESIGN
In this section, two effective solutions for P 1 based on iterative strategies are proposed.The successive geometric program approximation (SGPA) is a more general framework, which could be adapted to more kinds of constraints easily.However, it is more computationally intensive and time consuming.The SEP is more efficient for solving the basic problem but less extendable for various constraints.

A. SGPA Solution
Problem P 1 consists of maximizing a posynomial subject to a power constraint.Similar problem has been tackled in [34] for WPT by geometric program (GP), which is actually a popular technique in wireless communication [45].Therefore, the method is adapted to solve problem P 1 and set the benchmark.In (21), J 1 can be expanded as the posynomial with L monomials when the tone number N is specified, which can be inferred from the following equation: (The derivation is shown in Appendix B.) , N is odd , N is even. ( As we can see, the number of monomials grows cubically with the tone number, which implies a huge complexity of problem transformation by this method for large N. Nevertheless, suppose that the lth monomial takes the following form: where α l is the coefficient for the lth monomial.And d i,l is the ith exponent index for the lth monomial.Then, J 1 = L l=1 g l (a).And P 1 can be transformed to the standard minimization problem by inversion of the objective.To make the new optimization problem coincide with the standard GP form, an arithmetic-geometric mean inequality (AGMI) is applied to transform the minimization objective into a monomial where {γ l } satisfies L l=1 γ l = 1, which has a significant impact on the tightness of the upper bound.By iteratively updating {γ l } and solving the standard following GP problem P 2 , the approximated solution of P 1 is obtained: One popular strategy for updating {γ l } at each iteration is to set γ l = g l (a ( ) )/J 1 , where a ( ) is the feasible solution from previous iteration [45].The standard GP problem could be solved efficiently and robustly with the convex optimization method, like the penalty method, interior-point method, and dual method.Besides, existing toolbox, e.g., CVX [46], could also be used as the submodule conveniently.In general, Algorithm 1 summarizes the whole procedure.
The solution could be extended to more constraints.The easiest way is to reformulate them into the equivalent posynomial inequality, f i (a) ≤ 1, in line with standard GP and append them in (25).For example, the band-limited transmit waveform design problem could be tackled by adding the constraint as follows: where β i is the mask in the corresponding frequency.And the waveform design with PAPR limitation could also be handled by applying the similar technique as that in [34], which first samples the signal in the time domain and then transform the denominator form inequality to standard monomial by the AGMI.In general, the SGPA solution sets an extendable framework and benchmark suitable for the complicated tasks.

