{"id":2138,"date":"2023-04-07T21:01:25","date_gmt":"2023-04-07T21:01:25","guid":{"rendered":"https:\/\/vibromera.eu\/?p=2138"},"modified":"2026-07-02T19:45:53","modified_gmt":"2026-07-02T19:45:53","slug":"application-of-the-fourier-transform-to-the-analysis-of-vibration-signals","status":"publish","type":"post","link":"https:\/\/vibromera.eu\/bn\/example\/application-of-the-fourier-transform-to-the-analysis-of-vibration-signals\/","title":{"rendered":"\u0995\u09ae\u09cd\u09aa\u09a8 \u09b8\u0982\u0995\u09c7\u09a4 \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3\u09c7 \u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u09b0\u09c2\u09aa\u09be\u09a8\u09cd\u09a4\u09b0\u09c7\u09b0 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u0997\u0964"},"content":{"rendered":"

\u0995\u09ae\u09cd\u09aa\u09a8 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2 \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3\u09c7 \u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u099f\u09cd\u09b0\u09be\u09a8\u09cd\u09b8\u09ab\u09b0\u09cd\u09ae\u09c7\u09b0 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u0997<\/h1>\n

Andrei Shelkovenko\u0964 Vibromera-\u098f\u09b0 \u09a1\u09c7\u09ad\u09c7\u09b2\u09aa\u09be\u09b0\u09a6\u09c7\u09b0 \u098f\u0995\u099c\u09a8 \u098f\u09ac\u0982 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b7\u09cd\u09a0\u09be\u09a4\u09be\u0964
\n\u09aa\u09cd\u09b0\u09ac\u09a8\u09cd\u09a7\u099f\u09bf\u09b0 \u0985\u09a8\u09c1\u09ac\u09be\u09a6\u09c7 \u0995\u09bf\u099b\u09c1 \u0985\u09b8\u0999\u09cd\u0997\u09a4\u09bf \u09a5\u09be\u0995\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7\u0964<\/p>\n

\u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u09b0\u09c2\u09aa\u09be\u09a8\u09cd\u09a4\u09b0 \u098f\u09ac\u0982 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0<\/h2>\n

\u0985\u09a8\u09c7\u0995 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u098f\u0995\u099f\u09bf \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u09c7\u09b0 \u09b8\u09cd\u09aa\u09c7\u0995\u099f\u09cd\u09b0\u09be\u09ae<\/a> \u09b8\u09cd\u09aa\u09c7\u0995\u099f\u09cd\u09b0\u09be\u09ae \u09aa\u09be\u0993\u09af\u09bc\u09be\u09b0 (\u0997\u09a3\u09a8\u09be \u0995\u09b0\u09be\u09b0) \u0995\u09be\u099c\u099f\u09bf \u09a8\u09bf\u09ae\u09cd\u09a8\u09b0\u09c2\u09aa\u0964 \u098f\u0995\u099f\u09bf ADC \u0986\u099b\u09c7, \u09af\u09be \u09b8\u09cd\u09af\u09be\u09ae\u09cd\u09aa\u09b2\u09bf\u0982 \u09a6\u09bf\u09af\u09bc\u09c7 \u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf<\/a> Fd \u09a4\u09be\u09b0 \u0987\u09a8\u09aa\u09c1\u099f\u09c7 \u09b8\u09ae\u09af\u09bc T \ub3d9\uc548 \u0986\u09b8\u09be \u09a7\u09be\u09b0\u09be\u09ac\u09be\u09b9\u09bf\u0995 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u0995\u09c7 \u09a1\u09bf\u099c\u09bf\u099f\u09be\u09b2 \u09b8\u09cd\u09af\u09be\u09ae\u09cd\u09aa\u09b2\u09c7 \u09b0\u09c2\u09aa\u09be\u09a8\u09cd\u09a4\u09b0 \u0995\u09b0\u09c7 – N\u099f\u09bf \u0985\u0982\u09b6\u0964 \u09a4\u09be\u09b0\u09aa\u09b0 \u098f\u0987 \u09b8\u09cd\u09af\u09be\u09ae\u09cd\u09aa\u09b2\u0997\u09c1\u09b2\u09cb\u09b0 \u0985\u09cd\u09af\u09be\u09b0\u09c7 \u0995\u09cb\u09a8\u09cb \u09aa\u09cd\u09b0\u09cb\u0997\u09cd\u09b0\u09be\u09ae\u09c7 \u09a6\u09c7\u0993\u09af\u09bc\u09be \u09b9\u09af\u09bc (\u0989\u09a6\u09be\u09b9\u09b0\u09a3\u09b8\u09cd\u09ac\u09b0\u09c2\u09aa FourierScope<\/a><\/span>) \u09af\u09be \u0986\u0989\u099f\u09aa\u09c1\u099f \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 N\/2\u099f\u09bf \u09b8\u0982\u0996\u09cd\u09af\u09be\u09ae\u09be\u09a8 \u09a6\u09c7\u09af\u09bc\u0964<\/p>\n

program\u099f\u09bf \u09b8\u09a0\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u0995\u09be\u099c \u0995\u09b0\u099b\u09c7 \u0995\u09bf \u09a8\u09be \u09a4\u09be \u09af\u09be\u099a\u09be\u0987 \u0995\u09b0\u09a4\u09c7 \u0986\u09ae\u09b0\u09be \u09a6\u09c1\u0987\u099f\u09bf sine-\u098f\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 sin(10*2*pi*x)+0.5*sin(5*2*pi*x) \u09a6\u09bf\u09af\u09bc\u09c7 sample array \u09a4\u09c8\u09b0\u09bf \u0995\u09b0\u09c7 program-\u098f \u09a6\u09bf\u0987\u0964 program\u099f\u09bf \u09a8\u09bf\u099a\u09c7\u09b0 \u09ab\u09b2 \u09a6\u09c7\u0996\u09be\u09af\u09bc:
\n

\"\u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0

Fig.1 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u09c7\u09b0 \u09b8\u09ae\u09af\u09bc-\u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0997\u09cd\u09b0\u09be\u09ab<\/span><\/i><\/em><\/p><\/div><\/p>\n

 <\/p>\n

\"Fig.2

Fig.2 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0\u09b0 \u0997\u09cd\u09b0\u09be\u09ab<\/p><\/div>\n

 <\/p>\n

\u09a6\u09c1\u099f\u09bf \u09b0\u09af\u09bc\u09c7\u099b\u09c7 \u09b9\u09be\u09b0\u09ae\u09cb\u09a8\u09bf\u0995\u09cd\u09b8<\/a> \u09b8\u09cd\u09aa\u09c7\u0995\u099f\u09cd\u09b0\u09be\u09ae \u0997\u09cd\u09b0\u09be\u09ab\u09c7 – 0.5 V \u0985\u09cd\u09af\u09be\u09ae\u09aa\u09cd\u09b2\u09bf\u099f\u09bf\u0989\u09a1\u09c7 5 Hz \u098f\u09ac\u0982 1 V \u0985\u09cd\u09af\u09be\u09ae\u09aa\u09cd\u09b2\u09bf\u099f\u09bf\u0989\u09a1\u09c7 10 Hz, \u09b8\u09ac\u0995\u09bf\u099b\u09c1\u0987 \u09ae\u09c2\u09b2 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u09c7\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0\u09c7\u09b0 \u09ae\u09a4\u09cb\u0964 \u09b8\u09ac \u09a0\u09bf\u0995 \u0986\u099b\u09c7, grogram \u09b8\u09a0\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u0995\u09be\u099c \u0995\u09b0\u099b\u09c7\u0964<\/p>\n

