Enhancing Weather Target Detection with Non-Uniform Pulse Repetition Time (NPRT) Waveforms

Abstract

The velocity/distance trade-off poses a fundamental challenge in pulsed Doppler weather radar systems and is known as the velocity/distance dilemma. Techniques such as multiple-pulse repetition frequency, staggered pulse repetition time (PRT), and pulse phase coding are commonly used to mitigate this issue. The current study evaluates the adaptability/capability of a specific type of low-capture signal called the non-uniform PRT (NPRT) through analyzing the weather target characteristics of typical velocity distributions. The spectral moments estimation (SME) signal-processing algorithm of the NPRT weather echo is designed to calculate the average power, velocity, and spectrum width of the target. A comprehensive error analysis is conducted to ascertain the efficacy of the NPRT processing algorithm under influencing factors. The results demonstrate that the spectral parameters of weather target echo with a velocity of $[-50,50]$ m/s through random-jitter NPRT signals align with radar functionality requirements (RFRs). Notably, the NPRT waveform resolves the inherent conflicts between the maximum unambiguous distance and velocity and elevates the upper limit of the maximal observation velocity. The evaluation results confirm that nonlinear radar signal processing technology can improve a radar’s detection performance and provide a new method for realizing the multifunctional observation of radar in different applications.

1. Introduction

Weather radar has been proven to have a considerable number of applications in various sectors, including meteorology, aviation, the military, forest fire prevention, and environmental protection, since the inception of Doppler frequency weather radar networks, such as the WSR-88D radar in the United States and the CINRAD radar in China [1,2]. In particular, the primary outputs of weather radars are spectral moment data, which encompass average power, average velocity, the velocity’s spectral width, and polarimetric data including differential reflectivity, correlation coefficients, and specific differential phases. These data contain valuable insights into weather dynamics and thermodynamics, forming the foundation for quantitative rainfall estimations, hail observations, and severe convective weather/flood forecasting and early warning [3,4]. Thus, the advancements in and extensive use of weather radar technology have significantly improved our capacity to safeguard lives and property from weather and hydrological disasters [5].

To enhance the capabilities of weather forecasting and warnings, the spectral data and polarimetric variables from weather radar must be of an adequate quality. Currently, countries like the United States and China have established functional requirements for operational radars, specifying clear regulations regarding radar coverage, angular resolution, the minimum detectable signal, the estimation accuracy of polarimetric variables, and maximum unambiguous velocity. For instance, the WSR-88D must support a maximum unambiguous velocity of ±32 m/s, as indicated in the literature [6]. However, radial velocities often exceed 32 m/s in the presence of tornadic supercells and strong mid-latitude cyclones, and even in certain moderate-intensity rainfall situations [4,5]. Additionally, new waveform designs and signal-processing techniques are necessary to sustain the current high level of forecasting and warning, extend the warning time for tornadoes and other high-impact events, lower the false alarm rate for tornadoes, and improve our overall understanding of weather conditions. Thus, to optimize the operation of radars, they must adapt to the use of complex radar waveform transmission and signal processing in order to minimize range–velocity ambiguities. The future (∼2030) needs to cope with a proposed maximum unambiguous velocity of 50 m/s, according to the literature [5].

The maximum unambiguous velocity is an intrinsic characteristic of a radar waveform, linked to the radar’s wavelength and pulse repetition frequency (PRF). On the one hand, for a constant-PRF radar waveform, the maximum unambiguous velocity is given by $v_a=\lambda ·\mathrm{PRF}/4$, where $\lambda$ is the radar wavelength [3,7,8,9,10]. On the other hand, the design of the PRF parameter for radar waveforms is constrained by the range/multi-frequency dilemma, $v_a{r}_a=c\lambda /8$ [11]. While increasing the PRF can effectively improve the maximum unambiguous velocity within certain ranges, it may also result in significant range ambiguity problems. In addition, when multiple storms are present along the same radial or when the area of rainfall is extensive, as is often the case during typhoons, plum rains in East Asia, and hurricanes in the Americas, increasing the PRF to improve the maximum unambiguous velocity usually leads to greater range ambiguity, resulting in a notable decline in the overall quality of the data [12,13].

In order to ensure a sufficient maximum range for the ambiguity distance, dual/multiple PRF waveforms [4,9,14] and staggered PRT waveforms [15,16,17] are frequently employed, which are detailed by authors Cho and Torres, respectively [17,18]. This method is commonly utilized in C-band radars that use a TDWR, as well as in certain short-range X-band radars. The dual PRF and staggered PRT techniques, when combined with low PRF waveforms, generally resolve both range and velocity ambiguity issues effectively [1,19,20,21,22]. Multiple PRFs, segmented PRT, and phase coding technologies can be utilized to partially address, and in specific scenarios even provide better solutions to, the speed/distance dilemma. However, regardless of whether staggered PRT or dual PRF techniques are used, an issue of radio-frequency blockage, commonly referred to as the dead zone problem, exists. In response to this issue, academic groups such as the NOAA National Severe Storms Laboratory (NSSL) have proposed technologies like Auto-PRF, which effectively mitigate the problems of radio-frequency blockages and dead zones. Nevertheless, in complex weather situations, it is still necessary to re-scan specific radials [5,7,8].

