Meteor trail echoes are associated with very small Doppler shifts due to the neutral wind. Given a short enough time period, the complex target scatter coefficient can be assumed to be constant. Even though a single meteor is a point target in range, multiple meteor echoes are often observed simultaneously. There are also many cases where meteor echoes can spread in range. Therefore, it is beneficial to treat our unknown as a range spread coherent radar target. The coherence assumption is made by assuming that the target scatter as constant over L samples. We will refer to this as the coherence time.

In time domain, a radar measurement equation for a radar target that changes amplitude as a function of time is mt=∑r=0Rϵt-rζr,i+ξt, where mt∈C is the measured complex baseband voltage seen by the receiver, ϵt∈C is the radar transmission waveform, and ζr,i∈C is the target backscatter coefficient at a given range gate r and coherence time i. Receiver noise is modeled with ξt∈C. We assume that it is a proper complex Gaussian random process ξt∼NC(0,Σ), with Σ the covariance matrix describing the correlation structure of the receiver noise.

In a multi-static measurement with multiple transmitters operating on the same center frequency, there can be multiple different propagation paths that are simultaneously actively probed by a certain receiver. In order to differentiate between transmitters, each transmitter should have a different transmit waveform ϵt,k∈C. We use k∈[0,N]∈N to index the transmitters. In this case the measurement equation is a sum of convolutions, with one convolution for each transmitter–receiver pair received by a single station mt=∑k=0N∑r=0Rϵt-r,kζr,i,k+ξt, where also the scatter is different for each transmitter–receiver propagation path ζr,i,k∈C.

Let us first write the inner sum of this equation in matrix form: mi=∑k=0NAi,kxi,k+ξi, where the unknown parameters – i.e., radar echoes from transmitter k and time t∈[iL,(i+1)L-1] – are xi,k=[ζ0,i,k,⋯,ζR,i,k]T∈CR, and the measurement vector is mi=[miL,miL+1,⋯,m(i+1)L-1]T∈CL. Here the theory matrix Ai,k∈CL×R providing the linear relation between echoes for a single transmitter k and the measurements is Ai,k=ϵiL,kϵiL-1,k⋯ϵiL-R,kϵiL+1,kϵiL+1-1,k⋯ϵiL+1-R,k⋮⋮⋱⋮ϵ(i+1)L-1,kϵ(i+1)L-1-1,k⋯ϵ(i+1)L-1-R,k. This is depicted in the form of a range–time diagram in Fig. .

Equation  can be expressed as an equation with just a single matrix mi=Aixi+ξi, where the theory matrix is a set of stacked convolution matrices: Ai=[Ai,0,⋯,Ai,N], and the unknown vector is a set of stacked parameter vectors xi=[xi,0T,⋯,xi,NT]T.

The maximum likelihood estimate for the unknown parameters (i.e., echoes for all the range gates and all transmit–receiver pairs) can be obtained using the standard maximum likelihood formula for complex linear-least-squares problems with proper complex normal random errors: xi,ML=(Ai†Σ-1Ai)-1Ai†Σ-1mi, which is solvable as long as Ai†Σ-1Ai is not singular. Here † is the Hermitian operation, i.e., a conjugate and transpose. With just a few transmitters using independent pseudorandom transmit waveforms, there are more measurements than unknowns L≥NR and the problem is overdetermined and solvable using Eq. ().

Code optimality is determined by how good the a posteriori covariance matrix is Σpost=(Ai†Σ-1Ai)-1.

For a single transmitter–receiver case, Frank and Chu codes are known to be perfect: they result in the smallest possible constant diagonal value of Σpost. However, pseudorandom codes are also known to have near-optimal covariance matrix structure .

If each station utilizes a different pseudorandom transmit waveform that is statistically orthogonal from one another, i.e., Eϵt,kϵt′,k′*=0 for all t and t′ when k≠k′ and Eϵt,kϵt′,k′*=0 for all k and k′ when t≠t′, one can analyze the multi-static measurements by assuming that there is only one transmitter, and the signals from other transmitters are additive noise mi=Ai,kxi,k+ξi′, where ξi′=ξi+∑k′≠kAi,k′xi,k′. If the radar echoes have a low signal-to-noise ratio, then we can assume that ξi′∼NC(0,Σ) and that ξi′≈ξi because the echoes from other transmitters do not significantly contribute to the receiver noise. This allows us to treat each transmit–receive pair as an independent radar measurement. This significantly simplifies the estimation problem for large-scale multi-static networks. In this case, the maximum likelihood estimate for each transmitter–receiver pair is obtained with theory matrix Ai,k used in Eq. ().

