Waveform-Domain Complementary Signal Sets for Interrupted Sampling Repeater Jamming Suppression

W a v ef or m-Domain Complementar y Signal Sets f or Interr upted Sampling Repeater J amming Suppression Hanning Su National Uni versity of Defense T echnology , Changsha 410073, China Qinglong Bao National University of Defense T echnology , Changsha 410073, China Jiameng Pan National University of Defense T echnology , Changsha 410073, China Fucheng Guo National University of Defense T echnology , Changsha 410073, China W eidong Hu National University of Defense T echnology , Changsha 410073, China Abstract— The interrupted-sampling repeater jamming (ISRJ) is coherent and has the characteristic of suppression and deception to degrade the radar detection capabilities. The study focuses on anti-ISRJ techniques in the wavef orm domain, primarily capital- izing on wav eform design and and anti-jamming signal processing methods in the wavef orm domain. By exploring the relationship between wa veform-domain adaptive matched filtering (WD-AMF) output and wav eform-domain signals, we demonstrate that ISRJ can be effectiv ely suppressed when the transmitted wavef orm exhibits wav eform-domain complementarity . W e introduce a phase-coded (PC) waveform set with wavef orm-domain complementarity and propose a method for generating such wavef orm sets of arbitrary code lengths. The performance of WD-AMF are further dev eloped due to the designed wav eforms, and simulations affirm the superior adaptive anti-jamming capabilities of the designed wavef orms compared to traditional ones. Remarkably , this improv ed perfor - mance is achieved without the need for prior knowledge of ISRJ interference parameters at either the transmitter or receiv er stages. Index T erms— Interrupted-sampling repeater jamming (ISRJ), wavef orm-domain adaptive matched filtering (WD-AMF), Manuscript receiv ed XXXXX 00, 0000; revised XXXXX 00, 0000; accepted XXXXX 00, 0000. (Corr esponding author: Hanning Su and Qinglong Bao) . Hanning Su, Qinglong Bao, Jiameng Pan, Fucheng Guo and W ei- dong Hu are with the College of Electronic Science and T echnology , National Uni versity of Defense T echnology , Changsha 410073, China (e-mail: hanningsu18@nudt.edu.cn ; baoqinglong@nudt.edu.cn ; panjia- meng@nudt.edu.cn ; gfcly@21cn.com ; wdhu@nudt.edu.cn ). This work was supported by the National Science Foundation of China under Grant 62231026. 0018-9251 © 2022 IEEE wav eform-domain, wav eform design, complementary wav eform sets. I. INTRODUCTION I N T E R R U P T E D - S A M P L I N G repeater jamming (ISRJ) is a form of coherent jamming, capable of swiftly and accurately generating jamming signals using digital radio frequency memory (DRFM) [ 1 ], [ 2 ]. The jamming signals retransmitted by interference machines based on DRFM de vices are coherent with the radar transmission signals, leading to the presence of both genuine and false target peaks in the range profile acquired through matched filtering. T o suppress ISRJ and bolster radar detection capabilities, the anti-ISRJ methodology is studied in this work. The leading theory concerning ISRJ was initially pro- posed by W ang et al. [ 2 ], attracting considerable attention and e xtensiv e research o ver more than a decade. In [ 3 ] and [ 4 ], ISRJ was employed to create multiple false targets in the range profile, with an analysis of the characteristics of these false targets. T o generate false targets ahead of the true target, ISRJ was enhanced by modulating the ISRJ signal using frequenc y-shifting jamming in [ 5 ]. Non- periodic ISRJ was explored in [ 6 ], capable of producing a comprehensiv e jamming coverage ov er the radar’ s true target. In [ 7 ], a linear weighted optimization approach was adopted to rectify the unev en energy distribution of non-periodic ISRJ, resulting in the creation of a square jamming strip with more effecti ve co verage. As per publicly accessible literature, it is evident that existing techniques for mitigating ISRJ encompass both recei ver -side signal processing methods and transmitter- side wa veform design strategies. In the realm of receiver - side signal processing, ISRJ-induced false tar gets ha ve been ef fecti vely eliminated through the skillful application of time-frequency analysis and band-pass filtering [ 8 ]– [ 11 ]. The critical parameters associated with ISRJ hav e been deduced using time-frequency analysis and decon- volution processing. In the context of transmitter-side wa veform design methodologies, certain researchers have sought to disrupt the Doppler continuity of interfering signals by constructing sparse Doppler wav eforms [ 12 ], thereby facilitating the identification and suppression of interfering signals. Despite endea vors such as mismatch filtering [ 13 ] and complementary wa veforms [ 14 ] to jointly optimize the design of transmission wa veforms and mismatch filters through algorithmic means, the fun- damental nature of these approaches still hinges on prior kno wledge of segments related to the jamming signals. It is not arduous to discern that existing means of countering ISRJ e xhibit a pronounced reliance on pattern recognition of jamming and the estimation of pi votal jamming parameters. Consequently , the primary step in most anti-jamming measures often in volv es estimating ISRJ key parameters through cogniti ve methods [ 15 ]– [ 19 ]. These approaches, howe ver , exhibit a certain de gree of constraint in their adaptability to complex interference IEEE TRANSACTIONS ON AEROSP A CE AND ELECTRONIC SYSTEMS V OL. XX, No. XX XXXXX 2022 1 scenarios since they hinge upon the accuracy of cognitive models. This restriction impedes their further practical applicability in real-world interference scenarios. Further- more, certain scholars have sought to employ deep neural networks for the extraction of jamming-free segments and the subsequent generation of band-pass filters [ 20 ]. This adaptiv e approach to some extent diminishes the necessity for prior ISRJ information. Nonetheless, the training of neural networks demands copious data, and the performance of networks trained with simulated data ne- cessitates further validation through radar -measured data. Additionally , in our own work [ 21 ], we put forth a wa veform-domain adaptiv e matched filtering (WD-AMF) technique using linear frequency modulated (LFM) wav e- forms as an illustration. This method achie ved effecti ve ISRJ suppression in the absence of prior ISRJ informa- tion. Ho we ver , it is important to note that LFM wav eforms may not be the most optimal choice for WD-AMF . This is because when the real target echo is overwhelmed by ISRJ in the wa veform domain, it results in a loss of the main lobe lev el and an increase in sidelobe le vel in the WD-AMF output. Relati ve to our pre vious publication discussed in [ 21 ], this article delves deeper into the tenets of ISRJ suppres- sion by de vising the most suitable w av eform for WD- AMF . Consistent with the principles we adhered to in our pre vious work, the wa veform is fashioned without any antecedent knowledge of the ISRJ parameters. The contributions of this article can be succinctly summarized as follows. 