{"id":4706,"date":"2012-05-28T01:40:34","date_gmt":"2012-05-27T17:40:34","guid":{"rendered":"http:\/\/www.andrewsun.net\/panta_rhei\/?p=4706"},"modified":"2020-02-04T23:50:51","modified_gmt":"2020-02-04T15:50:51","slug":"%e6%b3%a2%e5%bd%a2%e9%87%8d%e6%9e%84%e4%bb%a3%e7%a0%81","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/4706","title":{"rendered":"\u6ce2\u5f62\u91cd\u6784\u4ee3\u7801"},"content":{"rendered":"<div class=\"kcite-section\" kcite-section-id=\"4706\">\n<p>\u5bf9\u771f\u5b9e\u4e16\u754c\u7684\u4fe1\u53f7\u8fdb\u884c\u6c42\u5bfc\uff0c\u9700\u8981\u5148\u5927\u5e45\u63d0\u9ad8\u4fe1\u53f7\u7684\u4fe1\u566a\u6bd4\uff0c\u5426\u5219\u6c42\u5bfc\u8fd0\u7b97\u5c31\u662f\u4e00\u4e2a\u566a\u97f3\u653e\u5927\u5668\u3002\u5bf9\u4e8eLAOS\u5b9e\u9a8c\u7684\u4fe1\u53f7\uff0c\u7531\u4e8e\u5047\u8bbe\u5176\u4e3a\u4e00\u7cfb\u5217\u5947\u6b21\u8c10\u6ce2\u7684\u548c\uff1a<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Csigma%5Cleft%28t+%5Cright+%29%3D%5Csum_%7Bn%3D1%3A%5Ctextup%7Bodd%7D%7D%5E%5Cinfty%5Csigma_%5Ctextup%7Bn%7D%5Csin%5Cleft%28n%5Comega+t%2B%5Cdelta_%5Ctextup%7Bn%7D+%5Cright+%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;sigma&#92;left(t &#92;right )=&#92;sum_{n=1:&#92;textup{odd}}^&#92;infty&#92;sigma_&#92;textup{n}&#92;sin&#92;left(n&#92;omega t+&#92;delta_&#92;textup{n} &#92;right )\" class=\"latex\" \/>\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u636e\u6b64\u76f4\u63a5\u8fdb\u884c\u6ce2\u5f62\u91cd\u6784\u3002\u5373\u901a\u8fc7FFT\u5f97\u5230\u4fe1\u53f7\u7684\u8c10\u6ce2\u632f\u5e45\u548c\u76f8\u4f4d\u89d2\uff0c\u6309\u7167\u6b64\u5f0f\u91cd\u6784\u4e00\u4e2a\u5e73\u6ed1\u7684\u6ce2\u578b\uff0c\u518d\u62ff\u53bb\u505a\u5fae\u5206\u3002\u539f\u7406\u5f88\u7b80\u5355\uff0c\u4f46\u662f\u5728MATLAB\u91cc\u6574\u4e86\u534a\u5929\uff0c\u5f3a\u5ea6\u662f\u5bf9\u7684\uff0c\u4f46\u76f8\u4f4d\u89d2\u6c38\u8fdc\u662f\u9519\u7684\u3002\u7814\u7a76\u4e86\u4e00\u665a\u4e0a\uff0c\u624d\u53d1\u73b0FFT\u5f97\u5230\u7684\u76f8\u4f4d\u89d2\u603b\u662f\u5c11\u4e8690\u00b0\uff0c\u4e5f\u4e0d\u77e5\u9053\u4ec0\u4e48\u539f\u56e0\u3002\u603b\u4e4b\u52a0\u4e0a\u8fd990\u00b0\u4e4b\u540e\u5c31\u5b8c\u7f8e\u4e86\u3002\u4e3a\u4e86\u51cf\u5c11\u65e5\u540e\u6478\u7d22\uff0c\u628a\u4ee3\u7801\u8d34\u5728\u8fd9\u91cc\uff1a<\/p>\n<p><code lang=\"matlab\"><br \/>\nclear all;close all;clc;<\/code><\/p>\n<p><code lang=\"matlab\">%============ Generate the signal for reconstruction ============%<\/p>\n<p>fs=57; % sampling frequency<br \/>\nfi0=10\/3; % input frequency<\/p>\n<p>t=0:1\/fs:123456\/fs; % the length of signal is 12345.<br \/>\nt=t';<br \/>\ndelta=[1 3 2 1];<br \/>\nx1=sin(2*pi*fi0*t); % the input signal is a sinusoidal wave;<br \/>\nx2=12*sin(2*pi*fi0*t+1) +3*sin(3*2*pi*fi0*t+3)+0.8*sin(5*2*pi*fi0*t+2)+0.02*sin(7*2*pi*fi0*t+1) +2*rand(size(t)); % the output is a sum of odd harmonics<\/p>\n<p>x=[x1 x2];<\/p>\n<p>%============ Reconstruction begins ============%<br \/>\nfi=sinfapm(x(:,1),fs); % the frequency is obtained by sinewave fitting.<\/p>\n<p>Nr_cycles=floor(fi*length(x)\/fs); % number of comlete cycles<\/p>\n<p>N=round(Nr_cycles*fs\/fi); % cutoff length for coherent sampling<br \/>\nx=x(1:N,:);<\/p>\n<p>t=t(1:N); % cut the time axis too<\/p>\n<p>Xraw=fft(x(:,2));<br \/>\nphase=angle(Xraw);<\/p>\n<p>freq_bin=0:N-1; % frequency bin<br \/>\nfreq_res=fs\/N; % frequency resolution<br \/>\nfreq_i=freq_bin*freq_res; % frequency axis<\/p>\n<p>cutoff=ceil(N\/2); % cut the first half of FFT<br \/>\nXraw=2*Xraw(1:cutoff)\/N;<br \/>\nphase=phase(1:cutoff);<\/p>\n<p>max_nr_harm=7; % in practice the max number of harmonics is determined by the Nyquist frequency<\/p>\n<p>I_n=zeros((max_nr_harm+1)\/2,1); % amplitudes of the odd harmonics<br \/>\ndelta_n=zeros((max_nr_harm+1)\/2,1); % phase angles of the odd harmonics<\/p>\n<p>% extract the harmonic information<br \/>\nfor n=1:(max_nr_harm+1)\/2<br \/>\nI_n(n)=abs(Xraw(round((2*n-1)*fi\/freq_res)+1));<br \/>\ndelta_n(n)=phase(round((2*n-1)*fi\/freq_res)+1)+0.5*pi; % for the phase angles to be correct 0.5*pi must be added. The reason is unknown.