Radar distance-Doppler joint sparse super-resolution method for inter-pulse non-repetitive waveforms

Abstract

The invention discloses an inter-pulse non-repetitive waveform radar distance-Doppler joint sparse super-resolution method. The method comprises the steps of 1, initializing system parameters; step 2, reading measurement data from a radar receiver; step 3, constructing a fast-slow time data matrix through a time domain translation pulse string dynamic alignment mechanism; 5, constructing a two-dimensional joint sparse signal model; and step 6, sparse reconstruction: solving the measurement vector by using an orthogonal matching pursuit algorithm, and completing radar distance-Doppler joint sparse super-resolution estimation based on the inter-pulse non-repetitive waveform. According to the method, the problems that target energy diffuses to a distance-Doppler plane, the adjacent target resolution capability is limited, measurement and sparse representation dictionary matrixes are mismatched and the like under a radar system adopting inter-pulse non-repetitive waveforms in a traditional method are solved; the super-resolution estimation of the target distance and Doppler under the radar system adopting the inter-pulse non-repetitive waveform is realized, and the resolution performance and the detection precision of the radar system are remarkably improved.

Description

Technical Field

[0001] The invention belongs to the technical field of radar target parameter estimation, and in particular relates to a radar range-Doppler joint sparse super-resolution method of an inter-pulse non-repetitive waveform. Background Art

[0002] The inter-pulse non-repetitive waveform has excellent anti-clutter and anti-interference capabilities because its parameters such as repetition rate, initial phase, frequency and amplitude show pseudo-random jumps between pulses. At the same time, it can achieve unambiguous ranging and speed measurement. It has become an important development direction of radar technology and has been widely used in fields such as airborne radar and stealth target detection. It has broad development space in both civilian and military fields.

[0003] Traditional methods for processing such waveforms typically rely on operations such as matched filtering, range-gate alignment, phase compensation, and non-uniform discrete Fourier transform (NDFT). Specifically, matched filtering is first used to achieve intra-pulse coherence accumulation. Range-gate alignment is then used to ensure synchronization of the pulse echoes in the slow-time dimension. Phase compensation is then used to eliminate inter-pulse phase mismatch. Finally, NDFT is used to complete inter-pulse coherence accumulation and obtain the range-Doppler power spectrum. However, these methods have significant limitations. First, range-gate alignment and NDFT cause target energy to diffuse in the range and Doppler dimensions, respectively. The combined effect of these two factors significantly weakens the target energy accumulation and degrades parameter estimation performance. Second, as a linear processing framework, the method's resolution depends on the signal bandwidth and the number of pulses. When the signal bandwidth is limited or the number of pulses is insufficient, it is difficult to effectively distinguish adjacent targets in the two-dimensional range-Doppler space. Moreover, the energy diffusion caused by NDFT further limits the resolution performance in the Doppler dimension. These problems highlight the current need to optimize inter-pulse non-repetitive waveform processing technology. Achieving super-resolution estimation of target range-Doppler is still a key difficulty that needs to be overcome in this type of radar system.

[0004] In recent years, compressed sensing theory, leveraging the sparse distribution of targets in observation scenes, has become an important method for radar super-resolution parameter estimation and has been widely studied in the field of range-Doppler estimation. The paper "Joint Sparse Super-Resolution Method for Azimuth-Range-Velocity Estimation on Array Radar, in Proceedings of 2024 13th International Conference on Control, Automation and Information Sciences (ICCAIS), 2024, pp. 1-6" proposes a three-dimensional joint sparse reconstruction method for azimuth, range, and Doppler based on compressed sensing for traditional uniform waveforms, significantly improving the accuracy and resolution of target parameter estimation. However, this method only addresses sparse super-resolution estimation of targets in uniform waveforms and does not consider non-repeating waveform signals with pulse-to-pulse parameter agility modulation characteristics. The document "High-precision parameter estimation of frequency-agile radar targets based on sparsity adaptation and iterative weighting, Signal Processing, 2025, 41(3): 437-447" establishes a range-Doppler sparse signal processing framework for inter-pulse frequency agile waveforms, and proposes a sparse reconstruction algorithm combining sparsity adaptation with iterative weighting, which improves the accuracy and resolution performance of target parameter estimation under frequency agile modulation. However, this method is limited to the frequency single parameter agile system and does not consider the multi-dimensional modulation characteristics caused by the joint jump of parameters such as repetition frequency and initial phase. It is difficult to deal with the problem of complex changes in signal structure under the situation of joint agility of multiple parameters between pulses. Patent application CN118393449A discloses a frequency agile radar target tracking and detection method based on an adaptive sparsity matching pursuit algorithm. By combining a dimensionality reduction dictionary with adaptive sparse reconstruction, the efficiency and accuracy of target detection and tracking in the frequency agile radar system are improved. However, this method is also only for inter-pulse frequency agile modulation and cannot meet the demand for high-precision parameter estimation under complex modulation of non-repeating waveforms between pulses. In summary, although existing research has made positive progress in sparse super-resolution estimation under the inter-pulse frequency agile regime, how to construct a robust, flexible and generalizable sparse super-resolution estimation method under the more widespread inter-pulse non-repetitive waveform regime remains a key issue that needs to be solved urgently. Summary of the Invention