In principle, the speed ambiguity of a uniform PRF signal stems from the constraints of conventional (Shannon–Nyquist) sampling and reconstruction under uniform sampling conditions [13,23,24,25]. The primary reason that dual PRF and staggered PRT can improve the maximum unambiguous range lies in their equivalent use of non-uniform sampling techniques with lower degrees of freedom. This indicates that when utilizing radar waveforms with higher PRF degrees of freedom, the speed ambiguity issue is anticipated to be effectively resolved. In fact, from a theoretical standpoint, uniform PRF radar waveforms are neither essential nor optimal in radar engineering. The significant enhancement in their processing capability enables a fresh look into radar signal processing and, in particular, into the use of nonlinear and iterative radar signal processing techniques. These methods hold the promise of greatly enhancing radar performance such that smaller radar apertures and thus smaller platforms can provide equivalent capabilities [26,27,28,29,30,31]. Currently, the design and signal processing techniques used for non-uniform PRF (NPRT) waveforms are receiving increased attention [32]. While their role in weather observation has not yet been definitively assessed, they have various and valuable applications in weather-aiding channels in radar systems. Moreover, the combination of non-repetitive waveforms and nonlinear processing is expected to be used to overcome losses due to blind ranges or speeds and to reduce the need for fill pulses [26].

The characteristics of random non-uniform PRF pulse sequences have been analyzed in terms of the method used for extended unambiguous distance and velocity measurements in the presence of dead zones [33]. Non-uniform PRF solves the problem of traditional azimuth resolution reduction to obtain a wide swath in conventional Synthetic aperture radar (SAR) systems [34,35]. One paper shows that the blind zone is minimized by changing the PRF, randomly selecting the number of pulses to improve the noise floor, and obtaining more accurate data products [36]. Based on the theory of compressed sensing, a spectrum analysis and estimation algorithm for multiple PRF radar targets is proposed in [37]. Non-uniform PRF has the characteristic of a low probability of detection (LPD) [38]. Combined with compressed sensing, it can achieve high-resolution Doppler range reconstruction and achieve the aim of a low sidelobe and anti-interference [32,39,40,41].

The purpose of these works is to solve the dilemma that exists within a single PRF. To improve the speed ambiguity or solve the speed ambiguity, Cho of MIT’s Lincoln Laboratory used the 7/9 PRF waveform of the ASR-9 air traffic control radar to carry out related work and successfully applied it to weather observations of an airport terminal area [42]. Similar to the starting point of this paper, weather observation is not the main function of the radar used in the above applications, but an auxiliary function. Compared with the above NPRT signal, the PRT randomness of the detection waveform in this paper is stronger, its potential to deal with sparse echoes is greater, and the design of its corresponding processing algorithm is more flexible.

In fields such as multifunctional radars, air defense radars, and air traffic control radars, the NPRT waveform represents a competitive new technological direction. It has advantages in resisting main lobe interference and multitasking, and it is expected to address common radio-frequency blockage issues in weather radars, such as multiple frequencies and segmented PRT. In areas where meteorological observation is not the primary task, the optimization criteria for waveform design are based on other primary tasks. At this point, the task of processing weather radar signals is to define the waveform and design signal processing algorithms to obtain high-quality base data.

In the current study, we demonstrated the characteristics of weather echo and NPRT weather echo modeling. Based on this, we proposed an average power, velocity, and spectral width estimation algorithm based on spectral moment estimation (SME). The results indicate that the NPRT waveform can achieve speed measurements of over ±50 m/s when the average PRF is around 1 kHz using an S-band radar. We also analyzed the time–frequency-domain signal characteristics of weather echoes in Section 2 and introduced the Sliding Window Adaptive (SWA) and SME algorithms for NPRT weather signal processing in Section 3. We further presented our numerical simulation results in Section 4, including a performance comparison of the algorithms under various influencing variables.

2. Signal Model

2.1. Weather Model

A radar system receives a complex random signal when the frequency domain characteristics of deterministic signals typically determined by their spectrum are matched. Similarly, the frequency domain properties of random signals are described by their power spectral density (PSD). The PSD of a random signal is directly linked to its autocorrelation function through a Fourier transformer [4].

In the single-phase coherent pulse interval (CPI) observation of a weather radar, a specific implementation of the random process of the weather signal is captured. The spectral correlated power obtained by the Fourier transform is known as the PSD of the weather signal [43]. By placing the radial velocity on the transverse axis of the PSD, the result is the velocity spectral density. It is assumed that the normalized velocity spectrum of the weather echo is expressed as

$$S\left(\nu \right)={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\nu }}}exp\left(-{\displaystyle \frac{{\left(\nu -\overline{\nu }\right)}^2}{2{\sigma }_{\nu }^2}}\right)$$

(1)

where $\overline{\nu }$ denotes the mean radial Doppler velocity and ${\sigma }_{\nu }$ is the width of the velocity spectrum of the weather echo. The corresponding normalized PSD, also known as the true PS, can be expressed as

$$S\left(f\right)={\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_f}}exp\left(-{\displaystyle \frac{{\left(f-\overline{f}\right)}^2}{2{\sigma }_f^2}}\right)$$

(2)

where $\overline{f}$ is the mean Doppler frequency, ${\sigma }_f$ is the spectrum width of the weather echo, and it holds that $f_d=-2\nu /\lambda$ and ${\sigma }_f=2{\sigma }_v/\lambda$, where $\lambda$ denotes the radar’s operating wavelength. Equation (2) introduces concentrated power around the average Doppler frequency to improve the representation of weather signals.

