## 1. Introduction

The phase coding method tags the transmit waveform with a phase code. The same code is used in demodulation to retrieve the pulse parameters (amid interference from different pulses like second trips or cross-polar coupling) in the weather radar system. The phase code can be a single tag used for a pulse (interpulse coding) to separate multiple trips. This can help suppress the superposition of other trips on the first trip and recover these multiple trips under certain restrictions. Here, the first trip refers to the unambiguous range defined by the pulse repetition time (PRT). However, the echoes due to strong weather events after the PRT’s unambiguous range can be classified as second, third, and so on trips depending upon the multiple of PRT’s range that they correspond to. The suppression effect of the phase code could be observed over the full cycle of coherent processing, which is the coherent interval to generate polarimetric moments from raw data.

In pulse Doppler weather radars, the pulses are transmitted at the pulse repetition interval of *τ* seconds, the maximum unambiguous range is given by *r*_{unb} = *cτ*/2 and the maximum unambiguous velocity is given by *υ*_{unb} = *λ*/4*τ*, where *λ* is the wavelength of operation. Hence, both *r*_{unb} and *υ*_{unb} are inversely proportional to each other. Due to this, the unambiguous range and velocity cannot be simultaneously optimized. If one is increased, the other one is inversely affected. This is termed as a range–Doppler dilemma. A high pulse repetition frequency (PRF = 1/PRT) radar system would be able to measure high Doppler velocity but the unambiguous range would suffer. Whereas in a low PRF system, it would detect weather events going on at much farther range but the velocity would be folded in the Doppler domain. For a medium PRF system, it would be difficult to measure both range and velocity without ambiguity. This problem is more prominent for weather radar systems as the scatterers are continuously distributed throughout the beam illumination volume, with broad dynamic range (Bringi and Chandrasekar 2001).

Different phase coding schemes exist in literature to overcome this fundamental limitation and retrieve polarimetric moments from the second, third, and so on, trip echoes for weather radar. These codes can be broadly classified into intrapulse (phase changes on a subpulse basis; Kumar and Chandrasekar 2020) and interpulse (phase change on a pulse basis) codes. The random phase codes and systematic phase codes are examples of interpulse phase codes, popular to retrieve first-trip and overlaid second-trip echoes (Sachidananda and Zrnić 1999).

Staggered and dual PRF techniques are other methods described in Torres and Warde (2017) and Cho and Chornoboy (2005), which are generally used to improve *r*_{unb} and *υ*_{unb}. In particular, they are very effective in increasing the *υ*_{unb}. This is typically accomplished through the use of PRF diversity by playing two PRF’s one after the other or in batches. However, it takes more scan time and uses *N* times the time required for the constant PRF, *N* being the number of stagger pulses. Additionally, it requires more processing to perform range or Doppler unfolding compared to a uniform PRF radar.

Interpulse codes have been explored extensively in the weather radar community for second-trip echo suppression (Sachidananda and Zrnić 1999) and for orthogonal channel coding as in Chandrasekar and Bharadwaj (2009) for dual-polarization weather radars. The retrieval of moments for the first and second trips is based on spectral processing of weather echoes in batches of coherent intervals (depending upon the antenna rotation rate and the decorrelation time of weather echoes). For first and second trips, the code’s orthogonality is achieved over the coherent interval as the second trips get modulated by a cyclic shift of the phase code. The separation between first and second trips is achieved by having these cyclic shifts orthogonal to each other. The systematic codes use the derivatives of Chu codes, which present deep nulls in the cross-correlation function for time-delayed versions of themselves. However, the rejection of echoes from the unwanted trips is a function of the spectral width of the weather echoes and would degrade in case of multimodal distribution, broad spectral width of the weather echoes and the presence of phase noise.

This paper introduces a novel frequency diversity interpulse scheme and discusses its implementation on the NASA dual-frequency, dual-polarization, Doppler radar (D3R). In the proposed scheme, we change the intermediate frequency (IF) from pulse to pulse. For example, for second-trip suppression, we use two frequencies, *f*_{1} and *f*_{2} for alternate pulses and beat them separately with two digital down-converters to recover both first and second trips. If the frequencies are properly selected, it gives first- and second-trip retrieval abilities. However, if these frequencies are far apart by more than the transmit waveform bandwidth, then power returns from hydrometeors become uncorrelated between these two sets of pulses. So we discuss a novel method to retrieve velocity and spectrum width for the first and second trips under these constraints of the pulsing scheme with *f*_{1} and *f*_{2} for alternate pulses. This method can recover velocity and spectral width, from a batch of coherent processing time (128 pulses, in case of D3R), without compromising the unambiguous velocity range. The improvement in the second-trip recovery region, based on the ratio of |*P*_{1}/*P*_{2}|, where *P*_{1} is first-trip power and *P*_{2} is second-trip power, is observed. A block diagram of this scheme is shown in Fig. 1. The carrier generator outputs two different oscillations at frequencies, *f*_{1} and *f*_{2}, for odd and even pulses, respectively. If the sequence at the down-converter is *f*_{1} and *f*_{2} for odd and even pulses, we would recover the first trip. However, if the sequence is *f*_{2} and *f*_{1}, then we would be recovering second-trip parameters.

