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    <p>The basic idea is to use two separate loops. The outer loop is over the <code>chi</code> variable values, whereas the inner loop is over the <code>i</code> variable values. Referring to the above diagram in the original question, the <code>i</code> variable corresponds to the x-axis (time), and the <code>j</code> variable corresponds to the y-axis (frequency). Assuming that the <code>chi</code>, <code>i</code>, and <code>j</code> variables can take on any real number, <a href="http://en.wikipedia.org/wiki/Bilinear_interpolation" rel="nofollow">bilinear interpolation</a> is then used to find an amplitude corresponding to an element in matrix <code>M</code>. The integration is just an averaging over elements of <code>M</code>.</p> <p>The following code snippet provides an overview of the basic algorithm to express elements of a matrix as a vector using the spectral collapsing from 2D to 1D. I can't find any reference for this, but it is a solution that works for me.</p> <pre><code>% Amp = amplitude vector corresponding to Figure 11.5(b) in book reference % M = matrix corresponding to the absolute value of the complex Gabor transform % matrix in Figure 11.5(a) in book reference % Nchi = number of chi in chi vector % prod = product of timestep and frequency step % dt = time step % domega = frequency step % omega_max = maximum angular frequency % i = time array element along x-axis % j = frequency array element along y-axis % current_i = current time array element in loop % current_j = current frequency array element in loop % Nchi = number of chi % Nivar = number of i variables % ivar = i variable vector % calculate for chi = 0, which only occurs when % t = 0 and omega = 0, at i = 1 av0 = mean( M(1,:) ); av1 = mean( M(2:end,1) ); av2 = mean( [av0 av1] ); Amp(1) = av2; % av_val holds the sum of all values that have been averaged av_val_sum = 0; % loop for rest of chi for ccnt = 2:Nchi % 2:Nchi av_val_sum = 0; % reset av_val_sum current_chi = chi( ccnt ); % current value of chi % loop over i vector for icnt = 1:Nivar % 1:Nivar current_i = ivar( icnt ); current_j = (current_chi / (prod * (current_i - 1))) + 1; current_t = dt * (current_i - 1); current_omega = domega * (current_j - 1); % values out of range if(current_omega &gt; omega_max) continue; end % use bilinear interpolation to find an amplitude % at current_t and current_omega from matrix M % f_x_y is the bilinear interpolated amplitude % Insert bilinear interpolation code here % add to running sum av_val_sum = av_val_sum + f_x_y; end % icnt loop % compute the average over all i av = av_val_sum / Nivar; % assign the average to Amp Amp(ccnt) = av; end % ccnt loop </code></pre>
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    1. POAlgorithm to express elements of a matrix as a vector
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    1. USNicholas Kinar
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