\u098f\u09b0 \u09ae\u09be\u09a8\u09c7 \u09b9\u09b2\u09cb, \u09af\u09a6\u09bf \u0986\u09ae\u09b0\u09be \u09a6\u09c1\u0987\u099f\u09bf sinusoid-\u098f\u09b0 \u09ae\u09bf\u09b6\u09cd\u09b0\u09a3 \u09a5\u09c7\u0995\u09c7 \u0997\u09a0\u09bf\u09a4 \u098f\u0995\u099f\u09bf \u09ac\u09be\u09b8\u09cd\u09a4\u09ac signal ADC \u0987\u09a8\u09aa\u09c1\u099f\u09c7 \u09a6\u09bf\u0987, \u09a4\u09ac\u09c7 \u098f\u0995\u0987 \u09a7\u09b0\u09a8\u09c7\u09b0 \u09a6\u09c1\u0987\u099f\u09bf harmonic-\u09b8\u09ae\u09c3\u09a6\u09cd\u09a7 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0 \u09aa\u09be\u09ac\u0964<\/p>\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982, \u0986\u09ae\u09be\u09a6\u09c7\u09b0 real <\/span><\/b><\/strong>\u09aa\u09b0\u09bf\u09ae\u09be\u09aa\u0995\u09c3\u09a4 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2 5 sec. \u09b8\u09ae\u09af\u09bc\u0995\u09be\u09b2\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f<\/span><\/b><\/strong>, \u09af\u09be ADC \u09a6\u09cd\u09ac\u09be\u09b0\u09be digitized \u09b9\u09af\u09bc\u09c7\u099b\u09c7, \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09bf\u09a4 \u09b9\u09af\u09bc\u09c7\u099b\u09c7 \u09ac\u09bf\u099a\u09cd\u099b\u09bf\u09a8\u09cd\u09a8 <\/span><\/b><\/strong>sample \u09a6\u09cd\u09ac\u09be\u09b0\u09be, \u098f\u09b0 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u099a\u09cd\u099b\u09bf\u09a8\u09cd\u09a8 \u0985-\u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u09ac\u09c3\u09a4\u09cd\u09a4 <\/span><\/b><\/strong>\u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0\u0964
\n\u0997\u09a3\u09bf\u09a4\u09c7\u09b0 \u09a6\u09c3\u09b7\u09cd\u099f\u09bf\u0995\u09cb\u09a3 \u09a5\u09c7\u0995\u09c7 – \u098f\u0987 \u09ac\u09be\u0995\u09cd\u09af\u09be\u0982\u09b6\u09c7 \u0995\u09a4\u099f\u09bf \u09ad\u09c1\u09b2 \u0986\u099b\u09c7? <\/span><\/i><\/em><\/p>\n

\u098f\u0996\u09a8 \u098f\u0995\u0987 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u0995\u09c7 0.5 sec. \u09b8\u09ae\u09af\u09bc\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u09ae\u09be\u09aa\u09be\u09b0 \u099a\u09c7\u09b7\u09cd\u099f\u09be \u0995\u09b0\u09bf\u0964<\/p>\n

\"Fig.3

Fig.3 0.5 sec. measurement period-\u098f\u09b0 \u099c\u09a8\u09cd\u09af sin(10*2*pi*x)+0.5*sin(5*2*pi*x) \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0997\u09cd\u09b0\u09be\u09ab<\/p><\/div>\n

 <\/p>\n

\"Fig.4

Fig.4 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0<\/p><\/div>\n

 <\/p>\n

\u098f\u0996\u09be\u09a8\u09c7 \u0995\u09bf\u099b\u09c1 \u098f\u0995\u099f\u09be \u09ad\u09c1\u09b2 \u0986\u099b\u09c7! 10 Hz-\u098f\u09b0 harmonic \u09b8\u09cd\u09ac\u09be\u09ad\u09be\u09ac\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u09a6\u09c7\u0996\u09be \u09af\u09be\u099a\u09cd\u099b\u09c7, \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 5 Hz-\u098f\u09b0 harmonic-\u098f\u09b0 \u099c\u09be\u09af\u09bc\u0997\u09be\u09af\u09bc \u0995\u09bf\u099b\u09c1 \u0985\u09b8\u09cd\u09aa\u09b7\u09cd\u099f harmonic \u09a6\u09c7\u0996\u09be \u09af\u09be\u099a\u09cd\u099b\u09c7\u0964<\/p>\n

\u0987\u09a8\u09cd\u099f\u09be\u09b0\u09a8\u09c7\u099f\u09c7 \u09ac\u09b2\u09be \u09b9\u09af\u09bc, sample-\u098f\u09b0 \u09b6\u09c7\u09b7\u09c7 zero \u09af\u09cb\u0997 \u0995\u09b0\u09b2\u09c7 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0 \u09b8\u09cd\u09ac\u09be\u09ad\u09be\u09ac\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09ac\u09c7\u0964<\/p>\n

\"Fig.5

Fig.5 \u0986\u09ae\u09b0\u09be sample-\u098f zero \u09af\u09cb\u0997 \u0995\u09b0\u09c7 5 sec \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 \u09ac\u09be\u09a1\u09bc\u09bf\u09af\u09bc\u09c7\u099b\u09bf<\/p><\/div>\n

 <\/p>\n

\"Fig.6

Fig.6 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0\u0964<\/p><\/div>\n

 <\/p>\n

\u098f\u099f\u09bf \u09ae\u09cb\u099f\u09c7\u0993 \u09b8\u09a0\u09bf\u0995 \u09a8\u09af\u09bc\u0964 \u098f\u0996\u09a8 \u09a4\u09a4\u09cd\u09a4\u09cd\u09ac\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09af\u09c7\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964 \u099a\u09b2\u09c1\u09a8 \u09af\u09be\u0987 wikipedia<\/b><\/strong><\/a><\/span>\u00a0– \u099c\u09cd\u099e\u09be\u09a8\u09c7\u09b0 \u0989\u09ce\u09b8\u0964<\/p>\n

\u09a7\u09be\u09b0\u09be\u09ac\u09be\u09b9\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8 \u098f\u09ac\u0982 \u09a4\u09be\u09b0 Fourier series \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09a8<\/h2>\n

\u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995\u09ad\u09be\u09ac\u09c7, T \u09b8\u09c7\u0995\u09c7\u09a8\u09cd\u09a1 \u09b8\u09cd\u09a5\u09be\u09af\u09bc\u09c0 \u0986\u09ae\u09be\u09a6\u09c7\u09b0 signal \u09b9\u09b2\u09cb interval {0, T}-\u098f \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4 \u098f\u0995\u099f\u09bf function f(x) (\u098f\u0996\u09be\u09a8\u09c7 X \u09b9\u09b2\u09cb \u09b8\u09ae\u09af\u09bc)\u0964 \u098f \u09a7\u09b0\u09a8\u09c7\u09b0 function-\u0995\u09c7 \u09b8\u09ac\u09b8\u09ae\u09af\u09bc \u09a8\u09bf\u09ae\u09cd\u09a8\u09b0\u09c2\u09aa harmonic function-\u098f\u09b0 (sine \u09ac\u09be cosine) \u09af\u09cb\u0997\u09ab\u09b2 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc:<\/p>\n

