File:DFT approximation to Hilbert filter.png
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DFT_approximation_to_Hilbert_filter.png (665 × 523像素,文件大小:24 KB,MIME类型:image/png)
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[编辑]描述DFT approximation to Hilbert filter.png | The green graph is a section of the infinitely long Hilbert transform filter impulse response. The blue graph is a shorter section, produced by the Scilab function hilb() for use as an FIR filter. hilb() apparently just applies a simple rect() window, but other windows are also possible. When the filtering (convolution) is performed by multiplication of DFTs in the frequency domain (circular convolution), people sometimes replace the DFT of hilb() with samples of the DTFT (discrete-time Fourier transform) of h[n] = 2/(πn), whose real and imaginary components are all just 0 or ±1, thereby avoiding actual multiplications. But in that case, the convolution is actually being done with the periodic summation of h[n], shown in red. Unlike hilb(), it never goes to zero, which means that the "edge effects" of circular convolution affect (distort) every output sample. They can't simply be eliminated by discarding a few corrupted samples. That effect is generally worse than the distortion caused by windowing the h[n] sequence, even with the crude rectangular window. (example) | |||
日期 | ||||
来源 | 自己的作品 | |||
作者 | Bob K | |||
授权 (二次使用本文件) |
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PNG开发 InfoField | 本PNG 位图使用LibreOffice创作。 |
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Scripts InfoField | Do it in two languages (for fun).
Scilab codeN = 512; % DFT size
L = N/2; % length of plots
odd = 1:2:L;
// Create a segment of the IIR filter and its periodic summation
h_IIR = zeros(1,L);
h_periodic = zeros(1,L);
for n = odd
h_IIR(n) = 2/(%pi*n);
h_periodic(n) = 2/(N*tan(%pi*n/N)); % periodic summation
end
// Equivalent method
// M = 2*L+1;
// h_IIR = hilb(M); //513-tap, FIR Hilbert transform filter
// h_IIR = h_IIR(L+2:M);
// Create a 65-tap, FIR Hilbert transform filter
minimum_display_value = 0.0001;
M = 65;
M2 = (M-1)/2;
// The next 2 statements are equivalent to the one commented out below them
h_65 = hilb(M);
h_65 = [h_65(M2+2:M) ones(1,L-M2)*minimum_display_value]; // align with h_IIR
//h_65 = [h_IIR(1:M2) ones(1,L-M2)*minimum_display_value]; // align with h_IIR
// Create another filter (equivalent to h_periodic) by sampling the DTFT
H_DFT = %i*[0 -ones(1,L-1) ones(1,L)];
h_DFT = real(fft(H_DFT, 1)); // inverse FFT
h_DFT = h_DFT(2:$); // align with h_IIR
// Display the results
r=5; g=3; b=2; // based on a call to getcolor()
plot2d(odd', [h_IIR(odd)' h_DFT(odd)' h_65(odd)'], logflag="nl", style=[g r b],..
rect=[0,minimum_display_value,256,1], frameflag=1, axesflag=1);
title("Hilbert filter (green) and two approximations", "fontsize", 4);
ylabel("impulse response (for n > 0)", "fontsize", 3);
xlabel("n (odd values only)", "fontsize", 3);
a = gca();
//a.box = "on"; included in frameflag=1
a.font_size=3; //set the tics label font size
a.x_ticks = tlist(["ticks", "locations", "labels"], [1 51 101 151 201 251],..