1) Through an analysis of the interaction between WD-AMF and the transmitted wa veform, it be- comes e vident that the primary reason for the main lobe lev el loss and sidelobe lev el increase in the WD-AMF output is largely due to the absence of waveform-domain complementarity in the radar waveform. This complementarity can be ef fectiv ely achieved by using phase-coded (PC) wa veforms. 2) T o achieve wa veform-domain complementarity for the phase-coded (PC) w av eform, a definition of wa veform-domain complementary signal sets has been introduced, and their properties have been demonstrated. A method for generating wav eform- domain complementary signal sets of arbitrary code lengths has been proposed. It has been pro ven that this complementary signal set retains its com- plementarity in the wav eform domain, unaffected by Doppler effects. Furthermore, real target echo signals and ISRJ interference in the waveform domain can be treated as non-overlapping mono- component signals. 3) An enhanced version of WD-AMF , improved with respect to wa veform-domain complementary sig- nal sets, has been developed to bolster anti-ISRJ performance. Simulation results validate the effec- ti veness of the proposed techniques, af firming that the wa veform-domain complementary signal sets represent the optimal choice for WD-AMF . The subsequent sections of this article are structured as follo ws. In Section II, the fundamental aspects of ISRJ are elucidated, and an analysis is conducted to explore the relationship between WD-AMF and the waveform- domain signals. Section III is dedicated to the definition, properties, and generation methods of wa veform-domain complementary signal sets. Section IV provides the per- tinent expressions for wa veform-domain complementary signal sets in WD-AMF , along with the underlying anti- jamming principles. The simulations which validate the proposed techniques and present research findings, are detailed in Section V . Lastly , this article culminates in Section VI with a brief discussion of the study’ s conclu- sions. II. PROBLEM FORMULA TION In this section, we will commence by analyzing the model of ISRJ, followed by an analysis of the relationship between WD-AMF output and wa veform-domain signals. W e will then deri ve the necessary constraints for the optimal wa veform adaptation to WD-AMF . A. ISRJ model Assuming s ( t ) represents the transmitted w av eform and ȷ ( t ) represents the interference wav eform, the echo signal x ( t ) can be expressed as follo ws [ 2 ]: x ( t ) = A s s ( t − τ s ) + A ȷ ȷ ( t − τ ȷ ) (1) where, A s and A ȷ represent the amplitudes of the target echo signal and the interference signal, while τ s denotes the delay of the target echo signal, and τ ȷ signifies the delay of the jamming signal. If we consider ȷ ( t ) as an example of interrupted sampling direct repeater jamming (ISDRJ), where the interrupted sampling frequency is f J = 1 T J and the jamming slice width is T ȷ . Hence, the matched filtering output of x ( t ) can be expressed as: x o ( t ) = A s χ ( t − τ s , 0) + Q X q = − Q A ȷ A q χ ( t − τ ȷ , − q f J ) = s o ( t − τ s ) + ȷ d ( t − τ ȷ ) (2) where, χ ( t, f d ) = R + ∞ −∞ s ( τ ) s ∗ ( t + τ ) e j 2 π f d τ dτ represents the ambiguity function of s ( t ) , A q = T ȷ f J sinc ( π q f J T ȷ ) denotes the modulation function of the harmonics, Q represents the number of transitions between jammer acquisition and transmission modes. s o ( t ) = A s χ ( t, 0) signifies the matched filtering output of the target echo signal, and ȷ d ( t ) = Q P q = − Q A ȷ A q χ ( t, − qf J ) represents the matched filtering output of ISDRJ. Eq. ( 2 ) illustrates that the output of ISDRJ can equiv a- lently be vie wed as the superposition of multiple weighted target matched filtering outputs. The number of false 2 IEEE TRANSACTIONS ON AEROSP A CE AND ELECTR ONIC SYSTEMS V OL. XX, No. XX XXXXX 2022 targets in the range profile is determined by the Doppler tolerance of s ( t ) . When the Doppler tolerance of s ( t ) is high, multiple f alse targets will be generated in the range profile. When the interference system operates in an in- terrupted sampling repetitive repeater jamming (ISRRJ) mode, the matched filtering result of ISRRJ can be equi valently expressed as multiple time-domain shifts of ISDRJ. In the range profile, it exhibits multiple clusters of false targets, with the interference characteristics within each cluster resembling those in the ISDRJ mode. The matched filtering output of ISRRJ can be expressed as: ȷ r ( t ) = P − 1 X p =0 ȷ d ( t − pT ȷ ) (3) where P denotes the number of repetitiv e repeater . Ho wev er , when the jamming system employs a inter- rupted sampling cyclic repeater jamming (ISCRJ) mode, due to the fact that different slices have identical trans- mission delays only during the initial relay , during the second and subsequent relays, their delays are delayed by ( q − 1) T ȷ . Consequently , the pulse compression result will contain multiple clusters of f alse tar gets. The distrib ution characteristics of the f alse target clusters can similarly be described using ISDRJ. The matched filtering output of ISCRJ can be expressed as: ȷ c ( t ) = Q − 1 X q =0 ȷ d ( t − q T ȷ − q T J ) (4) B. W av ef or m-domain adaptive matched filtering In our previous work [ 21 ], we introduced an extended domain within the signal matched filtering process, re- ferred to as the wav eform domain. This domain represents the integral interval of the impulse response h ( µ ) of the matching filter, specifically , µ ∈ U =  − T 2 , T 2  , where T denotes the w av eform pluse width. The product of the signal x ( t − µ ) and the impulse response h ( µ ) through the matching filter is referred to as the wav eform response function at time t : v ( t ) ( µ ) = x ( t − µ ) h ( µ ) = A s s ( t − µ − τ s ) h ( µ ) + A ȷ ȷ ( t − µ − τ ȷ ) h ( µ ) = v ( t − τ s ) s ( µ ) + v ( t − τ ȷ ) ȷ ( µ ) (5) where v ( t ) s ( µ ) = A s s ( t − µ ) h ( µ ) denotes the w av eform response function of s ( t ) , and v ( t ) ȷ ( µ ) = A ȷ ȷ ( t − µ ) h ( µ ) denotes the wa veform response function of ȷ ( t ) . In the wa veform domain, we hav e formulated an adapti ve threshold function ˆ E ( t ) ( µ ) with the purpose of integrating v ( t ) ( µ ) ov er the set containing ef fectiv e integration elements U ( t ) s , while masking the ineffecti ve integration element set U ( t ) ȷ . Simultaneously , the process adapti vely compensates for the masked elements in the unbiased estimation ˆ v ( t ) ( µ ) . This compensation is derived from a randomly selected continuous subset Ψ ( t ) ⊆ U ( t ) s , with a length matching that of U ( t ) ȷ . This process is re- ferred to as wa veform-domain adapti ve matched filtering (WD-AMF), and its output z o ( t ) can be articulated