B. SEP Solution
Problem P 1 shows a unique homogeneous structure, i.e., quartic posynomial objective and quadratic posynomial constraint.Therefore, exploiting the potential inner structure of the solution space, we further propose a more Algorithm 2: SEP-MLCC: Successive Eigenvector Pursuit for Multitone LCC Waveform Design.
Input: H, W, I max , ξ 1 , ξ 2 Output: a , 1: Calculate by (18) 2: Initiate a (0) , set i = 0 3: repeat 4: Calculate c (i) = Ha (i) 5: Calculate C (i) by (28) and R (i) by ( 29) 6: repeat 7: Set k = 0, a (i) 0 = a (i) 8: 10: where Then, the objective J 1 could be rewritten as follows: where R = H T C T WCH is referred to as the generalized autocorrelation matrix (GACM) below.R is a Hermitian matrix in general and becomes a Toeplitz matrix when W is flat (with constants for all entries).The objective J 1 is referred to as the quasi Rayleigh quotient (QRQ) in this article since it is similar to traditional Rayleigh quotient (RQ) in form.However, the essential difference is that the numerator of RQ is quadratic, while that of QRQ is quartic since R is dependent on the variables a.The concise expression in (29) actually implies the existence of a well-structured solution space and a more efficient searching strategy.First, we prove that the QRQ is upper bounded by the maximum eigenvalue of the GACM.According to the Lagrange multiplier method, the equivalent unconstrained optimization problem could be reformulated as follows: where 2λ is the Lagrange multiplier and the extra coefficient 2 is for the compactness of the final result.The gradient of J 3 could be obtained by the chain rule of vector derivatives where c T J p c is a number and, therefore, could be swapped backward in the second step.Then, the following Karush-Kuhn-Tucker (KKT) condition gives the necessary requirement for the optimal solution: Therefore, the optimal solution a would be located among the eigenvectors of the GACM.And substituting (32) into (29), QRQ becomes λ in (29), thus being upper bounded by the maximum eigenvalue of the GACM, λ max .And a becomes the corresponding normalized eigenvector of λ max .Second, we propose an efficient algorithm, SEP, to iteratively approach the upper bound.The detailed procedures are summarized in Algorithm 2. The intuitive idea is to alternatively construct R and a with proper initial vector a (0) .At the ith iteration, R (i) is constructed from a (i−1) by the definition in (29), and then, a (i) is updated by finding the eigenvector corresponding to the maximum eigenvalue (referred to as maximum eigenvector (ME) later) of R (i) .The power iteration method is adopted here to solve eigenvector since only the ME is needed.In general, the inner and outer loops in the algorithm are for the update the ME and GACM, respectively.The accuracy and the convergent time of the algorithm could be controlled by the hyperparameters ξ 1 and ξ 2 .The subproblem for each iteration is the RQ problem, which preserves the original QRQ structure, thus leading to a much more efficient searching process.
The SEP solution could also be extended to some constraints.For example, the band-limited transmit waveform design problem could be tackled by adding a virtual mask filter before h(t ), i.e., substituting H with Diag(β ) H, where β = [β 1 , . . ., β N ].However, it is not an easy task to directly extend the method for those nonstructural constraints, like PAPR constraint.As we can see, the SEP algorithm only consists of basic algebraic operations and matrix calculations, without depending on the commercial convex optimization package.Therefore, it could be easily implemented in the edge devices, which is promising for online waveform design based on updated real-time CSITC parameters.

C. Convergent Condition
The convergent condition is one of the most critical aspects for the analysis of iterative algorithms.The SGPA algorithm strictly follows the framework of SCA, the property of which has been intensively studied.It is guaranteed to converge to a local optimal point fulfilling the KKT condition as long as the initiation starts from one point within the feasible region [47].Therefore, it is enough to randomly initiate a 0 in the SGPA solution within the disk around the origin, i.e., ||a|| ≤ 1.
As for the SEP algorithm, we find that its convergence could also be guaranteed as long as two successive entries in a 0 are nonzero.To further state that, we first need to introduce a useful lemma named Perron-Frobenius (PF) theorem [48], which is defined as follows.
LEMMA 1 (PF THEOREM) Suppose that A ∈ R n×n is nonnegative and regular, i.e., A k > 0 for some k. 1 Then, there is an eigenvalue λ pf of A that is real and positive, with associated positive left and right eigenvectors, which is called the PF eigenvalue.And for any other eigenvalue λ of A, we have |λ| ≤ λ pf .
When a (0) contains at least two successive nonzero entries, it can be proved that a (i) > 0 for all i > 0, although no explicit constraint for that is exerted in the algorithm.According to the definition, each entry of R (i) actually corresponds to a posynomial of the entry of a (i) .Therefore, R (i) > 0 when a (i) > 0 since H and W are all positive.Then, the PF theorem guarantees that R (i) has one unique realvalued maximum eigenvalue, λ (i+1)  max , and the corresponding eigenvector a (i+1) would also consist of positive entries strictly.In other words, the proper initialization could guarantee that the solution {a (i) } locates always in the feasible region.When a (0) contains at least two successive entries, R (0) has nonzero main and second main diagonal entries.Then, it is easy to note that (R (0) ) N > 0, i.e., R (0) would be a nonnegative regular matrix.Then, the recursion process above starts to get under way.
Then, on the one hand, the sequence of J 1 (or {λ (i)  max }) of N-tone input is upper bounded On the other hand, the updated sequence by algorithm SEP is nondecreasing.By definition, we have where the inequality is due to the fact that a (i+1) is the ME of R (i) .Therefore, {λ (i) max } is convergent in the field of real number.In summary, the convergence of both the proposed algorithms could be guaranteed when the initialization of a (0) is given properly as mentioned above.