With high signal-to-noise ratio, using the signal propagation path model in Eq. () is not as accurate. But even then, the effect of not modeling the echoes from other transmissions properly only results in an increase in the variance of the estimates. If the radar transmit waveform is considered as a zero-mean random variable, then the error term ξ′ is also a zero-mean random process, and the linear-least-squares estimator still provides an unbiased estimator, albeit not the most optimal one.

In the previous sections, we did not restrict the waveform in any way, although we mention that pseudorandom waveforms are good waveforms to use. In practice, from a numerical point of view, there are good reasons to design a waveform in such a way that it repeats after a short enough period of time.

If our continuous code repeats exactly at the coherence length L, it is possible to precalculate the computationally intensive portion of the linear-least-squares estimator: B=(A†Σ-1A)-1A†Σ-1.

The maximum likelihood estimate for target backscatter can then be obtained with xML=Bm. If the coherence assumption L is shorter than the length of the repeating pseudorandom sequence Ls, the maximum likelihood can still be precalculated, but there have to be Ls/L precalculated matrices Bi∈CNR×L that cycle through, assuming that there are an integer number of coherence lengths in the pseudorandom cycle length i∈[1,Ls/L].

With a precalculated maximum likelihood estimation matrix, the deconvolution can be performed significantly faster than real time with a modern personal computer. Our deconvolution software is currently implemented with Python script that uses NumPy for computationally intensive portions of the code. This software can easily keep up with five receive channels at receive bandwidths of around 100 kHz and 10 µs range gates up to 1500 km.

When using the matrix method for analyzing target echoes, it is possible to deal with bursty interference from nearby power-line interference or a nearby radar using short transmit pulses. The rows corresponding to bad measurements can be detected using a statistical outlier test and simply removed from the theory matrix and measurement vector. This method works as long as A†Σ-1A is not singular. This is typically the case if there are more measurements than unknown parameters. Leaving out rows with time localized interference however prevents the use of a precalculated maximum likelihood estimation matrix, which results in a numerically less efficient solution for the target scatter estimate.

When detecting targets, we assume that each target is point-like and has a Doppler shift. Specular echoes have very small range migration, which means that we can assume them to be at a fixed range gate. The following equation can be used for a model when detecting targets: ma,t=ϵt-rσγa(α,ϕ)eiωt+ξt, where ma,t∈C is the complex baseband voltage received with antenna a∈N, σ∈C the complex scattering coefficient, γa(α,ϕ)∈C is the direction of arrival-dependent delay as complex phase with azimuth α∈[0,2π]⊂R and elevation ϕ∈[0,π/2]⊂R, and ω∈R is Doppler shift. The term ξt∼NC(0,bI) is measurement noise, which is assumed to be proper complex Gaussian white noise with covariance matrix bI. Here b∈R is a constant that determines the receiver noise power.

In matrix form this is m=Aθx+ξ, where m is a vector of measured complex baseband voltages from all antennas. The moving point target model parameters are θ=[ω,r,α,ϕ]T. The vector x=[σ]T is a single element vector containing the complex target scatter coefficient.

This is a non-linear statistical inverse problem with a likelihood density function proportional to p(m|θ,x)∝exp⁡-‖Aθx-m‖2.

The peak of this distribution (i.e., the maximum likelihood estimator for target parameters θ) is θML=argmaxθ‖Aθ†m‖2, which has been shown for moving point targets by, e.g., Markkanen et al. () for space debris. In practice, the maximum likelihood parameters are found using a sufficiently dense grid search in parameter space, which in this case includes range, Doppler, and angle of arrival.

For this experiment we used a 1000-baud-long binary phase code with pseudorandom bits. The baud length was 10 µs, which results in a 1.5 km range resolution. Due to the repetition of the code after 1000 bits, the range aliasing of this experiment occurs at 3000 km. Note that this is not an inherent limitation of the method. In order to increase this aliasing range, a longer code can be used. The code was sent continuously in a repeated fashion to allow the use of a numerically efficient precalculated maximum likelihood estimation matrices in the analysis as described in Sect. .

The transmitted waveform was generated with a PC, with Gaussian pulse shaping performed with a FIR filter. The digital transmit waveform is then transferred over a 1 Gb Ethernet connection to a USRP-N200 software-defined radio, which converts the digital baseband signal to an analog signal centered at 32.55 MHz with a 100 kHz bandwidth. The USRP N200 was synchronized to the global reference clock using a Trimble Thunderbolt GPS disciplined oscillator (GPSDO). The signal generated by the USRP N200 was then amplified to 30 W of continuous power and fed through a low-pass filter to suppress harmonic emissions. The 30 W signal was fed to the three-element Yagi transmit antenna of the Juliusruh SKiYMET meteor radar. Only one linear polarization component was used.