<br \/>\nend<br \/>\nDC=abs(Xraw(1))*cos(angle(Xraw(1)))\/2; % Normalization of the DC component need not multiply with 2, so the division by 2. The cosine factor determine the sign of DC offset.<\/p>\n<p>fs2=300; % reconstruct the signal at a higher sampling frequency<\/p>\n<p>t2=0:1\/fs2:t(end);<br \/>\nt2=t2';<\/p>\n<p>reconst=zeros(length(t2),2);<br \/>\nreconst(:,1)=sin(2*pi*fi*t2); % the input signal at high sampling frequency<\/p>\n<p>% reconstruct the output signal by summing up the sine components<br \/>\nfor n=1:2:max_nr_harm<br \/>\nreconst(:,2)=reconst(:,2)+I_n((n+1)\/2)*sin(n*2*pi*fi*t2+delta_n((n+1)\/2));<br \/>\nend<br \/>\nreconst(:,2)=reconst(:,2)+DC; % DC shift<\/p>\n<p><\/code><code lang=\"matlab\">plot(x(:,1),x(:,2),'k',reconst(:,1),reconst(:,2),'r')<br \/>\n<\/code><\/p>\n<p>\u4e3a\u4e86\u5c55\u793a\uff0c\u8be5\u4ee3\u7801\u751f\u6210\u4e00\u4e2a\u793a\u8303\u4fe1\u53f7\u3002\u8fd9\u4e2a\u793a\u8303\u4fe1\u53f7\u7684\u9891\u7387\u3001\u53d6\u6837\u9891\u7387\u548c\u957f\u5ea6\u90fd\u5c3d\u53ef\u80fd\u7684\u4efb\u610f\uff08\u9891\u7387\u662f\u4e2a\u9664\u4e0d\u5c3d\u7684\u5c0f\u6570\uff0c\u53d6\u6837\u9891\u7387\u662f\u4e2a\u8d28\u6570\uff0c\u957f\u5ea6\u662f12345\uff09\uff0c\u4e5f\u589e\u52a0\u4e86\u6bd4\u8f83\u660e\u663e\u7684\u566a\u97f3\u3002\u6ce2\u5f62\u91cd\u6784\u65f6\uff0c\u9891\u7387\u662f\u901a\u8fc7\u5bf9\u8f93\u5165\u4fe1\u53f7\u8fdb\u884c<a href=\"http:\/\/www.andrewsun.net\/panta_rhei\/archives\/3862\">\u6b63\u5f26\u6ce2\u62df\u5408<\/a>\u6765\u83b7\u5f97\u7684\u3002\u8fd9\u4e00\u6b65\u5bf9\u539f\u4fe1\u53f7\u7684\u566a\u97f3\u8fd8\u662f\u5341\u5206\u654f\u611f\u7684\u3002\u5982\u679c\u5728\u4ee5\u4e0a\u4ee3\u7801\u7ed9\u539f\u4fe1\u53f7\u52a0\u4e0a\u9ad8\u4e8e\u5343\u5206\u4e4b\u4e00\u7684\u566a\u97f3\uff0c\u6574\u4e2a\u4ee3\u7801\u7684\u7ed3\u679c\u5c31\u5b8c\u5168\u4e0d\u53ca\u683c\u4e86\u3002\u8fd9\u5c31\u8981\u7531\u4e8e\u662f\u540e\u9762\u51b3\u5b9a\u8c10\u6ce2\u6240\u5904\u7684frequency bin\u90a3\u6b65\u5bf9\u4fe1\u53f7\u9891\u7387\u7684\u51c6\u786e\u6027\u5341\u5206\u654f\u611f\u3002<\/p>\n<p>\u6709\u4e86\u4fe1\u53f7\u7684\u9891\u7387\uff0c\u5c31\u53ef\u4ee5\u8fdb\u884c\u76f8\u5e72\u53d6\u6837\u3001FFT\u2026\u2026\u4e00\u5207\u90fd\u6309\u6b63\u5e38\u8fdb\u884c\u3002<\/p>\n<p>\u4ee5\u4e0a\u4ee3\u7801\u672c\u8eab\u4f1a\u7ed9\u51fa\u4e00\u4e2a\u7ed3\u679c\u6bd4\u8f83\u3002\u6211\u7528\u4ee5\u4e0a\u65b9\u6cd5\u5bf9\u4e00\u4e2a\u771f\u5b9e\u7684LAOS\u4fe1\u53f7\u8fdb\u884c\u91cd\u6784\uff0c\u5f97\u5230\u7684\u7ed3\u679c\u5982\u4e0b(\u9ed1\u7684\u662f\u5b9e\u9a8c\u7ed3\u679c\uff0c\u7ea2\u7684\u662f\u91cd\u6784\u66f2\u7ebf\uff09\uff1a<\/p>\n<div id=\"attachment_4710\" style=\"width: 570px\" class=\"wp-caption alignright\"><a href=\"http:\/\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2012\/05\/Untitled1.