The radar transmission signal can be given by

$$s\left(t\right)=\sum _{n=0}^{N-1}Arect\left[t-T\left(n\right)\right]exp\left(j2\pi f_0\right).$$

(3)

Here, $\mathit{A}$ represents the signal’s amplitude, $f_0$ is the carrier frequency, and $T\left(n\right)$ denotes the NPRT time interval, which is generated randomly within the range of $[{\mathrm{PRT}}_{\min },{\mathrm{PRT}}_{\max }]$ (based on a certain uniform PRT). Then, $T(n+1)$ is the time interval from the nth pulse to the $(n+1)$th pulse, where $T\left(n\right)=0$, and it holds that

$$\mathrm{rect}\left(t\right)=\left\{\begin{array}{c}\hfill 1,0<t<\tau \hfill \\ \hfill 0,\mathrm{otherwise}\hfill \end{array}\right.$$

(4)

where $\tau$ is the transmitted pulse’s width.

When the radar signal in (3) is used to detect the weather, the radar echo seen after conventional processing generates N pulse sampling points and the time series (random process) of the weather signal is obtained, which is expressed as

$$V\left(t_n\right)\equiv V\left(n\right),n=0,\cdots ,N-1.$$

(5)

Here, $V\left(n\right)$ is the sample value of the nth pulse-matching filtered output signal. Assuming that the transmitted signal is a uniform PRT, $t_n=nT-t_d$ is the sampling time of each transmitted pulse and $t_d$ is the weather echo delay. However, in the case of sending the NPRT signal, the $t_n$ should be given by

$$t_n=\sum _{n=0}^{N-1}T\left(n\right)-t_d.$$

(6)

We perform a Fourier transform on $V\left(n\right)$ and compute the PSD to obtain the instantiation of Equation (2). Due to the coherent synthesis of precipitation echoes from a large number of raindrops, their radar echoes exhibit a complex Gaussian distribution. The components of different frequencies present a Gaussian distribution after the Fourier transform, and their power density obeys the ${\kappa }^2$ distribution of two degrees of freedom, i.e., has an exponential distribution. In addition, the phase follows a uniform distribution. Based on the above analysis, the time series of weather signals that can be simulated by PSD are as follows:

$$V\left(n\right)={\int }_{-\infty }^{\infty }\sqrt{S\left(f\right)y\left(f\right)}exp\left[j2\pi \phi \left(f\right)\right]exp\left(j2\pi f{t}_n\right)df$$

(7)

where $y\left(f\right)$ represents a unit of exponential distribution and $\phi \left(f\right)$ is the phase after a random distribution. When digitally simulating weather echoes, Equation (7) can be discretized as

$$V\left(n\right)={\displaystyle \sum _{l=-L}^L}[\sqrt{S(l\Delta f)y(l\Delta f)}exp\left[j2\pi \phi (l\Delta f)\right]\times exp\left[j2\pi (l\Delta f)t_n\right]\Delta f]$$

(8)

where $\Delta f$ is the quantification interval for the PSD of the weather echo and $2L+1$ is the number of frequency-domain sampling points used for simulating the echo sequence (set to 1024). Through the analysis of these equations, it can be seen that in the process of echo simulation, the PSD of the weather echo is truncated, resulting in the discarding of frequency components below a certain threshold $(-\infty ,-L\Delta f)\cup (L\Delta f,\infty )$. Given the properties of the Gaussian function, when $L\Delta f>\overline{f}+3{\sigma }_f$ is chosen appropriately, the discarded signal components become irrelevant.

For the weather target of interest in this study, the maximum average velocity is $\pm 50$ m/s. In the case of a spectrum width of 4 m/s, the maximum and minimum frequencies in Equation (8), which exceed the corresponding frequency range of $\pm 62$ m/s, can meet the $\pm 2800$ Hz ($\pm 70$ m/s) standard determined in this paper.

2.2. Simulation of NPRT Echo

In this study, the operation of an S-band radar with a pulse width $\tau$ of 1 microsecond was studied. It produces 64 pulse data points, and the PRT randomly jitters in the range of 0.7 to 1.2 ms, causing the interval between each pulse to change. The actual power of the meteorological echo target is 0 dB, its spectral width is 4 m/s, and its average Doppler radial velocity range is $v\in [-50,50]$ m/s. According to the established signal model, the simulation of an NPRT weather echo signal can be realized in the frequency domain and time domain. Figure 1 shows the velocity spectrum and the corresponding NPRT signal amplitude spectrum of the time domain of a weather echo with average radial velocities of 50 m/s and 15 m/s. In addition, the time interval value of the 64-point NPRT waveform generated under random jitter PRT at one point in time and the statistical average of the frequency spectrum of the NPRT waveform, simulated 5000 times, are also contained in the model.

The results highlight the characteristics and differences in the weather echo’s frequency domain at different velocities. The frequency spectrum simulation outcomes under the NPRT conditions provide new insights for radar signal processing under random pulse intervals.

2.3. Spectral Analysis of NPRT Echo

The frequency spectrum of an NPRT weather echo can be acquired through a Fourier transform [44], as follows:

$$V\left(f\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }V\left(t\right)exp\left(-j2\pi ft\right)dt.$$

(9)

When $t=t_n$, the discretized frequency spectrum of NPRT is represented as

$$V\left(f_k\right)={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}[V\left(t_n\right)exp(-j2\pi f_k{t}_n)]$$

(10)

where $f_k={\displaystyle \sum _{k=-K}^K}k\Delta f$ denotes different frequency points, $2K+1$ represents the number of NPRT sampling points in the simulation’s frequency domain (here it is 180), and the variables K and L can assume varying values. It is important to note that in this context, the spectral resolution $\Delta f$ is constant. If randomly jittered NPRT time sequences N are generated based on a specific uniform PRF, the resultant spectral resolution should be $\mathrm{PRF}/N$, and then a Doppler velocity spectrum can be produced that covers $[-70,70]$ m/s.

