Software Chopping with ``infinite'' Frequency

Astronomical observations in the mm/submm bands are usually made by chopping, that is, by rapidly (1-10 Hz) moving the telescope beam between points on the sky. By differencing the measured flux from two or more nearby points, much of the atmospheric emission can be subtracted from the data. It is the residual atmospheric emission after this subtraction that appears as sky noise in these observations.

At Pico Veleta a chopping frequency of $\nu_{chop}\sim$ 2Hz is used for observations with bolometer receivers. This is usually not sufficient to remove all of the sky noise down to the receiver noise (the statistics of weather conditions at Pico Veleta can be found in the IRAM Data Base). Figure3 shows two examples of sky noise visible in data chopped with 2Hz during very good (i.e. the sky noise difference is below the bolometer noise) and moderate weather conditions (i.e. sky noise difference is $\sim$200 mJy/beam/2Hz).

Here I describe a method to achieve an ``infinite'' chopping frequency using software. This method was first introduced for mapping with bolometer receivers in October 2003, and during the following year also for all other observing modes. In principle it could be used for any receiver, providing the data streams obtained at both chopper positions are recorded independently. For observations with heterodyne receivers the chopping frequency is far below the $\sim$2Hz used with bolometers, therefore the gain in data quality described below should be even higher than for bolometer data.

The chopping via software is possible if the scanning velocity on the sky $v$ fulfills the condition

$v < 0.5\times$HPBW $\times \nu_{chop}$

where $\nu_{chop}$ is the chopping frequency. In this case the data streams from both chopper positions are spatially Nyquist sampled, therefore they can be resampled to different time values. Note that the frequency of the atmospheric variations (i.e. the sky noise) and the sampling frequency of the data are unimportant for this approach. Resampling the data streams from both chopper positions to the same time values and subtracting them afterwards gives mathematically infinite chopping frequency for one data pair.

The total number of data pairs, however, cannot be changed because the data rate is given by the data acquisition system.

Further practical limitations are:

• the receiver might be insensitive to variations of high frequency; e.g. the bolometers are insensitive to variations shorter than $\sim 10$ msec,
• the electronics might have even higher time constants; for MAMBO arrays e.g. it is above 1sec, the data streams obtained at both wobbler positions are therefore not fully independent.

Figure 4 compares the correlation plots of the difference signal obtained in conventional way (Sequential Signal Difference, SSD) and with the new method (Resampled Signal and then the Difference, RSD). The gain in data quality using different modes of data processing including the Correlated Signal Filter (CSF) in MOPSIC is further shown in Fig.5.

Although for on-off observations the scanning velocity is exactly 0, i.e. the condition given above is fulfilled, precisely these data usually can't be processed this way. The problem is the very short integration time per nodding positions (usually 10 sec), therefore the data stream obtained at one nodding position of the telescope has just 20 values. This is not enough to ensure really robust resampling, although a significant improvement of data quality can usually be obtained (Fig.6). More important, the sky noise difference can be very efficiently filtered out using the CSF in MOPSIC. The robustness of the CSF for on-off observations is much higher than that of the resampling of such short time series, therefore I do not suggest to use the new method for on-offs as long as the nodding cycle is shorter than $\sim 30$sec.

Figure 3: Correlation plots of chopped data obtained with MAMBO2 at very good (top - sky noise below the bolometer noise in data chopped with 2Hz) and moderate weather conditions (bottom - sky noise $\sim$200mJy/beam/2Hz). The noise of various Receiver Channels (RC, number is given in the upper left-hand corner) is plotted against the noise of the RC #53.

Figure 4: Correlation plots of chopped data obtained with MAMBO2. Top: conventional SSD. Bottom: The same data with RSD. The correlation coefficient of the noise is considerably lower in the second case (0.4 vs. 0.63), i.e. a large fraction of the correlated noise was removed. The labeling is as in Fig.3.

Figure 5: Upper row: maps reduced without the sky noise filter CSF; left - the difference signal calculated in the conventional way (SSD), right - the difference signal calculated using the new method (RSD). Bottom row: as above, but the sky noise was filtered using the CSF. The grainy structure in the map is due to the fact that MAMBO2 is not a ``filled array'', i.e. the spacing between the bolometer receivers ist $\sim 2\times$HPBW. As long as a correlated signal exists in the data, i.e. all bolometers show roughly the same signal in the same moment, the grainy structure is visible with the spacings of the bolometers in the final map. The ``grains'' have a size comparable to the HPBW, appear therefore as true sources for the CSF and can hardly be removed.

Figure 6: Upper row: on-offs reduced without the CSF algorithm; left - the difference signal calculated in the conventional way (SSD), right - the difference signal calculated using the new method (RSD). Bottom row: as above, but the sky noise was filtered using the CSF. Within the errors the CSF processed on-offs show the same signal for both methods of calculating the difference signal (the results are written above each plot). In all plots the red line shows the signal of the on-bolometer as a function of integration time, the blue continuous line the mean off all other bolometers; the dashed line the rms value of this mean.

Robert W. ZYLKA