IV. NUMERICAL ANALYSIS AND SIMULATION
In this section, extensive simulations are carried out to demonstrate the effectiveness of the system model and superiority of the proposed solutions.The simulations can be broadly categorized into three sets.First, the predicted RRCE from the proposed system model is compared with the result obtained from the harmonic balance (HB) simulator to validate the accuracy of the model.Fast and accurate predictions allow for the statistical comparison with various baselines.Second, several waveform design baselines, initially intended to maximize RDCE, are statistically contrasted, considering the scarcity of existing literature on subject of optimizing RRCE.Third, the optimal waveform under flat CSITC conditions is investigated.The simulation predicts that waveforms with a truncated Gaussian spectrum are preferable for RRCE.
In the simulations, we generate the general frequencyselective CSITC, H (ω) and W (ω), by randomly initializing "nonflat" linear dynamic filters.These filters are modeled by 18 taps whose delays and phases are uniformly distributed over [0 0.1] μs and [0 2π ], as described in [49].The bandwidth and center frequency are set to 10 and 905 MHz, respectively.The upper half of Fig. 3 shows a realization for input and output CSITC.The red star and circle-shaped dots represent the two input frequency bands corresponding to the large and small perturbation signals, respectively.The cyan dots represent the frequency band of the output signal.
The methods for comparison are presented in Table I, which can be classified into single-tone and multitone strategies.The single-tone localized conjugate component (SLCC) method is developed from our previous work [40], which selects the large-signal tone with maximum RRCE by searching all available frequency channels in band.Therefore, other single-tone methods, like adaptive single

TABLE I Brief Summary of Static and Adaptive Waveform Design Methods for
Performance Comparison sine (ASS) [34], are not included since SLCC would be the optimal solution for the single-tone case.Multitone approaches are mainly recast from WPT waveform designs.Uniform power (UP) represents the heuristically predetermined waveform, neglecting the CSITC, which features a high peak envelope.Maximal ratio transmission (MRT) [50] and scaled match filter (SMF) [36] methods are two representatives for adaptive waveform design with approximated closed-form solution in WPT tasks.The lower half of Fig. 3 shows the magnitude of the optimal waveform in the spectrum for different methods.In general, most approaches tend to allocate more power to the frequencies corresponding to the input CSITC with less transmission loss, although the degree of emphasis varies among different methods.

A. Validation of the System Model
First, the effectiveness of the system model is verified using the HB simulator.HB-based simulations are widely used in the analysis of the nonlinear system, but it is time consuming and relies on the powerful processors.And the proposed system model provides a fast prediction for the RRCE when the CSITCs are given, allowing extensive evaluation of different waveform designs.
Fig. 4(a) shows the schematic of the circuit used in the simulation, similar to the one in [34].The CSITCs in Fig. 3 are included by the Touchstone file (.s2p), which is followed by a common nonlinear rectifying circuit.The diode, SMS7630, is modeled by the Shockley model with SPICE parameters in the datasheet [51].The average power of the large signal is set to −35 dBm, and the power backoff is 10 dB to ensure the validity of the weak nonlinearity hypothesis.For the HB simulator, the first seventh-order harmonic bands with a total of 2448 mixed frequencies are considered to ensure accuracy.In Fig. 4(b), the scatter and histogram plots are the results predicted by (13) and by the HB simulator, which shows good agreement for all configurations.And even some details are fully predicted by the proposed model.For example, the small performance difference between UP and SLCC in four and eight tone cases is clearly predicted.Therefore, the proposed model would be quite useful for rapid verification and comparison of different designs.