as follo ws: z o ( t ) = R U ( t ) s v ( t ) ( µ )d µ + R Ψ ( t ) ˆ v ( t ) ( µ )d µ (6) In WD-AMF , the sets U ( t ) s and U ( t ) ȷ can be obtained through the following equation: U ( t ) ȷ = n µ    | ˆ v ( t ) ( µ ± γ d µ ) | > ˆ E ( t ) ( µ ) o (7a) U ( t ) s = C u U ( t ) j (7b) where γ d µ denotes the protectiv e interval. C . ISRJ suppression analysis for WD-AMF Eq. ( 6 ) illustrates that the two components of z o ( t ) are interrelated. When v ( t ) ( µ ) represents a non-periodic sig- nal, z o ( t ) experiences a significant energy accumulation, ex emplified by z o ( τ s ) . Con versely , when v ( t ) ( µ ) embodies a periodic signal, z o ( t ) does not attain substantial ampli- fication. Specifically , when | τ ȷ − τ s | < T , and if t  = τ s , U ( t ) s is no longer a continuous interval. As a result, the integration of v ( t ) ( µ ) over U ( t ) s and Ψ ( t ) may no longer yield a periodic function. This situation could lead to a certain degree of ener gy accumulation in z o ( t ) , potentially resulting in an elev ation of side-lobe le vels. Furthermore, o wing to the inclusion relation Ψ ⊆ U ( t ) s , when    U ( τ s ) s    < T 2 , where ∥·∥ represents the length of the set, z o ( τ s ) will encounter a loss of energy from the genuine echo signal. Assuming ISRJ to be a form of self-defensiv e repeater interference, wherein sampling and repeater are temporally asynchronous. The jamming slice width to the interrupted sampling period ratio, is expressed as ε = T ȷ T J ⩽ 1 2 . Under dif ferent interference scenarios, ISRJ displays a marked discrepancy in its duty cycle. For instance, in the conte xt of ISDRJ, the duty cycle is given by η = ε . Con versely , in the case of ISRRJ, the duty cycle is expressed as η = P · ε . In the scenario of ISCRJ, the duty c ycle tak es the form η = ( Q +1) 2 · ε . If τ s = τ ȷ , when η > 1 2 , it occasionally verifies that    U ( τ s ) s    < T 2 , and this circumstance is comparativ ely feasible in the context of ISRRJ. The majority of constant modulus w av eforms in WD- AMF processing encounter the aforementioned circum- stances, presenting an urgent challenge for WD-AMF’ s adaptability to div erse interference scenarios. From the analysis abov e, it becomes evident that the performance deterioration of WD-AMF results from the wa veform-domain overlapping when v ( t ) ( µ ) is mis- matched. Therefore, the ideal w av eform adaptation for WD-AMF should meet the following criteria: v ( t ) ( µ ) =    v ( t − τ s ) s ( µ ) , t = τ s v ( t − τ ȷ ) ȷ ( µ ) , t = τ ȷ 0 , else (8) A UTHOR ET AL.: SHOR T AR TICLE TITLE 3 Hence, this study , with an emphasis on the perspective of transmit wav eform design, does not endeavor to solely resolve the issue through a single pulse wa veform but seeks a combination of wav eforms that can effecti vely suppress interference signals in the w av eform domain. This enhancement is achie ved through inter-pulse pro- cessing to bolster the effecti veness of ISRJ interference mitigation across a broader range of scenarios. III. W A VEFORM DOMAIN COMPLEMENT AR Y SIGNAL SETS Since Eq. ( 8 ) can be seen as a constraint on the cross- correlation properties before signal inte gration, we can start by designing wa veforms that meet the constraint conditions based on phase-coded (PC) wav eforms with good cross-correlation properties. A. Definitions and characteristics of wa vef or m domain complementar y sequence sets Assuming the transmitted D -sets sequence is rep- resented as A = { a 0 , a 1 , . . . , a i , . . . , a D − 1 } T , where a i = { a i (0) , a i (1) , . . . , a i ( n ) , . . . , a i ( N − 1) } , the follo w- ing equation holds to satisfy Eq. ( 8 ): b ( m ) ( n ) = D − 1 X i =0 a i ( n + m ) a ∗ i ( n ) (9a) b ( m ) ( n ) =  D , m = 0 0 , m  = 0 (9b) where b ( m ) = { b ( m ) (0) , b ( m ) (1) , . . . , b ( m ) ( N − 1) } rep- resents the wav eform response sequence at time m . As per Eq. ( 9b ), it is evident that A is a column orthogonal matrix. Since for m  = 0 , all elements of b ( m ) are zeros, we refer to this characteristic as wa veform domain complementarity . A sequence set that satisfies wa veform domain complementarity is termed a wa veform domain complementary sequence set. Consequently , the following lemmas and theorems hold: Theor em 1 : The length N of sequences a i that satisfy Eq. ( 9 ) does not exceed the number of sequence sets, represented as D . Pr oof : As A constitutes a column orthogonal matrix, it ensues that the dimension of the column space of A coincides with its rank, designated as N . Since the dimension of the row space of a matrix is equi valent to the dimension of its column space, the ro w space dimension of matrix A is also N . The number of rows D in a matrix must be greater than or equal to the dimension of its row space. Therefore, we hav e the inequality: N ⩽ D (10) Lemma 1 : Let matrix A satisfies Eq. ( 9 ), then A constitutes a set of complementary sequence. Pr oof : Let ψ a i , a i denote the autocorrelation function of the sequence a i , and let ψ a i , a i ( m ) denote the m -th element in the sequence. D − 1 X i =0 ψ a i , a i ( m ) = N − 1 X n =0 b ( m ) ( n ) = 0 , m  = 0 (11) Since when A satisfies complementarity in the wa ve- form domain, it necessarily fulfills time domain comple- mentarity . Considering that in radar systems, transmitted wa veforms often adhere to constant modulus constraints, in our subsequent discussions, we will focus on the case where A is a symbolic matrix. Therefore, when A forms a w av eform domain com- plementary sequence set, it inevitably satisfies the corre- lation properties of complementary sequence sets [ 22 ]: Lemma 2 : A wav eform domain complementary se- quence set comprises an ev en number of sequences. Lemma 3 : If the length N of binary sequences that compose a wa veform domain complementary sequence set is odd, then the number D of sequences must be a multiple of 4 . B. Generation of wa vef orm domain complementar y sequence sets There are currently man y methods to generate binary column orthogonal matrices [ 23 ]. In this question, we introduce a method to generate matrices of any size 2 r +1 × N . Since N ⩽ D , we can first generate a D × D W alsh- Hadamard matrix [ 22 ], and then select any N columns to form a D × N wa veform domain complementary sequence set A . A D × D W alsh-Hadamard matrix can be extended to ha ve longer lengths through two typical iterati ve ex- pansion methods: interleaving and cascading. This paper briefly introduces the cascading method [ 22 ]. The follo w- ing operations are defined: SS represents the cascading operation on S , ˜ S represents the operation of reversing the ro w sequence of ˜ S , and − S denotes the ne gation operation on S . T o carry out the initial step of the recursi ve procedure, we construct first an orthogonal set ( s 0 , s 1 ) , D = 2 , (see Golay [ 24 ]). Next, the initial matrix ∆ is constructed: ∆ =  s 0 ˜ s 1 s 1 − ˜ s 0  (12) If we consider the matrix ∆ as a