By employing the concept of periodogram estimation, the power spectral density of the NPRT echo sequence obtained from Equation (10) can be denoted as

$$S\left(f_k\right)={\left|V\left(f_k\right)\right|}^2.$$

(11)

Here, $S\left(f_k\right)$ signifies the simulated PS under the NPRT weather signal for the estimation of subsequent spectral moments. Figure 2a,b depict the power spectrum of NPRT weather echo samples under typical characteristics.

Comparing the power spectrum of the typical weather echoes in Figure 1a,c and Figure 2a,b, it can clearly be seen that the simulated PS of the NPRT weather echo displays stochastic fluctuation in each frequency component in contrast to the true PS. Figure 1f illustrates that these fluctuations primarily stem from the spectral leakage induced by randomly jittered NPRT sequences.

The spectral leakage resulting from random PRTs gives rise to the two distinct spectra aliasing in the NPRT PS. One spectrum closely resembles the true PS (referred to as the target PS), originating from the primary frequency spectral components of the non-uniform sampling signal (component 1). The other spectrum, termed the non-target PS, significantly diverges from the true PS due to spectral leakage from a random PRT (components 2 and 3). It can be seen from the theory that the frequency spectrum of the weather echo of a uniform PRT has a Gaussian distribution. Additionally, the frequency spectrum of a non-uniform PRT will produce several replicas, which is essentially a spectrum leakage problem. However, the NPRT frequency spectrum will not produce a replica of the same amplitude. Therefore, as long as the weather echo satisfies the spectral representation of the signal model under specific P, V, and W attributes, there will be a distinction between the main component and the secondary component, and there will be no result for which the main component is equivalent to other secondary components. Therefore, the NPRT signal processing algorithm we designed performs spectral moment estimation by capturing the main component as the target PS, as shown in Figure 2.

Further, we simulate the power spectrum seen under an NPRT weather echo with an actual power of 0 dB and an average velocity of 50 m/s. A statistical analysis of the power spectrum of weather echo samples over 5000 Monte Carlo simulations reveals that the target PS surpasses the non-target PS in strength with a probability of 1. On average, the most dominant power (component 1) is $0.1595$ W stronger than the second most potent power (component 2) and $0.4116$ W more robust than the weakest power (component 3). Directly estimating the moments of this aliasing spectrum may yield inaccurate results, necessitating the preprocessing of the aliasing spectrum before an effective SME method can be carried out.

3. Signal Processing of NPRTs

3.1. The Sliding Window Adaptive Algorithm

The SWA algorithm has been developed to address the aliasing challenge in the power spectrum of NPRT weather echo samples. It defines a window width’s velocity range (WIN) in meters per second. After a certain window width is selected, the aliasing PS is scanned in sequence with a step size of 1 m/s, calculating the cumulative power within the fixed window range. Subsequently, the spectral interval with the highest power is identified as the target PS, while the power at other positions in the aliased PS is set to zero. In this way, the algorithm aims to find the target PS to facilitate an accurate spectral parameter estimation, and this is formalized as

$$S_{\mathrm{target}}\left(f_k\right)\stackrel{max}{=}\sum _{\mathrm{WIN}}S\left(f_k\right).$$

(12)

As discussed in Section 2.1, the power spectrum of weather echoes predominantly falls within the $\pm 3\sigma$ interval centered around the mean frequency. Therefore, a carefully chosen window can effectively capture the characteristics of the true PS. For typical weather echoes, a window selection within the range of $[9,25]$ m/s is considered appropriate, assuming that the signal model parameters are set as above, the average power of the weather echo is 0 dB, and the average speeds are 50 m/s and 15 m/s. Figure 2a,b show the schematic process of the SWA algorithm searching for the target PS under a window width of 22 m/s. Different color boxes represent different power spectra. Figure 2c,d are the target PS obtained by the SWA algorithm for typical weather echo samples.

In terms of the radar’s functional requirements, the WSR-88D requires a PRF combination between 318 and 1310 Hz to achieve a maximum speed of 32 m/s and observe targets at a distance of 460 km. The uniform PRT of a radar system determines the unambiguous velocity, $v_a$, which is given by

$$v_a=\lambda /4T.$$

(13)

If transmitted pluses are within the uniform pulse repetition interval, T can be replaced by a uniform PRT. When the $\lambda$ is $0.1$ m and the uniform PRT is selected to be 1 ms, the $v_a$ is only 25 m/s.

In this study, we refer to a single PRT at 0.7 ms as the lower limit corresponding to a maximum unambiguous speed of 35 m/s. We assume the reference PRT is centered at 1 ms, and thus choose an upper limit of 1.2 ms. The asymmetric gap further enhances the randomness. Therefore, the selected random-jitter PRT range of the NPRT designed in this paper is constrained between 0.7 and 1.2 milliseconds when the reference PRT is 1 ms. It must be admitted that the choice of the NPRT’s random jitter range contains a certain amount of randomness and uncertainty. Further, following this idea, we have similarly chosen and analyzed cases where the reference PRT is 2 ms and 3 ms. This choice ensures that the measurement of the maximum unambiguous distance can be continuously enhanced under the basic requirement of achieving a maximum unambiguous speed of 35 m/s.

