Narrowband spacecraft signal processing

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range of weights available for the individual data points. Usually, during the first couple of iterations, when the calibration is poor, it is best to have nearly equal weights on all baselines. In this way, the best advantage is taken from the (𝑢, 𝑣) coverage, allowing weak baselines to help the process converge. When the calibration improves, the weights can be adjusted according to the antennas sensitivities. Otherwise, if the natural weights are kept the noise limit will be determined by the weak baselines, possibly limiting the quality of the final image.

After a number of CLEAN iterations the procedure will stop having visible improvements on the output image. At this point, the dynamic range of the map should be improved with self-calibration.

3. Self-calibration of phase and amplitude

The method of self-calibration is used to calibrate the remaining errors of the amplitude and phase components of the antenna gain. The idea be-hind the self-calibration algorithm is to minimize the differences between the observed visibilities and the model visibilities, which are theoretically calculated. The residual is minimized by adjusting the antenna gain val-ues in the model visibilities in an iterative process. The solution of the self-calibration will be the set of antenna gains that lead to the minimum residuals of the difference of the models. Subsequently, the imaging pro-cedure can start again, this time using the new solution given by the self-calibration. By alternating between CLEAN and self-calibration for a certain number of times, the obtained solution should converge to the true bright-ness distribution.

The self-calibration operation can solve amplitude components and phase components either separately or simultaneously. Usually, the phase ponents are calibrated first, and then both the phase and amplitude com-ponents simultaneously.

When the process converges, the final map (CLEANed map) is obtained.

The final map of the calibrator sources should be examined to corroborate whether the amplitude and phase calibration solutions have been correctly de-termined; by obtaining a point-like source image and by comparing it with the images presented in VLBI calibrator catalogs (for instance, the VLBA calibra-tor list). Values such as peak brightness should be comparable to the catalog images, taking into account the size of the synthesized beam.

The output of the broadband processing of the calibrator data, within the PRIDE pipeline, are the delay and phase corrections derived in the process of obtaining the calibrator image: the group delay and delay-rate residuals estimated by means of the fringe-fitting technique, and the phase errors extracted from the self-calibration process.

3.4. Narrowband spacecraft signal processing

The first step in the narrowband data processing part of the pipeline is to retrieve the topocentric Doppler detections from the single dish open-loop data collected by

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each of the participating stations. The software used to process these data is the SWSpec/SCtracker/DPLL package, developed between Metsähovi Radio Observatory (MRO) and JIVE.

3.4.1. Software spectrometer (SWSpec)

In PRIDE experiments, multiple wide sub-bands (typically 8 or 16 MHz) are necessary for a successful detection of the phase-referencing background radio source calibra-tors, while the spacecraft signal itself can be represented in a band of several kHz (taking into account the range of possible Doppler shifts). Therefore, the initial step for the spacecraft signal processing is to extract from the Mark-5 data the channel corresponding to the sub-band containing the spacecraft signal. Since the nominal frequency of the spacecraft is known, the channel where the spacecraft carrier signal is expected to have been recorded is known. Subsequently, a Window-OverLapped Add (WOLA) Discrete Fourier Transform (DTF) is applied on the data extracted and a time integration over the obtained spectra is performed. This procedure is carried out using the high resolution software spectrometer SWSpec [43], which allows the initial detection of the spacecraft carrier and sub-ranging tones, and the temporal evolution of their frequencies throughout the scans. This software, which was developed at MRO, supports different input formats; from raw data to formatted data like Mark-5A/B/C (developed by Haystack/MIT), PC-EVN (developed by Metsähovi/Aalto) or the VLBI Data Interchange Format (VDIF) (developed by a collaboration of international institutes).

Figure 3.8 shows a typical spacecraft signal spectrum, consisting of a carrier line with the highest spectral density power, subcarriers, usually separated by 100 -200 kHz from the carrier, data band with effective width of -200-500 kHz, and several ranging tones separated by∼ 1 MHz from the carrier and spread over 5-10 MHz, with a power decreasing with the harmonic number.

Figure 3.8: Example of the spectrum of a spacecraft signal. This is the power spectrum of ESA’s Mars Express spacecraft signal (X-band), obtained during the experiment GR035 (treated in Chapter 4).

3.4.Narrowband spacecraft signal processing

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Figure 3.9 shows an example of the resulting spectrum after the raw spacecraft signal has been processed with Software Spectrometer (SWSpec).

Figure 3.9: Left panel: SWSpec detection 10 Hz resolution over an 8 MHz bandwidth, 19 minutes averaged spectrum. Right panel: temporal evolution of the carrier line spectrum. Both figures correspond to detections with Wettzell station of the Venus Express.