three-dimensional matrix, where each column contains various binary se- quences, it forms an orthogonal sequence set. Applying the cascading method to the third dimension of matrix ∆ as follows allows us to obtain a matrix ∆ ′ where each column’ s binary sequences also form an orthogonal sequence set. ∆ ′ =  ∆∆ ( − ∆)∆ ( − ∆)∆ ∆∆  (13) By iterating this method r times, the resulting matrix ∆ ′ contains a total of R = 2 r +1 W alsh-Hadamard matri- ces, each of size 2 r +1 × 2 r +1 . Therefore, we can choose 4 IEEE TRANSACTIONS ON AEROSP A CE AND ELECTR ONIC SYSTEMS V OL. XX, No. XX XXXXX 2022 any one of the W alsh-Hadamard matrices and select any N columns from that matrix to obtain the wa veform domain complementary sequence set A . C . W av ef or m domain complementarity of the WDCSS wa vef or m The wav eform domain complementary sequence set necessitates the baseband modulation of each individual bit within the binary sequence to acquire the baseband wa veform-domain complementary signal set, S ( t ) = { s i ( t ) } D − 1 i =0 , in the time domain: s i ( t ) = N − 1 X n =0 a i ( n )Ω ( t − nT c ) (14) where Ω( t ) denotes the baseband modulation signal, and T c represents the symbol duration. For the sake of clarity , in all subsequent sections, we shall refer to S ( t ) as the WDCSS wa veform. Therefore, the wav eform response function of the WDCSS wa veform S ( t ) can be expressed as: w ( t ) s ( µ ) = D − 1 X i =0 s i ( t + µ ) s ∗ i ( µ ) (15) and its numerical solution can be represented as: w ( t ) s ( µ ) =        N − 1 P n =0 D · Ω ( t − nT c + µ ) Ω ∗ ( µ − nT c ) , | t | ⩽ T c 0 , | t | > T c (16) If there e xists a Doppler component e j 2 π f d t , defin- ing the ( µ, f d ) plane at time t as ℧ ( t ) ( µ, f d ) = w ( t ) ( µ ) e j 2 π f d ( t + µ ) , substituting into Eq. ( 16 ) yields: ℧ ( t ) ( µ, f d ) =            N − 1 P n =0 D · Ω ( t − nT c + µ ) Ω ∗ ( µ − nT c ) e j 2 π f d ( t + µ ) , | t | ⩽ T c 0 , | t | > T c (17) Eq. ( 17 ) indicates that when | t | > T c , the Doppler shift cannot alter the complementary properties of the wa veform set S ( t ) in the wa veform domain. If ȷ ( t ) represents an example of JSDRJ, its w av eform response function can be represented as: w ȷ d ( t ) = g ( t + µ ) w ( t ) s ( µ ) (18) where g ( t ) denotes the intermittent sampling signal of the interference. And its numerical solution can be repre- sented as: w ( t ) ȷ d ( µ ) =    g ( t + µ ) w ( t ) s ( µ ) , | t | ⩽ T c 0 , | t | > T c (19) Eq. ( 19 ) indicates that when ȷ ( t ) is ISDRJ, w ( t ) ȷ d ( µ ) satisfies wa veform domain complementarity . If ȷ ( t ) represents an e xample of JSRRJ, its w av eform response function can be represented as: w ( t ) ȷ r ( µ ) = P − 1 X p =0 w ( t − pT ȷ ) ȷ d ( µ ) (20) if T ȷ ⩾ T c , its numerical solution can be represented as: w ( t ) ȷ r ( µ ) =    w ( t − pT ȷ ) ȷ d ( µ ) , | t − pT ȷ | ⩽ T c 0 , else (21) Eq. ( 21 ) indicates that when ȷ ( t ) is ISRRJ, and T ȷ ⩾ T c , w ( t ) ȷ r ( µ ) can be considered as a linear shift of w ( t ) ȷ d ( µ ) , and at each moment t = pT ȷ , the signal duty cycle in the wa veform domain is gi ven by η = ε . If ȷ ( t ) represents an e xample of JSCRJ, its w av eform response function can be represented as: w ( t ) ȷ c ( µ ) = Q − 1 X q =0 w ( t − q T ȷ − q T J ) ȷ d ( µ ) (22) if T ȷ + T J ⩾ T c , its numerical solution can be represented as: w ( t ) ȷ c ( µ ) =    w ( t − q T ȷ − q T J ) ȷ d ( µ ) , | t − q T ȷ − q T J | ⩽ T c 0 , else (23) Eq. ( 23 ) indicates that when ȷ ( t ) is ISCRJ, and T ȷ + T J ⩾ T c , w ( t ) ȷ r ( µ ) can be considered as a linear shift of w ( t ) ȷ d ( µ ) , and at each moment t = q T ȷ + qT J , the signal duty cycle in the wa veform domain is gi ven by η = ε . Therefore, when the time delay difference between the ISRJ and the true echo signal, | τ ȷ − τ s | > T ȷ ⩾ T c , the follo wing equation holds: w ( t ) ( µ ) = w ( t − τ s ) s ( µ ) + w ( t − τ ȷ ) ȷ ( µ ) (24a) w ( t ) ( µ ) =    w ( t − τ s ) s ( µ ) , | t − τ s | ⩽ T c w ( t − τ ȷ ) ȷ ( µ ) , | t − τ ȷ | ⩽ T c 0 , else. (24b) Eq. ( 24 ) demonstrates that when | τ ȷ − τ s | > T ȷ ⩾ T c , the non-zero elements of w ( t − τ s ) s ( µ ) and w ( t − τ ȷ ) ȷ ( µ ) do not ov erlap at any time t . This ensures that w ( t − τ s ) s ( µ ) and w ( t − τ ȷ ) ȷ ( µ ) do not cross interference. This characteristic of wa veform domain complementarity provides significant con venience for subsequent wav eform domain processing. IV . THE WD-AMF FOR THE WDCSS W A VEFORM As per the analysis from the previous section, when the condition | τ ȷ − τ s | > T ȷ ⩾ T c is satisfied, all inter - ference signals generated by different ISRJ modes can be considered as standard ISDRJ signals with different delays in the w av eform domain. Therefore, we only need to discuss ho w to suppress ISDRJ using the WDCSS A UTHOR ET AL.: SHOR T AR TICLE TITLE 5 wa veform and WD-AMF , which can handle all modes of ISRJ. Thus, this section will in vestigate the suppression of ISDRJ through WD-AMF and the WDCSS w av eform. A. W av eform domain adaptive threshold function T o describe the v ariation of w ( t ) ( µ ) during the wav e- form domain integration process, we define the cumula- ti ve wa veform coherence function as: y ( t ) ( ρ ) = Z ρ −∞ w ( t ) ( µ )d µ (25) It is evident that when ρ → ∞ , y ( t ) ( ρ ) corresponds to the output of the matched filter at time t . Combining Eq. ( 16 ), ( 19 ), ( 24 ) and ( 25 ), we obtain: y ( t ) ( ρ ) =    y ( t − τ s ) s ( ρ ) , | t − τ s | ⩽ T c y ( t − τ ȷ ) ȷ ( ρ ) , | t − τ ȷ | ⩽ T c 0 , else. (26) where y ( t ) s ( ρ ) = R ρ −∞ w ( t ) s ( µ )d µ , and y ( t ) ȷ ( ρ ) = R ρ −∞ w ( t ) ȷ ( µ )d µ . Especially , when t = τ s , or t = τ ȷ , we hav e: y ( τ s ) ( ρ ) = y (0) s ( ρ ) = A s D  ρ + T 2  (27a) y ( τ ȷ ) ( ρ ) = y (0) ȷ ( ρ ) =            A ȷ D  ρ + T 2 − q T J  + q D T ȷ , q T J ⩽ ρ − T 2 ⩽ q T J + T ȷ ( q + 1) A ȷ D T ȷ q T J + T ȷ < ρ − T 2 < ( q + 1) T J (27b) Eq. ( 27 ) indicates that y ( τ s ) ( ρ ) is a linear function, implying that the energy accumulation of w ( τ s ) ( µ ) in the wa veform domain grows linearly . On the other hand, y ( τ ȷ ) ( ρ ) is a piece wise linear function, indicating that the energy accumulation of w ( τ ȷ ) ( µ ) in the waveform domain gro ws in a piecewise linear manner . Hence, we can define a linear objecti ve function O ( t ) ( ρ ) to characterize the av erage linear growth process of y ( t ) ( ρ ) . O ( t ) ( ρ ) = o ( t ) ρ (28) where o ( t ) represents the av erage rate of energy growth for y ( t ) ( ρ ) . As the matched filter is designed to maximize the output signal-to-noise ratio(SNR), the follo wing equation must hold: o ( t ) = y ( t )  T 2  T (29a)    o ( t ) ⩽ A s D = | w ( t ) ( µ ) | , | t − τ s | ⩽ T c o ( t ) ⩽ εA ȷ D = ε | w ( t ) ( µ ) | , | t − τ ȷ | ⩽ T c 0 , else (29b) In Eq. ( 29b ), the equality in the inequality concerning o ( t ) holds if and only if the equality in the inequality concerning t is satisfied. Eq. ( 29 ) demonstrates a natural adapti ve