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4710\" data-attachment-id=\"4710\" data-permalink=\"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/4706\/untitled-9\" data-orig-file=\"https:\/\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2012\/05\/Untitled1.png\" data-orig-size=\"560,420\" data-comments-opened=\"0\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"\u5b9e\u9645\u4fe1\u53f7\u793a\u8303\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;\u5b9e\u9645\u4fe1\u53f7\u793a\u8303&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2012\/05\/Untitled1.png\" src=\"http:\/\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2012\/05\/Untitled1.png\" alt=\"\u5b9e\u9645\u4fe1\u53f7\u793a\u8303\" title=\"\u5b9e\u9645\u4fe1\u53f7\u793a\u8303\" width=\"560\" height=\"420\" class=\"size-full wp-image-4710\" srcset=\"https:\/\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2012\/05\/Untitled1.png 560w, https:\/\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2012\/05\/Untitled1-300x225.png 300w\" sizes=\"auto, (max-width: 560px) 100vw, 560px\" \/><\/a><p id=\"caption-attachment-4710\" class=\"wp-caption-text\">\u5b9e\u9645\u4fe1\u53f7\u793a\u8303<\/p><\/div>\n<!-- kcite active, but no citations found -->\n<\/div> <!-- kcite-section 4706 -->","protected":false},"excerpt":{"rendered":"<p>\u5bf9\u771f\u5b9e\u4e16\u754c\u7684\u4fe1\u53f7\u8fdb\u884c\u6c42\u5bfc\uff0c\u9700\u8981\u5148\u5927\u5e45\u63d0\u9ad8\u4fe1\u53f7\u7684\u4fe1\u566a\u6bd4\uff0c\u5426\u5219\u6c42\u5bfc\u8fd0\u7b97\u5c31\u662f\u4e00\u4e2a\u566a\u97f3\u653e\u5927\u5668\u3002\u5bf9\u4e8eLAOS\u5b9e\u9a8c\u7684\u4fe1\u53f7\uff0c\u7531\u4e8e\u5047\u8bbe\u5176\u4e3a\u4e00\u7cfb\u5217\u5947\u6b21\u8c10\u6ce2\u7684\u548c\uff1a\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u636e\u6b64\u76f4\u63a5\u8fdb\u884c\u6ce2\u5f62\u91cd\u6784\u3002\u5373\u901a\u8fc7FFT\u5f97\u5230\u4fe1\u53f7\u7684\u8c10\u6ce2\u632f\u5e45\u548c\u76f8\u4f4d\u89d2\uff0c\u6309\u7167\u6b64\u5f0f\u91cd\u6784\u4e00\u4e2a\u5e73\u6ed1\u7684\u6ce2\u578b\uff0c\u518d\u62ff\u53bb\u505a\u5fae\u5206\u3002\u539f\u7406\u5f88\u7b80\u5355\uff0c\u4f46\u662f\u5728MATLAB\u91cc\u6574\u4e86\u534a\u5929\uff0c\u5f3a\u5ea6\u662f\u5bf9\u7684\uff0c\u4f46\u76f8\u4f4d\u89d2\u6c38\u8fdc\u662f\u9519\u7684\u3002\u7814\u7a76\u4e86\u4e00\u665a\u4e0a\uff0c\u624d\u53d1\u73b0FFT\u5f97\u5230\u7684\u76f8\u4f4d\u89d2\u603b\u662f\u5c11\u4e8690\u00b0\uff0c\u4e5f\u4e0d\u77e5\u9053\u4ec0\u4e48\u539f\u56e0\u3002\u603b\u4e4b\u52a0\u4e0a\u8fd990\u00b0\u4e4b\u540e\u5c31\u5b8c\u7f8e\u4e86\u3002\u4e3a\u4e86\u51cf\u5c11\u65e5\u540e\u6478\u7d22\uff0c\u628a\u4ee3\u7801\u8d34\u5728\u8fd9\u91cc\uff1a clear all;close all;clc; %============ Generate the signal for reconstruction ============% fs=57; % sampling frequency fi0=10\/3; % input frequency t=0:1\/fs:123456\/fs; % the length of signal is 12345. t=t&#8217;; delta=[1 3 2 1]; x1=sin(2*pi*fi0*t); % the input signal is a sinusoidal wave; x2=12*sin(2*pi*fi0*t+1) +3*sin(3*2*pi*fi0*t+3)+0.8*sin(5*2*pi*fi0*t+2)+0.02*sin(7*2*pi*fi0*t+1) +2*rand(size(t)); % the output is a sum of odd harmonics x=[x1 x2]; [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[212,264,303],"tags":[102],"class_list":["post-4706","post","type-post","status-publish","format-standard","hentry","category-tagged","category-264","category-rheo","tag-laos"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p1aPvF-1dU","jetpack-related-posts":[{"id":5297,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5297","url_meta":{"origin":4706,"position":0},"title":"Fourier\u53d8\u6362\uff08\u4e8c\uff09","author":"Andrew","date":"2013\u5e744\u670823\u65e5","format":false,"excerpt":"[mathjax] \u4e0a\u4e00\u7bc7\u4ecb\u7ecd\u4e86\u7528Fourier\u53d8\u6362\u6765\u91cd\u6784\u4e00\u4e2a\u672a\u77e5\u6b63\u5f26\u4fe1\u53f7\u7684\u65b9\u6cd5\u3002\u672c\u7bc7\u4ecb\u7ecd\u5bf9\u4e8e\u672a\u77e5\u975e\u6b63\u5f26\u4fe1\u53f7\u7684\u60c5\u51b5\u3002 \u5927\u5e45\u632f\u8361\u526a\u5207\u7684\u53c2\u6570 \u5927\u5e45\u632f\u8361\u526a\u5207\u5b9e\u9a8c\u4ea7\u751f\u7684\u6ce2\u5f62\u662f\u975e\u6b63\u5f26\u4fe1\u53f7\uff0c\u975e\u6b63\u5f26\u662f\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7279\u70b9\u3002\u6240\u4ee5\u5bf9\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7a0b\u5ea6\u7684\u8868\u5f81\uff0c\u5c31\u662f\u5bf9\u6ce2\u5f62\u6b63\u5f26\u5f62\u72b6\u504f\u79bb\u7a0b\u5ea6\u7684\u8868\u5f81\u3002\u5173\u4e8e\u5927\u5e45\u632f\u8361\u526a\u5207\u548c\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u539f\u7406\u4e0e\u5b9e\u8df5\uff0c\u53ef\u89c1Hyhn\u7b49\u7684\u7efc\u8ff0\uff0c\u6216\u8005\u53c2\u9605\u6211\u7684\u535a\u58eb\u5b66\u4f4d\u8bba\u6587\u7b2c1.2.1\u8282\u3001\u7b2c1.3\u8282\u548c\u7b2c2\u7ae0\u3002\u672c\u6587\u53ea\u89e3\u51b3MATLAB\u7f16\u7a0b\u5b9e\u73b0\u95ee\u9898\u3002 \u5927\u5e45\u632f\u8361\u526a\u5207\uff08LAOS\uff09\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u4ece\u6d41\u53d8\u4eea\u540c\u65f6\u83b7\u53d6\u4e24\u4e2a\u4fe1\u53f7\u3002\u4e00\u4e2a\u662f\u5e94\u53d8\u7684\u4fe1\u53f7\uff0c\u4e00\u4e2a\u662f\u5e94\u529b\u7684\u4fe1\u53f7\uff08\u4e8b\u5b9e\u4e0a\uff0c\u4e00\u4e2a\u662f\u9a6c\u8fbe\u4f4d\u79fb\u4fe1\u53f7\uff0c\u53e6\u4e00\u4e2a\u662f\u529b\u77e9\u4fe1\u53f7\uff0c\u5b83\u4eec\u90fd\u8981\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u624d\u662f\u5e94\u53d8\u548c\u5e94\u529b\u3002\u6b64\u6587\u5148\u7565\u53bb\u8fd9\u4e00\u4e8b\u5b9e\u4e86\uff09\u3002\u5728\u5e94\u53d8\u63a7\u5236\u7684\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u65bd\u52a0\u4e00\u4e2a\u6b63\u5f26\u7684\u5e94\u53d8\uff0c\u6d4b\u91cf\u5e94\u529b\u7684\u54cd\u5e94\uff0c\u5373 $$\\begin{cases}\\gamma\\left(t\\right)=\\sin\\omega t\\\\ \\sigma\\left(t\\right)=\\sum_{n=0:\\text{odd}}^\\infty\\sigma_n\\sin\\left(\\omega t+\\delta_n\\right)\\end{cases}$$\u7531\u6b64\u5f0f\u53ef\u5b9a\u4e49LAOS\u5b9e\u9a8c\u7ed3\u679c\u7684\u8868\u5f81\u53c2\u6570\uff0c\u4ee5\u53ca\uff0c\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\u504f\u79bb\u6b63\u5f26\u51fd\u6570\u7684\u7a0b\u5ea6\u3002\u6b64\u5916\uff0c\u6211\u4eec\u8fd8\u4f1a\u4ee5\u5bf9\u4f5c\u56fe\uff08\u53c2\u6570\u65b9\u7a0b\u4f5c\u56fe\uff09\uff0c\u5f97\u5230Lissajous\u66f2\u7ebf\u3002\u7136\u540e\u7528\u66f2\u7ebf\u7684\u5f62\u72b6\u7279\u5f81\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\u66f2\u7ebf\u504f\u79bb\u692d\u5706\u7684\u7a0b\u5ea6\uff09\u3002\u63cf\u8ff0\u66f2\u7ebf\u5f62\u72b6\u7279\u5f81\uff0c\u5f80\u5f80\u5c31\u662f\u5404\u79cd\u5404\u6837\u7684\u5207\u7ebf\u659c\u7387\uff08\u53ef\u81ea\u884c\u4e0a\u7f51\u641c\u7d22\u590d\u4e60\u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc\uff09\u3002\u5728LAOS\u4e2d\u5e38\u7528\uff1a$$G_\\text{M}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\gamma=0}$$$$G_\\text{L}=\\frac{\\sigma{\\gamma_0}}{\\gamma_0}$$$$G_\\text{k}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\sigma=0}$$ MATLAB\u7f16\u7a0b\u6b65\u9aa4 \u5728\u8be6\u7ec6\u4ecb\u7ecdMATLAB\u4ee3\u7801\u4e4b\u524d\uff0c\u5148\u603b\u7ed3\u4e00\u4e0b\u6211\u4eec\u8981\u5b8c\u6210\u4ec0\u4e48\u4efb\u52a1\uff0c\u8ddf\u4ece\u54ea\u4e9b\u6b65\u9aa4\u3002\u6211\u4eec\u7684\u4efb\u52a1\u5c31\u662f\u8981\u4ece\u5e94\u529b\u548c\u5e94\u53d8\u7684\u4fe1\u53f7\u8ba1\u7b97\u51fa\u4e0a\u8ff0\u7684\u8fd9\u4e48\u591a\u53c2\u6570\u3002\u5b9e\u9a8c\u65f6\uff0c\u6211\u4eec\u4f1a\u89c4\u5b9a\u5e94\u53d8\u7684\u5e45\u5ea6\u548c\u9891\u7387\uff0c\u4f46\u662f\uff0c\u7531\u4e8e\u4eea\u5668\u8bef\u5dee\uff0c\u5b9e\u9645\u65bd\u52a0\u5728\u6837\u54c1\u4e0a\u7684\u5e94\u53d8\u5e45\u5ea6\u548c\u9891\u7387\uff0c\u672a\u5fc5\u5b8c\u5168\u7b49\u4e8e\u7684\u547d\u4ee4\u503c\u3002\u5c3d\u7ba1\u8fd9\u4e00\u8bef\u5dee\u975e\u5e38\u5c0f\uff0c\u4f46\u4ece\u505a\u6cd5\u4e0a\u6211\u4eec\u8fd8\u662f\u8981\u4ee5\u5b9e\u9645\u5e94\u53d8\u4fe1\u53f7\u4e3a\u51c6\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u9996\u5148\u8981\u5bf9\u5b9e\u9645\u5e94\u53d8\u4fe1\u53f7\u8fdb\u884c\u5224\u65ad\uff0c\u83b7\u5f97\u5b9e\u9645\u7684\u5e94\u53d8\u5e45\u5ea6\u548c\u9891\u7387\u3002\u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u672a\u77e5\u6b63\u5f26\u4fe1\u53f7\u7684\u632f\u5e45\u3001\u9891\u7387\u548c\u76f8\u4f4d\u7684\u65b9\u6cd5\u5df2\u7ecf\u5728\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4e2d\u4ecb\u7ecd\u4e86\u3002 \u7136\u540e\uff0c\u5c31\u8981\u5904\u7406\u5e94\u529b\u7684\u4fe1\u53f7\u3002\u4e3a\u4e86\u83b7\u5f97\u548c\u540c\u6837\u9700\u8981\u5bf9\u8fdb\u884cfft\u5904\u7406\uff0c\u65b9\u6cd5\u8ddf\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4ecb\u7ecd\u7684\u7c7b\u4f3c\uff0c\u533a\u522b\u53ea\u5728\u4e8e\u6211\u4eec\u9664\u4e86\u8981\u83b7\u5f97\u57fa\u9891\u7684\u5e45\u503c\u548c\u76f8\u4f4d\u4e4b\u5916\uff0c\u8fd8\u8981\u83b7\u5f97\u9ad8\u6b21\u8c10\u6ce2\u7684\u5e45\u503c\u548c\u76f8\u4f4d\u3002\u7406\u8bba\u4e0a\u6765\u8bf4\uff0c\u4f46\u662f\u5b9e\u9645\u4e0a\u6211\u4eec\u6839\u636e\u9700\u8981\u53d6\u6709\u9650\u7684n\u4e2a\u9ad8\u6b21\u8c10\u6ce2\uff0c\u5c31\u53ef\u4ee5\u6bd4\u8f83\u51c6\u786e\u5730\u91cd\u6784\u51fa\u7684\u6ce2\u5f62\u4e86\u3002\u63d0\u9192\u4e00\u4e0b\uff0c\u5b9a\u4e49\u5f0f\u91cc\u5e94\u53d8\u662f\u4e00\u4e2a\u96f6\u76f8\u4f4d\u7684\u6807\u51c6\u6b63\u5f26\u51fd\u6570\uff0c\u4f46\u5b9e\u9645\u91cd\u6784\u7684\u5e94\u53d8\u6ce2\u5f62\u53ef\u80fd\u4f1a\u6709\u975e\u96f6\u7684\u76f8\u4f4d\uff0c\u8fd9\u65f6\uff0c\u5e94\u529b\u7684\u5404\u4e2a\u8c10\u6ce2\u7684\u76f8\u4f4d\u8981\u90fd\u51cf\u53bb\u5e94\u53d8\u7684\u76f8\u4f4d\uff0c\u624d\u662f\u6211\u4eec\u8981\u6c42\u7684\u3002 Lissajous\u66f2\u7ebf\u7684\u5404\u79cd\u5bfc\u6570\uff0c\u8981\u540c\u65f6\u5728\u548c\u90fd\u91cd\u6784\u4e86\u4e4b\u540e\uff0c\u518d\u8fdb\u884c\u6c42\u5bfc\u8ba1\u7b97\u5f97\u5230\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u8981\u5b8c\u6210\u4ee5\u4e0b\u4e09\u4e2a\u6b65\u9aa4\uff1a \u91cd\u6784\u5e94\u53d8\uff0c\u83b7\u5f97\u3001\u548c\u76f8\u4f4d \u91cd\u6784\uff0c\u83b7\u5f97\u548c\uff0c\u8ba1\u7b97 \u7528\u91cd\u6784\u7684\u548c\u4f5cLissajous\u66f2\u7ebf\uff0c\u4ee5\u53ca\u8ba1\u7b97\u3001\u548c \u91cd\u6784\u548c MATLAB\u5e2e\u52a9\u6587\u4ef6\u91ccfft\u547d\u4ee4\u7684\u6761\u76ee\u6709\u4e00\u4e2a\u8303\u4f8b\uff0c\u5e94\u8be5\u80fd\u89e3\u51b3\u5927\u90e8\u5206\u95ee\u9898\u4e86\u3002\u5176\u4e2d\u5728coherent sampling\uff08\u89c1\u4e0a\u7bc7\u6587\u7ae0\uff09\u5904\uff0c\u8303\u4f8b\u7528next power of 2\uff0c\u5176\u5b9e\u6ca1\u6709\u5fc5\u8981\u3002\u4e0b\u9762\u7684\u4ee3\u7801\uff0c\u6211\u5c31\u4e0d\u591a\u4f5c\u89e3\u91ca\u4e86\u3002 clear all clc % the strain signal fs=1000; % sampling frequency t=0:1\/fs:30.2; % the time series of the strain signal t=t(:); % convert to colume vector w=5; %\u2026","rel":"","context":"In &quot;\u5de5\u4f5c\u7b14\u8bb0&quot;","block_context":{"text":"\u5de5\u4f5c\u7b14\u8bb0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/personal\/%e5%b7%a5%e4%bd%9c%e7%ac%94%e8%ae%b0"},"img":{"alt_text":"Figure 1 Parts of the strain and the stress signals","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/fig1.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/fig1.