\"\u09a7\u09be\u09b0\u09be\u09ac\u09be\u09b9\u09bf\u0995

\u00a0(1), \u09af\u09c7\u0996\u09be\u09a8\u09c7:<\/p>\n

<\/p><\/div>\n

k \u09b9\u09b2\u09cb \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b8\u0982\u0996\u09cd\u09af\u09be (\u0985\u09b0\u09cd\u09a5\u09be\u09ce harmonic component-\u098f\u09b0 \u09b8\u0982\u0996\u09cd\u09af\u09be, harmonic-\u098f\u09b0 \u09b8\u0982\u0996\u09cd\u09af\u09be)
\nT – \u09af\u09c7 segment-\u098f function \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4 (\u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u09c7\u09b0 \u09b8\u09cd\u09a5\u09be\u09af\u09bc\u09bf\u09a4\u09cd\u09ac\u0995\u09be\u09b2)
\nAk - k-\u09a4\u09ae harmonic component-\u098f\u09b0 amplitude,
\n\u03b8k - k-\u09a4\u09ae harmonic component-\u098f\u09b0 \u09aa\u09cd\u09b0\u09be\u09b0\u09ae\u09cd\u09ad\u09bf\u0995 phase
\n“function-\u0995\u09c7 series-\u098f\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be” \u09ac\u09b2\u09a4\u09c7 \u0995\u09c0 \u09ac\u09cb\u099d\u09be\u09af\u09bc? \u098f\u09b0 \u0985\u09b0\u09cd\u09a5 \u09b9\u09b2\u09cb, Fourier series-\u098f\u09b0 harmonic component-\u0997\u09c1\u09b2\u09cb\u09b0 \u09ae\u09be\u09a8 \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u09af\u09cb\u0997 \u0995\u09b0\u09b2\u09c7 \u0986\u09ae\u09b0\u09be \u09b8\u09c7\u0987 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0986\u09ae\u09be\u09a6\u09c7\u09b0 function-\u098f\u09b0 \u09ae\u09be\u09a8 \u09aa\u09be\u0987\u0964
\n(\u0986\u09b0\u0993 \u0995\u09a0\u09cb\u09b0\u09ad\u09be\u09ac\u09c7 \u09ac\u09b2\u09b2\u09c7, series \u098f\u09ac\u0982 function f(x)-\u098f\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 mean square deviation \u09b6\u09c2\u09a8\u09cd\u09af\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u09a7\u09be\u09ac\u09bf\u09a4 \u09b9\u09ac\u09c7; \u09a4\u09ac\u09c7 mean square convergence \u09a5\u09be\u0995\u09be \u09b8\u09a4\u09cd\u09a4\u09cd\u09ac\u09c7\u0993 Fourier series \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u099c\u09a8\u09c0\u09af\u09bc\u09ad\u09be\u09ac\u09c7 point-by-point function\u099f\u09bf\u09b0 \u09a6\u09bf\u0995\u09c7 converge \u0995\u09b0\u09ac\u09c7 \u098f\u09ae\u09a8 \u09a8\u09af\u09bc\u0964)
\n\u098f\u0987 series-\u099f\u09bf \u0986\u09b0\u0993 \u098f\u09ad\u09be\u09ac\u09c7\u0993 \u09b2\u09c7\u0996\u09be \u09af\u09c7\u09a4\u09c7 \u09aa\u09be\u09b0\u09c7:<\/p>\n

\"(2),\"

(2),<\/p><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

\u09af\u09c7\u0996\u09be\u09a8\u09c7 \"\u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0 , k-\u09a4\u09ae complex amplitude\u0964<\/p>\n

 <\/p>\n

\u09ac\u09be<\/p>\n

\"

(3)<\/p><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

coefficient (1) \u098f\u09ac\u0982 (3)-\u098f\u09b0 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995 \u09a8\u09bf\u09ae\u09cd\u09a8\u09b2\u09bf\u0996\u09bf\u09a4 \u09b8\u09c2\u09a4\u09cd\u09b0\u0997\u09c1\u09b2\u09cb \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09af\u09bc:<\/p>\n

\"\u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0<\/p>\n

 <\/p>\n

 <\/p>\n

\"\u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0<\/p>\n

 <\/p>\n

 <\/p>\n

\u09ae\u09a8\u09c7 \u09b0\u09be\u0996\u09ac\u09c7\u09a8 Fourier series-\u098f\u09b0 \u098f\u0987 \u09a4\u09bf\u09a8\u099f\u09bf \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09a8\u0987 \u09b8\u09ae\u09cd\u09aa\u09c2\u09b0\u09cd\u09a3 \u09b8\u09ae\u09a4\u09c1\u09b2\u09cd\u09af\u0964 \u0995\u0996\u09a8\u0993 Fourier series \u09a8\u09bf\u09af\u09bc\u09c7 \u0995\u09be\u099c \u0995\u09b0\u09be\u09b0 \u09b8\u09ae\u09af\u09bc sine \u0993 cosine-\u098f\u09b0 \u09ac\u09a6\u09b2\u09c7 imaginary argument-\u098f\u09b0 exponent \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09be, \u0985\u09b0\u09cd\u09a5\u09be\u09ce complex form-\u098f Fourier transform \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09be \u09ac\u09c7\u09b6\u09bf \u09b8\u09c1\u09ac\u09bf\u09a7\u09be\u099c\u09a8\u0995 \u09b9\u09af\u09bc\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af formula (1) \u09ac\u09cd\u09af\u09ac\u09b9\u09be\u09b0 \u0995\u09b0\u09be \u09b8\u09c1\u09ac\u09bf\u09a7\u09be\u099c\u09a8\u0995, \u09af\u09c7\u0996\u09be\u09a8\u09c7 Fourier series \u09b8\u0982\u09b6\u09cd\u09b2\u09bf\u09b7\u09cd\u099f amplitude \u0993 phase-\u09b8\u09b9 cosine-\u098f\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6\u09bf\u09a4\u0964 \u0995\u09a0\u09cb\u09b0\u09ad\u09be\u09ac\u09c7 \u09ac\u09b2\u09a4\u09c7 \u0997\u09c7\u09b2\u09c7, real signal-\u098f\u09b0 Fourier transform complex coefficient\u0987 \u09a6\u09c7\u09af\u09bc (form (3)): \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf coefficient \u09a4\u09be\u09b0 harmonic-\u098f\u09b0 amplitude \u0993 phase \u0989\u09ad\u09af\u09bc\u0987 \u09ac\u09b9\u09a8 \u0995\u09b0\u09c7\u0964 Real signal-\u098f\u09b0 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u098f\u0987 complex coefficient\u0997\u09c1\u09b2\u09cb\u09b0 conjugate (Hermitian) symmetry \u09a5\u09be\u0995\u09c7 \u2014 negative-frequency \u0985\u09b0\u09cd\u09a7\u09be\u0982\u09b6 \u0995\u09c7\u09ac\u09b2 positive-frequency \u0985\u09b0\u09cd\u09a7\u09be\u0982\u09b6\u0995\u09c7 mirror \u0995\u09b0\u09c7 \u098f\u09ac\u0982 \u0985\u09a4\u09bf\u09b0\u09bf\u0995\u09cd\u09a4 \u09a4\u09a5\u09cd\u09af \u09ac\u09b9\u09a8 \u0995\u09b0\u09c7 \u09a8\u09be\u0964 \u09a4\u09be\u0987 complex coefficient \u09a5\u09c7\u0995\u09c7 \u0986\u09ae\u09b0\u09be \u09b8\u09ac\u09b8\u09ae\u09af\u09bc formula (1)-\u098f\u09b0 real non-negative amplitude Ak \u0993 phase \u03b8k-\u09a4\u09c7 \u09af\u09c7\u09a4\u09c7 \u09aa\u09be\u09b0\u09bf \u2014 \u098f\u09ac\u0982 analysis program\u0997\u09c1\u09b2\u09cb \u09a0\u09bf\u0995 \u098f\u0987 amplitude spectrum-\u0987 plot \u0995\u09b0\u09c7\u0964<\/p>\n

 <\/p>\n

\u09b8\u09be\u09b0\u09be\u0982\u09b6:<\/span><\/b><\/u><\/strong>
\n<\/span><\/b><\/strong>\u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u09c7\u09b0 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0 \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3\u09c7\u09b0 \u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09ad\u09bf\u09a4\u09cd\u09a4\u09bf \u09b9\u09b2\u09cb Fourier transform\u0964<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>Fourier transform interval {0, T}-\u098f \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4 \u098f\u0995\u099f\u09bf \u09a7\u09be\u09b0\u09be\u09ac\u09be\u09b9\u09bf\u0995 function f(x) (signal)-\u0995\u09c7 \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f amplitude \u0993 phase-\u09b8\u09b9 \u0985\u09b8\u09c0\u09ae \u09b8\u0982\u0996\u09cd\u09af\u0995 \u09a4\u09cd\u09b0\u09bf\u0995\u09cb\u09a3\u09ae\u09bf\u09a4\u09bf\u0995 function-\u098f\u09b0 (sine \u098f\u09ac\u0982\/\u0985\u09a5\u09ac\u09be cosine) \u09af\u09cb\u0997\u09ab\u09b2 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09a4\u09c7 \u09a6\u09c7\u09af\u09bc\u0964 \u098f\u0987 series-\u0995\u09c7 Fourier series \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u0964<\/span><\/b><\/strong><\/p>\n