constraint relationship between the variable o ( t ) and | w ( t ) ( µ ) | . Therefore, an adapti ve threshold function E ( t ) ( µ ) can be designed to serve as the condition for dis- tinguishing effecti ve integration elements in the wa veform domain U ( t ) s from ineffecti ve integration elements U ( t ) ȷ : U ( t ) s = n µ    | w ( t ) ( µ ) | ⩽ E ( t ) ( µ ) o (30a) U ( t ) ȷ = n µ    | w ( t ) ( µ ) | > E ( t ) ( µ ) o (30b) Next, let’ s analyze the boundary conditions of the adapti ve threshold function E ( t ) ( µ ) that satisfy the de- cision criteria. If we set E ( t ) ( µ ) = λ    o ( t )    (31) to ensure that w ( t ) ( µ ) satisfies Eq. ( 30a ) when | t − τ s | ⩽ T c , we must hav e λ ⩾ 1 . Similarly , to ensure that w ( t ) ( µ ) satisfies Eq. ( 30b ) when | t − τ ȷ | ⩽ T c , we must have λε ⩽ 1 . Therefore, the boundary conditions for λ should be 1 ⩽ λ ⩽ 1 ε . T ypically , ε is unknown, so we can narrow do wn the boundary conditions to 1 ⩽ λ = 1 ε 0 ⩽ 1 ε (32) where ε 0 ⩾ ε . It is important to emphasize that ε 0 is an upper bound, meaning that E ( t ) ( µ ) has adaptiv e decision capability for all interference elements with duty cycles less than ε 0 . For example, in the case of self-jamming interference, where ε ⩽ 1 2 , ε 0 = 1 2 , and λ = 2 . B. State estimation of wav ef or m domain signals Eq. ( 29 ), ( 30 ) and ( 31 ) indicate that in the presence of time domain noise, a necessary condition for ensuring algorithm robustness is to obtain unbiased estimates of w ( t ) ( µ ) and y ( t ) ( T 2 ) . Assuming that time domain noise has a mean of 0 and a variance of σ 2 , due to the additivity property of Gaussian distrib utions, it is known that the Gaussian white noise wg n ( t ) ( µ ) distrib uted on w ( t ) ( µ ) within one co- herently processed interv al (CPI) follo ws the distribution characteristics: w g n ( t ) ( µ ) ∼  0 , D σ 2  (33) As the indefinite upper limit integral of wg n ( t ) ( µ ) is a Marko v random process, the noise distributed on y ( t ) ( µ ) is a Brownian noise bn ( t ) ( µ ) that follows the distribution: bn ( t ) ( µ ) ∼  0 ,  µ + T 2  D σ 2  (34) Considering the linear and piecewise linear relation- ships in Eq. ( 27 ), we can obtain unbiased estimates, ˆ y ( t ) ( T 2 ) , and ˆ w ( t ) ( µ ) using the Interacti ve Multiple Model Kalman Filter (IMM-KF) algorithm [ 25 ]. This relation- ship can be described by a linear model, ˆ M ( t ) 1 , and two impulse models, ˆ M ( t ) 2 and ˆ M ( t ) 3 : ˆ M ( t ) ( µ | µ ) = u 1 ˆ M ( t ) 1 ( µ | µ ) + u 2 ˆ M ( t ) 2 ( µ | µ ) + u 3 ˆ M ( t ) 3 ( µ | µ ) (35) 6 IEEE TRANSACTIONS ON AEROSP A CE AND ELECTR ONIC SYSTEMS V OL. XX, No. XX XXXXX 2022 in which the weights u 1 , u 2 , and u 3 are ascertained for each model based on the residuals and residual cov ari- ance acquired through the implementation of the Kalman filter . ˆ M ( t ) 1 ( µ | µ ) , ˆ M ( t ) 2 ( µ | µ ) , and ˆ M ( t ) 3 ( µ | µ ) denote the estimated state matrices of their respecti ve models, and their one-step prediction state equations are delineated as follo ws: ˆ M 1 ( µ + dµ | µ ) = F 1 ˆ M ( µ | µ ) =      1 dµ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1            ˆ y ( t ) ( µ ) ˆ w ( t ) ( µ ) ˆ δ ( t ) − ( µ ) ˆ δ ( t ) + ( µ ) w g n ( t ) ( µ )       (36a) ˆ M 2 ( µ + dµ | µ ) = F 2 ˆ M ( µ | µ ) =      1 dµ dµ 0 0 0 1 dµ 0 0 0 − 1 0 0 0 0 0 0 1 0 0 0 0 0 1            ˆ y ( t ) ( µ ) ˆ w ( t ) ( µ ) ˆ δ ( t ) − ( µ ) ˆ δ ( t ) + ( µ ) w g n ( t ) ( µ )       (36b) ˆ M 3 ( µ + dµ | µ ) = F 3 ˆ M ( t ) ( µ | µ ) =      1 dµ 0 dµ 0 0 1 0 dµ 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1            ˆ y ( t ) ( µ ) ˆ w ( t ) ( µ ) ˆ δ ( t ) − ( µ ) ˆ δ ( t ) + ( µ ) w g n ( t ) ( µ )       (36c) where F i , with i = 1 , 2 , 3 , signifies the matrices gov- erning state transitions. The entities ˆ δ ( t ) − ( µ ) and ˆ δ ( t ) + ( µ ) pertain to distinct impulse functions, influencing both the direction and magnitude of ˆ w ( t ) ( µ ) . Specifically , ˆ δ ( t ) − ( µ + dµ | µ ) = − ˆ w ( t ) ( µ ) and ˆ δ ( t ) + ( µ + dµ | µ ) = ˆ δ ( t ) + ( µ ) = K o ( t ) , K ⩾ 1 ε . It is note worthy that the state matrix has been expanded in this context due to the measurement value y ( t ) ( µ ) conforming to the Bro wnian noise model. And the probability transition matrix can be represented as: P ( t ) =   1 − 2 p 0 p 0 p 0 1 0 0 1 0 0   (37) wherein, p 0 designates the likelihood of an abrupt alter- ation in the magnitude of | w ( t ) ( µ ) | , and the zero entries on the diagonal of the matrix are not rigorously zero but typically denote exceedingly minute values to guarantee the in vertibility of the matrix. Follo wing the IMM-KF procedure, we can obtain unbiased estimates ˆ w ( t ) ( µ ) and ˆ y ( t ) ( µ ) for each time instant t . C . The output of WD-AMF with the WDCSS wa vef or m Upon obtaining the unbiased estimates ˆ w ( t ) ( µ ) and ˆ y ( t ) ( µ ) , we can derive an unbiased estimate for the adapti ve threshold function: ˆ E ( t ) = λ ˆ o ( t ) = λ · ˆ y ( t )  T 2  T (38) Gi ven that w ( t ) ȷ ( µ ) is not a continuous function, we must e xtend U ( t ) ȷ . Subsequently , Eq. ( 30 ) can be updated as follows: U ( t ) s = C u U ( t ) ȷ (39a) U ( t ) ȷ = n µ    | ˆ w ( t ) ( µ ± γ d µ ) | > ˆ E ( t ) o (39b) where γ d µ denotes the protectiv e interval. The output of WD-AMF at each time instant t can then be e xpressed as the definite integral of w ( t ) ( µ ) o ver U ( t ) s : w o ( t ) = Z U ( t ) s w ( t ) ( µ ) dµ (40) Let’ s analyze the numerical solutions for w o ( t ) . Since ˆ E ( t ) ⩽ λA s D with λ ⩾ 1 , let t λ denote the moment when ˆ E ( t λ ) = A s D . When | t − τ s | ⩽ | t λ − τ s | , taking into account the waveform domain complementarity described in Eq. ( 24b ), it is certain that | w ( t ) ( µ ) | ⩽ ˆ E ( t λ ) , and in this case, U ( t ) s spans the entire wav eform domain  − T 2 , T 2  . According to Lemma 1, we know that S ( t ) is also a complementary signal set. Therefore, t λ is unique, and consequently , for all other time instants t , we ha ve U ( t ) s = ∅ . Hence, the numerical solutions for w o ( t ) can be regarded as the numerical solutions for the main lobe width of the matched filter output s o ( t ) under a level of − s o (0) λ belo w the main lobe peak, as follows: w o ( t ) =  s o ( t − τ s ) , | t − τ s | ⩽ | t λ − τ s | 0 , else. (41) T o maintain robust sparsity in the distance domain of w o ( t ) and to facilitate subsequent target detection algorithms on w o ( t ) , we typically apply compensation to the elements on U ( t ) ȷ equi valent to their original noise en vironment. Consequently , Eq. ( 40 ) and Eq. ( 41 ) can be further generalized as follows: w o ( t ) = Z U ( t ) s w ( t ) ( µ ) dµ + Z U ( t ) ȷ w g n ( t ) ( µ ) dµ (42a) w o ( t ) =  s o ( t − τ s ) , | t − τ s | ⩽ | t λ − τ s | bn ( t )  T 2  , else. (42b) Through the previously described WD-AMF on the WDCSS wav eform, complete suppression of ISRJ in any operational mode can be achiev ed. V . NUMERICAL EAMPLES