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/fig1.png?resize=525%2C300 1.5x"},"classes":[]},{"id":5377,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5377","url_meta":{"origin":4706,"position":1},"title":"Fourier\u53d8\u6362\uff08\u4e09\uff09","author":"Andrew","date":"2013\u5e745\u67082\u65e5","format":false,"excerpt":"[mathjax] \u901a\u8fc7\u4e0a\u4e00\u7bc7\u6211\u4eec\u5df2\u7ecf\u5b66\u4f1a\u600e\u6837\u901a\u8fc7Fourier\u53d8\u6362\u91cd\u6784\u5e94\u53d8\u548c\u5e94\u529b\u7684\u6ce2\u5f62\u3002\u81f3\u5c11In\/1\u662f\u5df2\u7ecf\u53ef\u4ee5\u76f4\u63a5\u7528Fourier\u53d8\u6362\u7684\u7ed3\u679c\u8ba1\u7b97\u4e86\u3002\u672c\u6587\u5148\u4ecb\u7ed3\u5b9e\u9645\u5b9e\u9a8c\u6570\u636e\u7684\u5bfc\u5165\uff0c\u518d\u7ee7\u7eed\u4e0a\u4e00\u7bc7\u7684\u8bdd\u9898\u4ecb\u7ecd\u5982\u4f55\u8ba1\u7b97In\/1\u548c\u5176\u4f59\u7684LAOS\u53c2\u6570\u3002 LAOS\u5b9e\u9a8c\u662f\u600e\u4e48\u5b9e\u65bd\u7684 \u6211\u4eec\u662f\u5728TA ARES-RFS\u6d41\u53d8\u4eea\u4e0a\u505aLAOS\u5b9e\u9a8c\u3002\u516c\u53f8\u7684\u540d\u5b57\uff1aTA Instruments\u3002\u8fd9\u662f\u4e00\u53f0\u5e94\u53d8\u63a7\u5236\u578b\u6d41\u53d8\u4eea\uff08strain-controlled rheometer\uff09\uff0c\u76f8\u6bd4\u4e4b\u4e0b\u540c\u516c\u53f8\u7684AR-G2\u5219\u662f\u5e94\u529b\u63a7\u5236\u578b\uff08stress-controlled\uff09\u3002\u8fd9\u4e2a\u5c0f\u518c\u5b50\u4e0a\u4ecb\u7ecd\u4e86ARES\u548cAR-G2\u4e24\u53f0\u6d41\u53d8\u4eea\u7684\u7279\u70b9\uff0c\u5176\u4e2d\u7684\u539f\u7406\u56fe\u548c\u5177\u4f53\u53c2\u6570\u53ef\u4ee5\u7528\u5728\u6bd5\u4e1a\u8bba\u6587\u4e2d\u3002\u8fdb\u884cLAOS\u5b9e\u9a8c\u9700\u8981\u6d41\u53d8\u4eea\u6301\u7eed\u5bf9\u6837\u54c1\u65bd\u52a0\u6b63\u5f26\u5e94\u53d8\uff0c\u6211\u4eec\u662f\u5229\u7528\u6d41\u53d8\u4eea\u63a7\u5236\u8f6f\u4ef6TA Orchestrator 7.2.1\u81ea\u5e26\u7684\u6d4b\u8bd5\u6a21\u5f0fdynamic time sweep\u6765\u63a7\u5236\u6d41\u53d8\u4eea\u8fdb\u884c\u8fd9\u4e00\u64cd\u4f5c\u3002Dynamic time sweep\u662f\u56fa\u5b9a\u632f\u8361\u7684\u5e94\u53d8\u5e45\u5ea6\u03b30\u548c\u9891\u7387\u03c9\uff0c\u6d4b\u91cfG'\u548cG''\u968f\u65f6\u95f4\u53d8\u5316\u7684\u6d4b\u8bd5\uff0c\u6240\u4ee5\u6b63\u9002\u5408\u6211\u4eec\u3002\u5728Orchestrator\u8f6f\u4ef6\u4e2d\u8bbe\u5b9a\u597ddynamic time sweep\u7684\u5e94\u53d8\u548c\u9891\u7387\uff0c\u4eea\u5668\u5c31\u4f1a\u4ee5\u76f8\u5e94\u7684\u6b63\u5f26\u51fd\u6570\u53bb\u63a7\u5236\u9a6c\u8fbe\u4e86\u3002\u6b64\u65f6\uff0c\u6211\u4eec\u9700\u8981\u9a6c\u4e0a\u4ece\u4eea\u5668\u80cc\u540e\u7684\u6a21\u62df\u8f93\u51fa\u7aef\u53e3\u540c\u65f6\u8bfb\u51fa\u4eea\u5668\u7684\u6d4b\u91cf\u4fe1\u53f7\uff0c\u5e76\u7528\u4e00\u4e2a\u6570\u6a21\u8f6c\u6362\u5668\uff08analog-to-digital converter, ADC\uff09\u8f6c\u6362\u6210\u6570\u5b57\u4fe1\u53f7\uff0c\u8f93\u5165\u5230\u8ba1\u7b97\u673a\u3002\u6211\u4eec\u7528\u7684ADC\u662fNI USB-9215A\uff0c\u8981\u901a\u8fc7\u7535\u8111\u6765\u63a7\u5236\u5b83\u4f55\u65f6\u5f00\u59cb\u548c\u505c\u6b62\u91c7\u6837\uff0c\u4ee5\u53ca\u4ee5\u591a\u5927\u7684\u91c7\u6837\u9891\u7387\u8fdb\u884c\u91c7\u6837\u3002\u8bb8\u591a\u8f6f\u4ef6\u90fd\u53ef\u4ee5\u901a\u8fc7\u5b83\u7684\u9a71\u52a8\u7a0b\u5e8f\u63a5\u53e3\u6765\u5b9e\u65bd\u63a7\u5236\uff08MATLAB\u672c\u8eab\u5c31\u53ef\u4ee5\uff09\uff0c\u5728LAOS\u5b9e\u9a8c\u4e2d\u6211\u4eec\u7528\u4e00\u4e2aLabView\u5f00\u53d1\u7684\u5de5\u5177FT Meassurement\u6765\u91c7\u96c6\u6570\u636e\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u7528\u4e00\u53f0\u7535\u8111\uff0c\u7528\u4e24\u4e2a\u8f6f\u4ef6\u5206\u800c\u63a7\u5236\u4e0d\u540c\u7684\u8fc7\u7a0b\u3002Orchestrator\u63a7\u5236\u6d41\u53d8\u4eea\u7684\u8fd0\u4f5c\uff0c\u800cFT Meassurement\u63a7\u5236ADC\u3002\u8fd9\u4e00\u6d41\u7a0b\u56fe\u53ef\u53c2\u8003\u548c\u4e2d\u7684\u7528\u56fe\u3002\u5177\u4f53\u4fe1\u53f7\u7ebf\u63a5\u6cd5\u548c\u5904\u7406\u65b9\u5f0f\u89c1\u4e0b\u9762\u7684PPT\uff08\u53ef\u4e0b\u8f7d\uff09\u3002 ARES\u6a21\u62df\u8f93\u51fa View more presentations or Upload your own. \u4e0a\u7bc7\u6587\u7ae0\u4e5f\u8bf4\u8fc7\uff0c\u4ece\u6d41\u53d8\u4eea\u91c7\u96c6\u5230\u7684\u4e00\u4e2a\u662f\u9a6c\u8fbe\u4f4d\u79fb\u7684\u7535\u538b\u4fe1\u53f7\uff0c\u53e6\u4e00\u4e2a\u662f\u529b\u77e9\u7684\u7535\u538b\u4fe1\u53f7\uff0c\u5b83\u4eec\u90fd\u8981\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u624d\u662f\u5e94\u53d8\u548c\u5e94\u529b\u3002\u4eceARES\u6d41\u53d8\u4eea\u7684\u8bf4\u4e66\u4e2d\u53ef\u4ee5\u67e5\u5230\u80cc\u677f\u8f93\u51fa\u4fe1\u53f7\u7684\u5177\u4f53\u60c5\u51b5\uff1a