above: Table 1 from Performance Highlights of the ALMA Correlators

The ALMA receivers use 3-bit ADCs for what would seem to be to be a high dynamic range application needing much finer quantization to get anything useful.

Then I found these sentences within the abstract of ADC bit number and input power needed, in new radio-astronomical applications:

Abstract- For the most part, so far radio astronomy observations have been performed in protected frequency bands, reserved by ITU for scientific purposes. This means that, ideally, only the amplified equivalent system noise is present at the end of the receiver chain (i.e. the ADC input). So, typically, only a few bits are necessary to describe the signal (VLBI signals are digitised with only 2 bits), but today astronomers, in order to get more sensitivity and to boldly observe where no one has observed before, would like to study the radio sky even outside the protected bands...

And I even found a 1-bit ADC in Performance Measurements of 8-Gsps 1-bit ADCs Developed for Wideband Radio Astronomical Observations.

I think I am just missing something obvious, but I can't understand how a measurement requiring high dynamic range gets by using few-bit ADCs.

edit: Is it possible that the actual conversion of analog to digital is done to a far higher precision than suggested by the number of bits?

• As a casual non-astronomer enthusiast, I have no clue what this is asking. But +1 for the nerdiest & most impressively complex question I've ever seen on here. – iMerchant Aug 25 '16 at 5:29
• I'm not qualified but I suspect there's some Delta Sigma conversion (or similar) going on. A 1-bit ADC (really just a comparator) can be used at very high frequencies on an integrated signal, to give a high speed bitstream. (Instead of a much slower set of multi-bit samples.) Then the proportion of 1s in the bitstream indirectly gives you the analogue level. (I guess their 3-bit converter mentioned is some more exotic version of the common 1-bit method.) – Andy Aug 25 '16 at 11:58
• So that's what a $\Sigma \Delta$ ADC is - thanks! That's starting to make a little sense. I think the baseband is 0-2GHz (or 2-4GHz - it may be shifted up somewhere anyway, it's 2GHz of bandwidth), and the sample rate is only twice that - 4G samples/sec - so it's not oversampling enough for a simple $\Sigma \Delta$ but maybe that's where the 3 bits come in. – uhoh Aug 25 '16 at 12:21
• @Andy I've added a bounty. – uhoh Mar 14 '17 at 6:08
• What you are probably seeing here is using the ADCs in a pipeline. You can do very high speed conversion by using pipeline ADCs. Here you pass your signal to a number of low-bit ADCs that do fast comparisons like in a sieve. In its most simplistic incarnation, each ADC is 1-bit and it a simple comparator, so the first one looks at it and says "is it greater or less than this" and passes it on to the next comparator. – Dave Oct 14 '18 at 16:23

It is wasteful to sample with many bits because the signal to noise ratio at the ADC of a radio telescope is typically << 1, so using many bits would just be resolving noise. (An exception to this is when there is strong radio-frequency interference that needs to be resolved, but this is not a big problem for ALMA due to its location and observing frequencies).

High dynamic range measurements arise after averaging together many samples (or correlations of samples), which boosts the SNR to a meaningful level.

Using very few bits at the ADC does introduce quantization noise that reduces the efficiency of the instrument, but 3 bits is enough to achieve 96% efficiency [1].

[1] Convenient formulas for quantization efficiency

• Hey thank you for your attention to my long-lost question! Can you expand your explanation a bit so that I and other readers will be able to understand it better? I'll read the link about loss of efficiency due to quantization noise, but I can't stop worrying about possible loss of information or signal distortion due to quantization noise. Is there a simple way to understand why this doesn't introduce some kind of problem? As other systems use even 1-bit ADCs, there's something I'm totally missing here. Thanks!! – uhoh Dec 5 '16 at 0:08
• Your linked article (Thompson 2007) mentions "...Radio Research Laboratory Report 51 of Harvard University, dated 1943, at which time it was classified." I looked here thinking that an early report might contain some basic insight, but it seems it's still unavailable! – uhoh Dec 5 '16 at 0:32

The resolution of ADCs is inversely related to their conversion time. Getting more bits requires the signal to travel through more circuitry, which takes time. This is why you can have those high-quality audio ADCs with 18 or 20 bit resolution, which operate at frequencies in kHz range, meaning each conversion can take several milliseconds. At 4GS/s you only have 250 picoseconds at your disposal, so you can only get 3 bits (and only 1 bit at 8GS/s).

how a measurement requiring high dynamic range gets by using few-bit ADCs?

This depends on the nature of the measurement, but the typical solution is to make successive measurements and calculate the average.

• Thanks but I need something a little more specific than "it depends on the nature of the measurement." We know the nature of the measurement here. I don't need a full blown analysis, but some kind of mathematical outline of how a few-bit ADC can make high dynamic range measurements necessary to see a weak source in the presence of many strong radio sources. 3-bit, 2-bit, 1-bit??? – uhoh Aug 25 '16 at 11:28

Intuitively you think of quantization as something that discards information. That may be true in the end, but it is not a useful way to look at it. Think the other way around, quantization adds an error-signal. If you know what this error signal looks like, it gives you opportunity to analyze how digital processing transforms the error and if it ends up interfering with your desired signal (and how much that interference will be).

ALMA is a phased array, it gets its precision from the correlation of phases if multiple receivers (likewise, phase is typically more important than amplitude in recent modulation schemes). The error function for phase typically is a sawtooth, as the phasor (of a theoretical clean signal) rotates. How the function looks exactly and what the fundamental frequency is, depends on properties of the ADC (and sometimes on AGC settings). The error signal frequency will be n times the received frequency, n=12 or n=8 being typical values. I would have to look into the details of ALMA, I'm not familiar with this one.

Now consider how this error function is sampled. There is no way to attenuate it before sampling, so aliased images of harmonics of this sawtooth end up in your digital data. You can calculate where these harmonics are and how strong they are. And you can shift them by altering the sampling rate (with a given fixed signal frequency). If you want to observe a certain bandwidth and you do optimize sampling rate, you may find that you have f.e. the 11th harmonic (with amplitude 1/11) somewhere in your signal, but you can avoid all the lower (and stronger) harmonics.

Investing in more bits for quantization reduces the amplitude of errors, raising the fundamental freq of the error function at the same time. You may find, that the contribution of quantization errors is already in the magnitude of other noise sources, so there is not much to gain for overall system performance. This is typically the case for direct code spread spectrum applications like GNSS systems.

• I would want to move this question over to the dsp site, but I do not have enough reputation in astronomy to even suggest that (except commenting my own answer). – Andreas Sep 22 '16 at 12:31
• Why don't I ask a somewhat different question there, but please do not move this question!. Radio astronomy has some practical aspects that are specific to generating quantitative images of extended radio sources through Earth's atmosphere. See for example There is no "modulation scheme" here, and amplitude and phase are both important! – uhoh Sep 22 '16 at 13:02
• @uhoh Sorry, I was looking at this a problem from my perspective too much. This question is of course about astronomy, though it has some relations to signal processing. The methodology to look at quantization applies nevertheless. When it comes to amplitude, the integration of data from several antenna gives you more precision than just 3bit. And I would think, that power can be averaged over time, because there is no temporal structure in the observed signal. This too will add precision. – Andreas Sep 22 '16 at 16:27
• I've added a bounty - have another go? – uhoh Mar 14 '17 at 6:07