Matlab listing: zpfmin.m

function [zpmin] = zpfmin(window,MT,freqBiasMax);
% [zpmin] = zpfmin(window,MT,freqBiasMax)
% returns the minimum zero-padding factor to achieve prescribed
% tolerances in sinusoidal peak parameters measured using the QIFFT
% method with the  window.
%
% INPUTS:
%   window      = 'rect', 'hamming', 'hann', or 'blackman'
%   MT          = window length (seconds)
%   freqBiasMax = max frequency bias allowed in peak frequency (Hz)
%   ampBiasMax  = max peak-magnitude bias allowed (relative fraction)
%
%   If ampBiasMaxPct is not specified, amplitude bias is not considered.
%
% RETURNED:
%   zpmin = minimum zero-padding factor (1 means no zero padding)
%
% EXAMPLE
%   Assuming the window contains one cycle of a sinusoid whose
%   frequency f must be estimated to within 0.1% accuracy, we have
%   zpfmin('blackman',1/f,0.001*f) = 1.9 for min zero-padding factor.
%   Similarly,
%     2 cycles under the Blackman window yields zpf > 1.5
%     4 cycles under the Blackman window yields zpf > 1.2
%     6 cycles under the Blackman window yields zpf > 1.02
%
% Reference:
%    "Design Criteria for Simple Sinusoidal Parameter Estimation
%     based on Quadratic Interpolation of FFT Magnitude Peaks", by
%    Mototsugu Abe and Julius O. Smith,
%    Audio Engineering Society Convention preprint 6256", NY, 2004.

%!test
%! assert(zpfmin('blackman',1,0.001),2.3609,0.001);

% disp(sprintf('%s window',window));
switch window
 case 'rect'
  c0 = 0.4467;
  r0 = -0.3218;
  c1 = 0.8560;
  r1 = -0.2366;
 case 'hamming'
  c0 = 0.2456;
  r0 = -0.3282;
  c1 = 0.4381;
  r1 = -0.2451;
 case 'hann'
  c0 = 0.2436;
  r0 = -0.3288;
  c1 = 0.4149;
  r1 = -0.2456;
 case 'blackman'
  c0 = 0.1868;
  r0 = -0.3307;
  c1 = 0.3156;
  r1 = -0.2475;
 otherwise
  error(sprintf('Window type %s unknown - say help zpfmin',window));
end

zpmin = c0*(MT*freqBiasMax)^r0;

if nargin>3
  zpminamp = c1*(MT*freqBiasMax)^r1;
  zpmin = max(zpmin,zpminamp);
end

if strcmp(window,'rect')
  zpmin=max(1.7,zpmin); % See Abe & Smith 2004, Fig. 2 re. why this
else
  zpmin=max(1,zpmin);
end

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.