Many orthogonal polyphase coded systems have been proposed in literature, as in He et al. (2009), Song et al. (2016), Deng (2004) and Griep et al. (1995). They are part of a broad category of phase coding schemes called intrapulse coding elaborated by Kumar and Chandrasekar (2020). But it is challenging to obtain a peak cross-correlation function between different polyphase codes lesser than −40 dBc (with respect to the peak autocorrelation function of these polyphase codes). The frequency diversity scheme proposed in this paper is meant to give a higher level of isolation than is possible with polyphase or intrapulse coding. The limits on the peak autocorrelation and cross-correlation sidelobes of the sequences (which limits the orthogonality between these sequences in case of intrapulse codes) have been discussed in Sarwate (1979) and Welch (1974). These lower bounds on the cross correlation of polyphase codes tell us that there is a need to explore other techniques. In his correspondence in Sarwate (1979), the author points out that if a set of sequences has good autocorrelation properties, then the cross-correlation properties are not very good and vice versa. He also derived a result that shows the trade-offs between the maximum magnitudes of the correlation functions. He further derived an inequality that provides a lower bound on one of the maxima if the value of the other correlation function is specified. However, for weather-based radar applications, there is a need for lower peak sidelobe levels for both auto- and cross-correlation functions. Hence our proposed scheme, which works with alternating frequencies (that are apart by more than the transmit bandwidth), seeks to overcome this limitation. It can also address the ever-growing need for orthogonality in MIMO systems (Li et al. 2008). In MIMO radars, all transmit elements can radiate at the same time without interfering with each other using frequency diversity. And the same approach may work for CDMA-based communications too (Liu and Xu 1996).

Having said this, we also want to point out that frequency diversity has been existent and used in the past for weather radars in various extents and combinations. For example, in Karhunen et al. (2009), frequency diversity is used for range extension and by estimating the unknown phase due to frequency change pulse to pulse for velocity retrievals. However, the estimation of spectral width for weather echoes is not mentioned in this. Moreover, there is no quantification of results; for example, how does this technique compare against the interpulse Chu-based phase codes or any other scheme? And also no mention of practical phase noise conditions. Similar is the case with Lihua et al. (2019), where the inventors have used frequency diversity pulse pair methods by alternating the order of the pulse pair transmitted or of multiple pulses on different center frequencies. But the performance quantification for practical radar systems also lacks in this case.

This paper is organized as follows: A detailed description of the frequency diversity scheme and the Chu phase coded interpulse scheme are given in section 2. Specifically, section 2a gives introduction to Chu-based codes followed by section 2b which discusses the performance of these codes under phase noise. Next section 2c has the description of the proposed scheme. Later, section 3 presents the results from D3R weather radar on which this scheme was implemented, followed by the Conclusion in section 4. Finally, the appendix analyzes the effect of frequency change at the IF level on other dual-polarization moments.

## 2. Interpulse waveforms

### a. Generalized Chu codes for second-trip suppression and retrieval

The most popular interpulse codes are the systematic phase codes (SZ), detailed in Sachidananda and Zrnić (1999) for the retrieval of parameters of overlaid echoes. In this paper, the SZ code’s performance is based on the Chu phase codes that are analyzed for the second-trip suppression and retrieval. We use this as the reference performance and compare it with the proposed frequency diversity scheme. A point to be noted here is that under a broad spectral width of weather echoes, the SZ codes, which rely on replicating the other trip spectra multiple times (while cohering to one trip, which is being recovered), become “white” due to a considerable overlap between the replicas. Another interpulse code which introduces uniformly distributed random phase on pulses (known as random phase code), attempts to whiten the weaker trip and at the same time, it coheres the stronger trip signal. Whereas the SZ code produces replicas of the weaker signal spectrum. If a notch filter is used in a random phase code, it also removes some part of the second-trip spectrum, which cannot be recovered later on, thus generating a self-noise (Frush et al. 2002). Also, while notching out, the Gaussian spectrum of precipitation broadens due to ground clutter and phase noise, etc. Hence, to completely notch out the stronger trip, a much wider notch filter is essential (apart from the notch width required for the Gaussian spectrum width). The proposed frequency diversity scheme is immune to the overlaid echo’s spectrum width, which is also one of its significant advantages. Since the second-trip echo is filtered out in the fast time domain by the FIR baseband down-converter filter, the slow time spectral domain will not have any contamination when processing first-trip echoes for velocity and spectral width. This is also true when we are estimating second-trip velocity and spectral width by notching out the first-trip power in the fast range–time domain.

*N*and

*M*to form an SZ(

*N*/

*M*) code. The SZ code (simulated here) is designed using Eq. (1) with

*N*= 8 and

*M*= 64. Details on the formulation of SZ code and the parameters

*N*and

*M*can be found in Sachidananda and Zrnić (1999):

If the spectrum of the precipitation echoes lies within *M*/8 spectral coefficients in the spectral domain [for SZ(*N*/*M*) where *N* = 8 and *M* = 64], then there would be less overlap between successive replicas and better estimation can be performed. But in a broader spectral width scenario, the overlap region can pose a constraint for the recovery region of velocity and spectral width. In such a case, the SZ(4/64) code with *N* = 4, will most likely provide a better separation in exchange for allowing a much lesser notch width, to retain a minimum of two spectral replicas. Additionally, the phase noise leads to broadening of the spectrum, which can be linked to phase noise of the coherent oscillator used to synchronize various subsystems of the radar.