[0005] The purpose of the present invention is to overcome the shortcomings of the existing technology and provide a radar range-Doppler joint sparse super-resolution method using an inter-pulse non-repetitive waveform. The present invention effectively solves the problems of echo energy diffusion, insufficient parameter estimation resolution, and poor matching of sparse model construction in radar systems using inter-pulse non-repetitive waveforms, thereby achieving high-precision range-Doppler joint sparse super-resolution estimation and significantly improving the resolution performance and detection accuracy of the radar system.

[0006] The object of the present invention is achieved through the following technical solution: a radar range-Doppler joint sparse super-resolution method for pulse-interval non-repetitive waveform, which specifically comprises the following steps:

[0007] Step 1: Initialize system parameters: initialize pulse width T, number of pulses N; pulse emission time t m , m=0,1,2,…,N-1; pulse initial phase φ m , the repetition rate T of the uniform pulse train p , the repetition rate T of the non-uniform pulse train pm , the number of super-resolution distance units N f , the number of super-resolution Doppler units N s , sampling interval T s ;

[0008] Step 2: Read measurement data from the radar receiver: Assume that there are Q targets within the radar detection range, and consider the Doppler effect of the targets. The echo received by the radar is expressed as:

[0009]

[0010] Where, σ k , τ dk 、f dk are the scattering coefficient, time delay, and Doppler frequency of the kth target respectively; rect(·) is the rectangular function, T is the time width of the intra-pulse signal; x(t) is the baseband signal of each pulse; t m and φ m and are the emission time and initial phase of the m-th pulse respectively;

[0011] Step 3: Construct a fast-slow time data matrix through a pulse train dynamic alignment mechanism with time domain translation; left-shift the radar receiver measurements, retain the data to the left of the time 0, and then fill the right side with zeros;

[0012] The fast-slow time data matrix after dynamic alignment of the pulse train is expressed as follows:

[0013]

[0014] Where v = 0, 1, ..., N-1 is the slow time dimension unit sequence; is the two-dimensional target scattering coefficient matrix, T p(v-1) is the repetition frequency corresponding to the v-1th pulse. When v=0, assuming T p(-1) =0;

[0015] For the lth fast time sampling sequence after discretization of equation (2), the measurement at the vth slow time sequence is expressed as:

[0016]

[0017] Where l = l left , l left +1,…,l right -1, l right , is a fast time sampling sequence; l left and l right Represent the minimum sampling sequence number and the maximum sampling sequence number respectively; z(l, v) constitutes a two-dimensional fast-slow time data matrix Z N×M ;

[0018] Step 4: Construct a time-varying dictionary matrix;

[0019] make:

[0020]

[0021] but:

[0022]

[0023] Construct a time-varying range-Doppler joint steering vector:

[0024] α(τ di , f dj )=[y(0,0;τ di , f dj ), y(0, 1; τ di , f dj ),…,y(M-1,N-1;τ di , f dj )] (7)

[0025] Where, β(τ di , f dj ) represents the time delay τ di , Doppler is f dj The steering vector corresponding to the unit;

[0026] The time-varying two-dimensional joint dictionary matrix is obtained as:

[0027]

[0028] Step 5: Construct a two-dimensional joint sparse signal model; first traverse the rows and then traverse the columns to transform the fast-slow time data matrix Z N×M Extract into a column vector;