Fourier transform-\u0995\u09c7 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2 \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3\u09c7 \u09b8\u09a0\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u0997 \u0995\u09b0\u09a4\u09c7 \u0986\u09b0\u0993 \u0995\u09af\u09bc\u09c7\u0995\u099f\u09bf \u09ac\u09bf\u09b7\u09af\u09bc \u09ac\u09cb\u099d\u09be \u099c\u09b0\u09c1\u09b0\u09bf\u0964 \u09af\u09a6\u09bf \u0986\u09ae\u09b0\u09be \u09b8\u09ae\u0997\u09cd\u09b0 X-axis-\u098f Fourier series (sinusoid-\u098f\u09b0 \u09af\u09cb\u0997\u09ab\u09b2) \u09ac\u09bf\u09ac\u09c7\u099a\u09a8\u09be \u0995\u09b0\u09bf, \u09a4\u09ac\u09c7 \u09a6\u09c7\u0996\u09ac {0, T} interval-\u098f\u09b0 \u09ac\u09be\u0987\u09b0\u09c7 Fourier series function\u099f\u09bf \u0986\u09ae\u09be\u09a6\u09c7\u09b0 function-\u0995\u09c7 \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u0995\u09cd\u09b0\u09ae\u09c7 \u09aa\u09c1\u09a8\u09b0\u09be\u09ac\u09c3\u09a4\u09cd\u09a4\u09bf \u0995\u09b0\u099b\u09c7\u0964<\/p>\n

\u0989\u09a6\u09be\u09b9\u09b0\u09a3\u09b8\u09cd\u09ac\u09b0\u09c2\u09aa, Fig. 7-\u098f\u09b0 \u0997\u09cd\u09b0\u09be\u09ab\u09c7 \u09ae\u09c2\u09b2 function {-T\\2, +T\\2} interval-\u098f \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4, \u0986\u09b0 Fourier series \u09b8\u09ae\u0997\u09cd\u09b0 x-axis-\u098f \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4 \u098f\u0995\u099f\u09bf periodic function \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09c7\u0964<\/p>\n

\u0995\u09be\u09b0\u09a3 sinusoid \u09a8\u09bf\u099c\u09c7\u0987 periodic function, \u09a4\u09be\u0987 \u09a4\u09be\u09a6\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2\u0993 periodic function \u09b9\u09ac\u09c7\u0964<\/p>\n

\"Figure

Figure 7 Fourier series \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u098f\u0995\u099f\u09bf \u0985-\u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u09ac\u09c3\u09a4\u09cd\u09a4 source function-\u098f\u09b0 \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09a8<\/p><\/div>\n

Thus:<\/p>\n

\u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u09ae\u09c2\u09b2 function \u09b9\u09b2\u09cb \u098f\u0995\u099f\u09bf \u09a7\u09be\u09b0\u09be\u09ac\u09be\u09b9\u09bf\u0995, \u0985-\u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u09ac\u09c3\u09a4\u09cd\u09a4 function, \u09af\u09be \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af T-\u098f\u09b0 \u098f\u0995\u099f\u09bf segment-\u098f \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4\u0964
\nThe spectrum of this function is discrete, i.e. it is represented as an infinite series of harmonic components \u2013 a Fourier series.
\n\u0986\u09b8\u09b2\u09c7 Fourier series \u098f\u0995\u099f\u09bf periodic function \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4 \u0995\u09b0\u09c7, \u09af\u09be {0, T} interval-\u098f \u0986\u09ae\u09be\u09a6\u09c7\u09b0 function-\u098f\u09b0 \u09b8\u0999\u09cd\u0997\u09c7 \u09ae\u09bf\u09b2\u09c7 \u09af\u09be\u09af\u09bc; \u09a4\u09ac\u09c7 \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u098f\u0987 periodicity \u0997\u09c1\u09b0\u09c1\u09a4\u09cd\u09ac\u09aa\u09c2\u09b0\u09cd\u09a3 \u09a8\u09af\u09bc\u0964<\/p>\n

\u098f\u09ac\u09be\u09b0 \u09aa\u09b0\u09c7\u09b0 \u09ac\u09bf\u09b7\u09af\u09bc\u0964<\/p>\n

harmonic component-\u0997\u09c1\u09b2\u09cb\u09b0 period {0, T} interval-\u098f\u09b0 \u0997\u09c1\u09a3\u09bf\u09a4\u0995, \u09af\u09c7\u0996\u09be\u09a8\u09c7 \u09ae\u09c2\u09b2 function f(x) \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4\u0964 \u0985\u09a8\u09cd\u09af\u09ad\u09be\u09ac\u09c7 \u09ac\u09b2\u09b2\u09c7, harmonic-\u098f\u09b0 period \u09b9\u09b2\u09cb signal measurement duration-\u098f\u09b0 \u0997\u09c1\u09a3\u09bf\u09a4\u0995\u0964 \u0989\u09a6\u09be\u09b9\u09b0\u09a3\u09b8\u09cd\u09ac\u09b0\u09c2\u09aa, Fourier series-\u098f\u09b0 \u09aa\u09cd\u09b0\u09a5\u09ae harmonic-\u098f\u09b0 period \u09b8\u09c7\u0987 interval T-\u098f\u09b0 \u09b8\u09ae\u09be\u09a8, \u09af\u09c7\u0996\u09be\u09a8\u09c7 function f(x) \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4\u0964 \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc harmonic-\u098f\u09b0 period T\/2\u0964 \u098f\u09ad\u09be\u09ac\u09c7 \u099a\u09b2\u09a4\u09c7 \u09a5\u09be\u0995\u09c7 (Figure 8 \u09a6\u09c7\u0996\u09c1\u09a8)\u0964<\/p>\n

\"Fig.

Fig. 8 Fourier series-\u098f\u09b0 harmonic component-\u0997\u09c1\u09b2\u09cb\u09b0 period (frequency) (\u098f\u0996\u09be\u09a8\u09c7 T=2\u03c0)<\/p><\/div>\n

\u0985\u09a4\u098f\u09ac harmonic component-\u0997\u09c1\u09b2\u09cb\u09b0 frequency \u09b9\u09b2\u09cb 1\/T-\u098f\u09b0 \u0997\u09c1\u09a3\u09bf\u09a4\u0995\u0964 \u0985\u09b0\u09cd\u09a5\u09be\u09ce harmonic frequency Fk = k\\T, \u09af\u09c7\u0996\u09be\u09a8\u09c7 k-\u098f\u09b0 \u09ae\u09be\u09a8 0 \u09a5\u09c7\u0995\u09c7 \u221e \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4; \u09af\u09c7\u09ae\u09a8, k=0 \u09b9\u09b2\u09c7 F0=0; k=1 \u09b9\u09b2\u09c7 F1=1\\T; k=2 \u09b9\u09b2\u09c7 F2=2\\T; k=3 \u09b9\u09b2\u09c7 F3=3\\T; …. \u09b6\u09c2\u09a8\u09cd\u09af frequency-\u09a4\u09c7 \u098f\u0995\u099f\u09bf constant component \u09a5\u09be\u0995\u09c7\u0964<\/p>\n

\u09a7\u09b0\u09be \u09af\u09be\u0995, \u0986\u09ae\u09be\u09a6\u09c7\u09b0 initial function \u09b9\u09b2\u09cb T=1 sec. \u09b8\u09ae\u09af\u09bc\u09c7 \u09b0\u09c7\u0995\u09b0\u09cd\u09a1 \u0995\u09b0\u09be \u098f\u0995\u099f\u09bf signal\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 \u09aa\u09cd\u09b0\u09a5\u09ae harmonic-\u098f\u09b0 period \u09b9\u09ac\u09c7 T1=T=1 sec \u098f\u09ac\u0982 \u09a4\u09be\u09b0 frequency \u09b9\u09ac\u09c7 1 Hz\u0964 \u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc harmonic-\u098f\u09b0 period \u09b9\u09ac\u09c7 T2=T\/2=0.5 sec \u098f\u09ac\u0982 frequency \u09b9\u09ac\u09c7 2 Hz\u0964 \u09a4\u09c3\u09a4\u09c0\u09af\u09bc harmonic-\u098f\u09b0 \u099c\u09a8\u09cd\u09af T3=T\/3 sec \u098f\u09ac\u0982 frequency \u09b9\u09ac\u09c7 3 Hz\u0964 \u098f\u09ad\u09be\u09ac\u09c7 \u099a\u09b2\u09a4\u09c7 \u09a5\u09be\u0995\u09c7\u0964<\/p>\n