In this section, multiple simulations are pro vided to assess the proposed methodology . Initially , a numerical simulation is presented to substantiate the wav eform do- main complementarity and matched filtering performance of the WDCSS wav eform. Subsequently , the proposed methods’ ef fecti veness ag ainst anti-ISRJ is assessed in the presence of ISRJ under different operational modes. Finally , an analysis of the parametric sensitivity of the proposed approach is conducted. A UTHOR ET AL.: SHOR T AR TICLE TITLE 7 (a) (b) (c) Fig. 1. The variation of κ with respect to time t for different waveforms and their ISRJ characteristics. (a) LFM wa veform; (b) Golay wav eform; (c) WDCSS waveform. A. P erformance simulation of the WDCSS wa vef orm For radar , assuming a transmitted signal with a sam- pling frequency of F s = 10 MHz, and code length and number of pulses, N = 160 , D = 256 , the baseband modulation signal Ω( t ) is a rectangular pulse, and the chip duration is T c = 1 µ s , hence the pulse width is T = N T c = 160 µ s . For the ISRJ jamming device, consid- ering an intermittent sampling period of T J = 32 µ s and a jamming slice width to the interrupted sampling period ratio of ε = 10% , the width of an indi vidual jamming slice is thus T ȷ = 3 . 2 µ s . T o delineate the complementarity of distinct signals in the w av eform domain, we introduce κ ( t ) as the ratio of the length of non-zero elements in | w ( t ) s ( µ ) | to the total duration T in the waveform domain at time t : κ ( t ) =    n µ    | w ( t ) s ( µ ) | > 0 o    T (43) Fig. 1 illustrates the temporal evolution of κ ( t ) con- cerning various w av eforms and their corresponding op- erational modes under ISRJ conditions, where P = 9 and Q = 5 . The comparativ e e xperiments in v olve three distinct signals: an LFM wav eform featuring a 2 MHz instantaneous bandwidth, a complementary Golay wav e- form with parameters D = 2 and N = 160 , and the aforementioned WDCSS wa veform. All three signals possess an identical pulse width. Fig. 2. The ambiguity function of the WDCSS wa veform. Simulation results reveal that for the LFM wav eform and the Golay wa veform, κ ( t ) exhibits non-sparsity , lead- ing to unav oidable o verlapping between the tar get echo signal and the jamming signal in the wav eform domain. It is evident that for these two wav eforms, the jamming signal can occupy a duty c ycle of up to η in the wav eform domain of the target echo signal. When η > 1 2 , the output energy of the WD-AMF signal for the target echo signal will be compromised. Ho wev er , for the WDCSS wa veform, κ ( t ) is sparse. The interference signal will occupy at most an ε duty cycle of the waveform domain of the target echo signal. Specifically , as long as it satisfies τ ȷ − τ s > T c , the target echo signal and the jamming signal in the w av eform domain are completely orthogonal, and there will be no ov erlapping, regardless of the ISRJ mode and any η . In this case, the output energy of the tar get echo signal WD- AMF signal is always equal to its matched filter output, with no loss. The ambiguity function of the WDCSS wav eform, as sho wn in Fig. 2 , reveals that the matched filter output of the WDCSS wav eform is quite sensitiv e to Doppler frequency shifts. Howe ver , its complementary properties in the range profile are not sensitiv e to Doppler shifts. B. P erf or mance ev aluation f or ISRJ resistance In this section, we will conduct simulations to validate the ef fecti veness of the WD-AMF method for the WDCSS wa veform we ha ve designed. Considering an L-band radar system and an interference system, the simulation param- eters are as depicted in T ab . I . The wav eforms designed in Section V - A are employed herein, as their pulsewidth and bandwidth align with those discussed in Section V - A . The simulation scenario features a point tar get and an ISRJ jammer , and the positions of the targets and jammer are detailed in T ab . I . The subsection simulates the ISRJ without modulation, and the input jamming-to-noise ratio (JNR) is set at 20 dB for this simulation. ISRJ will be simulated in three different modes: ISDRJ, ISRRJ, and ISCRJ. All of them ha ve the same ε and T J , and their parameters are also displayed in T ab . I . It’ s important to 8 IEEE TRANSACTIONS ON AEROSP A CE AND ELECTR ONIC SYSTEMS V OL. XX, No. XX XXXXX 2022 note that in the simulation scenario as depicted in T ab . I , it holds that τ ȷ − τ s > T ȷ > T c . Consequently , the WDCSS wa veform exhibits w av eform-domain complementarity . For comparati ve analysis, we present the WD-AMF [ 21 ] outcomes, denoted as z o ( t ) (as referenced in Eq. ( 6 )), when the radar employs LFM or Golay wav eform sets. All wa veforms possess an identical coherently processing interv al, denoted as CPI = 256 . T ABLE I Simulation parameters Parameters V alue Radar carrier frequency f 0 = 2 GHz Pulse repetition interval PRI = 480 µ s T arget delay τ s = 0 µ s Jamming delay τ ȷ = 20 µ s Repetitiv e repeater number P = 9 Cyclic repeater number Q = 5 Input signal to noise ratio SNR = 0 dB Input jamming to noise ratio JNR = 20 dB Coherently processing interval CPI = 256 In order to better observe the wav eform processing in the waveform domain and obtain ef fective and reliable results, it may be advantageous to examine the labeled outcomes of U ( t ) s and U ( t ) ȷ at se veral time instances. Let’ s define moments as t 0 = 0 µ s , t 1 = 10 µ s , and t 2 = 20 µ s . At these three time instances, the labeled results for U ( t ) s and U ( t ) ȷ for the three wav eform types, accompanied by an ISRRJ with P = 9 , are presented in Fig. 3 . Fig. 3 (a)-Fig. 3 (c) depict the labeled outcomes of U ( t ) s and U ( t ) ȷ for w ( t ) ( µ ) at times t = t 0 , t 1 , t 2 when the radar transmits LFM wav eforms. It is evident that w ( t ) s and w ( t ) ȷ manifest discernible overlapping, with the length of interfering elements κ ( τ ȷ − t ) T in ISRRJ being non-zero, as depicted in the orange curve in Fig. 1 (a). At these time instances, the elements of U ( t ) s are partial wa veform domain elements after the remov al of interference signals, and these elements are discontinuous in the waveform domain. Fig. 3 (d)-Fig. 3 (f) illustrate the labeled outcomes of U ( t ) s and U ( t ) ȷ for w ( t ) ( µ ) at times t = t 0 , t 1 , t 2 when the radar transmits Golay wa veforms. Similarly , w ( t ) s and w ( t ) ȷ exhibit noticeable ov erlapping, with an interference interv al length of approximately κ ( τ ȷ − t ) T , in ISRRJ being non-zero, as depicted in the orange curv e in Fig. 1 (b). At these time instances, the elements of U ( t ) s are partial wa veform domain elements after the remo val of interference signals, and these elements are discontinuous in the wav eform domain. It is worth noting that the two sub-wa veforms of the Golay complementary wa veforms are orthogonal, but simulation results indicate that w ( t ) s in the waveform domain does not possess wav eform-domain complementarity