The cyclic version of the SZ(8/64) code (known as modulation code) has eight replicas of the second trip in the Nyquist interval and the original spectrum of the first-trip echo. Once the higher power echo of the first trip has been notch filtered, the remaining replicas aid in estimating second-trip parameters. A minimum of two replicas are essential for velocity computation and later magnitude deconvolution for spectral width computation (Sachidananda and Zrnić 1999). We simulated weather echoes, with a moderate rainfall scenario for a radial [method described in Galati and Pavan (1995)]. In this simulation, the first trip has the following parameters: velocity (*υ*_{1} = 10 m s^{−1}), spectral width (*w*_{1} = 1 m s^{−1}), and copolar correlation coefficient (*ρ*_{hv} = 0.995) and the second trip has the same parameters, except for velocity, which is taken as *υ*_{2} = −5 m s^{−1}. This was done to see the effectiveness of the SZ codes and later use it to compare with this paper’s proposed frequency diversity method. The simulation was carried out at S band with PRF = 1.2 KHz. In reception, the spectrum of the total received echo consists of the first trip’s spectrum and the second trip’s spectrum getting replicated eight times [with SZ(8/64)], shown in Fig. 2a.

(a) The combined first- and second-trip echo velocity spectrum while retrieving the first-trip echoes. The modulation code spreads out the power of second trip to eight replicas. (b) The combined first- and second-trip echo velocity spectrum while the first-trip echoes have been notched out with a normalized notch width of 0.75. (c) The combined first- and second-trip echo velocity spectrum while retrieving the second-trip echoes. The spectrum is recohered for the second trip with its six sidebands present in the spectrum.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The combined first- and second-trip echo velocity spectrum while retrieving the first-trip echoes. The modulation code spreads out the power of second trip to eight replicas. (b) The combined first- and second-trip echo velocity spectrum while the first-trip echoes have been notched out with a normalized notch width of 0.75. (c) The combined first- and second-trip echo velocity spectrum while retrieving the second-trip echoes. The spectrum is recohered for the second trip with its six sidebands present in the spectrum.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The combined first- and second-trip echo velocity spectrum while retrieving the first-trip echoes. The modulation code spreads out the power of second trip to eight replicas. (b) The combined first- and second-trip echo velocity spectrum while the first-trip echoes have been notched out with a normalized notch width of 0.75. (c) The combined first- and second-trip echo velocity spectrum while retrieving the second-trip echoes. The spectrum is recohered for the second trip with its six sidebands present in the spectrum.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

After estimating the first-trip parameters, the total spectrum is notch filtered, removing the power of the first trip, and the second-trip parameters can be estimated. Suppose a rectangular window is used for truncating the signal. In that case, the stronger trip will be contaminating the weaker trip spectrum through its spectral sidelobes, and the dynamic range, |*P*_{1}/*P*_{2}|, where the second trip could be retrieved, will get smaller. However, to reduce spectral leakage, if we use other window functions, that would lead to loss of signal-to-noise ratio (SNR) meaning reducing the number of independent samples. A Hann window is used in our simulations. It has a higher SNR loss (2.88 dB), but the spectral dynamic range gets substantially increased. Although this window function does not represent a healthy compromise between SNR and spectral leakage power but still, we have retained it for our simulations to get a fair comparison with the analysis done with SZ codes in Sachidananda and Zrnić (1999). The loss is incurred due to an aggressive amplitude taper to contain sidelobes’ energy but leading to mainlobe broadening. After the notching process, at least two replicas need to be retained for velocity and spectral width estimate. The spectrum after the notch process is shown in Fig. 2b.

The remaining signal (after filtering out first trip) has six symmetrical sidebands centered at the mean velocity of the second trip, shown in Fig. 2c.

### b. Effect of phase noise on SZ/Chu codes

The phase noise (also referred to as jitter) is a major limiting factor for the dynamic range of |*P*_{1}/*P*_{2}| using an interpulse SZ/Chu code. Phase noise leads to an overall broadening of the spectrum of both first and second trips. The overall phase noise is dominated by the oscillator’s phase noise (Underhill 1992), which is a reference for the entire system. Single-side band phase noise is usually measured in a 1 Hz bandwidth, and it can be defined as the ratio of noise in a 1 Hz bandwidth to the signal power at the center frequency.

*A*is the area under the phase noise curve.

It has been shown in Sachidananda and Zrnić (1999) that, if there is no phase noise and the Hann window function is used, the limit on retrieval of second-trip velocity spans around 90 dB of |*P*_{1}/*P*_{2}| power ratio, for spectral width smaller than 4 m s^{−1}. But it drastically reduces to 60 dB, if the RMS jitter is on the order of 0.2° RMS, and further reduces to 40 dB in case of 0.5° RMS jitter. This is also shown by the simulations of weather echoes presented in section 2a that the phase noise reduces the accuracy of the second-trip velocity retrieval. The range of |*P*_{1}/*P*_{2}| power ratios, in which the second-trip velocity can be recovered, with and without phase noise, is depicted in Figs. 3a and 3b, respectively.

(a) The dynamic range of |*P*_{1}/*P*_{2}|, in which the second-trip velocity can be recovered (with acceptable standard deviation limits), without phase noise. (b) The dynamic range of |*P*_{1}/*P*_{2}|, in which the second-trip velocity can be recovered (with acceptable standard deviation limits), with phase noise.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The dynamic range of |*P*_{1}/*P*_{2}|, in which the second-trip velocity can be recovered (with acceptable standard deviation limits), without phase noise. (b) The dynamic range of |*P*_{1}/*P*_{2}|, in which the second-trip velocity can be recovered (with acceptable standard deviation limits), with phase noise.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The dynamic range of |*P*_{1}/*P*_{2}|, in which the second-trip velocity can be recovered (with acceptable standard deviation limits), without phase noise. (b) The dynamic range of |*P*_{1}/*P*_{2}|, in which the second-trip velocity can be recovered (with acceptable standard deviation limits), with phase noise.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

### c. Frequency diverse chirp waveforms

Modern-day systems with higher computation power and embedded with field programmable gate arrays (FPGAs) are capable of high-speed signal processing architectures. The digital receiver system in D3R, which now has such an architecture, is capable of switching IF on a pulse by pulse basis (Kumar et al. 2018). This feature has been utilized to obtain frequency diversity at IF. The main factor limiting the amount of second-trip suppression is the IF filter, which is implemented digitally in the FPGA (based on its stopband suppression and rolloff in the frequency domain). That also can decide which frequencies to select while transmitting the pulses.