[0029] The measurement vector considering the noise term is expressed as:

[0030] Z=VEc[Z N×M ]+δ (9)

[0031] Where, δ∈C MN×1 is the noise column vector; Vec[Z M×N ]∈C MN×1 is the signal column vector, expressed as:

[0032]

[0033] make:

[0034]

[0035] In the formula is the two-dimensional target scattering coefficient matrix σ 2D The target scattering coefficient vector obtained by column expansion represents the sparse signal to be recovered; the signal column vector is further expressed as:

[0036] Vec[Z M×N ]=Aσ (12)

[0037] Then the measurement vector Z is expressed as:

[0038] Z=Aσ+δ (13);

[0039] Step 6: Sparse reconstruction: Use the orthogonal matching pursuit algorithm to solve the measurement vector (13) and complete the radar range-Doppler joint sparse super-resolution estimation based on the inter-pulse non-repetitive waveform.

[0040] The beneficial effects of the present invention are as follows: the method of the present invention utilizes the idea of compressed sensing theory. First, under the premise of ensuring the integrity of the pulse sequence, the pulse train dynamic alignment mechanism uses the transmission time of each pulse to accurately synchronize the echo, construct a fast-slow time matrix, and provide a basis for subsequent dictionary design. Secondly, for the non-repetitive waveform between pulses, a time-varying range-Doppler dictionary matrix is constructed to accurately characterize its dynamic characteristics. Subsequently, a two-dimensional joint sparse signal model is constructed based on the dictionary to achieve sparse representation of the target scattering coefficient. Finally, the range-Doppler scattering coefficient matrix of the target is restored by a sparse reconstruction method, thereby realizing range-Doppler joint sparse super-resolution estimation. The advantage of the present invention is that the constructed time-varying dictionary matrix and the two-dimensional joint sparse signal model can adapt to non-repetitive waveform signals with arbitrary parameter agility characteristics between pulses, realize joint sparse reconstruction of range-Doppler, thereby significantly improving the accuracy and resolution performance of target parameter estimation. This method effectively addresses issues such as echo energy diffusion, insufficient parameter estimation resolution, and poor matching of sparse models in radar systems using inter-pulse non-repetitive waveforms. It achieves high-precision range-Doppler joint sparse super-resolution estimation, significantly improving the radar system's resolution and detection accuracy. It has significant engineering application value and can be widely applied to radar systems using inter-pulse non-repetitive waveforms and related fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 Flowchart of the radar range-Doppler joint sparse super-resolution method for pulse-to-pulse non-repetitive waveforms.

[0042] Figure 2 Schematic diagram of dynamic alignment of pulse trains.

[0043] Figure 3 The distance-speed results of the proposed method and the comparison method in the global scenario, where (a) is the result of the matched filter-discrete Fourier transform method processing uniform waveforms; (b) is the result of the matched filter-non-uniform discrete Fourier transform method processing non-repetitive waveforms; (c) is the result of the two-dimensional sparse super-resolution method processing uniform waveforms; (d) is the result of the method proposed in this invention processing non-repetitive waveforms.

[0044] Figure 4 The distance-speed results of the proposed method and the comparison method in a local scene, where (a) is the result of the matched filter-discrete Fourier transform method processing uniform waveforms; (b) is the result of the matched filter-non-uniform discrete Fourier transform method processing non-repetitive waveforms; (c) is the result of the two-dimensional sparse super-resolution method processing uniform waveforms; (d) is the result of the method proposed in this invention processing non-repetitive waveforms.

[0045] Figure 5The root mean square error changes with the signal-to-noise ratio, where (a) is the root mean square error result in the global scenario; (b) is the root mean square error result in the local scenario. DETAILED DESCRIPTION

[0046] In order to facilitate the description of the present invention, the following terms are first explained:

[0047] Term 1: Uniform Waveform

[0048] A coherent pulse train consisting of multiple pulses uses conventional modulation schemes, such as linear frequency modulation (LFM) and phase encoding, and maintains a consistent signal structure across pulses. This waveform exhibits no random variation in inter-pulse parameters (such as repetition rate, initial phase, frequency, and amplitude), exhibiting high stability and predictability, making it suitable for traditional coherent integration processing frameworks.