\u098f\u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 harmonic-\u0997\u09c1\u09b2\u09cb\u09b0 \u09ae\u09a7\u09cd\u09af\u0995\u09be\u09b0 step \u09b9\u09b2\u09cb 1 Hz\u0964<\/p>\n

\u0985\u09a4\u098f\u09ac, 1 sec \u09b8\u09cd\u09a5\u09be\u09af\u09bc\u09c0 \u098f\u0995\u099f\u09bf signal-\u0995\u09c7 harmonic component-\u098f \u09ad\u09c7\u0999\u09c7 (\u0985\u09b0\u09cd\u09a5\u09be\u09ce spectrum \u09aa\u09c7\u09a4\u09c7) 1 Hz frequency resolution-\u098f \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964
\n\u09b0\u09c7\u099c\u09cb\u09b2\u09bf\u0989\u09b6\u09a8 2 \u0997\u09c1\u09a3 \u09ac\u09be\u09a1\u09bc\u09bf\u09af\u09bc\u09c7 0.5 Hz \u0995\u09b0\u09a4\u09c7 \u09b9\u09b2\u09c7, \u09aa\u09b0\u09bf\u09ae\u09be\u09aa\u09c7\u09b0 \u09b8\u09ae\u09af\u09bc\u0995\u09be\u09b2 2 \u0997\u09c1\u09a3 \u09ac\u09be\u09a1\u09bc\u09bf\u09af\u09bc\u09c7 2 sec \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964 10-second \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u0995\u09c7 0.1 Hz \u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf \u09b0\u09c7\u099c\u09cb\u09b2\u09bf\u0989\u09b6\u09a8\u09b8\u09b9 \u09b9\u09be\u09b0\u09ae\u09a8\u09bf\u0995 \u0989\u09aa\u09be\u09a6\u09be\u09a8 (\u09b8\u09cd\u09aa\u09c7\u0995\u099f\u09cd\u09b0\u09be\u09ae)-\u098f \u09ac\u09bf\u09b6\u09cd\u09b2\u09c7\u09b7\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964 \u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf \u09b0\u09c7\u099c\u09cb\u09b2\u09bf\u0989\u09b6\u09a8 \u09ac\u09be\u09a1\u09bc\u09be\u09a8\u09cb\u09b0 \u0985\u09a8\u09cd\u09af \u0995\u09cb\u09a8\u09cb \u0989\u09aa\u09be\u09af\u09bc \u09a8\u09c7\u0987\u0964 \u0986\u09aa\u09a8\u09bf \u0986\u09ae\u09be\u09a6\u09c7\u09b0 FFT \u09b0\u09c7\u099c\u09cb\u09b2\u09bf\u0989\u09b6\u09a8 \u0995\u09cd\u09af\u09be\u09b2\u0995\u09c1\u09b2\u09c7\u099f\u09b0<\/a>.<\/p>\n

sample array-\u09a4\u09c7 zero \u09af\u09cb\u0997 \u0995\u09b0\u09c7 signal duration \u0995\u09c3\u09a4\u09cd\u09b0\u09bf\u09ae\u09ad\u09be\u09ac\u09c7 \u09ac\u09be\u09a1\u09bc\u09be\u09a8\u09cb \u09af\u09be\u09af\u09bc\u0964 \u0995\u09bf\u09a8\u09cd\u09a4\u09c1 \u098f\u09a4\u09c7 \u09aa\u09cd\u09b0\u0995\u09c3\u09a4 frequency resolution \u09ac\u09be\u09a1\u09bc\u09c7 \u09a8\u09be\u0964<\/p>\n

\u09a1\u09bf\u09b8\u0995\u09cd\u09b0\u09bf\u099f \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2 \u098f\u09ac\u0982 \u09a1\u09bf\u09b8\u0995\u09cd\u09b0\u09bf\u099f \u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u099f\u09cd\u09b0\u09be\u09a8\u09cd\u09b8\u09ab\u09b0\u09cd\u09ae<\/h2>\n

\u09a1\u09bf\u099c\u09bf\u099f\u09be\u09b2 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf\u09b0 \u09ac\u09bf\u0995\u09be\u09b6\u09c7\u09b0 \u09b8\u0999\u09cd\u0997\u09c7 measurement data (signal) \u09b8\u0982\u09b0\u0995\u09cd\u09b7\u09a3\u09c7\u09b0 \u09aa\u09a6\u09cd\u09a7\u09a4\u09bf\u0993 \u09ac\u09a6\u09b2\u09c7\u099b\u09c7\u0964 \u0986\u0997\u09c7 signal tape recorder-\u098f analog form-\u098f tape-\u098f \u09b8\u0982\u09b0\u0995\u09cd\u09b7\u09a3 \u0995\u09b0\u09be \u09af\u09c7\u09a4, \u098f\u0996\u09a8 signal digitize \u0995\u09b0\u09c7 computer memory-\u09b0 file-\u098f \u09b8\u0982\u0996\u09cd\u09af\u09be (count)-\u098f\u09b0 \u09b8\u09c7\u099f \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09b8\u0982\u09b0\u0995\u09cd\u09b7\u09a3 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964<\/p>\n

\u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2 measurement \u0993 digitization-\u098f\u09b0 \u09aa\u09cd\u09b0\u099a\u09b2\u09bf\u09a4 \u09b8\u09cd\u0995\u09bf\u09ae\u099f\u09bf \u09b9\u09b2\u09cb:<\/p>\n

\u09ae\u09be\u09aa\u099c\u09cb\u0995 \u099f\u09cd\u09b0\u09be\u09a8\u09cd\u09b8\u09a1\u09bf\u0989\u09b8\u09be\u09b0 —- \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2 \u09a8\u09b0\u09ae\u09be\u09b2\u09be\u0987\u099c\u09be\u09b0 —- ADC —– \u0995\u09ae\u09cd\u09aa\u09bf\u0989\u099f\u09be\u09b0
\n(Fig.9 \u09ae\u09be\u09aa\u099c\u09cb\u0995 \u099a\u09cd\u09af\u09be\u09a8\u09c7\u09b2\u09c7\u09b0 \u09b8\u09cd\u0995\u09bf\u09ae\u09cd\u09af\u09be\u099f\u09bf\u0995<\/p>\n

<\/span><\/i><\/em><\/p>\n

measurement transducer \u09a5\u09c7\u0995\u09c7 signal \u09b8\u09ae\u09af\u09bc T \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 ADC-\u09a4\u09c7 \u09af\u09be\u09af\u09bc\u0964 \u09b8\u09ae\u09af\u09bc T-\u09a4\u09c7 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 signal reading (sampling) computer-\u098f \u09aa\u09be\u09a0\u09be\u09a8\u09cb \u09b9\u09af\u09bc \u098f\u09ac\u0982 memory-\u09a4\u09c7 \u09b8\u0982\u09b0\u0995\u09cd\u09b7\u09a3 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964<\/p>\n

\"Fig.10

Fig.10 Digitized signal – \u09b8\u09ae\u09af\u09bc T-\u09a4\u09c7 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 N\u099f\u09bf sample<\/p><\/div>\n

signal digitization parameter-\u098f\u09b0 \u099c\u09a8\u09cd\u09af \u0995\u09c0 \u0995\u09c0 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u099c\u09a8\u09c0\u09af\u09bc\u09a4\u09be \u0986\u099b\u09c7? \u09af\u09c7 device \u0987\u09a8\u09aa\u09c1\u099f analog signal-\u0995\u09c7 discrete code (digital signal)-\u098f \u09b0\u09c2\u09aa\u09be\u09a8\u09cd\u09a4\u09b0 \u0995\u09b0\u09c7 \u09a4\u09be\u0995\u09c7 analog-to-digital converter (ADC) \u09ac\u09b2\u09be \u09b9\u09af\u09bc (\u00a9 Wiki)\u0964<\/p>\n

ADC-\u098f\u09b0 \u098f\u0995\u099f\u09bf \u09ae\u09cc\u09b2\u09bf\u0995 parameter \u09b9\u09b2\u09cb maximum sampling rate – \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09b8\u09ae\u09af\u09bc\u09c7 \u09a7\u09be\u09b0\u09be\u09ac\u09be\u09b9\u09bf\u0995 \u098f\u0995\u099f\u09bf signal-\u098f\u09b0 sampling frequency\u0964 Sample rate hertz-\u098f \u09ae\u09be\u09aa\u09be \u09b9\u09af\u09bc\u0964<\/p>\n