because it does not meet the constraints of Theor em 1 . Fig. 3 (g)-Fig. 3 (i) represent the annotated results of U ( t ) s and U ( t ) ȷ for w ( t ) ( µ ) at times t = t 0 , t 1 , t 2 during WDCSS wa veform transmission by the radar . It is ob- served that w ( t ) s and w ( t ) ȷ do not demonstrate o verlapping, signifying their complementarity within the wa veform domain, as depicted in Fig. 1 (c). At t 0 , U ( t ) s spans the entire wav eform domain interv al, characterized by contin- uous elements in the wav eform domain, while U ( t ) ȷ is the null set. At t 1 , U ( t ) ȷ cov ers the entire wav eform domain interv al, whereas U ( t ) s remains unpopulated. At t 2 , U ( t ) ȷ encompasses all the slice interference signal elements, while U ( t ) s solely comprises the noise elements. Furthermore, the WD-AMF outputs for the three wa veform types, accompanied by different interference modes of ISRJ, are depicted in Fig. 4 . The gray curve represents the matched filter output x o ( t ) , while the black curve represents the WD-AMF output z o ( t ) and w o ( t ) . T o facilitate the description of the anti-jamming performance of different wav eforms under WD-AMF , T ab . II -T ab . IV provides the main lobe level (MLL), sidelobe lev el (SLL), and peak side lobe ratio (PSLR) for v arious wav eforms acquired using WD-AMF . Fig. 4 (a)-Fig. 4 (c) illustrate the range profiles of LFM wa veforms under ISDRJ, ISRRJ, and ISCRJ, respecti vely . The results indicate that when the interference is ISDRJ or ISCRJ, the MLL of z o ( t ) is equal to that of x o ( t ) , suggest- ing that WD-AMF can ensure the preserv ation of ener gy in the actual target echo signal through compensation when η < 50% . Howe ver , when the interference mode is ISRRJ, the MLL of z o ( t ) is smaller than that of x o ( t ) by 10 . 51 dB, indicating that ev en with compensation, WD- AMF cannot guarantee the preservation of energy in the actual tar get echo signal when η > 50% , resulting in a loss of approximately κ ( τ ȷ − t ) D T . Furthermore, upon careful observ ation of Fig. 4 (a)-Fig. 4 (c), it is noticeable that the SLL is higher than the noise floor . This is due to the discontinuity of elements in U ( t ) s , as sho wn in Fig. 3 (a)-Fig. 3 (c), leading to an increase in the SLL during signal mismatch. Fig. 4 (d)-Fig. 4 (f) depict the range profiles of Golay wa veforms under ISDRJ, ISRRJ, and ISCRJ, respecti vely . The WD-AMF results exhibit a similar performance to LFM wav eforms, with the distinction being that the side lobe le vel of WD-AMF is higher than that of LFM wa ve- forms. This is due to the time-domain complementary properties of Golay being sensiti ve to Doppler effects, and the discontinuous integration elements of w ( t ) ( µ ) , µ ∈ U ( t ) s disrupt this complementary nature, resulting in an increase in SLL. Fig. 4 (g)-Fig. 4 (i) illustrate the range profiles of WDCSS wa veforms under ISDRJ, ISRRJ, and ISCRJ, respecti vely . The results show that, regardless of the interference mode, the MLL of w o ( t ) is consistently equal to that of x o ( t ) , and the SLL of w o ( t ) is on par with the noise le vel. This is because WDCSS wav eforms exhibit waveform-domain complementarity , enabling w ( t ) s and w ( t ) ȷ to be treated as two single-component signals at dif ferent time instances, as shown in Fig. 3 (g)-Fig. 3 (i). Therefore, WDCSS wa veforms are the most suitable anti- jamming wa veforms for WD-AMF . A UTHOR ET AL.: SHOR T AR TICLE TITLE 9 (a) (b) (c) (d) (e) (f) (g) (h) (i) Fig. 3. Coupled Results of U ( t ) s and U ( t ) ȷ with ISRRJ ( P = 9 ) for different wav eforms. (a) LFM waveform at t 0 ; (b) LFM waveform at t 1 ; (c) LFM wa veform at t 2 . (d) Golay waveform at t 0 ; (e) Golay waveform at t 1 ; (f) Golay waveform at t 2 . (g) WDCSS waveform at t 0 ; (h) WDCSS waveform at t 1 ; (i) WDCSS waveform at t 2 . From the above analysis, it can be inferred that differ - ent waveforms exhibit varying wa veform domain comple- mentarity , leading to distinct PSLR values in their WD- AMF performance, particularly under ISRRJ conditions. Therefore, it is imperati v e to further validate the WD- AMF performance of different wa veforms in the context of T ab . I scenarios under various input SNR and input JNR conditions, particularly in the presence of ISRRJ. T o mitigate the stochastic effects introduced by noise, 500 Monte Carlo simulations were conducted for each input SNR and input JNR value. Fig. 5 represents the average PSLR results obtained from multiple simulations. It is note worthy that for Fig. 5 (a), the input JNR is fixed at JNR = 20 dB, while for Fig. 5 (b), the input SNR is fix ed at SNR = 0 dB. T ABLE II WD-AMF outputs of the LFM wav eform ISRJ mode MLL (dB) ( t = τ s ) SLL (dB) ( t = τ ȷ ) PSLR (dB) ISDRJ 0 − 35 . 34 35 . 34 ISRRJ − 10 . 51 − 30 . 40 19 . 89 ISCRJ 0 − 29 . 09 29 . 09 T ABLE III WD-AMF outputs of the Golay wav eform ISRJ mode MLL (dB) ( t = τ s ) SLL (dB) ( t = τ ȷ ) PSLR (dB) ISDRJ 0 − 26 . 01 26 . 01 ISRRJ − 57 . 00 − 33 . 25 − 23 . 75 ISCRJ 0 − 26 . 42 26 . 42 10 IEEE TRANSACTIONS ON AEROSP ACE AND ELECTRONIC SYSTEMS V OL. XX, No. XX XXXXX 2022 (a) (b) (c) (d) (e) (f) (g) (h) (i) Fig. 4. WD-AMF outputs for three wa veform types under various ISRJ interference modes. (a) LFM waveform with ISDRJ; (b) LFM wav eform with ISRRJ; (c) LFM wav eform with ISCRJ. (d) Golay waveform with ISDRJ; (e) Golay waveform with ISRRJ; (f) Golay waveform with ISCRJ. (g) WDCSS wa veform with ISDRJ; (h) WDCSS wav eform with ISRRJ; (i) WDCSS wav eform with ISCRJ. T ABLE IV WD-AMF outputs of the WDCSS wav eform ISRJ mode MLL (dB) ( t = τ s ) SLL (dB) ( t = τ ȷ ) PSLR (dB) ISDRJ 0 − 55 . 62 55 . 62 ISRRJ 0 − 52 . 46 52 . 46 ISCRJ 0 − 53 . 58 53 . 58 From the simulation results in Fig. 5 (a), it can be observed that when SNR > − 20 dB, the PSLR of LFM and Golay wav eforms remains relatively constant as SNR increases. This is because their WD-AMF output’ s SLL is greater than the noise output level. Ho we ver , when SNR < − 20 dB, the SLL becomes smaller than the noise output level, resulting in a nearly linear increase in PSLR with rising SNR. Similarly , it is noted that when SNR > − 35 dB, the PSLR of the WDCSS wa veform linearly increases with SNR. This is attributed to the complementary nature of the WDCSS wav eform in the wav eform domain, causing its WD-AMF output’ s SLL to match the noise output lev el. Con v ersely , when SNR < − 35 dB, the PSLR experiences a steep decline as SNR decreases. This is because the WDCSS wav eform also exhibits time-domain complementarity , leading to the ˆ E ( τ s ) of the real target echo not benefiting from the sidelobe gain of ISRJ, resulting in a smaller ˆ E ( τ s ) at lo w SNR. Consequently , U ( τ s ) s loses some elements in the waveform domain, leading to a decrease in MLL and PSLR. As anticipated, the WDCSS wav eform exhibits higher PSLR, indicating superior interference resistance performance. A UTHOR ET AL.: SHOR T AR TICLE TITLE 11 (a) (b) Fig. 