In this study, in order to illustrate the prominent benefit of the NPRT sequence in improving the limit of velocity estimations, a baseline reference PRT (RPRT) of 1 ms is utilized, and the NPRT sequence is randomly generated within the interval $[0.7,1.2]$ ms to enable velocity detection within the range of $\pm 50$ m/s. Furthermore, the SWA algorithm is used to determine the optimal window width of NPRT spectral moment estimations in this fixed PRT range (details in Section 4.1). A specific window width is considered to achieve the optimal estimation of weather detection. Based on this window, performance analyses and comparisons of the results obtained under other influencing factors are carried out. Figure 2c,d are the target PS obtained by the SWA algorithm for typical weather echo samples. The main component 1 is obtained with a window width of 22 m/s.

3.2. Spectral Moment Estimation Algorithm

Upon obtaining weather echo samples with an NPRT based on Equation (8), spectral moment estimators are applied. The computation of average power is carried out directly using the PPP method, i.e., Equation (14). Subsequently, the estimation of average velocity and spectrum width relies on the target PS of the NPRT echo signal, as delineated in Equation (12). The calculation of the first-order moment Equation (15) and the second-order moment Equation (16), based on the normalized PS, facilitates the estimation of the mean velocity and spectrum width.

$$\widehat{P}={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{V}^{*}\left(n\right)V\left(n\right)$$

(14)

$$\widehat{f}={\displaystyle \frac{{\displaystyle \sum _k}{f}_{k}S\left(f_k\right)}{{\displaystyle \sum _k}S\left(f_k\right)}}$$

(15)

$${\widehat{\sigma }}_f^2={\displaystyle \frac{{\displaystyle \sum _k}{(f_k-\widehat{f})}^{2}S\left(f_k\right)}{{\displaystyle \sum _k}S\left(f_k\right)}},$$

(16)

where $\widehat{P}$ is the estimated average power, $\widehat{f}$ denotes the estimated average frequency, and ${\widehat{\sigma }}_f^2$ is the squared estimate of the frequency’s spectrum width.

The estimation outcomes for the mean velocity and spectrum width of the NPRT echo signal are deduced from the correlation between frequency and velocity:

$$\widehat{v}={\displaystyle \frac{\lambda }{2}}\widehat{f}$$

(17)

$${\widehat{\sigma }}_v={\displaystyle \frac{\lambda }{2}}{\widehat{\sigma }}_f.$$

(18)

Here, $\widehat{v}$ stands for the estimation of the mean radial velocity and ${\widehat{\sigma }}_v$ is the Doppler velocity’s spectrum width estimation.

4. Performance Analysis and Discussion

The specifications of China’s CINRAD radar system are designed to align with the RFR, as delineated in

Table 1. Functional requirements for the national weather radar, including bias and standard deviation.

Variable	Bias	Standard Deviation	Conditions
Power	≤1 dB	≤1 dB	Spectrum width $=4$ m/s; SNR > 10 dB
Radial Velocity	≤1 m/s	<2 m/s	Spectrum width $=4$ m/s; SNR > 8 dB
Spectrum Width	≤1 m/s	<1 m/s	Spectrum width $=2$ m/s; SNR > 10 dB
		<2 m/s	Spectrum width $=4$ m/s; SNR >10 dB

. We have conducted a series of Monte Carlo simulations to verify the performance of NPRT weather echo processing algorithms in estimating spectral moments across various crucial parameters. These parameters encompass sliding window sizes, temporal pulse numbers, baseline RPRTs, ranges of observation velocities, spectrum widths, and SNRs. The evaluation process aims to quantify the SME errors related to different parameters and reveal the performance of the NPRT processing algorithm. Figure 3 shows the complete flow of NPRT weather echo signal processing.

4.1. Optimal SWA Window

In Section 3.1, the SWA algorithm demonstrated its pivotal role in isolating the target PS from aliased replicas for improving subsequent spectrum moment estimations. This section discusses the optimal window size based on the SME algorithm when using an NPRT weather echo. Figure 4 shows the statistical error of the variable estimation results (the average value at each speed) when varying window widths within the range of 9 to 25 m/s are used within the observation velocity range of $[-50,50]$ m/s. The goal is to identify the influence of different window intervals on NPRT moment estimation so as to find a universal window width.

Assuming the true weather echo has a power of 0 dB and a velocity spectrum width of 4 m/s, each row of the analysis portrays the estimation errors for the power, velocity, and spectrum width, while columns denote the bias and standard deviation. It is essential to note that a positive average bias value signifies an overestimation of the variable, while a negative value signals an underestimation. The Monte Carlo simulations are iterated 100 times to ensure the robustness and reliability of the experiments.

The outcomes underscore a compelling alignment with the RFR. Notably, the maximum bias in power estimation is observed to be $-0.141$ dB, with a standard deviation of $0.175$ dB. Furthermore, the results reveal that an optimal performance in velocity estimation is achieved, with the smallest bias of $-0.0003$ m/s, at a window interval of 18 m/s, while the lowest standard deviation of $0.074$ m/s is seen when using a 25 m/s window. Similarly, the spectrum width estimation exhibits a bias of $0.056$ m/s and a standard deviation of $0.073$ m/s with a 22 m/s window size. Based on the above analysis, and considering the error results of the SME obtained with the designed NPRT radar waveform, the 22 m/s window is selected for carrying out the performance analysis of subsequent simulation experiments under other influencing factors. Consequently, a 22 m/s window is selected as the optimal choice for estimating the power, velocity, and spectrum width in the aliasing PS of an NPRT weather echo when leveraging the SWA algorithm.

