Noise correlation between multiple analog-to-digital converters

[radar_macgyver]

I was wondering how well, if at all, noise correlates between independent data converters that are fed the same signal, and have a correlated sampling clock. The motivation is to be able to use multiple available channels in a multi-channel digital receiver to improve SNR. In my definition of 'noise', I'm including quantization noise, noise induced by sampling clock jitter (aperture uncertainty) and thermal noise.

Of these, thermal noise should be completely uncorrelated.

Various sources (such as [1]) I found seem to indicate that quantization noise will be uncorrelated as well, but I can't wrap my head around why this is the case when the same signal is present at the input to each converter. Quantization noise is a function of the input analog signal, so wouldn't they be highly correlated?

I have no intuition on the sampling clock jitter behavior across channels.

Finally, how would I go about testing this? If I terminate the inputs of a bunch of ADCs fed by the same sampling clock, I would only be digitizing the thermal noise present on the inputs, and get a low correlation between the converter channels. If I fed a signal, say a sinusoid, then the signal would correlate with itself and give an artificially high correlation between channels. Maybe take the FFT and notch out the signal? Would I also need to notch out any spurious tones that ADC non-linearity introduces?

Thanks.

[1] "Analog-to-Digital Converter Survey and Analysis", Robert H. Walden, IEEE Journal on Selected Areas in Communications, Vol 17, No. 4, April 1999.

[rhb]

Quantization noise will be uncorrelated under the assumption that the ADC linearity errors from device to device are not correlated.  That is probably largely true, but only within limits.

The canonical reference for your question is "Random Data" by Bendat and Piersol.  I strongly recommend getting a copy.

I need to think about this for a bit.  A good starting point would be to take the difference of pairs of channels and  compute the crosscorrelation of the differences.

I'm on my second craft beer and don't want to pull down Bendat and Piersol right now as supper will be ready shortly. I'll look at the problem in the morning and give you a proper answer.  It's not a difficult problem, but the details can be tricky.  I want to get it right on the first pass.

[JS]

Hi there... The assumptions here will depend on the selected converter.

For example, for a low bit depth converter I wouldn't assume quantification noise independency, for a high bit count I would... What is high or low, when the noise takes a few full bits is high, when the noise is few LSB is low. 10 bits or 24?

Jitter noise depends on sampling freq, signal BW, and jitter characteristics. Here again, bit depth will also be a factor. If you are working with low frequency stuff jitter is hardly a problem, in high freq can be a real problem, as it could even be the sync between channels. Few Hz or GHz?

Normally for better SNR oversampling is the usual choice, I don't know why you are not doing that. Now, you asked for this specific methods, there are others. In any case, without your target SNR, (or the order of magnitude at least) and BW is hard to point you in the right direction.

JS

[rhb]

Feed the output of your signal generator to all the inputs and record a long sample.  A white noise source is best as it will test all frequencies at once.  If you use a sine wave you'll need to sweep it to spot resonant coupling between channels.

Compute the mean of the channels in the time domain and subtract that from each channel

Compute the autocorrelation of each of the differenced  outputs

Compute the  pairwise crosscorrelations of the differenced  outputs

Plot all the above as a color intensity with frequency on one axis and correlation pair index on the other.

Compute the mean and variance at each frequency for all  correlations as a set and for the  autocorrelations and the crosscorrelations as separate sets.

If the mean and variance of the autocorrelations at each frequency are the same as those of the crosscorrelations, then all the residual errors are uncorrelated.  Signal correlated errors are attributed to the signal in the analysis presented.  As noted by JS, there is a component of the quantization error which is correlated with the signal.  Clock jitter will also correlate across all channels.

[radar_macgyver]

Thanks for the responses, that gives me something to chew on...

To clarify, the converter rates I'm considering here are in the 100+ MHz range, at 16 bit resolution (~12.5 ENOB). I would like to eventually move to the GSPS range (and do RF sampling), but the ENOB drops dramatically. I was wondering if it would be possible to get back a sufficient SNR by combining multiple channels. Since this is for a relatively narrow-band application, I do consider the effects of time-averaging (one gets log4(N) bits improvement in ENOB by time-averaging N samples).

One of my existing designs uses a downconverter feeding an ADS5485 running at 200 MHz, with an FPGA digital downconverter running between 1-5 MSPS (depending on the mode). The datasheet lists the SNR at -75 dBFS (=12.5 ENOB) for a ~70 MHz input. With the highest downconversion rate available, I can expect to get 3.8 bits by time averaging (which is what the downconverter's filters do) for a total of 16.3 bits. I can verify this on my setup. I was considering, for example, the ADC12DJ2700 that can operate at 5.4 GSPS, but only gives me -48 dBFS (=8 ENOB) SNR at 6 GHz input frequency. Assuming the same 1 MSPS output rate, I would expect an improvement of 6.2 bits, for a total of 14.2 ENOB at the output, considerably worse than the original design. Since this would be a beam-formed multichannel receiver, I was wondering if I could exploit that fact to improve the SNR some.

Rhb, you may be on to something regarding use of broadband noise as the input during a bench test. All of the sources at my disposal are low ENR (HP 346B) and won't put out enough to measure directly with the ADCs, but I could try amplifying them. Thanks for the reference, I'll look it up. Bendat and Piersol also have an "engineering applications" book that I have on hold at my library.


The article [1] by Reeder et al seems to indicate that averaging multiple converters does result in an SNR improvement, and ADI apparently even sold a standard product to do this. The details are a bit hazy (the assumption is that all sources of noise are uncorrelated), but they do show measurements from time-averaging four converters. If indeed the quantization noise is correlated, I would expect that once one gets below 14 ENOB (for the AD6645 used by Reeder et al), you'd no longer see any improvement in SNR, right?


[1] Pushing the State of the Art with Multichannel A/D Converters

[rhb]

If for your signal,  you crosscorrelate the ADC outputs, you can get whatever bit resolution you desire by lengthening the window.  The extreme example of this is sign bit recording.  Sam Allen was doing this back in the 70's with vibroseis data.  You're trading acquisition time for bits. Sam was producing excellent quality data of at least 16 bits from his one bit data.  It also has the property of rejecting noise very well bursts very well which made it useful for acquiring data in urban environments.

I spent 30 years in reflection seismic processing, so multichannel data analysis is my native habitat. The assumption that noise sources are uncorrelated is what I call "sprinkling Gauss water" on the problem.

[awallin]

FWIW I think many FFT-analyzers and phase-noise analyzers use multiple ADCs and cross-spectrum to go below the noise-floor of a single ADC when computing a statistical property like PSD or phase-noise density.

A general reference is:
https://arxiv.org/abs/1003.0113