File:Discrete Hilbert transforms of a cosine function, using piecewise convolution.jpg

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Captions

Captions

Compare a simple FIR Hilbert transform filter to the IIR version provided by Matlab/Octave function hilbert().

Summary

[edit]
Description
English: The blue graph shows a sine function that was created by computing the Discrete Hilbert transform of a cosine function. The cosine function was divided into 4 overlapping segments, which were individually convolved with an FIR Hilbert transform filter, and the 4 output segments were seamlessly pieced together. If the DFT of the FIR filter is replaced by the trivial samples of the DTFT of an IIR Hilbert transform filter, the cosine function segments are effectively convolved with a periodic summation of the IIR filter. That results in some frequency-dependent amplitude distortion and discontinuities at the segment boundaries. Examples of these effects are shown in the red graph.
Date
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Author Bob K
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Scilab script

[edit]
// Create a 64th-order Hilbert transform filter
M = 65;         
h = hilb(M);    // applies a rectangular window to the IIR function

// Derive overlap-save parameters
overlap         = M-1;
N               = 4*overlap;  // an efficient block-size
step_size       = N-overlap;
M2 = overlap/2; // length of the edge effects for a zero-phase (non-causal) filter
h = [h(1+M2:M) zeros(1,N-M) h(1:M2)];           // convert filter to zero-phase
H1 = fft(h, -1);                                // transfer function
H2 = %i*[0 -ones(1,N/2-1) ones(1,N/2)];         // or just sample the DTFT

// Create an input function
num_steps = 4;                                  // signal length, in steps
n = (0 : num_steps*step_size+overlap)-M2;       // sample indices (minus filter delay)
cycles_per_step = 5/3;                          // just a non-integer
cycles_per_sample = cycles_per_step / step_size;
x = cos(2*%pi*cycles_per_sample*n);             // transform a pure sinusoid

// Overlap-Save convolution
position = 0;
while position+N <= length(x)
        yt = real(fft(fft(x(position+(1:N)), -1) .* H1, 1));
        y1(position+(1:step_size)) = yt(1+M2 : N-M2);
// The next 2 lines are equivalent, so the 2nd one is commented out.
        yt = real(fft(fft(x(position+(1:N)), -1) .* H2, 1));
//      yt = imag(hilbert(x(position+(1:N))));
        y2(position+(1:step_size)) = yt(1+M2 : N-M2);
        position = position + step_size;
end

// Compare the results
y1 = y1 / max(abs(y1));
y2 = y2 / max(abs(y2));
plot(y1, 'b');
plot(y2, 'r.');

title("Discrete Hilbert transforms of a cosine function, using Overlap-save algorithm", "fontsize", 4);
ylabel("amplitude", "fontsize", 3);

a = gca();
a.x_location = "origin";
a.children.children.mark_size=4;	// size of red dots

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Date/TimeThumbnailDimensionsUserComment
current15:34, 29 March 2015Thumbnail for version as of 15:34, 29 March 20151,108 × 576 (305 KB)Bob K (talk | contribs)less gaudy
23:03, 21 March 2015Thumbnail for version as of 23:03, 21 March 2015796 × 461 (283 KB)Bob K (talk | contribs)User created page with UploadWizard

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