3.4.2. Phase-stop polynomial fit

Using as an input the time-integrated spectra generated by SWSpec, the goal of this step is to determine the frequency drift of the spacecraft detections along the series of spectra. For this purpose, a polynomial fit is used to model the moving phase of the spacecraft carrier tone frequencies (Fig. 3.9, right panel) along the time-integrated spectra per scan. The 𝑀-order phase polynomial is estimated using the following relation (in practice,𝑀 = 6 is usually used):

𝑃(𝑡) = 0 + ̃𝐶 (1) ⋅ 𝑡 + ̃𝐶 (2) ⋅ 𝑡 + ... + ̃𝐶 (𝑀 − 1) ⋅ 𝑡 (3.38) where 𝑃(𝑡) is the phase polynomial function, ̃𝐶 are the phase polynomial coeffi-cients to be estimated and𝑡 is the elapsed time of the scan. The phase polynomial fit is conducted using the Weighted Least Mean Square (WLMS) method, in which the weights assigned depend on the detection’s SNR and the nearby RFI characteristics:

(𝑇 𝑊_SNR𝑇) ̃𝐶 = 𝑇 𝑊_SNR𝐹 (3.39)

where the𝑇 is the time matrix, 𝐹 is the frequency matrix and 𝑊 is the weighted SNR matrix, all representing the data along the entire integrated spectra. The output of this step are the estimates of the phase-stopping polynomial coefficients, 𝐶 .̃ They will be used as input to the next processing step: phase stopping and narrow band tone filtering and extraction.

3.4.3. Spacecraft multi-tone tracking

In order to stop the moving phase of the carrier signal and retrieve the Doppler fre-quency detections of the spacecraft signal the spacecraft multi-tone detection and tracking software (SCtracker) is used. As shown in Figure 3.9, the carrier tone fre-quency shifts in time as the observation progresses. This Doppler shift is the instanta-neous Doppler observable to be retrieved. For this purpose, it is necessary to stop the

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moving phase of the carrier tone, and accumulate the corrections (Doppler shift cor-rections) until the spectral resolution of the carrier tone is within the desired (∼mHz) level. The algorithm implemented in SCtracker applies the double-precision polyno-mial coefficients obtained in the previous step to the baseband sample sequence𝑥[𝑛]

in order to stop the carrier tone phase, by means of the following relation:

̃𝑥[𝑛] = 𝑥[𝑛]𝑒^{(± ∑ ̃ ( )⋅( ) )} (3.40) where the ̃𝑥[𝑛] are the new samples, the 𝑥[𝑛] is the original raw samples and the 𝑇 is the time samples of the spectrum. The resulting time-integrated spectrum is thus corrected for an initial Doppler shift approximation along the corresponding scan, given by the polynomial fit. Subsequently, a narrow band is selected around the spacecraft carrier tone. The selected narrow band is extracted from the stopped baseband signal and then filtered and downconverted, using a2 order WOLA Direct Fourier Transform (DFT)-based algorithm of the Hilbert transform approximation. The output of this step is the Doppler-corrected (initial fit) carrier signal in a narrow band of 2 kHz bandwidth (in contrast to the initial 8/16 MHz bandwidth of the channel where it was recorded). The extracted complex time-domain signal is written with complex floating-point precision in an output file.

3.4.4. Digital Phase-Locked Loop

Having extracted the narrow band containing the carrier tone, the remaining residuals resulting from the initial polynomial fit will be corrected for with the Digital Phase-Locked-Loop (DPLL) software, which allows to track the carrier tone at a higher pre-cision along the scan. The DPLL will run high prepre-cision iterations from equation 3.38 to 3.40 on the filtered signal. At each iteration, the PLL first calculates a new time-integrated overlapped spectra, then it estimates a new set of phase polynomial fit, and finally it performs the phase stopping of the time-integrated spectra. After each iteration, the output of the DPLL is a new filtered and down-converted signal, with its corresponding residual frequency and phase. The number of iterations is dependent on the desired frequency precision, which is usually in the order of mHz. The output of the DPLL is the filtered down-converted signal and the accumulated residual fre-quency in the stopped band. The bandwidth of the output detections is 20 Hz with a frequency spectral resolution of 2 mHz. As an indication, when running the DPLL using 20000 FFT points and 10 s integration time on a three-way Doppler detection, a Doppler noise of 2 mHz, at X-band, translates into a range rate random error of

∼ 30 𝜇m/s . This is the total Doppler noise derived from the data reduction following the PRIDE approach, and gives the uncertainty to which the spacecraft carrier tone frequency is estimated at every sampled point.

3.4.5. Phase delay of the carrier line

After running the DPLL, the topocentric Doppler detections 𝑓 can be obtained by adding the base frequency of the selected channel to the obtained time averaged car-rier tone frequencies. Then, each of the topocentric Doppler detections are reduced to geocenter using the following relation:

𝑓 _, = 𝑓 _,(𝑡 − 𝜏 ) ⋅ (1 − ̇𝜏 ) (3.41)