[0049] Term 2: Pulse-to-pulse non-repetitive waveform

[0050] A coherent pulse train consisting of multiple pulses, whose intra-pulse parameters (such as duration, bandwidth, and frequency) or inter-pulse parameters (such as repetition rate, initial phase, frequency, and amplitude) exhibit pseudo-random jumps between pulses. This type of waveform is highly random and complex, significantly improving the system's anti-interference and anti-interception capabilities.

[0051] Term 3: Sparse Super-Resolution

[0052] It refers to the process of achieving super-resolution estimation of target parameters based on compressed sensing theory and utilizing the sparse characteristics of the target in the scene, thereby breaking through the resolution limitations of traditional methods.

[0053] Term 4: Pulse train dynamic alignment mechanism

[0054] It means synchronizing the target echo according to the emission time of each pulse while ensuring the integrity of the pulse train structure to ensure the effectiveness of inter-pulse coherence accumulation.

[0055] Term 5: Time-varying 2D joint dictionary matrix

[0056] It means that due to the agility of repetition frequency, the range phase term and the Doppler phase term are coupled with each other, making the dictionary matrix present a time-varying characteristic.

[0057] The technical solution of the present invention is further described below with reference to the accompanying drawings.

[0058] like Figure 1 As shown, the radar range-Doppler joint sparse super-resolution method of the present invention for pulse-to-pulse non-repetitive waveform comprises the following steps:

[0059] Step 1: Initialize system parameters: initialize pulse width T = 10μs, number of pulses N = 16; pulse emission time t0=1×10 -8 s, m=0,1,…,N-1; pulse initial phase φ m =zeros(N,1), uniform pulse train repetition rate T p =5×10 -5 s, repetition rate T of the non-uniform pulse train pm =T p +δ m ,m=0,1,2,…,N-1,δ m is a uniform distribution on (-μ, μ), that is, δ m ~U(-μ,μ),μ∈[0,0.5-T / T p ]; Number of super-resolution distance units N f = [56, 15], number of super-resolution Doppler units N s =[18, 19], sampling interval T s =1×10 -7 s;

[0060] Step 2: Read the measurement data from the radar receiver:

[0061] Assume that there are Q targets within the radar detection range, and consider the Doppler effect of the target, then the echo received by the radar is expressed as:

[0062]

[0063] Where, σ k , τ dk 、f dk are the scattering coefficient, time delay, and Doppler frequency of the kth target respectively; rect(·) is a rectangular function, T is the time width of the intra-pulse signal; x(t) is the baseband signal of each pulse, which can be any modulation waveform such as linear frequency modulation, frequency coding, or phase coding; t m and φ m and are the emission time and initial phase of the mth pulse respectively; the time delay τ dk Determine the target distance. To simplify the expression, all the time delays τ dk The term is considered as a term related to distance, then the distance term of the mth pulse is

[0064] Step 3: construct a fast-slow time data matrix through a pulse train dynamic alignment mechanism of time domain translation; perform pulse train dynamic alignment to construct the fast-slow time data matrix.

[0065] In the process of dynamic pulse train alignment, the radar receiver obtains the measurement and shifts it to the left, and retains the data on the left side of the time 0, and then fills it with zero on the right side; Figure 2 As shown; the fast-slow time data matrix after dynamic alignment of the pulse train is expressed as follows:

[0066]

[0067] Where v = 0, 1, ..., N-1 is the slow time dimension unit sequence; is the two-dimensional target scattering coefficient matrix, T p(v-1) is the repetition frequency corresponding to the v-1th pulse. When v=0, assuming T p(-1) =0;

[0068] For the lth fast time sampling sequence after discretization of equation (2), the measurement at the vth slow time sequence is expressed as:

[0069]

[0070] Where l = l left , l left +1,…,l right -1, l right , is a fast time sampling sequence; let M = l right -l left +1 is the number of sampling points in the fast time dimension. The sampling sequences on the leftmost and rightmost sides of the observation interval are:

[0071]

[0072] Where, τ min and τ max They represent the time delay corresponding to the minimum observation distance and the time delay corresponding to the maximum observation distance respectively; l left and l right Represent the minimum sampling sequence number and the maximum sampling sequence number respectively; z(l, v) constitutes a two-dimensional fast-slow time data matrix Z N×M ;

[0073] Step 4: Construct a time-varying dictionary matrix;

[0074] make:

[0075]

[0076] but:

[0077]

[0078] Construct a time-varying range-Doppler joint steering vector:

[0079] α(τ di , f dj )=[y(0,0;τ di , f dj ), y(0, 1; τ di , f dj ),…,y(M-1,N-1;τ di , f dj )] (7)

[0080] In the formula, α(τ di , f dj ) represents the time delay τ di , Doppler is f dj The steering vector corresponding to the unit.

