[evlatests] Effect of digitizer on synchronous detector

[Barry Clark]

Not exactly our case, but, I think, fairly close.

Consider a zero mean gaussian noise feeding a digitizer with levels at 
... -3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5 ....

The variance of the digitized signal is
1*(prob signal between -1.5 and -0.5 or signal between 0.5 and 1.5) +
  4*(prob signal between -2.5 and -1.5 or signal between 1.5 and 2.5) 
  9*(prob signal between -3.5 and -2.5 or signal between 2.5 and 3.5) + etc.

In terms of the usual error function, if the variance of the signal is a,

variance of digitized signal = (erf(1/2a) - erf(3/2a)) 
   + 4(erf(3/2a) - erf(5/2a)) + 9 (erf(5/2a) - erf(7/2a)) + ....

A ALC loop and synchronous detector will provide a response proportional
to the derivative of the logarithm of that variable.

Numerical evaluation follows.  Conclusion: things get weird at very small
signal levels, then there is a minimum in the sync detector response, of 
about 10% when the signal variance is 1; otherwise the sync detector behaves
as one might expect.

Inp. volts  Varience   Sync Det
0.200000  0.000407    2153.608674
0.400000  0.077100    119.135181
0.600000  0.239814    69.811658
0.800000  0.400838    68.584571
1.000000  0.583225    71.543083
1.200000  0.803331    73.720022
1.400000  1.063333    75.173698
1.600000  1.363333    76.174689
1.800000  1.703333    76.888321
2.000000  2.083333    77.412891
2.200000  2.503333    77.808778
2.400000  2.963333    78.114384
2.600000  3.463333    78.354948
2.800000  4.003333    78.547547
3.000000  4.583333    78.704044
3.200000  5.203333    78.832873
3.400000  5.863333    78.940158
3.600000  6.563333    79.030423
3.800000  7.303333    79.107073
4.000000  8.083333    79.172702
4.200000  8.903333    79.229320
4.400000  9.763333    79.278498
4.600000  10.663333    79.321482
4.800000  11.603333    79.359266
5.000000  12.583333    79.392654
5.200000  13.603333    79.422302
5.400000  14.663333    79.448746
5.600000  15.763333    79.472432
5.800000  16.903333    79.493729
6.000000  18.083333    79.512947
6.200000  19.303333    79.530349
6.400000  20.563333    79.546155
6.600000  21.863333    79.560554
6.800000  23.203333    79.573709
7.000000  24.583333    79.585759
7.200000  26.003333    79.596823
7.400000  27.463333    79.607007
7.600000  28.963333    79.616401
7.800000  30.503333    79.625085
8.000000  32.083333    79.633129
8.200000  33.703333    79.640593
8.400000  35.363333    79.647533
8.600000  37.063333    79.653995
8.800000  38.803333    79.660024
9.000000  40.583333    79.665657
9.200000  42.403333    79.670927
9.400000  44.263333    79.675865
9.600000  46.163333    79.680499
9.800000  48.103333    79.684853