Typically, when the transition of the filter frequency response from passband to stopband is steep, it requires many digital multiply–accumulate (MAC) units. However, with the advent of increased processing power and FPGA nodes optimized for DSP application, we can quickly obtain very sharp rolloff filters working in real time. The analog filter before the analog to digital converter has a wide enough passband to accommodate both *f*_{1} and *f*_{2}. Finally, it is difficult to have high bandwidth stages because of spurious and intermodulation products in the mixing process, which may show up in the passband. This can also lead to a reduced spurious-free dynamic range (SFDR).

*μ*s and a bandwidth of 1 MHz (pulse repetition frequency of 0.5 kHz) is selected to maximize sensitivity, minimize blind range, and have a reasonable data rate for the signal processor. The

*f*

_{1}and

*f*

_{2}will be selected based on the digital filter characteristics and will be dealt with in more detail in section 2c. If

*A*

_{1}and

*A*

_{2}denote the amplitude of first- and second-trip echoes, then the output at mixer out port would be

For recovering first trip, the frequency component to be filtered out is *A*_{2} exp[−*j*(*ω*_{2} − *ω*_{1})*t*]. Similarly, for the next pulse, we will retain *A*_{1} and filter out the second-trip frequency component, *A*_{2} exp[−*j*(*ω*_{1} − *ω*_{2})*t*]. Finally, we average out similar pulse sets to retrieve the first trip and the process for second-trip retrieval would be very similar as well.

To gain a better understanding of the proposed frequency diversity system, we go back to our time series simulation of moderate rainfall case with the following parameters: *υ*_{1} = 10 m s^{−1}, *w*_{1} = 1 m s^{−1}, and *ρ*_{hv} = 0.995. The second trip has the same spectral width and copolar correlation coefficients, except velocity which is *υ*_{2} = −5 m s^{−1}.

The simulated time series spectrum is centered on *f*_{1} for the first pulse and *f*_{2} for the next pulse in a frame. A frame is a basic unit of two pulse echoes, which repeats in time. A digital filter then filters the pulse returns at baseband and then after the pulse compression process, the power of the echo signal is calculated. In the simulation, we can vary the first-trip power over the second-trip one, and obtain the dynamic range of values, |*P*_{1}/*P*_{2}|, where the parameter retrievals of the second trip are within an acceptable range (based on measured bias and standard deviation). This concept of weather time series simulation is elaborated in Fig. 4.

The time series simulation of weather echoes at IF frequency.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The time series simulation of weather echoes at IF frequency.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The time series simulation of weather echoes at IF frequency.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

Figure 5 depicts both the first and second trips, with equal power, such that |*P*_{1}/*P*_{2}| = 0 dB, and both have RMS phase jitter of 0.5°. The waveforms are upconverted to IF frequency. The odd-numbered pulse first-trip echoes are upconverted to *f*_{1} and then combined with the even-numbered second-trip echoes (upconverted to *f*_{2}). The spectrum after this process, where, *f*_{1} = 60 MHz and *f*_{2} = 70 MHz, is shown in Fig. 6. This simulated first- and second-trip echoes are down-converted with appropriate IF frequency for even and odd-numbered pulses. Under this setting, we have various options available to work with:

Suppose we have one down converter and pulse compression system available. In that case, we can either retrieve first-trip echoes by switching the frequency to a sequence 1:

*f*_{1},*f*_{2}for a frame of two pulse echoes, based on odd or even pulse numbers. For retrieving of the second-trip echoes, we would need to switch to a frequency sequence 2:*f*_{2},*f*_{1}for a two-pulse echo frame. This will require less resources but would also take twice the amount of time for retrieval of first and second trips as compared to the parallel scheme described next.If we have enough resources to perform two parallel down converters and pulse compression systems, then by programming frequencies to a sequence 1 or 2, we should be able to retrieve both trips simultaneously.

Another approach could be to use alternate pulse-pair echo frames for the first trip and in-between frame for second-trip retrievals. In this scenario, the sequence of the IF frequency would be

*f*_{1},*f*_{2},*f*_{2}, and*f*_{1}for a frame of four pulse echoes. This scheme would have an advantage of saving resources, and computational complexity is decreased.

Both the first- and the second-trip echoes are generated with equal power such that |*P*_{1}/*P*_{2}| = 0 dB and with parameters: *υ*_{1} = 10 m s^{−1}, *w*_{1} = 1 m s^{−1}, and *ρ*_{hv} = 0.995 while the second trip has the same parameters, except velocity of *υ*_{2} = −5 m s^{−1}.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

Both the first- and the second-trip echoes are generated with equal power such that |*P*_{1}/*P*_{2}| = 0 dB and with parameters: *υ*_{1} = 10 m s^{−1}, *w*_{1} = 1 m s^{−1}, and *ρ*_{hv} = 0.995 while the second trip has the same parameters, except velocity of *υ*_{2} = −5 m s^{−1}.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

Both the first- and the second-trip echoes are generated with equal power such that |*P*_{1}/*P*_{2}| = 0 dB and with parameters: *υ*_{1} = 10 m s^{−1}, *w*_{1} = 1 m s^{−1}, and *ρ*_{hv} = 0.995 while the second trip has the same parameters, except velocity of *υ*_{2} = −5 m s^{−1}.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The spectrum of up-converted first- and second-trip echoes with *f*_{1} = 60 MHz and *f*_{2} = 70 MHz.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The spectrum of up-converted first- and second-trip echoes with *f*_{1} = 60 MHz and *f*_{2} = 70 MHz.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The spectrum of up-converted first- and second-trip echoes with *f*_{1} = 60 MHz and *f*_{2} = 70 MHz.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The down-converter’s stage overall frequency response is set to allow a passband ripple of 0.2 dB and provide stop-band attenuation of nearly 60 dB. The amplitude and phase response of such a filtering stage is shown in Fig. 7.