5. V ariations in PSLR of WD-AMF outputs under dif ferent SNR and JNR conditions, with a specific focus on ISRRJ conditions. (a) Curves depicting PSLR variation with SNR; (b) Curves illustrating PSLR variation with JNR. From Fig. 5 (b), it can be observed that when JNR < 0 dB, the PSLR of LFM and Golay wa veforms remains relati vely constant as SNR increases. This is because with a small JNR, A ȷ + A s < ˆ E ( τ s ) , and in such cases, U s ( τ s ) cov ers the entire wa veform domain, thus incurring no MLL loss. Conv ersely , when JNR > 0 dB, the PSLR of LFM and Golay wav eforms gradually decreases as SNR increases, and then remains constant. This is because with A ȷ + A s > ˆ E ( τ s ) , MLL and PSLR suf fer losses. For the WDCSS wa veform, due to its complementary nature in the waveform domain, the real target echo signal and ISRJ do not exhibit ov erlapping in the wav eform domain. Therefore, its PSLR is always equi valent to MLL. Consequently , the performance of the WDCSS wav eform in WD-AMF is unaffected by JNR. In conclusion, based on the abov e analysis, in the simulation scenarios as presented in T ab . I , for the WD- CSS wav eform, under all parameter configurations, PSLR exceeds 18 dB when SNR > − 35 dB, thus meeting the detection requirements of this scenario. C . P arameter sensitivity analysis In the preceding section, we observed that the PSLR of WD-AMF output for WDCSS wav eforms remains un- af fected by the ISRJ parameter η due to their wa veform- domain complementarity . Howe ver , for LFM and Golay wa veforms, which lack such complementarity , the PSLR of their WD-AMF outputs is significantly influenced by η . T o in vestigate the impact of this critical ISRJ parameter η on PSLR for different wa veforms, this section emplo ys ISRRJ as an example and conducts simulation and com- parati ve experiments by adjusting η . W e will maintain the simulation parameters as out- lined in T ab . I , while v arying the range of repetitions, denoted as P ∈ [1 , 9] , and the corresponding duty cycle η ∈ [0 . 1 , 0 . 9] . Fig. 6 illustrates the variation in the PSLR of WD-AMF output for the three wav eforms with respect to the changes in η . Fig. 6. V ariations in PSLR of WD-AMF outputs under dif ferent η conditions, with a specific focus on ISRRJ conditions. Simulation results indicate that the PSLR of the WDCSS wav eform remains consistently at its maximum output SNR across the entire range of parameter variation for η . This implies that the WDCSS wav eform is insensi- ti ve to changes in η . Con versely , the PSLR of LFM and Golay wa veforms exhibits significant fluctuations with v ariations in η due to changes in MLL and the fluctuation of SLL. As anticipated, the WDCSS waveform, pos- sessing waveform-domain complementarity , demonstrates superior and robust interference resistance performance. Furthermore, all PSLR v alues for the WDCSS wa veform are above 55 dB, further validating the practicality of the proposed method in engineering applications. VI. CONCLUSION This paper introduces a wav eform-domain comple- mentary signal set (the WDCSS wav eform), to address the issue of ISRJ resistance. Through an analysis of ISRJ and the w av eform response functions of the transmitted wa veforms, it is rev ealed that wa veform-domain non- sparsity is the primary factor causing the degradation of interference resistance performance in wav eform-domain adapti ve matched filtering (WD-AMF) when there is signal mismatch. Improved WD-AMF , adapted to the WDCSS wa veform, is also considered because the WD- CSS wa veform does not necessitate additional wav eform- domain signal compensation. The anti-ISRJ problem is formulated, and the WDCSS waveform is designed by introducing the W alsh-Hadamard matrix. 12 IEEE TRANSACTIONS ON AEROSP ACE AND ELECTRONIC SYSTEMS V OL. XX, No. XX XXXXX 2022 Se veral simulations are conducted to demonstrate the ef fectiv eness of the proposed method. Simulation results indicate that WD-AMF using the WDCSS wav eform achie ves superior anti-ISRJ performance compared to WD-AMF using separately designed wav eforms. Para- metric sensiti vity analysis shows that the WDCSS wav e- form is insensiti ve to the signal duty cycle, regardless of the interference mode. Since both the design and processing of WD-AMF with the WDCSS wa veform do not require prior informa- tion about ISRJ-related parameters, the transmitter only needs a single operating mode to achiev e ISRJ suppres- sion for dif ferent scenarios. F or future w ork, attention may be directed tow ards the rapid implementation of WD-AMF and the design of WDCSS wa veforms with a lar ger Doppler tolerance, enabling the proposed method to be applied to real-time tar get tracking for ISRJ suppres- sion and improving anti-ISRJ performance for high-speed moving tar gets. REFERENCES [1] Dejun Feng, Letao Xu, Xiaoyi Pan, and Xuesong W ang. 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Hanning Su received the B.Sc degree in electronic engineering from Xidian Univ ersity in 2018. He is currently working tow ards the Ph.D. degree in signal and information processing with the National Key Lab of Science and T echnology on A TR, National Univ ersity of Defense T echnology . His current research interests include radar signal process- ing, target tracking, and radar anti-jamming. Qinglong Bao received his B.Sc and Ph.D degrees from the National Univ ersity of Defense T echnology , Changsha, China, in 2003 and 2010, respectiv ely . Currently , he is an Associate Professor with the School of Electronic Science, National University of Defense T echnology . His current research interests include radar data acquisition and signal processing. A UTHOR ET AL.: SHOR T AR TICLE TITLE 13 Jiameng Pan received the B.E. degree in Zhejiang University in 2013, and the Ph.D. degree in National University of Defense T echnology in 2020. He is currently a lecturer with the College of Electronic Science and T echnology , National Uni versity of Defense T echnology . His main research interests include radar signal processing, tar get tracking, and radar anti-jamming. Fucheng Guo receiv ed the Ph.D. degree in information and communi- cation engineering from the National University of Defense T echnology (NUDT), Changsha, Hunan, China, in 2002.,He is no w a Professor in the School of Electronic Science, NUDT . His research interests include source localization, target tracking, and radar/communication signal processing. W eidong Hu was born in September 1967. He received the B.S. degree in microwa ve technology and the M.S. and Ph.D. degrees in communication and electronic system from the National University of Defense T echnology , Changsha, China, in 1990, 1994, and 1997, respectiv ely . He is currently a Full Professor in the A TR Laboratory , National Univ ersity of Defense T echnology , Changsha. His research interests include radar signal and data processing. 14 IEEE TRANSACTIONS ON AEROSP ACE AND ELECTRONIC SYSTEMS V OL. XX, No. XX XXXXX 2022