In Kühlungsborn, the signals were received with five Yagi antennas arranged in a so-called Jones configuration . These are the same antennas that are used routinely as a bistatic receiver for the Juliusruh pulsed SKiYMET meteor radar. The receiving antennas were configured to receive circular polarization. The five-channels were recorded using a five channel USRP N200 receiver setup, which was synchronized to a global reference using a Trimble Thunderbolt GPSDO. The receivers were synchronized to one another using a 1 PPS and 10 MHz signal from the GPSDO. The five channels were recorded at a 1 MHz bandwidth and further FIR filtered and decimated to 100 kHz using a Gaussian pulse shaping filter to reduce out-of-band emissions. The signals were recorded to an external hard drive for offline processing.

In order to ascertain the performance of our low-power coded CW meteor radar concept, we compare the results from our bistatic campaign with existing SKiYMET operational pulsed specular meteor radar (SMR) measurements. First, we compare the distributions of some selected parameters obtained during 24 h of operations between our CW and a standard pulsed experiment, operating on the same multi-static configuration operating in the same bistatic configuration. Since the same antennas were used, a simultaneous comparison is not possible. However, as we show below, the comparisons are meaningful. Second, we compare the winds obtained with the coded CW measurements with those obtained from the monostatic SMR system located in Collm , ∼ 350 km south of our Juliusruh–Kühlungsborn observations.

In Fig. , we show selected results of the CW experiment conducted on 11 June 2015: (a) meteor count as a function of time of the day, (b) signal-to-noise (SNR) distributions, (c) Doppler velocity distribution, (d) echo location, (e) bivariate distribution of the precedence of echoes, as function of direction cosines in x (west to east) and y (south to north) directions; and (e) altitude distribution. These parameters have been obtained on the detected echoes, which were detected with the detection schemes described in Sect. . After detection, the parameter estimation has been done followed by the procedures similar to those described in and . We show the results of all the possible meteor-like echoes detected above 140 km total range (i.e., 70 km for a round-trip monostatic range), since range-aliased meteor echoes are not expected in this coded CW experiment.

The results obtained a few days earlier (4 June 2015) with a pulsed SMR, using the same bistatic configuration, are shown in Figs.  and . These measurements have been obtained with a 15 kW peak power transmitter, 625 Hz pulse repetition frequency (i.e., maximum total unambiguous range of 480 km), and 10 µs sampling. The results are obtained using the same identification procedure used with the CW data, while the detection is the standard detection used in our operational pulsed-Doppler systems. In Fig. , we show the results of all meteor-like echoes before cleaning. The results after a cleaning procedure including the selection of echoes with underdense characteristics are shown in Fig. . The cleaning procedure consists mainly in removing those echoes that do not present clear underdense characteristics , or are outside the expected altitude range between 70 and 120 km. Most of the echoes that have been removed are caused either by pulsed external radio interference, non-specular meteor echoes, and airplane echoes. Note that specular meteor echoes coming from ranges above 480 km will be range-aliased into lower ranges, where ground and airplane echoes occur.

By comparing Figs.  and , we can clearly see that many non-meteor echoes were originally detected. They are clearly visible as narrow distributions in all the plots, but more pronounced in the bivariate direction cosine distributions (i.e., coming from specific locations) and in the velocity distributions (showing larger Doppler shifts than expected). The origin for these echoes is most likely man-made radio interference.

The results of the CW experiment are in very good quantitative agreement with the results of the pulsed configuration after cleaning. Both show a similar diurnal pattern, with a maximum count of ∼ 200 meteors in half-hour bins; the Doppler velocity distributions are also comparable, as well as the bivariate direction cosine and altitude distributions. In the case of the SNR, qualitatively the comparison is good and as expected, i.e., with increasing count as the SNR decreases. Since SNR is a relative quantity (depending on how the noise is treated), we do not focus on the actual number. Instead we observe that with the detection procedures used, specular meteor echoes are detected when SNRs are greater than ∼ 0 and ∼ -6 dB in the CW and pulsed systems, respectively. A fairer comparison would have been to use the same detection procedures in both data sets (e.g., using the Doppler grid search used in the CW experiment also in the pulse system). However, we have preferred to compare existing procedures in a standard pulsed-system, against the improvements in both transmission as well as in signal processing in the proposed coded CW system.

To further demonstrate the quality of our coded CW system results, in Fig.  we present the derived winds obtained with the CW experiment: zonal and meridional (bottom) wind. The winds are derived using the approach outlined in , , and , i.e., assuming a horizontal homogeneity of the horizontal wind in the observed volume. The resulting winds show typical summer conditions for a northern mid-altitude station with the expected wind reversal at around 90 km altitude in the zonal flow. Further, both wind components show a strong semidiurnal oscillation.

The winds observed during the same time with the Collm system are shown in Fig. . As there is no direct overlap between both observation volumes, a one-to-one correspondence of the measured zonal and meridional winds is not expected. However, the main general features observed with the CW system are also observed with the Collm system.