["1" "51" "101" "151" "201" "251"]);
// Set line thickness of plots
a.children.children.thickness=3;
// This works too
//f = gcf();
//f.children.children.children.thickness=3;
// Can do it this way when the thicknesses are not all the same:
// pb = a.children.children(1); // Note that the order (compared to "style") is LIFO
// pr = a.children.children(2);
// pg = a.children.children(3);
// pg.thickness = 3;
// pr.thickness = 3; // equivalent to set(pr,'thickness',3);
// pb.thickness = 3;
Octave codepkg load signal
N = 512; % DFT size
L = N/2; % length of plots
odd = 1:2:L;
% Create a segment of the IIR filter and its periodic summation
h_IIR = zeros(1,L);
h_periodic = zeros(1,L);
for n = odd
h_IIR(n) = 2/(pi*n);
h_periodic(n) = 2/(N*tan(pi*n/N)); % periodic summation
endfor
% Create a 65-tap, FIR Hilbert transform filter
minimum_display_value = 0.0001;
M = 65;
M2 = (M-1)/2;
h_65 = [h_IIR(1:M2) ones(1,L-M2)*minimum_display_value]; % align with h_IIR
% Create another filter (equivalent to h_periodic) by sampling the DTFT
H_DFT = i*[0 -ones(1,L-1) ones(1,L)];
h_DFT = real(ifft(H_DFT)); % inverse FFT
h_DFT = h_DFT(2:end); % align with h_IIR
% Display the results
figure
semilogy(odd', h_IIR(odd)', 'color', 'green', 'linewidth', 2)
hold on
% The next two statements are eqivalent
semilogy(odd', h_DFT(odd)', 'color', 'red', 'linewidth', 2)
% semilogy(odd', h_periodic(odd)', 'color', 'red', 'linewidth', 2)
semilogy(odd', h_65(odd)', 'color', 'blue', 'linewidth', 2)
xlim([0 256])
ylim([minimum_display_value 1])
set(gca, 'xtick', [1:50:251]);
title("Hilbert filter (green) and two approximations", "fontsize", 14);
ylabel("impulse response (for n > 0)", "fontsize", 12);
xlabel("n (odd values only)", "fontsize", 12);
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LaTex
[编辑]
La bildo estas kopiita de wikipedia:en. La originala priskribo estas (The image is copied from wikipedia: en. The original description is):
date/time | username | edit summary | source |
---|---|---|---|
23:55, 7 December 2005 | en:User:Bob K | (I created this image myself, using Matlab tools.) | http://en.wikipedia.org/wiki/Image:DFT_approximation_to_Hilbert_filter.png |
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点击某个日期/时间查看对应时刻的文件。
日期/时间 | 缩略图 | 大小 | 用户 | 备注 | |
---|---|---|---|---|---|
当前 | 2015年3月28日 (六) 20:52 | 665 × 523(24 KB) | Bob K(留言 | 贡献) | This one depicts the truncated part of the FIR filter as on the x-axis, which is slightly above zero. The actual value of the truncated part is exactly zero., of course. | |
2015年3月28日 (六) 20:27 | 659 × 518(24 KB) | Bob K(留言 | 贡献) | Improvements to the script caused some minor changes to the figure. | ||
2015年3月28日 (六) 02:37 | 587 × 559(27 KB) | Bob K(留言 | 贡献) | Add periodic summation formulation of the DFT approximation. Also, overlay an FIR approximation computed with the Scilab hilb() function. Surprisingly, it appears to be just a truncated version of the IIR function, with no windowing. | ||
2007年9月28日 (五) 00:49 | 560 × 420(6 KB) | Bob K(留言 | 贡献) | {{Information |Description= approximation error when Hilbert transform is implemented in frequency domain |Source=self-made |Date=27-Sep-2007 |Author= Bob K }} | ||
2006年3月18日 (六) 21:04 | 546 × 409(6 KB) | Maksim(留言 | 贡献) | La bildo estas kopiita de wikipedia:en. La originala priskribo estas: == Summary == I created this image myself, using Matlab tools. == Licensing == {{PD-self}} {| border="1" ! date/time || username || edit summary |---- | 23:55, 7 December 2005 || [[:e |
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水平分辨率 | 28.35 dpc |
垂直分辨率 | 28.35 dpc |
文件修改日期时间 | 2015年3月28日 (六) 20:46 |