\u0995\u09cb\u099f\u09c7\u09b2\u09a8\u09bf\u0995\u09ad\u09c7\u09b0 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7, \u09af\u09a6\u09bf \u098f\u0995\u099f\u09bf \u09a7\u09be\u09b0\u09be\u09ac\u09be\u09b9\u09bf\u0995 \u09b8\u0982\u0995\u09c7\u09a4\u09c7\u09b0 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0 Fmax \u0995\u09ae\u09cd\u09aa\u09be\u0999\u09cd\u0995 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c0\u09ae\u09be\u09ac\u09a6\u09cd\u09a7 \u09a5\u09be\u0995\u09c7, \u09a4\u09be\u09b9\u09b2\u09c7 \u0394t \u2264 1\/(2*Fmax) \u09b8\u09ae\u09af\u09bc\u09be\u09a8\u09cd\u09a4\u09b0\u09c7 \u09a8\u09c7\u0993\u09af\u09bc\u09be \u098f\u09b0 \u09ac\u09bf\u099a\u09cd\u099b\u09bf\u09a8\u09cd\u09a8 \u09a8\u09ae\u09c1\u09a8\u09be \u09a5\u09c7\u0995\u09c7, \u0985\u09b0\u09cd\u09a5\u09be\u09ce Fd \u2265 2*Fmax \u09a8\u09ae\u09c1\u09a8\u09be\u09af\u09bc\u09a8 \u0995\u09ae\u09cd\u09aa\u09be\u0999\u09cd\u0995\u09c7, \u098f\u0995\u09c7 \u09b8\u09ae\u09cd\u09aa\u09c2\u09b0\u09cd\u09a3\u09b0\u09c2\u09aa\u09c7 \u0993 \u0985\u09a6\u09cd\u09ac\u09bf\u09a4\u09c0\u09af\u09bc\u09ad\u09be\u09ac\u09c7 \u09aa\u09c1\u09a8\u09b0\u09cd\u0997\u09a0\u09a8 \u0995\u09b0\u09be \u09b8\u09ae\u09cd\u09ad\u09ac, \u09af\u09c7\u0996\u09be\u09a8\u09c7 Fd – \u09a8\u09ae\u09c1\u09a8\u09be\u09af\u09bc\u09a8 \u0995\u09ae\u09cd\u09aa\u09be\u0999\u09cd\u0995; Fmax – \u09b8\u0982\u0995\u09c7\u09a4 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0\u09b0 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u0995\u09ae\u09cd\u09aa\u09be\u0999\u09cd\u0995\u0964 \u0985\u09a8\u09cd\u09af \u0995\u09a5\u09be\u09af\u09bc, \u09b8\u0982\u0995\u09c7\u09a4 \u09a1\u09bf\u099c\u09bf\u099f\u09be\u0987\u099c\u09c7\u09b6\u09a8\u09c7\u09b0 \u0995\u09ae\u09cd\u09aa\u09be\u0999\u09cd\u0995 (ADC-\u098f\u09b0 \u09a8\u09ae\u09c1\u09a8\u09be\u09af\u09bc\u09a8 \u0995\u09ae\u09cd\u09aa\u09be\u0999\u09cd\u0995) \u0986\u09ae\u09b0\u09be \u09af\u09c7 \u09b8\u0982\u0995\u09c7\u09a4 \u09aa\u09b0\u09bf\u09ae\u09be\u09aa \u0995\u09b0\u09a4\u09c7 \u099a\u09be\u0987 \u09a4\u09be\u09b0 \u09b8\u09b0\u09cd\u09ac\u09cb\u099a\u09cd\u099a \u0995\u09ae\u09cd\u09aa\u09be\u0999\u09cd\u0995\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09a4 \u09a6\u09cd\u09ac\u09bf\u0997\u09c1\u09a3 \u09b9\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964<\/p>\n

\u0986\u09b0 \u09af\u09a6\u09bf \u0986\u09ae\u09b0\u09be Kotelnikov-\u098f\u09b0 theorem-\u098f \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u099c\u09a8\u09c0\u09af\u09bc \u09ae\u09be\u09a8\u09c7\u09b0 \u099a\u09c7\u09af\u09bc\u09c7 \u0995\u09ae frequency-\u09a4\u09c7 sample \u09a8\u09bf\u0987, \u09a4\u09be\u09b9\u09b2\u09c7 \u0995\u09c0 \u09b9\u09ac\u09c7?<\/p>\n

\u098f\u0987 \u0995\u09cd\u09b7\u09c7\u09a4\u09cd\u09b0\u09c7 \u098f\u0995\u099f\u09bf “ \u0986\u099b\u09c7\u0985\u09cd\u09af\u09be\u09b2\u09bf\u09af\u09bc\u09be\u09b8\u09bf\u0982<\/a>In this case there is an \u201caliasing\u201d effect (aka stroboscopic effect, moir\u00e9 effect), in which a high frequency signal after digitization turns into a low frequency signal, which in fact does not exist. In Fig. 11 the red sine wave of high frequency is the real signal. The blue sine wave of lower frequency is a fictitious signal, arising due to the fact that during the sampling time has time to pass more than half a period of the high-frequency signal.<\/p>\n

\"Fig.

Fig. 11 \u0985\u09aa\u09b0\u09cd\u09af\u09be\u09aa\u09cd\u09a4 sampling rate-\u098f\u09b0 \u0995\u09be\u09b0\u09a3\u09c7 spurious low frequency signal-\u098f\u09b0 \u0989\u09a6\u09cd\u09ad\u09ac<\/p><\/div>\n

 <\/p>\n

\u0985\u09cd\u09af\u09be\u09b2\u09bf\u09af\u09bc\u09be\u09b8\u09bf\u0982 \u09aa\u09cd\u09b0\u09ad\u09be\u09ac \u098f\u09a1\u09bc\u09be\u09a4\u09c7, \u098f\u0995\u099f\u09bf \u09ac\u09bf\u09b6\u09c7\u09b7 \u0985\u09cd\u09af\u09be\u09a8\u09cd\u099f\u09bf-\u0985\u09cd\u09af\u09be\u09b2\u09bf\u09af\u09bc\u09be\u09b8 \u09ab\u09bf\u09b2\u09cd\u099f\u09be\u09b0 (\u09b2\u09cb-\u09aa\u09be\u09b8 \u09ab\u09bf\u09b2\u09cd\u099f\u09be\u09b0<\/a>) ADC-\u098f\u09b0 \u0986\u0997\u09c7 \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09be \u09b9\u09af\u09bc\u0964 \u098f\u099f\u09bf ADC \u09b8\u09cd\u09af\u09be\u09ae\u09cd\u09aa\u09b2\u09bf\u0982 \u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf\u09b0 \u0985\u09b0\u09cd\u09a7\u09c7\u0995\u09c7\u09b0 \u0995\u09ae \u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf \u09af\u09c7\u09a4\u09c7 \u09a6\u09c7\u09af\u09bc \u098f\u09ac\u0982 \u098f\u09b0 \u099a\u09c7\u09af\u09bc\u09c7 \u09ac\u09c7\u09b6\u09bf \u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf \u0995\u09c7\u099f\u09c7 \u09a6\u09c7\u09af\u09bc\u0964<\/p>\n