[0081] The time-varying two-dimensional joint dictionary matrix is obtained as:

[0082]

[0083] Where,

[0084] Step 5: construct a two-dimensional joint sparse signal model;

[0085] By traversing the rows first and then the columns, the fast-slow time data matrix Z N×M Extract into a column vector; in practice, the noise term should also be considered, assuming that the noise δ = a + bj is zero-mean complex Gaussian white noise, where Assuming that the real part a and the imaginary part b are uncorrelated, the complex number The measurement vector considering the noise term is expressed as:

[0086] Z=Vec[Z N×M ]+δ (9)

[0087] Where Z∈C MN×1 is the measurement vector, δ∈C MN×1 is the noise column vector; Vec[Z M×N ]∈C MN×1 is the signal column vector, expressed as:

[0088]

[0089] make:

[0090]

[0091] In the formula is the two-dimensional target scattering coefficient matrix σ 2D The target scattering coefficient vector obtained by column expansion represents the sparse signal to be recovered. The signal column vector can be further expressed as:

[0092] Vec[Z M×N ]=Aσ (12)

[0093] Then the measurement vector Z can be expressed as:

[0094] Z=Aσ+δ (13)

[0095] The above formula is the constructed two-dimensional joint sparse signal model.

[0096] Step 6, sparse reconstruction: The measurement vector (13) is solved using the orthogonal matching pursuit algorithm to complete the radar range-Doppler joint sparse super-resolution estimation based on the inter-pulse non-repetitive waveform. For solving equation (13), traditional methods (such as the least squares method) may amplify noise and are therefore undesirable. Many algorithms have been proposed to solve this problem. Considering the high dimensionality of σ, sparse reconstruction requires a large amount of calculation. In order to improve the reconstruction speed in practical applications, the present invention adopts the orthogonal matching pursuit algorithm, which is a representative greedy algorithm.

[0097] Figure 3 (a)-(d) compare the performance of the proposed method with other methods in the entire range-velocity domain. Figure 3 (a) and (c) correspond to uniform waveforms, Figure 3 (b) and (d) correspond to inter-pulse non-repeating waveforms. It is clear that the matched filter-non-uniform discrete Fourier transform method based on inter-pulse non-repeating waveforms suffers from significant energy diffusion in the range-velocity domain, with diffusion along the velocity direction being particularly severe, significantly impacting the accuracy of target parameter estimation. In contrast, the proposed inter-pulse non-repeating waveform combined with sparse super-resolution method effectively suppresses the energy diffusion problem and achieves accurate reconstruction of target range and velocity. Its performance is comparable to that of the two-dimensional sparse super-resolution method using uniform waveforms, demonstrating superior accuracy and resolution.

[0098] Figure 4 (a)-(d) compare the performance of the proposed method with other methods in the local range-velocity domain. Figure 4 (a) and (c) correspond to uniform waveforms, Figure 4 (b) and (d) correspond to inter-pulse non-repeating waveforms. The results show that the proposed inter-pulse non-repeating waveform combined with sparse super-resolution method can effectively resolve adjacent targets in the range-velocity domain, significantly outperforming the matched filter-non-uniform discrete Fourier transform method. Its resolution performance is comparable to that of the two-dimensional sparse super-resolution method using uniform waveforms, demonstrating excellent target resolution capabilities.