The SWA algorithm with the optimal window size provides an efficient solution that alleviates the aliasing phenomenon in the power spectrum of NPRT weather echo sequences. This algorithm significantly improves the target detection capabilities of the radar under adverse weather conditions by effectively identifying the target PS and promoting accurate SME processing. It should be noted that Section 4.1 displays the numerical outcomes of spectral moment estimation errors for various windows. This is to illustrate that the theoretical basis on which we designed the SME algorithm of NPRT waveform weather echoes means that 22 m/s is the optimal sliding window for radar systems currently in operation at home and abroad in typical weather echo states. However, through our analysis, it can be concluded that the window selection can have a certain randomness, as long as the selected window can meet the radar’s demand for situation awareness indicators in weather echoes. Therefore, the performance analysis of different factors of the observation speed range, the number of time-domain pulses, and the SNRs is based on the optimal window of 22 m/s in this paper. The simulation experiments of the various spectrum widths and reference PRTs only provide more possibilities for different NPRT waveform designs and processing algorithms in weather detection. See the performance analyses conduced under different factors in the fourth part of this article.

Based on the analysis in this paper, when operating an actual radar system, a window of 22 m/s can be applied to assist in the detection of typical weather conditions for the NPRT radar waveform we designed. Therefore, when real weather conditions are unknown, the 22 m/s sliding window algorithm can still be used to accurately estimate these conditions. However, in certain cases, if the weather perceived is extreme, such as when the speed of bad weather exceeds the speed range of ± 70 m/s, then the sliding window algorithm in this paper will fail; that is, a correct period range for the principal value will not be able to be determined, which will lead to a completely wrong estimation of the weather’s state.

4.2. Target Mean Velocity

To highlight the advantages of NPRT in expanding the observable velocity range, a comprehensive analysis of the estimation errors for mean power, velocity, and spectrum width was carried out with a 1 m/s increment in the velocity variation range of $[-50,50]$ m/s. Figure 5 shows the accuracy analysis of the SME algorithm over the entire velocity spectrum. The given window is 22 m/s, the power is 0 dB, and the spectrum width is 4 m/s.

Our analysis reveals a minimal power estimation bias of $-0.1231$ dB, accompanied by a standard deviation of $0.1412$ dB. Similarly, the average velocity estimation bias is $0.0015$ m/s, with a standard deviation of $0.0613$ m/s. Furthermore, the average spectral width estimation bias is $0.056$ m/s, with a standard deviation of $0.0616$ m/s. These results signify the efficacy of NPRT in facilitating weather target detection across the entire observable velocity range, meeting the stringent error criteria of weather radar systems.

By enabling a clear velocity estimation in the range of $\pm 50$ and mitigating the interplay between the maximum unambiguous distance and velocity constraints that are common in the unified PRT setting, NPRT emerges as a promising approach to enhancing the performance of radars in weather monitoring applications.

In this paper, for the simulation experiment we select an SPRT waveform with k = 2/3 for comparison and assume that $\lambda =0.1$ m. According to Equation (13), when the PRFs are 800 Hz and 1200 Hz, we find that the maximum unambiguous velocities corresponding to them under a single PRF are 20 m/s and 30 m/s. However, using SPRT technology, the maximum unambiguous speed is determined by PRT₁-PRT₂. Hence, for the same parameters, the measurable radial velocity range is ±60 m/s when using a staggered PRF expansion. For the simulation experiment of the waveform of NPRT, we similarly select a random jitter of the PRT between 0.7 ms and 1.2 ms to generate NPRT time series, while the window width is 22 m/s. The weather echo attribute assumes that the average power is 0 dB, the average radial velocity is ±50 m/s, and the average spectrum width is 4 m/s. The spectral moment estimation errors of the two weather echo signals are compared over 100 Monte Carlo simulations. The results are shown in Figure 6.

From the simulation results, it can be seen that both NPRT and SPRT technologies meet the functional requirements of the spectral moment estimation error for weather radar, especially as the speed meets the detection requirement of ±50 m/s and the expansion of the maximum unambiguous speed is realized. However, whether SPRT or dual PRF technology is used, they inevitably face the problem of radio-frequency occlusion. This radio-frequency occlusion distance loop, which cannot be eliminated, will bring hidden dangers to severe weather detection. The advantage of NPRT is that the area of the radio-frequency occlusion is relatively small, and the number of pulses per distance bin is not large and can be skillfully avoided. Therefore, it is more advantageous to use the NPRT waveform in complex environments.

4.3. Temporal Pulse Number

The relationship between pulse repetition intervals and temporal pulse counts during each coherent processing interval (CPI) significantly impacts the estimation accuracy of spectral moments in radar systems. To delve into this crucial relationship, we conducted a comprehensive investigation to explore the influence of the number of time pulses on the accuracy of NPRT weather echo moment estimations. The simulation experiments include a number of time pulses, from 16 to 71, that increase in increments of 5, as well as other previously defined parameters, with the aim of analyzing the estimation error through 100 Monte Carlo simulations. Figure 7 shows the power, velocity, and spectrum width estimation errors of the NPRT weather echo at varying input speeds and under different pulse numbers.

The results indicate that the absolute bias in the power, velocity, and spectrum width are less than $0.5$ dB, $0.03$ m/s, and $0.5$ m/s, respectively, while the standard deviations are all less than $0.5$ dB, $0.18$ m/s, and $0.5$ m/s. Moreover, the results illustrate that an increase in the number of transmitted pulses leads to enhanced estimation accuracy. Notably, the errors observed within the specified pulse count range all align with the radar system’s specifications, indicating the potential for achieving accurate estimations with reduced pulse counts when using NPRT. This finding underscores the feasibility of rapid radar scanning capabilities facilitated by optimized temporal pulse numbers. Furthermore, Figure 8 visually presents the estimation bias for different pulse numbers within the velocity observation range.