TORQUE OUT\uff1a\u4ece0\u5230\u6ee1\u91cf\u7a0b\uff0c\u5bf9\u5e94\u7535\u538b\u662f0 ~ 5 V\u3002\u7531\u4e8e\u8f6c\u77e9\u53ef\u4ee5\u6709\u4e24\u4e2a\u65b9\u5411\uff0c\u6240\u4ee5\u603b\u7684\u7535\u538b\u8303\u56f4\u5c31\u662f-5 ~ +5 V STRAIN OUT\uff1a\u867d\u7136\u6807\u8bb0\u662fSTRAIN\uff0c\u4f46\u5176\u5b9e\u53ea\u662f\u9a6c\u8fbe\u8f6c\u52a8\u7684\u89d2\u4f4d\u79fb\u3002\u4ece0 ~ 0.5 rad\u5bf9\u5e94\u7535\u538b\u662f0~5V\uff0c\u8003\u8651\u4e24\u4e2a\u8f6c\u52a8\u65b9\u5411\uff0c\u603b\u7535\u538b\u8303\u56f4\u662f-5~+5 V TORQUE OUT\u7684\u201c\u6ee1\u91cf\u7a0b\u201d\u5230\u5e95\u662f\u591a\u5c11\u5462\uff1f\u7531\u4e8e\u6211\u4eec\u7684ARES\u6709\u4e24\u4e2a\u6863\u4f4d\u7684\u529b\u77e9\u4f20\u611f\u5668\u3002Xducer 1\u7684\u4f20\u611f\u5668\u53eb\u505a\u201c20g FRT\u201d\u300220g\u662f\u6307\u8be5\u6863\u4f4d\u7684\u6700\u5927\u91cf\u7a0b\u662f20 g\u00b7cm\uff0c\u529b\u77e9\u7684\u4e00\u4e2a\u975eSI\u5355\u4f4d\uff0cSI\u5355\u4f4d\u662fN\u00b7m\u30021 g\u00b7cm =\u2026","rel":"","context":"In &quot;\u5de5\u4f5c\u7b14\u8bb0&quot;","block_context":{"text":"\u5de5\u4f5c\u7b14\u8bb0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/personal\/%e5%b7%a5%e4%bd%9c%e7%ac%94%e8%ae%b0"},"img":{"alt_text":"Figure 1 Cone-plate geometry","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/05\/cap3.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/05\/cap3.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/05\/cap3.png?resize=525%2C300 1.5x"},"classes":[]},{"id":5244,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5244","url_meta":{"origin":4706,"position":2},"title":"Fourier\u53d8\u6362\uff08\u4e00\uff09","author":"Andrew","date":"2013\u5e744\u67088\u65e5","format":false,"excerpt":"[mathjax] \u4e3a\u4e86\u5e26\u5e08\u5f1f\u505a\u5b9e\u9a8c\uff0c\u7b80\u8981\u603b\u7ed3\u4e00\u4e0b\u5728MATLAB\u91cc\u8fdb\u884cFourier\u53d8\u6362\u7684\u4e00\u4e9b\u4ee3\u7801\uff0c\u4ee5\u4f9b\u5927\u5bb6\u53c2\u8003\u3002 Discrete Fourier transform (DFT): The sequence of N complex numbers x0, ..., xN-1 is transformed into an N-periodic sequence of complex numbers according to the DFT formula . \u4e00\u822c\u628a\u53d8\u6362\u524d\u7684\u4fe1\u53f7xn\u5f53\u6210\u65f6\u57df\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u7684Xk\u5f53\u6210\u9891\u57df\u4fe1\u53f7\u3002\u53ef\u89c1\uff0c\u65f6\u57df\u662fN\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u9891\u57df\u8fd8\u662fN\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\u3002\u4ec0\u4e48\u5468\u671f\u554a\u9891\u7387\u554a\u8fd9\u4e9b\u4fe1\u606f\u662f\u6682\u65f6\u6ca1\u6709\u7684\u3002\u4f8b\u5982xn\u4e3a\u4ee5\u4e0b\u8fd9\u4e2a\u5e8f\u5217\uff08Fig. 1\uff09\uff0c\u6a2a\u5750\u6807\u5c31\u662f\u5e8f\u53f7n\u3002\u8fd9\u6bb5\u5e8f\u5217\uff0c\u53ef\u4ee5\u662f\u4e00\u6bb510\u79d2\u949f\u7684\u7684\u4fe1\u53f7\uff0c\u4e5f\u53ef\u4ee5\u662f\u4e00\u6bb5100\u79d2\u949f\u7684\u4fe1\u53f7\u3002\u5f53\u7136\uff0c\u5b9e\u9645\u60c5\u51b5\u4e0b\u4e00\u6bb5\u4fe1\u53f7\u662f\u591a\u5c11\u79d2\u662f\u4e8b\u5148\u77e5\u9053\u7684\uff0c\u51fa\u6765\u7684\u4fe1\u53f7\u4e00\u5171\u6709\u591a\u5c11\u4e2a\u5355\u5143\u503c\uff08N\uff09\u5219\u53d6\u51b3\u4e8e\u6211\u4eec\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\u3002\u5982\u679c\u4e00\u6bb510\u79d2\u949f\u7684\u4fe1\u53f7\uff0c\u6211\u4eec\u6bcf\u79d2\u53d6\u4e00\u6b21\u6837\uff0c\u8fd9\u4e2a\u5e8f\u5217\u5c31\u670910\u4e2a\u5355\u5143\uff08N = 10\uff09\u3002\u53cd\u4e4b\uff0cFig. 