The filter response (amplitude and phase) of the down-converter stages.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The filter response (amplitude and phase) of the down-converter stages.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The filter response (amplitude and phase) of the down-converter stages.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The spectrum after down-conversion and filtering, for the frequency sequence set for the second-trip retrieval, is shown in Fig. 8. The bandwidth of the chirp used for modulation is 0.5 MHz. Finally, we try to compute the velocity of second-trip echoes, with a power ratio (*P*_{1}/*P*_{2}) = 0 dB over a set of coherent processing pulses and PRF of 1.2 kHz at S band. This is shown in Fig. 9.

The spectrum after the down-conversion stages (retrieving second trip).

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The spectrum after the down-conversion stages (retrieving second trip).

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The spectrum after the down-conversion stages (retrieving second trip).

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The velocity spectrum of the second trip, recovered after frequency switching between *f*_{1} and *f*_{2}.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The velocity spectrum of the second trip, recovered after frequency switching between *f*_{1} and *f*_{2}.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The velocity spectrum of the second trip, recovered after frequency switching between *f*_{1} and *f*_{2}.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The spectral noise floor is dominated by the phase noise of the echoes received, which negatively impacts the dynamic range of |*P*_{1}/*P*_{2}|. This fact has been demonstrated before using Chu (SZ) interpulse codes. Recall that the second trip could be recovered for the power ratio (|*P*_{1}/*P*_{2}|) spanning up to 40 dB, under 0.5° RMS phase jitter. However, under similar phase noise (jitter) conditions, the frequency diversity scheme, proposed and developed in this paper, can recover second trip for the power ratios spanning 60 dB, an improvement of about 20 dB over the Chu interpulse code. This is one of the achievements of this research, and this has been substantiated here with time series simulation of weather echoes. With this simulation, the mean bias and standard deviation of second-trip velocity as a function of power ratios for frequency diversity scheme is shown in Figs. 10a and 10b. The simulation steps given before have been repeated for various power ratios of |*P*_{1}/*P*_{2}|, and these values have been given in the figures. It can be easily observed that the standard deviation of the error increases significantly as the ratio exceeds 60 dB. These two figures imply that the proposed frequency diversity scheme can successfully reconstruct the second trip even if it is submerged in the stronger first-trip power (by 60 dBs). This is a substantial improvement over any other interpulse schemes (random or SP Chu phase codes) published in the literature.

(a) The mean bias in the measurement of the second-trip velocity, after frequency switching between *f*_{1} and *f*_{2}. (b) The standard deviation in the measurement of the second-trip velocity, after frequency switching between *f*_{1} and *f*_{2}.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The mean bias in the measurement of the second-trip velocity, after frequency switching between *f*_{1} and *f*_{2}. (b) The standard deviation in the measurement of the second-trip velocity, after frequency switching between *f*_{1} and *f*_{2}.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The mean bias in the measurement of the second-trip velocity, after frequency switching between *f*_{1} and *f*_{2}. (b) The standard deviation in the measurement of the second-trip velocity, after frequency switching between *f*_{1} and *f*_{2}.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

We would also like to point out that the oscillator technology has been rapidly advancing and better phase jitter reference sources for radars are rapidly becoming available. For example, in Stagliano et al. (2005), the authors relate phase noise and coherency to clutter rejection ability of the radar, and it is mentioned that WSR-88D radar is capable of 0.18° RMS jitter and D3R radar is within 0.1° RMS jitter. Thus it looks like the interpulse SZ codes would approach frequency diversity performance at lower phase noise conditions. As the systems become less noisy, the two techniques are comparable, and frequency diversity will have major benefit at jitter greater than 0.2° RMS. But we would like to highlight that the performance of frequency diversity can be directly linked to the stopband attenuation of the baseband digital filter in the downconverter and the frequencies *f*_{1} and *f*_{2}. The farther these frequencies, the better the performance which can be expected thanks to an increase of stopband attenuation. We have also observed from simulations that our proposed scheme gives 10 dB advantage at jitter conditions lesser than 0.1° RMS if a better stop-band attenuation of 80 dB is utilized in down-converter.

But the most significant advantage is in the velocity recovery region at broader spectral widths. This is basically because the SZ code replicates the spectrum of the second trip eight times [for SZ(8/64)] as compared to our scheme where sideband appears at *π* radians away from the primary velocity spectrum, for both first- and second-trip recovery. Thus there is only one replica of the original spectrum at *π* radians apart instead of eight replicas of the second trip as in the case of SZ(8/64) codes. Now it is easy to observe that the recovery region of our scheme at broader spectral width for second-trip velocity will be higher than these SZ codes. To quantify, the recovery region (defined for spectral width) will be at least twice of what is offered by SZ codes [SZ(4/16), SZ(8/64), and SZ(8/128)] in velocity retrievals. Hence our scheme will tolerate twice as wide spectral width and correct estimate of velocity compared to the specific SZ codes mentioned.