Acknowledging the practical constraints inherent in radar operation is imperative, as dwell time limits the actual number of transmitted pulses. Therefore, this study chooses the time pulse number 64 as the standard to analyze the influence of other variables on the estimation accuracy.

4.4. Various Spectrum Widths

We investigated the spectral moment estimator accuracy of the NPRT weather echo under different spectral width conditions. The spectrum width was systematically manipulated from 1 m/s to 10 m/s and categorized into small spectral width (SSW: 1∼3 m/s), weather spectral width (WSW: 4∼6 m/s), and large spectral width (LSW: 7∼10 m/s) segments. The SWA algorithm was used to determine the optimal window size in the range of 9∼25 m/s under different spectrum widths. Figure 9 shows the estimation bias and standard deviation of the power, velocity, and spectrum width after using the SWA algorithm to find the best window for NPRT weather echo estimation under different true spectral width values. The power level was 0 dB, and 100 Monte Carlo simulations were performed.

The error analysis denotes that the power and velocity estimations, irrespective of the spectral width classification, adhere to the specified RFR criteria, notably showcasing minimal bias in their velocity estimation, averaging around 0 m/s (with an upper threshold not surpassing $0.1$ m/s). The bias in the spectral width estimation remains within the $\pm 1$ m/s range for actual spectral widths spanning 1∼6 m/s, with the most accurate estimates observed at a true value of 4 m/s.

Further, we analyze the bias of the SME across the entire observed velocity range. The observations indicate that for SSWs, the velocity bias can be effectively contained to within 1 m/s at varied velocities of $[-50,50]$ m/s. However, the utilization of excessively large fixed sliding windows may result in spectral width overestimation. It is worth noting that for LSWs, a large number of velocity estimation biases (overestimation and underestimation) are obvious. When the actual spectrum width is set to 10 m/s, some biases exceed 1 m/s, while the rest are still below 1 m/s.

In the NPRT weather echo under an LSW, due to the fact that the target PS is truncated by the fixed window being too small, the spectral width estimation is often underestimated. Consequently, within the parameter-setting range of this study, the SME algorithm of NPRT is effective for processing typical WSW echoes. However, in order to achieve an accurate estimation of targets with diverse spectral widths, it is necessary to identify the optimal window based on the target’s attributes and SWA technology.

4.5. SNR Impact Assessing

The SNR significantly affects the accuracy of the moment estimation method, and it is necessary to scrutinize the performance of the NPRT weather echo algorithm under diverse SNRs. Figure 10 shows the error analysis of the SME algorithm, which changes in increments of 5 dB from 5 to 50 dB of SNR, while the other parameters are kept consistent with those in the previous explanation.

The simulation results underscore the pivotal role of maintaining a favorable SNR in NPRT weather echoes to meet the requisite RFRs across the observed velocity spectrum. Based on these findings, it is recommended that the SNR be sustained at a threshold of no less than 15 dB to ensure optimal signal fidelity and promote accurate moment estimation in the NPRT weather detection field. In addition, the estimated bias does not change significantly beyond this threshold. Therefore, in this study, all experiments are simulated under high SNR conditions; that is, no greater noise is introduced to better demonstrate the theoretical effectiveness of the SME algorithm of NPRT weather echoes.

4.6. Varying the Reference PRT

In this study, the SWA algorithm detailed in Section 3.1 introduces the generation of time sequences of NPRT random jitters predicated on a specific RPRT. The simulations conducted in this study have hitherto employed an RPRT of 1 ms. Through the frequency spectral analysis of NPRF signals, it can be observed that the number of replicas varies under different RPRT spectral aliases within a given speed detection range. This variation impacts the SME results obtained from the designed NPRT weather signal processing algorithm. This underscores the critical role of the random jitter range of the RPRT. Therefore, by altering the RPRT to generate different NPRT waveforms, the performance of the SME algorithm for weather detection is presented, and the influence of different RPRTs on the estimation results of typical weather echoes in NPRT is discussed. Finally, considerations for future research on the radar transmission of NPRT waveforms to assist in weather detection signal processing are provided.

By examining PRT random jitter ranges spanning from $1.7$ to $2.2$ ms (first row) and $2.7$ to $3.2$ ms (second row) as exemplars, Figure 11 showcases the power spectrum samples of typical NPRT weather echoes. The left side embodies an average velocity of 50 m/s, while the right side depicts 15 m/s, with all other parameters aligned with those elucidated in Figure 1. Comparing Figure 2 and Figure 11, it is found that the frequency ratio of the maximum observable velocity to the standard RPRT determines the number of replicas in the aliasing power spectra of the NPRT echo samples. For an RPRT of 1 ms, approximately three aliasing spectral components emerge, as depicted in Figure 2. In contrast, RPRTs of 2 ms and 3 ms yield around 6 and 11 aliased spectral components, respectively.

Figure 12 presents the estimation error of the NPRT weather spectral moment using a benchmark RPRT of 2 ms and diverse sliding windows. The outcomes indicate that only the standard deviations of the windows below 21 m/s satisfy the RFRs for velocity estimation, whereas the spectral width estimation complies with the criteria within a window range of 12 to 22 m/s. This accentuates the adjustable nature of optimal window sizing. The simulation experiments propose a fixed window size of 17 m/s as suitable when the RPRT is 2 ms.

The results show that in order to ensure the best estimation of the SME, the window size corresponding to the random jitter range of the rNPRT needs to be adjusted. The larger the RPRT, the more copies there are within the specified speed observation range, which indicates that a smaller optimal window size is required. Generally speaking, when the RPRT is increased by n times (n = RPRT/1kHz), the width of the sliding window width can be set as (22/n) m/s, which will meet the needs of weather detection. In the future, our research will emphasize to optimize the SWA algorithm to meet these estimation requirements effectively.