\u09a1\u09bf\u09b8\u0995\u09cd\u09b0\u09bf\u099f \u09b8\u09cd\u09af\u09be\u09ae\u09cd\u09aa\u09b2 \u09a5\u09c7\u0995\u09c7 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u09c7\u09b0 \u09b8\u09cd\u09aa\u09c7\u0995\u099f\u09cd\u09b0\u09be\u09ae \u0997\u09a3\u09a8\u09be \u0995\u09b0\u09a4\u09c7 \u09a1\u09bf\u09b8\u0995\u09cd\u09b0\u09bf\u099f \u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u099f\u09cd\u09b0\u09be\u09a8\u09cd\u09b8\u09ab\u09b0\u09cd\u09ae (DFT)<\/a> \u09ac\u09cd\u09af\u09ac\u09b9\u09c3\u09a4 \u09b9\u09af\u09bc\u0964 \u0986\u09ac\u09be\u09b0 \u09b2\u0995\u09cd\u09b7 \u0995\u09b0\u09c1\u09a8, \u098f\u0995\u099f\u09bf \u09a1\u09bf\u09b8\u0995\u09cd\u09b0\u09bf\u099f \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u09c7\u09b0 \u09b8\u09cd\u09aa\u09c7\u0995\u099f\u09cd\u09b0\u09be\u09ae “\u09b8\u0982\u099c\u09cd\u099e\u09be \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7” \u09b8\u09cd\u09af\u09be\u09ae\u09cd\u09aa\u09b2\u09bf\u0982 \u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf Fd-\u098f\u09b0 \u0985\u09b0\u09cd\u09a7\u09c7\u0995\u09c7\u09b0 \u099a\u09c7\u09af\u09bc\u09c7 \u099b\u09cb\u099f \u098f\u0995\u099f\u09bf \u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf Fmax-\u098f \u09b8\u09c0\u09ae\u09be\u09ac\u09a6\u09cd\u09a7\u0964 \u09a4\u09be\u0987 \u098f\u0995\u099f\u09bf \u09a1\u09bf\u09b8\u0995\u09cd\u09b0\u09bf\u099f \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u09c7\u09b0 \u09b8\u09cd\u09aa\u09c7\u0995\u099f\u09cd\u09b0\u09be\u09ae\u0995\u09c7 \u09af\u09cb\u0997\u09ab\u09b2 \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc \u09b8\u09b8\u09c0\u09ae <\/u>\u09b8\u0982\u0996\u09cd\u09af\u0995 harmonic \u09a6\u09bf\u09af\u09bc\u09c7, \u09af\u09be continuous signal-\u098f\u09b0 Fourier series-\u098f\u09b0 \u0985\u09b8\u09c0\u09ae \u09af\u09cb\u0997\u09ab\u09b2\u09c7\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4\u0964 Kotelnikov-\u098f\u09b0 theorem \u0985\u09a8\u09c1\u09af\u09be\u09af\u09bc\u09c0 harmonic-\u098f\u09b0 maximum frequency \u098f\u09ae\u09a8 \u09b9\u09a4\u09c7 \u09b9\u09ac\u09c7 \u09af\u09be\u09a4\u09c7 \u09aa\u09cd\u09b0\u09a4\u09bf harmonic-\u098f \u0985\u09a8\u09cd\u09a4\u09a4 \u09a6\u09c1\u0987\u099f\u09bf sample \u09aa\u09a1\u09bc\u09c7; \u09a4\u09be\u0987 harmonic-\u098f\u09b0 \u09b8\u0982\u0996\u09cd\u09af\u09be discrete signal-\u098f\u09b0 sample \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0985\u09b0\u09cd\u09a7\u09c7\u0995, \u0985\u09b0\u09cd\u09a5\u09be\u09ce N\/2\u0964<\/p>\n

\u098f\u0996\u09a8 discrete Fourier transform (DFT) \u09ac\u09bf\u09ac\u09c7\u099a\u09a8\u09be \u0995\u09b0\u09bf\u0964<\/p>\n

\"\u09ac\u09bf\u099a\u09cd\u099b\u09bf\u09a8\u09cd\u09a8<\/p>\n

\u098f\u099f\u09bf\u0995\u09c7 Fourier series-\u098f\u09b0 \u09b8\u0999\u09cd\u0997\u09c7 \u09a4\u09c1\u09b2\u09a8\u09be \u0995\u09b0\u09b2\u09c7<\/p>\n

 <\/p>\n

\"\u09ac\u09bf\u099a\u09cd\u099b\u09bf\u09a8\u09cd\u09a8<\/p>\n

As we can see, they coincide, except for the fact that time in the FFT is discrete and the number of harmonics is limited to N\/2, which is half the number of samples.<\/p>\n

DFT-\u098f\u09b0 \u09b8\u09c2\u09a4\u09cd\u09b0 dimensionless integer variable k \u0993 s-\u098f \u09b2\u09c7\u0996\u09be \u09b9\u09af\u09bc, \u09af\u09c7\u0996\u09be\u09a8\u09c7 k \u09b9\u09b2\u09cb signal sample-\u098f\u09b0 \u09b8\u0982\u0996\u09cd\u09af\u09be \u098f\u09ac\u0982 s \u09b9\u09b2\u09cb spectral component-\u098f\u09b0 \u09b8\u0982\u0996\u09cd\u09af\u09be\u0964
\ns-\u098f\u09b0 \u09ae\u09be\u09a8 \u09a6\u09c7\u0996\u09be\u09af\u09bc period T-\u098f (signal measurement duration) \u0995\u09a4\u0997\u09c1\u09b2\u09cb \u09aa\u09c2\u09b0\u09cd\u09a3 harmonic oscillation \u09b0\u09af\u09bc\u09c7\u099b\u09c7\u0964 discrete Fourier transform \u09b8\u0982\u0996\u09cd\u09af\u09be\u0997\u09a4\u09ad\u09be\u09ac\u09c7 harmonic-\u098f\u09b0 amplitude \u0993 phase \u09ac\u09c7\u09b0 \u0995\u09b0\u09a4\u09c7 \u09ac\u09cd\u09af\u09ac\u09b9\u09c3\u09a4 \u09b9\u09af\u09bc, \u0985\u09b0\u09cd\u09a5\u09be\u09ce “\u0995\u09ae\u09cd\u09aa\u09bf\u0989\u099f\u09be\u09b0\u09c7”\u0964<\/p>\n

\u0986\u0997\u09c7\u0987 \u09ac\u09b2\u09be \u09b9\u09af\u09bc\u09c7\u099b\u09c7, \u09af\u0996\u09a8 \u098f\u0995\u099f\u09bf \u0985-\u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u09ac\u09c3\u09a4\u09cd\u09a4 function (\u0986\u09ae\u09be\u09a6\u09c7\u09b0 signal)-\u0995\u09c7 Fourier series-\u098f \u09ad\u09be\u0999\u09be \u09b9\u09af\u09bc, \u09a4\u0996\u09a8 \u09ab\u09b2\u09b8\u09cd\u09ac\u09b0\u09c2\u09aa series\u099f\u09bf \u0986\u09b8\u09b2\u09c7 period T-\u09b8\u09b9 \u098f\u0995\u099f\u09bf periodic function-\u098f\u09b0 \u09b8\u09ae\u09a4\u09c1\u09b2\u09cd\u09af \u09b9\u09af\u09bc (Fig.12)\u0964<\/p>\n

 <\/p>\n

\"Fig.12.

Fig.12 period T0-\u09b8\u09b9 periodic function f(x), \u09af\u09c7\u0996\u09be\u09a8\u09c7 period T>T0<\/p><\/div>\n