### d. Velocity and spectral width retrieval

The proposed pulse-to-pulse alternate IF frequency scheme gives great benefit in terms of suppressing/retrieving of the second-trip echoes but it would require spectral processing to retrieve the velocity and spectral width information. This is because differing frequencies in alternate pulses make the data samples uncorrelated from one pulse to another. However, the echoes are correlated in alternate pulses. If we try to retrieve the velocity and spectral width information using the alternate pulse echoes, then the velocity range that could be resolved would become half. In this section, we describe a new process, which can still recover the original range of velocities, with some constraints.

Basically, the uncorrelated data from these two frequencies manifest with a different amplitude and phase in adjacent pulse echoes. This phase term is in addition to the Doppler associated with the motion of the weather echoes. Thus, even for a stationary target, the amplitude and phase would keep cycling between two states. Because of this, there would be a fixed amplitude and phase modulation with a periodic repetition rate of the PRI, and the overall spectrum would have another sideband at *V*_{1or2} − *π*. This spectrum looks as in Fig. 11. Moreover, you can observe that both the original and sideband looks identical, and there is a necessity for some other means to figure out the original velocity. For correct spectral width retrieval, the sideband would need to be filtered; otherwise there is going to be an overestimation of spectral width. We propose a mechanism here to correctly estimate velocity and spectral width with this type of fixed gain and phase state variation over multiple pulses. This would work under narrow spectral width constraint, and we define a narrow spectral width echo to be around one-tenth of the unambiguous velocity range. For S band, it would be approximately 5 m s^{−1}, and for the Ku band, this turns out to be close to 2 m s^{−1}.

Fixed gain and phase modulation due to uncorrelated frequencies in alternate pulses.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

Fixed gain and phase modulation due to uncorrelated frequencies in alternate pulses.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

Fixed gain and phase modulation due to uncorrelated frequencies in alternate pulses.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

*f*

_{1}. Hence, we can write the autocorrelation function for

*f*

_{1}at various lags as

*R*

_{n}is a single-lag autocorrelation function at

*f*

_{1}.

*π*/

*N*) (Oppenheim and Schafer 2009) having a periodicity of

*N*= 2. Thus the Fourier transform of the overall autocorrelation function is the power spectral density of the weather echoes convolved with an impulse train spaced apart by

*π*radians. Now it becomes easy to understand that the spectrum of weather echoes with odd and even pulses modulated at different frequencies will have a sideband at

*V*

_{1/2}−

*π*within the Nyquist interval. Let us look at how we can get rid of this additional sideband in the spectral domain.

#### Method

First, we process the upper half of the spectrum at a particular radial and range, having SNR > 10 dB, with a weather echo present. The assumption is that either the original or side-band velocity would fall in this region of the spectrum. This is a fair assumption because the original and sideband velocity spectrum is separated by *π* radians. We run a pulse-pair autocorrelation algorithm to estimate velocity on this half of the spectrum (making the other half zero) to get an initial crude estimate of velocity. As a next step, we use a notch filter with a normalized notch width equal to 0.5, on the original spectrum with its passband centered around the estimate of velocity from the pulse-pair algorithm. This would notch out the assumed sideband velocity component. We then run the pulse-pair autocorrelation algorithm, once again, to get an accurate estimate of velocity and spectral width. An example velocity spectrum from D3R radar at a radial with two frequencies used in alternate pulses is shown in Fig. 12.

The velocity spectrum of one range cell at a certain radial (azimuth), from D3R weather radar, after frequency switching between *f*_{1} and *f*_{2} in adjacent pulses. The number of pulses considered is 128.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The velocity spectrum of one range cell at a certain radial (azimuth), from D3R weather radar, after frequency switching between *f*_{1} and *f*_{2} in adjacent pulses. The number of pulses considered is 128.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The velocity spectrum of one range cell at a certain radial (azimuth), from D3R weather radar, after frequency switching between *f*_{1} and *f*_{2} in adjacent pulses. The number of pulses considered is 128.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

After the estimate of velocity has been obtained in one range cell of a radial, this can be propagated to other neighboring range cells below, above it, and to the one left and right of it, as an initial crude estimate of velocities for that range and radial. The assumption is that of spatial continuity of weather signals, which is good enough for most weather events. With the crude estimate, the notch filter passband would be centered around the crude velocity estimates in the adjoining range bins, and their sideband velocities could be notched out. The estimate of velocities obtained from these ranges and radials are passed to the neighboring cells, and thus, we continue to get a better estimate of velocity and spectral width progressively in the same and adjoining radials. However, we need to verify our original assumption of retaining the upper half of the spectrum in the very first range cell that we started from. With this assumption, the velocity profile was obtained at other ranges and radials. The verification process can be done by comparing with other radars or with different bands on the same radar. If we observe that the velocities are not matching for the same radials and elevations, we need to subtract out *υ*_{unb} from our computed velocities. This would construct a velocity profile in a way if we started with the other sideband in the first place. This whole method is summarized in algorithm 1 for better clarity.

Algorithm 1: Retrieval of velocity and spectral width for frequency diversity scheme

Initialization: Start with upper half of the spectrum for the range cell under consideration.

Run pulse pair to get a crude estimation of velocity.

With this estimate, notch out the other sideband in the original spectrum.

Propagate this estimate of velocity to neighboring range cells.

Use notch filter to get rid of other sideband in neighboring cells as well.

Likewise, reconstruct the velocity profile of the event by using velocity beliefs from neighboring cells.