B. Performance Comparison for Different Designs
Second, we make a statistical comparison between our proposed methods and those in Table I.The results show that the waveform design problems for maximizing RDCE and RRCE typically favor different solutions.Then, we study the convergence time between the SGPA and SEP methods.
Fig. 5 shows the predicted RRCE of the compared methods under different tone numbers.500 random initializations of CSITC are performed, and then, the mean normalized RRCE is calculated.Normalization is performed to prevent one specific realization from dominating the final result, since the RRCE for different realizations may differ by orders of magnitude.All approaches and all tone configurations are subject to the same unit power constraint, i.e., a 2 ≤ 1.In statistical sense, the proposed CSITC adaptive algorithms, SEP and SGPA, for maximizing RRCE achieve the best performance for all configurations.The single-tone adaptive method, SLCC, shows comparatively worse performance, which reaches a plateau after the optimal frequency is already included in the available searching channels.As the number of tones increases, all multitone methods show a remarkable improvement.Compared with the proposed algorithms, the adaptive methods, SMF and MRT, that maximize RDCE always shows secondary performance in the middle level, with around 25% degradation.The performance of the nonadaptive method, UP, increases much slower than that of other multitone counterparts, with around 50% degradation.In the detection scenario, the multitone signal is generally advantageous for boosting the RRCE, which is also consistent with the observation.Fig. 6 shows the running time of the two proposed algorithms as the number of tones increases.500 sets of Monte Carlo experiments are performed, and then, the mean and variation of the results are calculated.Although the SEP and SGPA algorithms eventually converge to similar solutions, as shown in Fig. 5, the SEP algorithm takes significantly less time to run.Notable increase in run time for SGPA is observed as N exceeds 28 in the simulation, since it takes up too much memory for the laptop to construct the GP problem with so many constraints.Therefore, the SGPA Fig. 7. Spectrum magnitude over normalized band for the optimal waveform by proposed methods with the increase in the number of tones.
algorithm is probably more suitable for research purpose due to its ability to handle more types of constraints, while the SEP is suitable for real-time online applications in the closed-loop system.

C. Optimal Waveform Under Flat CSITC
The frequency-flat situation is one of the basic cases of CSITC, which typically occurs in two situations.One is when the memoryless nonlinear targets are in a frequencyflat channel, whose net characteristics are frequency independent over the entire band.The other is when the signal bandwidth is significantly smaller than the fluctuation of target's characteristics in the frequency.The proposed algorithm can easily handle the flat CSITC case by setting H and W as constants.Then, the GACM becomes the autocorrelation matrix of the variable a. Two adaptive methods, MRT and SMF, degenerate to the same solution as that of the UP method.Fig. 7 shows the normalized spectrum magnitude over normalized bandwidth for the optimal waveform under different N. The magnitude is normalized to 1 by the maximal value for each N.It is observed that the spectrum of the optimal waveform gradually approaches a truncated Gaussian shape as N grows large.As we can see, the spectrum magnitude for N = 64 case almost coincides with a Gaussian function, the fitted formula of which can be written as where f and ā are the normalized frequency and magnitude, respectively.It has been known that waveform with rectangle spectrum magnitude is optimal for maximizing RDCE in the frequency-flat channel [34].Our simulation results imply that waveform with Gaussian spectrum magnitude is optimal for maximizing RRCE in frequency-flat CSITC.
The conclusion is still speculative because a rigorous mathematical proof is required, which is beyond the scope of this article.(The conclusion is mathematically formulated in Appendix C.) Fig. 8 shows the average RRCE per tone for different design strategies.The RRCE per tone is obtained by dividing the RRCE by N.Although the rectangular strategy is not optimal, the performance difference is small compared to the Gaussian one, the maximum of which is about 3% in relative.Moreover, the average RRCE per tone for both the cases reaches a plateau after N becomes larger than 10.Therefore, it is more cost effective to choose N < 10 if the number of carrier imposes huge burden on the system.
In summary, the proposed model achieves RRCE predictions that are consistent with the HB simulator, which enables extensive performance evaluation.Then, two proposed algorithms are expected to outperform various baselines statistically.And the SEP is more efficient than SGPA in terms of solving the basic power-constraint problem.Besides, the waveform with Gaussian shape spectrum is expected to be preferable for boosting RRCE in the flat CSITC case.Simulations above have already provided plenty of insights, while some important ones would be For the compactness of demonstration and implementation, mainly SEP, UP, and SLCC solutions would be implemented for comparison.

