Accumulator Error Feedback

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[[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms:
[[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms:
z<sup>-1</sup> is a unit delay, Q is a floating-point quantizer to 64 bits,
z<sup>-1</sup> is a unit delay, Q is a floating-point quantizer to 64 bits,
-
q<sub>i</sub> represents error due to quantization (additive by definition). <br>-Jon Dattorro]]
+
q<sub>i</sub> represents error due to quantization (additive by definition). &nbsp;&nbsp;<math>-</math> Jon Dattorro]]
<pre>
<pre>
function s_hat = csum(x)
function s_hat = csum(x)

Revision as of 22:49, 28 November 2017

CSUM() in Digital Signal Processing terms:  z-1 is a unit delay, Q is a floating-point quantizer to 64 bits,  qi represents error due to quantization (additive by definition).     Jon Dattorro
CSUM() in Digital Signal Processing terms: z-1 is a unit delay, Q is a floating-point quantizer to 64 bits, qi represents error due to quantization (additive by definition).   LaTeX: - Jon Dattorro
function s_hat = csum(x)
% CSUM Sum of elements using a compensated summation algorithm.
%
% For large vectors, the native sum command in Matlab does 
% not appear to use a compensated summation algorithm which 
% can cause significant roundoff errors.
%
% This code implements a variant of Kahan's compensated 
% summation algorithm which often takes about twice as long, 
% but produces more accurate sums when the number of 
% elements is large. -David Gleich
%
% Also see SUM.
%
% % Matlab csum() example:
% clear all
% csumv = 0;
% while ~csumv
%    v = randn(13e6,1);
%    rsumv = abs(sum(v) - sum(v(end:-1:1)));
%    disp(['rsumv = ' num2str(rsumv,'%18.16f')]);
%    [~, idx] = sort(abs(v),'descend'); 
%    x = v(idx);
%    csumv = abs(csum(x) - csum(x(end:-1:1)));
%    disp(['csumv = ' num2str(csumv,'%18.16e')]);
% end

s_hat=0; e=0;
for i=1:numel(x)
   s_hat_old = s_hat; 
   y = x(i) + e; 
   s_hat = s_hat_old + y; 
   e = (s_hat_old - s_hat) + y;  %calculate difference first (Higham)
end
return

links

Accuracy and Stability of Numerical Algorithms 2e, ch.4.3, Nicholas J. Higham, 2002

For multiplier error feedback, see:

Implementation of Recursive Digital Filters for High-Fidelity Audio

Comments on Implementation of Recursive Digital Filters for High-Fidelity Audio

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