Signal statistics for beamformer and correlator

Author: Eric Kooistra, Aug - Dec 2022

Purpose: Model the SNR of a beamformer and a correlator

Description:

Remarks:

Usage:

References:

  1. Understanding digital signal processing, R.G. Lyons

1 Statistics basics:

Signal statistics

If mean = 0 then var = power and std = rms.

For a complex signal (like subbands and beamlets), assume mean complex = 0 so rms = std and power = var (= std^2):

Coherent and incoherent signals. With S signals, the std of their sum signal:

Coherent averaging by summing voltage signals improves the 'coherent' SNR of a signal by a factor N^2 / N = N, because the coherent signal power increases by a factor N^2, while the incoherent noise adds as powers, so the noise power increases by a factor N.

Incoherent averaging by summing power signals does not improve the 'coherent' SNR, because the phase information of the signal is lost in the powers. Incoherent averaging does reduce the std of the signal power estimate by a factor sqrt(N), so incoherent averaging does inprove the 'incoherent' SNR, because it makes the signal power measurement more accurate.

2 Summator

Two types:

  1. Coherent summation in voltage beamformer (e.g. digital BF in LOFAR2 Station, tied array beamformer = TAB in ARTS)
  2. Incoherent summation in power statistics (e.g. SST, BST), power beamformer (e.g. IAB in ARTS)

2.1 Coherent summation (voltages beamformer)

Two signal input types:

  1. Coherent signals, add up as voltages
  2. Incoherent signal, add up as powers

In the voltage beamformer the sky signal in the beamlet direction adds coherently and the sky signals from other directions and the signals from the receivers noise add incoherently.

Conclusion: The voltage beamformer improves the 'coherent' SNR of the beamlet signal by factor S, because the coherent signal power increases by S^2 while the incoherent noise power increases by S. For very weak astronomical signals this 'coherent' SNR improvement is not enough to make them appear above the system noise, so then additional voltage beamforming is needed or integration in time using a correlator is needed.

2.2 Incoherent summation (powers beamformer)

Incoherent summation of powers from S inputs is equivalent to incoherent summation of S powers in time from a single input. Incoherent summation does not improve the 'coherent' SNR of the signal, but it does improve the accuracy of the power measurement by a factor S. Hence instead of measuring with one dish for S intervals it is equivalent to sum the powers of S dishes for 1 interval. Hence the field of view of the summed incoherent array power beam (IAB) is the same as the field of view of one signal input.

3 Correlator

3.1 Auto powers

Conclusion: The summation of power values does not improve the 'coherent' SNR, but it does improve the 'incoherent' SNR, so the accuracy of the power measurement. Therefore the SNR for the auto correlation is defined as the accuracy of the mean power measurement. This 'incoherent' SNR improves by N, so by 5 dB / decade, and applies to:

As a side effect the model (using len_factor = 1) also shows that 1 / (requested power - actual power) is proportional to N, so the accuracy of the power of the generated noise itself improves by N, so 10 dB / decade.

3.2 Cross powers

The expected cross correlation power is the power of the coherent input, because the other terms are uncorrelated and become zero:

E{(x+a)(y+a)} = E{xy} + E{xa} + E{ya} + E{a^2} = sigma_a^2 = var(a)

where:

The std of E{xy} is a measure of how close E{xy} is to zero. The var of E{xy} reduces with N_samples, so the std(E{xy}) reduces with sqrt(N_samples). Similar for E{xa} and E{ya}. The var(a) is constant and the std of E{xy} has the same (power) units as E{xy}, so therefore the 'coherent' SNR of the correlator depends on the accuracy of E{xy}, and therefore the 'coherent' SNR improves with sqrt(N_samples). This agrees with cross_SNR in the simulation.

Note that var(E{xy}) is not E{xyxy}.

Conclusion: The expected coherent cross power of the cross correlator is pow_coh. The measurement of cross_coh_mean = pow_coh becomes more accurate when N_samples increases. The incoherent cross power is cross_incoh_mean and goes to zero. The cross power is a power statistics, but the two inputs are voltages so their phase information is preserved and therefore the correlator also has a 'coherent' SNR improvement. The 'coherent' SNR of the coherent correlator is proportional to 1 / cross_incoh_mean. Dividing by almost zero causes the 'coherent' SNR to fluctuate, but on average the 'coherent' SNR of the coherent signal improves by sqrt(N_samples), so by 5 dB / decade.