V. EXPERIMENT VALIDATION
A prototype system is built to verify the effectiveness of the proposed system model and waveform design methodology.Both the nonlinear targets and the wireless channel exhibit frequency-selective characteristics under clutter conditions, which is crucial prior information for waveform design.However, the identification of CSITC from the measured data requires a systematic algorithm and function block [38], [52], which is a departure from the main work of this article.Therefore, two sets of experiments are carefully designed to equivalently validate the performance of the proposed methodology under flat and nonflat CSITC situations.
The schematic and the picture of the system are shown in Fig. 9.The signal generation and acquisition are handled by two daisy-chained vector signal transceivers (VSTs), which are synchronized with each other and controlled by the computer.The large signal is generated by one channel of the VST in the test mainly to simplify the implementation, which requires the amplifier to operate in the linear region.However, multiple channels with a spatial combining technique [43] could also be applied to increase the efficiency with higher transmit power.Moreover, instead of building an additional cancellation link similar to that in [40], a customized duplexer is adopted to combat self-leakage, thus making the RF front-end of the validation system more compact.The receivers labeled "R1" and "R2" are used to inspect the reflected and transmitted signals, respectively.Nevertheless, the system shown is not an optimal configuration, but it is sufficient to validate the proposed waveform design methodology.

A. Frequency-Flat Situation
The first set of experiments is for flat CSITC, which uses the lower intermodulation band in an over-the-air manner.To obtain a quasi-flat frequency characteristic, we restrict the bandwidth of the large signal within a small scale, e.g., kHz.The preliminary observation of vector network analyzer (VNA) shows that the characteristic fluctuation at this scale could be ignored for the indoor scenario.The simulated target of interest is made by connecting a low-noise amplifier (LNA), Mini-circuit ZX6083LN, to an antenna.Here, a log periodic antenna is used instead of a dipole to mitigate the high propagation loss, since the maximum peak power provided by the VST is limited to about 18 dBm, and an additional power back-off is required to guarantee the linearity of the transmitter.The simulated electronic target is placed about 2.2 m away to satisfy the far-field condition.The large signal is a multitone waveform with 10-kHz bandwidth at 847 MHz, while the small signal is the sinusoidal signal with −10 dB power back-off at 899 MHz.The lower intermodulation (IM) band at approximately 895 MHz is then collected and processed to obtain the total in-band power.
Preliminary simulations suggest that the number of carriers is a critical factor in improving performance in the flat CSITC case.Fig. 10 shows the received in-band LCC power at different transmit powers as the number of tones increases.The black dots represent the responses from the corner reflector under four-tone case, which serves as a reference to guarantee that the detected LCC power comes from nonlinear targets rather than the system itself.The blue dots and the solid line are the measured and fitted results by the SLCC scheme in [40], which are used as the baseline for performance comparison.The proposed model predicts an expected improvement in RRCE of approximately 1.76 and 4.39 dB for the two-and four-tone cases, respectively, as shown by the two dashed lines.Then, ten measurements were conducted to account for the fluctuation in the wireless channel.As shown, the results agree well with the prediction.Therefore, the multitone signal excites the target into a nonlinear state more efficiently than its single-tone counterpart and, thus, can increase detection performance at the same transmit power level.However, it is worth noting that the weak nonlinear hypothesis may be violated when the carrier number becomes quite large.

