File:DFT approximation to Hilbert filter.png

没有更高的分辨率。

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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摘要

描述

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

2007年9月27日

来源

自己的作品

作者

Bob K

授权
(二次使用本文件)

我，本作品著作权人，释出本作品至公有领域。这适用于全世界。
在一些国家这可能不合法；如果是这样的话，那么：
我无条件地授予任何人以任何目的使用本作品的权利，除非这些条件是法律规定所必需的。

PNG开发

本PNG 位图使用LibreOffice创作。

Scripts

Do it in two languages (for fun).

Scilab

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
  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;

Now do it in Octave.

Octave

pkg 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);

LaTex

$\color {OliveGreen}{\boldsymbol {h[n]={\frac {2}{\pi n}}}}$

$\color {Blue}{\boldsymbol {\operatorname {hilb} (65)}}$

$\color {Red}{\boldsymbol {\sum _{m=-\infty }^{\infty }h[n-m\cdot 512]}}$

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

en:Image:DFT approximation to Hilbert filter.png

文件历史

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) 21:52		665 × 523（24 KB）	wikimediacommons>Bob K	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.

文件用途

以下页面使用本文件：

希爾伯特轉換

File:DFT approximation to Hilbert filter.png

摘要

Scilab

Octave

LaTex

说明

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著作权持有者释出至公有领域 ^{简体中文（已转写）}

文件来源

上传者的原创作品

创作或成立時間

27 9 2007

多媒體型式

image/png

核對和

95ee1586c240f8bb958d25b4147898071aea7ae4

資料大小

25,042 字节

高度

523 像素

寬度

665 像素

文件历史

文件用途

导航菜单

File:DFT approximation to Hilbert filter.png

摘要

Scilab

Octave

LaTex

说明

此文件中描述的项目

某些值没有维基数据项目

著作权持有者释出至公有领域 简体中文（已转写）

27 9 2007

image/png

95ee1586c240f8bb958d25b4147898071aea7ae4

25,042 字节

523 像素

665 像素

文件历史

文件用途

导航菜单

搜索

著作权持有者释出至公有领域 ^{简体中文（已转写）}