Effect of zero-padding a DC signal

README

What happens when we take a non-zero but constant signal and upsample it by inserting zeros?

applysos.m

function y = applysos(soscoef, x)
% Apply a matrix of SOS cofficients to some time series.
%
% Tobin Fricke
% 2012-02-07

if size(soscoef,2) ~= 6
  error('soscoef is not the right size');
end

y = x;

for ii=1:size(soscoef,1)
  y = filter(soscoef(ii,1:3), soscoef(ii,4:6), y);
end

DACtimeseries.txt

           0          101
 1.52588e-05          100
 3.05176e-05          100
 4.57764e-05           99
 6.10352e-05           99
7.6294004e-05           98
 9.15528e-05           96
0.0001068116           95
0.0001220704          127
0.0001373292           92
0.00015258801           95
0.0001678468           97
0.0001831056           99
0.0001983644          100
0.00021362321          101
 0.000228882          101
0.0002441408          101
0.00025939959          101
0.00027465841          100
0.0002899172          100
0.00030517601          100
0.0003204348          100
0.00033569359          100
0.00035095241          100
0.0003662112          100
0.00038146999          100
0.00039672881          100
0.0004119876          100
0.00042724641          100
0.0004425052          100
0.00045776399          101
0.00047302281          101
0.0004882816          101
0.00050354039          100
0.00051879918          100
0.00053405802           99
0.00054931681           99
0.0005645756           98
0.00057983439           96
0.00059509318           95
0.00061035203          127
0.00062561082           92
0.00064086961           95
0.0006561284           97
0.00067138718           99
0.00068664597          100
0.00070190482          101
0.00071716361          101
0.0007324224          101
0.00074768119          101
0.00076293998          100
0.00077819882          100
0.00079345761          100
0.0008087164          100
0.00082397519          100
0.00083923398          100
0.00085449283          100
0.00086975162          100
0.00088501041          100
0.00090026919          100
0.00091552798          100
0.00093078677          100
0.00094604562          101
0.00096130441          101
0.0009765632          101
0.00099182199          100
0.0010070808          100
0.0010223396           99

simulate.m

% What is the effect of upsampling by inserting zeros 
% on a constant nonzero (DC) signal?
%
% Tobin Fricke 2012-10-08

% timeseries with 2048 sampling rate
x = ones(16, 1);
tx = (0:length(x)-1) * 1/2048;

% upsample to 64kHz (factor of 32) by inserting zeros
y = upsample(x, 32);
ty = (0:length(y)-1) * 1/(2048*32);

% Here is the digital anti-imaging filter used in CDS:
gain = 0.010581064947739;

sos = [ -1.90444302586137,    0.91078434629894,   -1.96090276933603,    0.99931924465090; ...
        -1.92390910024681,    0.93366146580083,   -1.84652529182276,    0.99866506867980];

% Put it into standard form
%
% The coefficients stored in the C code assume a form of the second order
% section where the leading coefficient in the numerator and the
% denominator are both 1.  Here we insert those two 1's into the matrix,
% and also swap the numerator and denominator.

rows = size(sos,1);
sos = [ ones(rows,1) sos(:,3:4) ones(rows,1) sos(:,1:2) ];

Y = applysos(sos, y) * gain;
plot(ty, Y, '-o');

%% Load the experimental data for comparison, and plot it

data = dlmread('DACtimeseries.txt');
N=121+10*32;  % time offset to line up properly and avoid startup transient 
plot(data(:,1), 100*ones(size(data(:,2))), 'o-r', 'linewidth',2);
hold all
plot(data(:,1), data(:,2), 'o-', 'linewidth',3);
plot(ty(1:68), Y(N:N+67)*3220, 'x-g','linewidth',2);

hold off
axis tight
legend('what I expected', 'observed DAC timeseries', 'predicted timeseries \times arbitrary scaling');
title('Effect of zero-padded upsampling on a constant DC offset');
grid on
xlabel('time [seconds]');
ylabel('counts');

orient landscape
print -dpdf zeropadding_effect.pdf
orient portrait
print -r100 -dpng zeropadding_effect.png

zeropadding_effect.pdf

zeropadding_effect.png