B. Frequency-Selective Situation
The second set of experiments is for nonflat CSITC, which uses the upper intermodulation band with the cablebased system.In this set of experiments, both the bandwidth and power sweeps are carried out to demonstrate the validity of the methodology under various configurations.As shown in Fig. 11(a), a linear dynamic subsystem is built in front of the device under test (DUT) by passive components (the power combiner, circulator, and long coaxial cable), which forms multiple signal paths and leads to nonflat characteristics in the megahertz scale.The transmission response of the subsystem, i.e., S 21 and S 12 , could be directly measured by the VNA and used as prior information for CSITC.
In Fig. 11(b), the subsystem shows significant frequency-selective characteristics in the multimegahertz scale, while the reflection characteristics of the nonlinear DUT1 are negligible in this scale according to the preliminary test.Based on the measured S-parameters, the optimal waveform is designed by the proposed method and loaded into VST for transmission.The large signal is centered at 904 MHz with varying bandwidths and input powers, while the small signal is sinusoidal with 10 dB power back-off at 865 MHz.Then, the upper IM band at about 942 MHz is collected and processed to obtain the total in-band power.
The markers indicate the approximate transmit and receive bands for two typical bandwidths.First, the bandwidth is one of the critical factors for RRCE because the CSITC varies under different bandwidths.Fig. 12 shows the RRCE gain with the increase of signal bandwidth, from 100 kHz to 10 MHz.The center frequency and tone number of the large signal are kept fixed during the experiment.The RRCE is normalized by that of SLCC in narrow bandwidth.The lines are the prediction from the model, while the dashed lines with symbols are the results from the measurement.As we can see, when the bandwidth is sufficiently small, the performance of the UP method is very close to that of the optimal one with only a slight degradation, which is consistent with the conclusion of the previous simulation.The RRCE becomes independent of the bandwidth because the CSITC in this small interval can be roughly considered as flat.However, as the bandwidth increases, the proposed method quickly outperforms others by dynamically adapting to the frequency-selective characteristics.Besides, an interesting phenomenon to note is that there seems to exist an optimal bandwidth for RRCE under the given configuration.At about 4-MHz bandwidth, the waveform matches the net dynamic characteristics, which promotes a kind of "resonance" that leads to around 3-dB improvement in RRCE compared with UP method.And around 4-8-dB improvement is observed compared with the traditional SLCC method.Nevertheless, the influence of the bandwidth is fully predicted by the proposed model and validated by the measured data.Second, the input power is another key factor because it affects the validity of the model assumption.Fig. 13 shows the measured RRCE for two DUTs under three design methods as the input power increases.The LNA in DUT1 is replaced by another two mixers, ZX05-63LH and ZX05-2, to demonstrate the wide applicability in terms of target types.The proposed waveform outperforms other methods before the input power reaches the critical points, which are approximately −10 and −6 dBm for DUT2 and DUT3, respectively.At higher input powers, the proposed method gradually becomes nonoptimal because the effectiveness of the model actually requires the target to be dominated by the low-order nonlinear effect.When the weakly nonlinear hypothesis is violated, the model's mismatch leads to performance degradation.Then, the improvement in RRCE becomes closely related to input power.In general, before the mismatch, around 4-and 8-dB improvements in RRCE are observed compared with UP and SLCC methods for the given configurations.It is possible to include higher orders of nonlinearity, but at the cost of increasing model complexity.There is a tradeoff between model complexity and the effectively applicable power range.Nevertheless, the results in Fig. 13 show that the optimal waveform designed under the simplified assumption actually works over a fairly wide power range, which is probably sufficient for RF electronics detection applications.
In general, the two types of designed experiments verify the feasibility and effectiveness of the proposed model and waveform design methodology under both flat and nonflat CSITC cases.The experiments show that the proposed model can predict the RRCE under various bandwidths and power levels, given appropriate prior information.For both the flat and nonflat CSITC cases, the multitone waveform is much more efficient than its single-tone counterpart.For larger bandwidths, the CSITC can hardly be considered constant, which leads to the necessity for adaptive waveform design.In general, the designed waveforms achieve around 4-8-dB improvement in RRCE under the feasible configurations, and the proposed algorithms can handle waveform design under both the flat and nonflat CSITC situations.