5. Conclusions

The signal processing and algorithm performance of NPRT sequences in weather detection were studied. Essentially, the NPRT time series consists of randomly varying pulse intervals based on an RPRT for each transmission. A theoretical time–frequency formula is introduced, and the algorithm steps are derived. It is worth pointing out that the observed velocity range of a typical meteorological target is $[-50,50]$ m/s, the power is 0 dB, the spectral width is 4 m/s, and the NPRT sequence’s jitter range is 0.7∼1.2 ms. Thus, through a signal analysis, we found that numerical simulations produce approximately three copies of this NPRT echo in the frequency domain.

We proposed an SWA algorithm that can accurately locate the target PS. On this basis, we next designed an SME method for estimating the average power, velocity, and spectrum width based on NPRT weather echoes. Note that the SWA method can be applied to an optimal window size of 22 m/s. It is true that the performance of the SME algorithm is based on RFRs, where were used as a reference for its comprehensive evaluation.

Through Monte Carlo simulations, we evaluated the estimation bias and standard deviation of typical weather echoes as a function of the observed velocity, emphasizing the benefits of NPRT in expanding the maximum unambiguous velocity. We observed deviations of −0.1231 dB, 0.0015 m/s, and 0.056 m/s for the mean power, speed, and spectral width, and standard deviations of 0.1412 dB, 0.0613 m/s, and 0.0616 m/s for these, respectively. These results prove that the evaluation error of an NPRT echo conforms to the RFR criteria and improves the effectiveness of weather target detection.

Therefore, we conclude that the advantages of using NPRT signals for weather target observation over uniform PRTs emphasize their ability to achieve a maximum unambiguous velocity estimation over a velocity variation range of $[-50,50]$ m/s. This method is designed to solve the range–velocity ambiguity dilemma that is prevalent in weather radar systems, which is a key aspect of detecting weather targets from complex radar waveforms. In addition, compared with traditional speed ambiguity resolution technology, NPRT has the advantage of effectively avoiding radio-frequency occlusion.

In addition, we used Monte Carlo simulations to evaluate the relationship between the estimation error and the number of time pulses within NPRT weather echoes. The results show that the absolute deviations of the power, speed, and spectral width are less than 0.5 dB, 0.03 m/s, and 0.5 m/s, respectively, and the standard deviations are less than 0.5 dB, 0.18 m/s, and 0.5 m/s, respectively, when the number of pulses is minimized. Thus, from a practical point of view, the NPRT waveform has the potential to accelerate the detection of weather targets with fewer emitted pulses, which implies an acceleration of radar scanning. In addition, integrating these proposed methods with frequency diversity is expected to enhance the detection capabilities of radars [45].

In this study, we also compared the estimation performance of NPRT weather echoes at different spectral widths. We showed that the power and speed estimates meet the required criteria for all three spectral width specifications, with a velocity deviation of about 0 m/s. In the case of spectral widths, the bias is minimal at a true spectral width of 4 m/s. However, estimates in SSWs or LSWs are prone to overestimation or underestimation because the improperly fixed window in the NPRT echo truncates the target PS. Therefore, the NPRT algorithm is best suited to targeting typical weather echoes with a spectral width of 4∼6 m/s. However, it is necessary to admit that the NPRT waveform shows a potential to detect other targets with different properties.

In addition, we added noise to the numerical simulations to change the signal-to-noise ratio (SNR) and found that it is best to maintain the signal-to-noise ratio of the NPRT weather echo at $\ge 15$ dB in the $[-50,50]$ m/s velocity range to meet the RFRs.

Finally, we demonstrated that different RPRTs result in different copies of the NPRT weather echo aliasing PS. Consistent with the numerical simulation results, RPRTs of 1 ms, 2 ms, and 3 ms yielded approximately 3, 6, and 11 copies, respectively. In general, there is a correlation between the number of replicas within a specified velocity observation range and the optimal window size. Specifically, simulation experiments show that a fixed window of 22 m/s is recommended when the RPRT is 1 ms. The window width can be directly applied to actual radar weather detection to achieve the variable estimation of different moments.

In conclusion, the use of NPRT waveforms is considered to have substantial advantages in distance and velocity de-ambiguation, with the potential to achieve ambiguity-free distance and velocity measurements. Through Monte Carlo simulation experiments, the effectiveness of the NPRT spectral moment estimation algorithm under different parameter changes and influencing factors was evaluated. Experimental results have confirmed that the nonlinear radar signal processing technique exhibits a significant improvement in the radar performance of weather detection, thereby opening up a new avenue for multifunctional radar observations across a range of operational scenarios.

Despite significant progress in this area, challenges remain, particularly in mitigating the estimation bias and performing exhaustive analyses in complex background environments such as ground clutter. The next steps may be to scrutinize error sources, optimize signal processing using specific techniques such as compressed sensing, and further investigate the effectiveness of the NPRT algorithm in clutter filtering [46,47]. The proposed algorithm will be further verified by real radar data.

Finally, combining NPRT with multi-dimensional coded waveforms for weather target detection may be of interest and unlock versatile radar applications. The merging of NPRT and SZ phase coding shows promise in two-way (or multi-way) overlapping echo scenarios, improving the accuracy of spectral parameter estimation and reducing the coverage of the purple zone. In addition, we believe that by designing an adaptive approach, NPRT waveforms can eliminate and compensate for blind spots caused by high PRFs [22].