Cross compare with NEXRAD or interleaved normal scans.

If velocity looks to be of opposite sign, subtract

*π*radians from the whole reconstructed profile.

### e. Limitations of the proposed scheme

It can be observed from the discussions in prior sections that due to the uncorrelated data samples in adjacent pulses, velocity, and spectral width need to be reconstructed with a new spectral-based method. But it works under the assumption of narrow spectral width. Such assumptions also hold for SZ code-based retrievals, which tend to behave more like random phase codes under broad spectral width. We want to emphasize that although both schemes work under narrowband assumption, our proposed scheme can tolerate twice as wide spectral width for reconstruction of second-trip velocity compared to SZ(4/16), SZ(8/64), or SZ(8/128) codes.

Another aspect which we have not dealt here is the multi-PRF situation like dual PRF techniques and staggered PRF modes. All of the illustrations carried out in this work have been tested with constant PRF mode. We feel that the extension for dual PRF-based radar might be straightforward with IF frequency switching happening within the constant PRF blocks, where dual PRF system may have two blocks of different PRFs. However, we need to do further research on the staggered mode of operation, considering how the velocities are unfolded in the staggered system and embedding our framework of different IFs. Clutter suppression is also easier to deal with in constant PRF mode but might need additional processing for multi-PRF schemes. For constant PRF mode, the original clutter spectrum and its sideband would lie around the zero Doppler and removed by the notch filter in our proposed frequency diversity system.

Another area requiring attention and is a limitation in finding out a way to differentiate between original and sideband spectra during the retrieval of velocity and spectral width. This is currently accomplished by comparing a few range cell velocities with other radar velocities. This step would require an offline processing step of comparison of few range cells in a plan position indicator (PPI) with a different radar velocity field to know if we got latched to the wrong sideband. If we find that is the case, we need to subtract out *π* radians from all radials and range cells in that PPI. This can also be managed by interleaving normal operation mode (without frequency change pulse to pulse) to verify velocity from the same radar. For applications where more trips need to be recovered, we need more IF frequencies with adequate isolation (for the digital filters cutoff and rolloff requirements). We would like to point out that these limitations do not exist with retrieval using interpulse SZ codes.

## 3. Performance test on D3R

With D3R weather radar, we can make coaligned Ku- and Ka-band echo observations for a precipitation event. It is a handy ground validation tool for the Global Precipitation Measurement (GPM) mission satellite with dual-frequency radar. D3R uses short and medium pulses with a pulse duration of 1 and 20 *μ*s, respectively. The short pulse is used to provide adequate sensitivity for the medium pulse duration and mitigate blind range of the medium pulse. The radar has been in numerous field campaigns (see Chandrasekar et al. 2017a,b, 2016). Recently, the D3R radar was upgraded with a new version of digital receiver hardware and firmware, which supports larger filter length and multiple phase coded waveforms, change of frequencies pulse to pulse, and newer IF subsystems (Kumar et al. 2018, 2017; Kumar and Chandrasekar 2017; Kumar et al. 2019). With these new subsystems, D3R was deployed for observing snow at the 2018 Winter Olympics in the Pheongchang region of South Korea (Chandrasekar et al. 2019, 2018). With a 500 *μ*s PRI, D3R’s unambiguous range is 60 km. Beyond 60 km, it is the second-trip range. In this section, we demonstrate the effectiveness of the pulse-to-pulse change of IF frequency on the D3R Ku band and also the retrieval of the second trip after 60 km of range. The first case that is demonstrated in Fig. 13, has all of the second-trip echoes and no weather echoes in the first-trip range. The normal transmission uses a chirp waveform for medium pulse centered at 65 MHz and a short pulse at 55 MHz. For the frequency diversity cases, the frequencies used are 55 and 65 MHz for short and medium (odd-numbered pulses). For the even-numbered pulses, the frequencies used are 60 and 70 MHz for short and medium pulse, respectively. The suppression of the second trip can be observed in the southeast sector. All of the second trips have been removed, and the remaining echoes are the clutter echoes. The D3R was installed at the CHILL radar site in Greeley, Colorado, when this experiment for IF change pulse to pulse was accomplished. We did a cross comparison with the nearby KFTG (NEXRAD at Denver, Colorado) and KCYS (NEXRAD at Cheyenne, Colorado) for confirming that there was indeed no first trip present for D3R unambiguous range, and all of the echoes do correspond to the second trip for D3R. For the KTFG radar, we found the common volume for the scans of D3R operating at 2° elevation and the NEXRAD at 0.48°. We can see this cross comparison in Figs. 14a and 14b. In these plots, it can be easily verified that second trip was present for the D3R range (above 60 km) which was folded into the first-trip range. The D3R was not collocated with these NEXRAD radars used for verification. Still we used the location, geometry, and scan information to find a common volume between the two radars (D3R and NEXRAD) and these common volumes are depicted with red circles in these figures. Another case is depicted in Fig. 15. Initially, we show normal chirp transmission as a reference with plots on reflectivity, velocity, and spectral width. This case has a first trip in the southwest sector, with the second-trip power overlaid. Figure 15e shows the velocity profile along 210° radial and the second-trip contamination can be clearly observed in 5 to 15 km of range. Also, Fig. 15f plots the velocity spectrum at a range of 10 km and at radial 210°, showing the second-trip velocity. The frequency diversity case is shown in Fig. 16, which is taken a couple of minutes later than the normal transmission. There were no second-trip signatures from 5 to 15 km of range at the same radial. Instead there is a replica of the original velocity spectrum as sideband, appearing at an offset of *π* radians from the original. The procedure to remove this undesired sideband is described in section 2d. After we go through the steps listed in that section, we can reconstruct velocity and spectral width by filtering out sideband power. The velocity and spectral width recovered after this process is shown in Figs. 16d and 16e, respectively.