VI. CONCLUSION
The nonlinear radar has been plagued by the low reradiated efficiency for many years.In this article, we study the waveform design problem for maximizing RF-to-RF conversion efficiency for nonlinear detection purposes.We demonstrate the full methodology for solving the intricate problem for the first time.The work includes deriving a tractable system model, formulating optimization problems, proposing efficient algorithms, and designing equivalent validation experiments.The results show that the optimal waveform for maximizing RRCE is typically different from that for RDCE.The block-oriented nonlinear model balances the complexity and performance, which could predict the RRCE across various bandwidths and power levels.Assuming known CSITC, the numerical simulations and real-world experiments show the effectiveness and superiority of our proposed solutions over various baseline methods from WPT.When the weakly nonlinear condition is fulfilled, around 4-8-dB improvement in RRCE has been observed in the experiments under various feasible configurations for several tested DUTs.
This study opens up a new door for extending the operational range of nonlinear radar, which is complimentary with traditional hardware approaches.However, this article does not address a number of avenues of research that could be further explored in future work.For instance, this article assumes a uniform spacing N-tone signal for the given bandwidth.Is it feasible to automatically select the optimal channels and tone numbers that are not necessarily evenly spaced?This would help to understand how to make the best of the limited RF spectrum for optimal nonlinear target detection.Besides, the experiments in this article demonstrate a simplified single-antenna detection system with comparative low power combining efficiency.A more complete architecture using multiple antennas with spatial combining technique could be further investigated.

APPENDIX A DERIVATION FOR THE CONVERSION EFFICIENCY OF LCC
According to the definition of the conversion efficiency where E{•} is the expectation or averaging operator.ỹLCC (t ) can be decomposed into the convolution between the virtual input z(t ) and shifted output CSITC, w(t )e jω s t , where z(t ) = c(t ) • c(t ) c(t ) = xL (t − τ ) h(t ).(37) According to the FT, c(t ) in the frequency domain is the dot product of xL (t ) and h(t ).Therefore, the discrete spectrum cn = A n H (n ω).Besides, z(t ) is the convolution of c(t )

Fig. 1 .
Fig. 1.Schematic of the general RF electronics detection using multitone LCC distortion.The large signal is designed for efficient nonlinearity excitation, while the auxiliary small signal is used as the perturbation for the generation of LCC responses.

Fig. 3 .
Fig. 3. (a) Magnitude of one realization of the frequency-selective CSITC.(b) Under the situation in (a), the spectrum magnitude of the designed optimal multitone waveform by the proposed and compared methods.

Fig. 4 .
Fig. 4. (a) Schematic of the circuit-level HB simulation for system model validation.(b) Normalized RRCE predicted by the proposed model and the HB simulations for different methods (P in = −35 dBm).

Fig. 6 .
Fig. 6.Statistical comparison of the running time for two proposed methods under 500 Monte Carlo tests with the increase of the tone number.

Fig. 8 .
Fig. 8. Average RRCE per tone as the number of tones, N, increases for different multitone methods under flat CSITC.

Fig. 9 .
Fig. 9. (a) Picture and (b) the block diagram of the over-the-air nonlinear detection system.(c) External RF link for the system.

Fig. 10 .
Fig. 10.Received in-band LCC power with the increase of the transmit power and the number of tones under flat CSITC.The result is measured ten times for the DUT1, Mini-circuit ZX6083.(power-ON state, N = 4, BW = 10 kHz).

Fig. 11 .
Fig. 11.(a) Block diagram of the cable-based measurement system with simulated nonflat CSITC targets.(b) Input CSITC and output CSITC of the simulated linear block measured by the VNA.

Fig. 12 .
Fig. 12. RRCE gain of the DUT1 for three methods under different bandwidths.The lines with and without markers are from measurement and model, respectively (N = 9, P in = −10 dBm).