[0099] In order to evaluate the root mean square error performance of the proposed method compared with other methods under different signal-to-noise ratio conditions, simulation experiments were designed, and the results are shown in Figure 2. Figure 5 shown. Figure 5 (a) and (b) show the trends of the root mean square error (RMS) of target range-velocity estimation as a function of signal-to-noise ratio (SNR) for global and local scenarios, respectively. It can be seen that the RMS error of the matched filter-discrete Fourier transform (MFT) method for uniform waveforms is consistently lower than that of the matched filter-non-uniform discrete Fourier transform (MFD) method for inter-pulse non-repeating waveforms, indicating that the latter has limitations when processing inter-pulse non-repeating waveforms, particularly regarding the diffusion of target energy in the range-velocity domain. In contrast, the proposed inter-pulse non-repeating waveform combined with sparse super-resolution can effectively suppress energy diffusion and exhibit lower RMS error under various SNR conditions. This significantly outperforms the matched filter-non-uniform discrete Fourier transform (MFD) method, and its performance is comparable to that of the two-dimensional sparse super-resolution method for uniform waveforms, demonstrating superior accuracy and resolution.

[0100] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and it should be understood that the scope of protection of the present invention is not limited to such specific descriptions and embodiments. Those skilled in the art can make various other specific variations and combinations based on the technical teachings disclosed in the present invention without departing from the essence of the present invention, and such variations and combinations are still within the scope of protection of the present invention.

Claims

1. A radar range-Doppler joint sparse super-resolution method for pulse-to-pulse non-repetitive waveforms, characterized in that: The specific steps include: Step 1: Initialize system parameters: initialize pulse width T, number of pulses N; pulse emission time t m ,m=0,1,2,…,N-1;Pulse initial phase φ m , the repetition rate T of the uniform pulse train p , the repetition rate T of the non-uniform pulse train pm , the number of super-resolution distance units N f , the number of super-resolution Doppler units N s , sampling interval T s ; Step 2: Read measurement data from the radar receiver: Assume that there are Q targets within the radar detection range, and consider the Doppler effect of the targets. The echo received by the radar is expressed as: Where, σ k , τ dk 、f dk are the scattering coefficient, time delay, and Doppler frequency of the kth target respectively; rect(·) is the rectangular function, T is the time width of the intra-pulse signal; x(t) is the baseband signal of each pulse; t m and φ m and are the emission time and initial phase of the m-th pulse respectively; Step 3: Construct a fast-slow time data matrix through a pulse train dynamic alignment mechanism with time domain translation; left-shift the radar receiver measurements, retain the data to the left of the time 0, and then fill the right side with zeros; The fast-slow time data matrix after dynamic alignment of the pulse train is expressed as follows: Where v = 0, 1, ..., N-1 is the slow time dimension unit sequence; is the two-dimensional target scattering coefficient matrix, T p(v-1) is the repetition frequency corresponding to the v-1th pulse. When v=0, assuming T p(-1) =0; For the lth fast time sampling sequence after discretization of equation (2), the measurement at the vth slow time sequence is expressed as: Where l = l left ,l left +1,…,l right -1,l right , is a fast time sampling sequence; l left and l right Represent the minimum sampling sequence number and the maximum sampling sequence number respectively; z(l, v) constitutes a two-dimensional fast-slow time data matrix Z N×M ; Step 4: Construct a time-varying dictionary matrix; make: but: Construct a time-varying range-Doppler joint steering vector: a(t di ,f dj )=[y(0,0;τ di ,f dj ),y(0,1;τ di ,f dj ),…,y(M-1,N-1;T di ,f dj )] (7) In the formula, α(τ di , f dj ) represents the time delay τ di , Doppler is f dj The steering vector corresponding to the unit; The time-varying two-dimensional joint dictionary matrix is obtained as: Step 5: Construct a two-dimensional joint sparse signal model. First traverse the rows and then traverse the columns to extract the fast-slow time data matrix ZN×M into a column vector. The measurement vector considering the noise term is expressed as: Z=Thing[Z N×M ]+δ (9) Where, δ∈C MN×1 is the noise column vector; Vec[Z M×N ]∈C MN×1 is the signal column vector, expressed as: make: In the formula is the two-dimensional target scattering coefficient matrix σ 2D The target scattering coefficient vector obtained by column expansion represents the sparse signal to be recovered; the signal column vector is further expressed as: Thing[Z M×N ]=Aσ (12) Then the measurement vector Z is expressed as: Z=Aσ+δ (13); Step 6: Sparse reconstruction: Use the orthogonal matching pursuit algorithm to solve the measurement vector (13) and complete the radar range-Doppler joint sparse super-resolution estimation based on the inter-pulse non-repetitive waveform.

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