Accumulator Error Feedback

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#REDIRECT [[PageRank]]
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[[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms:
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z<sup>-1</sup> is a unit delay, Q is a floating-point quantizer to 64 bits,
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q<sub>i</sub> represents error due to quantization (additive by definition). <br>-Jon Dattorro]]
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<pre>
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function s_hat=csum(x)
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% CSUM Sum of elements using a compensated summation algorithm.
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%
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% For large vectors, the native sum command in Matlab does
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% not appear to use a compensated summation algorithm which
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% can cause significant roundoff errors.
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%
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% This code implements a variant of Kahan's compensated
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% summation algorithm which often takes about twice as long,
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% but produces more accurate sums when the number of
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% elements is large. -David Gleich
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%
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% See also SUM
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%
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% Example:
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% v=rand(1e7,1);
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% sum1 = sum(v);
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% sum2 = csum(v);
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% fprintf('sum1 = %18.16e\nsum2 = %18.16e\n', sum1, sum2);
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s_hat=0; y=0; e=0;
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for i=1:numel(x)
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s_hat_old = s_hat;
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y = x(i) + e;
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s_hat = s_hat_old + y;
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e = (s_hat_old - s_hat) + y; %calculate difference first (Higham)
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end
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</pre>
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=== links ===
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[http://www.google.com/books?id=FJyBjjtHREQC&dq=Accuracy+and+Stability+of+Numerical+Algorithms&printsec=frontcover&source=bn#PPA92,M1 Accuracy and Stability of Numerical Algorithms, Nicholas J. Higham, 1996]
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For multiplier error feedback, see:
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[http://www.stanford.edu/~dattorro/HiFi.pdf Implementation of Recursive Digital Filters for High-Fidelity Audio]
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[http://www.stanford.edu/~dattorro/CorrectionsHiFi.pdf Comments on the above...]

Revision as of 15:24, 5 March 2009

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).
-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
%
% See also SUM
%
% Example:
%   v=rand(1e7,1);
%   sum1 = sum(v);
%   sum2 = csum(v);
%   fprintf('sum1 = %18.16e\nsum2 = %18.16e\n', sum1, sum2);

s_hat=0; y=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

links

Accuracy and Stability of Numerical Algorithms, Nicholas J. Higham, 1996

For multiplier error feedback, see:

Implementation of Recursive Digital Filters for High-Fidelity Audio

Comments on the above...

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