(a) The reflectivity and (b) the velocity with normal transmission. (c) The reflectivity with frequency change pulse to pulse.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The reflectivity and (b) the velocity with normal transmission. (c) The reflectivity with frequency change pulse to pulse.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The reflectivity and (b) the velocity with normal transmission. (c) The reflectivity with frequency change pulse to pulse.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The cross comparison of D3R first-trip data with the nearby Denver NEXRAD (1405:48 UTC 7 January 2019). (b) The validation of second-trip echoes present in the Cheyenne NEXRAD, which forms a second trip for D3R. In both cases, the location of D3R is also highlighted.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The cross comparison of D3R first-trip data with the nearby Denver NEXRAD (1405:48 UTC 7 January 2019). (b) The validation of second-trip echoes present in the Cheyenne NEXRAD, which forms a second trip for D3R. In both cases, the location of D3R is also highlighted.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) The cross comparison of D3R first-trip data with the nearby Denver NEXRAD (1405:48 UTC 7 January 2019). (b) The validation of second-trip echoes present in the Cheyenne NEXRAD, which forms a second trip for D3R. In both cases, the location of D3R is also highlighted.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The reflectivity, (b) the velocity and (c) the spectral width for an event observed by D3R with normal transmission. (d) The velocity spectrum vs range plot along a certain ray with traces of a second trip in the 5–15 km range. (e) Thus, bimodal Gaussian distribution is observed at a range bin at 10 km range.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The reflectivity, (b) the velocity and (c) the spectral width for an event observed by D3R with normal transmission. (d) The velocity spectrum vs range plot along a certain ray with traces of a second trip in the 5–15 km range. (e) Thus, bimodal Gaussian distribution is observed at a range bin at 10 km range.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The reflectivity, (b) the velocity and (c) the spectral width for an event observed by D3R with normal transmission. (d) The velocity spectrum vs range plot along a certain ray with traces of a second trip in the 5–15 km range. (e) Thus, bimodal Gaussian distribution is observed at a range bin at 10 km range.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) Reflectivity, (b) velocity spectrum vs range plot (frequency diversity scheme) and (c) velocity spectrum at a range of 10 km showing the sideband. (d) Recovered velocity and (e) spectral width after removal of the sideband with a narrow spectral width assumption.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) Reflectivity, (b) velocity spectrum vs range plot (frequency diversity scheme) and (c) velocity spectrum at a range of 10 km showing the sideband. (d) Recovered velocity and (e) spectral width after removal of the sideband with a narrow spectral width assumption.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

(a) Reflectivity, (b) velocity spectrum vs range plot (frequency diversity scheme) and (c) velocity spectrum at a range of 10 km showing the sideband. (d) Recovered velocity and (e) spectral width after removal of the sideband with a narrow spectral width assumption.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

Also, for this case, we recovered the second trip, which is shown after 60 km range in Fig. 17. For doing first-trip retrieval, we programmed the sequence, *f*_{1}, *f*_{2}, in a frame of two pulse echoes, while for second-trip recovery, the sequence used is *f*_{2}, *f*_{1}.

The recovered second trips are depicted on the PPI, which is taken at an elevation of 1°.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The recovered second trips are depicted on the PPI, which is taken at an elevation of 1°.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

The recovered second trips are depicted on the PPI, which is taken at an elevation of 1°.

Citation: Journal of Atmospheric and Oceanic Technology 37, 11; 10.1175/JTECH-D-19-0223.1

## 4. Conclusions

We have proposed a scheme using interpulse frequency diversity techniques for weather radar systems and utilized the orthogonality between two frequencies in the IF band to reject the second-trip echoes. This technique shows improvement in second-trip suppression and retrieval under higher phase noise condition compared with Chu phase codes (SZ codes). Extensive time series simulations were carried out to ascertain the performance of this technique. A comparison with Chu phase code–based interpulse system was presented and showed promising results with the recovery of the weather echoes under wider dynamic range of overlaid power contamination and broad spectral width.

However, the method of velocity and spectral width recovery has limitations of comparing computed velocity with other radar or using the normal mode in the same radar. This step was not present for the SZ interpulse codes. More research needs to go into making this recovery method more robust.

## Acknowledgments

This research was funded by the NASA GPM program. The authors acknowledge the support of the NASA GPM–GV team for this research.

## APPENDIX

### Effect of Frequency Diversity Scheme on Other Dual-Polarization Moments

*H*

_{1}and the second trip by

*H*

_{2}and the baseband filter matrix by

_{bb}, then the equivalent signal model for H-pol and V-pol, for the first-trip retrieval, can be written as

_{bbh}=

_{bbv}=

*ρ*

_{vh}(0) for the first- and second-trip echoes. This dissimilarity could arise due to a slight difference in filter characteristics on the receive [cumulative effects of antialiasing or baseband cascade integrate comb (CIC)/finite impulse response (FIR)-based filtering]. Now, we would try to analyze the impact of this scheme on differential reflectivity (

*Z*

_{dr}), when the first trip is being retrieved:

_{bbh}=

_{bbv}=

*Z*

_{dr}is through the middle two terms in numerator and denominator (getting multiplied by the first-trip voltage). The second-trip voltage, however, is always preceded by the filter matrix and is going to be low. The degree of dissimilarity between the filter response on the horizontal or vertical polarized echoes is also going to contribute toward bias in

*Z*

_{dr}.

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