1\u7684\u8fd9\u4e2a\u5e8f\u5217N = 93\uff0c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u96940.1\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u7684\u603b\u65f6\u957f\u5c31\u662f9.2\u79d2\uff1b\u800c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u969410\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u603b\u65f6\u957f\u5c31\u662f920\u79d2\u4e86\u3002\u6bcf\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\uff0c\u53ef\u4ee5\u7528\u53d6\u6837\u9891\u7387fs\u6765\u8868\u793a\u3002fs = 10 Hz\u5c31\u662f\u6bcf\u96940.1 s\u53d6\u4e00\u6b21\u6837\u3002\u6240\u4ee5\uff0c\u4e3a\u4e86\u7ed9\u4e00\u5bf9\u79bb\u6563Fourier\u53d8\u6362xn\u548cXk\u8d4b\u4ee5\u5468\u671f\u6216\u9891\u7387\u7684\u4fe1\u606f\uff0c\u5fc5\u987b\u7ed9\u65f6\u57df\u4fe1\u53f7\u89c4\u5b9a\u4e00\u4e2afs\u3002\u4e00\u65e6fs\u89c4\u5b9a\u4e86\uff0c\u5468\u671f\u548c\u9891\u7387\u4e5f\u5c31\u89c4\u5b9a\u4e86\u3002\u8fd8\u62ffFig. 1\u7684\u5e8f\u5217\u4e3a\u4f8b\uff0c\u8bbe\u8be5\u5e8f\u5217\u7684\u53d6\u6837\u9891\u7387fs = 10 Hz\uff0c\u5219\u8be5\u5e8f\u5217\u6bcf\u4e2a\u5355\u5143\u9694\u4e860.1 s\u3002\u6211\u4eec\u53ef\u4ee5\u6570\u4e00\u4e0b\u8fd9\u4e2a\u5e8f\u5217\u4e00\u4e2a\u5468\u671f\u6709\u591a\u5c11\u4e2a\u5355\u5143\u2014\u201420\u4e2a\uff0c\u90a3\u4e48\u8be5\u5e8f\u5217\u7684\u5468\u671fT = 2 s\uff0c\u9891\u7387f =\u2026","rel":"","context":"In &quot;\u5de5\u4f5c\u7b14\u8bb0&quot;","block_context":{"text":"\u5de5\u4f5c\u7b14\u8bb0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/personal\/%e5%b7%a5%e4%bd%9c%e7%ac%94%e8%ae%b0"},"img":{"alt_text":"Figure 1","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=525%2C300 1.5x, https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=700%2C400 2x"},"classes":[]},{"id":9177,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9177","url_meta":{"origin":4706,"position":3},"title":"\u5728\u7535\u5b50\u8ba1\u7b97\u673a\u4e4b\u524d\u7684Fourier\u5206\u6790","author":"Andrew","date":"2025\u5e743\u670830\u65e5","format":"aside","excerpt":"[latexpage] \u5982\u679c\u6211\u4eec\u8981\u5bf9\u4e00\u4e2a\u5468\u671f\u4fe1\u53f7\u8fdb\u884cFourier\u7ea7\u6570\u5206\u6790\uff0c\u90a3\u4e48\u6211\u4eec\u662f\u5047\u5b9a\u8be5\u4fe1\u53f7\u53ef\u4ee5\u8868\u793a\u6210\\[f\\left(x\\right)=\\sum_{n=-\\infty}^\\infty c_n e^{\\mathrm{i}2\\pi n x\/P}\\]\u5176\u4e2d\\[c_n=\\frac{1}{P}\\int_0^P f\\left(x\\right)e^{-\\mathrm{i} 2\\pi n x\/P}\\mathrm{d}x\\] \u5728\u4eca\u5929\uff0c\u6211\u4eec\u7684\u5b9e\u9a8c\u4fe1\u53f7\u57fa\u672c\u90fd\u5df2\u7ecf\u6570\u5b57\u5316\uff0c\u5728\u7535\u5b50\u8ba1\u7b97\u673a\u4e2d\u4e00\u79d2\u949f\u4e0d\u5230\u5c31\u505a\u5b8c\u4e86\u4e00\u4e2a\u5feb\u901fFourier\u53d8\u6362\u8ba1\u7b97\u3002\u4f46\u662f\u5728\u4e0a\u4e16\u7eaa50~60\u5e74\u4ee3\uff0c\u7535\u5b50\u8ba1\u7b97\u673a\u8fd8\u6ca1\u53d1\u660e\u4e4b\u524d\uff0c\u5b9e\u9a8c\u7ed3\u679c\u672c\u8eab\u5c31\u53ef\u80fd\u662f\u901a\u8fc7X-Y\u8bb0\u5f55\u4eea\u6216\u53cc\u7b14\u8bb0\u5f55\u4eea\u753b\u5728\u5750\u6807\u7eb8\u4e0a\u7684\u3002\u5f53\u65f6\u7684\u4eba\u662f\u600e\u6837\u5bf9\u8fd9\u4e48\u4e00\u6bb5\u58a8\u8ff9\u8fdb\u884cFourier\u7ea7\u6570\u5206\u6790\u7684\u5462\uff1f\u6211\u4eec\u53ef\u4ee5\u770b\u5230\uff0cFourier\u7ea7\u6570\u7684\u5404\u4e2a\u7cfb\u6570$c_n$\u662f\u4e00\u4e2a\u79ef\u5206\u3002\u5728\u90a3\u4e2a\u5e74\u4ee3\uff0c\u79ef\u5206\u53ef\u4ee5\u4f7f\u7528\u6b27\u62c9\u6cd5\u6570\u503c\u6c42\u89e3\u3002\u5728\u6b64\u4e4b\u524d\uff0c\u9700\u8981\u5c06\u88ab\u79ef\u51fd\u6570\u8fdb\u884c\u4eba\u5de5\u91c7\u6837\uff08sampling\uff09\u79bb\u6563\u5316\u3002\u56e0\u6b64\u8fd9\u662f\u4e00\u4e2a\u5f88\u7e41\u7410\u7684\u4efb\u52a1\u3002 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