 <\/p>\n

\u099a\u09bf\u09a4\u09cd\u09b0 12-\u098f \u09a6\u09c7\u0996\u09be \u09af\u09be\u09af\u09bc, f(x) \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf T0 \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u09b8\u09b9 \u09aa\u09b0\u09cd\u09af\u09be\u09ac\u09c3\u09a4\u09cd\u09a4\u0964 \u09a4\u09ac\u09c7 \u09ae\u09be\u09aa\u099c\u09cb\u0995\u09c7\u09b0 \u09a8\u09ae\u09c1\u09a8\u09be\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af T \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc T0-\u098f\u09b0 \u09b8\u09ae\u09be\u09a8 \u09a8\u09af\u09bc \u09ac\u09b2\u09c7, \u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u09b8\u09bf\u09b0\u09bf\u099c \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf\u09b0 T \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u098f\u0995\u099f\u09bf \u09ac\u09bf\u099a\u09cd\u099b\u09bf\u09a8\u09cd\u09a8\u09a4\u09be \u09a5\u09be\u0995\u09c7\u0964 \u098f\u09b0 \u09ab\u09b2\u09c7 \u098f\u0987 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b8\u09cd\u09aa\u09c7\u0995\u099f\u09cd\u09b0\u09be\u09ae\u09c7 \u09ac\u09bf\u09aa\u09c1\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u0995 \u0989\u099a\u09cd\u099a-\u09ab\u09cd\u09b0\u09bf\u0995\u09cb\u09af\u09bc\u09c7\u09a8\u09cd\u09b8\u09bf\u09b0 \u09b9\u09be\u09b0\u09ae\u09a8\u09bf\u0995 \u09a5\u09be\u0995\u09ac\u09c7\u0964 \u098f\u0987 \u0998\u099f\u09a8\u09be\u099f\u09bf \u09aa\u09b0\u09bf\u099a\u09bf\u09a4 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0 \u09ab\u09c1\u09b0\u09a8<\/a>\u09b9\u09bf\u09b8\u09c7\u09ac\u09c7, \u098f\u09ac\u0982 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac\u09c7 \u098f\u099f\u09bf \u0995\u09ae\u09be\u09a8\u09cb \u09b9\u09af\u09bc \u0989\u0987\u09a8\u09cd\u09a1\u09cb\u09af\u09bc\u09bf\u0982<\/a> \u09b0\u09c2\u09aa\u09be\u09a8\u09cd\u09a4\u09b0\u09c7\u09b0 \u0986\u0997\u09c7 \u09b8\u09bf\u0997\u09a8\u09cd\u09af\u09be\u09b2\u099f\u09bf\u0995\u09c7 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u0997 \u0995\u09b0\u09c7\u0964 \u09af\u09a6\u09bf \u09ae\u09be\u09aa\u099c\u09cb\u0995\u09c7\u09b0 \u09a8\u09ae\u09c1\u09a8\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09af\u09bc\u09bf\u09a4\u09cd\u09ac T \u09ab\u09be\u0982\u09b6\u09a8 T0-\u098f\u09b0 \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u09c7\u09b0 \u09b8\u0999\u09cd\u0997\u09c7 \u09ae\u09bf\u09b2\u09c7 \u09af\u09c7\u09a4, \u09a4\u09ac\u09c7 \u09ab\u09c1\u09b0\u09bf\u09af\u09bc\u09be\u09b0 \u099f\u09cd\u09b0\u09be\u09a8\u09cd\u09b8\u09ab\u09b0\u09cd\u09ae\u09c7\u09b0 \u09aa\u09b0\u09c7 \u09aa\u09cd\u09b0\u09be\u09aa\u09cd\u09a4 \u09b8\u09cd\u09aa\u09c7\u0995\u099f\u09cd\u09b0\u09be\u09ae\u09c7 \u09b6\u09c1\u09a7\u09c1 \u09aa\u09cd\u09b0\u09a5\u09ae \u09b9\u09be\u09b0\u09ae\u09a8\u09bf\u0995 \u09a5\u09be\u0995\u09a4 (\u09a8\u09ae\u09c1\u09a8\u09be\u09b0 \u09b8\u09cd\u09a5\u09be\u09af\u09bc\u09bf\u09a4\u09cd\u09ac\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8 \u09aa\u09b0\u09cd\u09af\u09be\u09af\u09bc\u09c7\u09b0 \u098f\u0995\u099f\u09bf \u09b8\u09be\u0987\u09a8\u09c1\u09b8\u09af\u09bc\u09c7\u09a1), \u0995\u09be\u09b0\u09a3 f(x) \u09ab\u09be\u0982\u09b6\u09a8\u099f\u09bf \u098f\u0995\u099f\u09bf \u09b8\u09be\u0987\u09a8\u09c1\u09b8\u09af\u09bc\u09c7\u09a1\u0964<\/p>\n

In other words, the DFT program \u201cdoes not know\u201d that our signal is a \u201cslice of a sine wave\u201d, but tries to represent as a series a periodic function which has a discontinuity due to the discontinuity of separate pieces of the sine wave.<\/p>\n

\u09ab\u09b2\u09c7 spectrum-\u098f \u098f\u09ae\u09a8 harmonic \u09a6\u09c7\u0996\u09be \u09af\u09be\u09af\u09bc, \u09af\u09c7\u0997\u09c1\u09b2\u09cb \u09b8\u09ae\u09cd\u09ae\u09bf\u09b2\u09bf\u09a4\u09ad\u09be\u09ac\u09c7 \u098f\u0987 discontinuity-\u09b8\u09b9 function-\u098f\u09b0 \u0986\u0995\u09c3\u09a4\u09bf \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09c7\u0964<\/p>\n

\u0985\u09a4\u098f\u09ac, \u09ac\u09bf\u09ad\u09bf\u09a8\u09cd\u09a8 period-\u09af\u09c1\u0995\u09cd\u09a4 \u098f\u0995\u09be\u09a7\u09bf\u0995 sinusoid-\u098f\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u09a8\u09bf\u09af\u09bc\u09c7 \u0997\u09a0\u09bf\u09a4 signal-\u098f\u09b0 “\u09b8\u09a0\u09bf\u0995” spectrum \u09aa\u09c7\u09a4\u09c7 \u09b9\u09b2\u09c7 \u09aa\u09cd\u09b0\u09af\u09bc\u09cb\u099c\u09a8 \u09af\u09c7 \u09aa\u09c2\u09b0\u09cd\u09a3\u09b8\u0982\u0996\u09cd\u09af\u0995 period <\/u>\u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf sinusoid-\u098f\u09b0 \u098f\u0995\u099f\u09bf \u09aa\u09c2\u09b0\u09cd\u09a3\u09b8\u0982\u0996\u09cd\u09af\u0995 period measurement period-\u098f\u09b0 \u09ae\u09a7\u09cd\u09af\u09c7 \u0989\u09aa\u09b8\u09cd\u09a5\u09bf\u09a4 \u09a5\u09be\u0995\u09a4\u09c7 \u09b9\u09ac\u09c7\u0964 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac\u09c7, \u09aa\u09b0\u09cd\u09af\u09be\u09aa\u09cd\u09a4 \u09a6\u09c0\u09b0\u09cd\u0998 signal measurement duration \u09a8\u09bf\u09b2\u09c7 \u098f\u0987 \u09b6\u09b0\u09cd\u09a4 \u09aa\u09c2\u09b0\u09a3 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/p>\n

 <\/p>\n

\"\u099a\u09bf\u09a4\u09cd\u09b0.13

\u099a\u09bf\u09a4\u09cd\u09b0.13 \u098f\u0995\u099f\u09bf \u0997\u09bf\u09af\u09bc\u09be\u09b0\u09ac\u0995\u09cd\u09b8\u09c7\u09b0 \u0995\u09be\u0987\u09a8\u09c7\u09ae\u09c7\u099f\u09bf\u0995 \u09a4\u09cd\u09b0\u09c1\u099f\u09bf \u09b8\u0982\u0995\u09c7\u09a4 \u09ab\u09be\u0982\u09b6\u09a8 \u098f\u09ac\u0982 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0\u09b0 \u0989\u09a6\u09be\u09b9\u09b0\u09a3<\/p><\/div>\n

 <\/p>\n

measurement duration \u0995\u09ae \u09b9\u09b2\u09c7 \u099a\u09bf\u09a4\u09cd\u09b0\u099f\u09bf \u0986\u09b0\u0993 “\u0996\u09be\u09b0\u09be\u09aa” \u09a6\u09c7\u0996\u09be\u09ac\u09c7:<\/p>\n

 <\/p>\n

\"Fig.14

Fig.14 \u09b0\u099f\u09be\u09b0 \u0995\u09ae\u09cd\u09aa\u09a8 \u09ab\u09be\u0982\u09b6\u09a8 \u0993 \u09ac\u09b0\u09cd\u09a3\u09be\u09b2\u09c0\u09b0 \u0989\u09a6\u09be\u09b9\u09b0\u09a3<\/p><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

In practice, it can be difficult to understand where the \u201creal components\u201d and where the \u201cartifacts\u201d caused by inconsistency of component periods and signal sampling durations or \u201cjumps and breaks\u201d in the waveform. Of course, the words \u201creal components\u201d and \u201cartifacts\u201d are put in quotes for a reason. The presence of many harmonics on the spectrum graph does not mean that our signal actually consists of them. It is like thinking that number 7 \u201cconsists\u201d of numbers 3 and 4. The number 7 can be thought of as the sum of 3 and 4 \u2013 that is correct.<\/p>\n

So also our signal\u2026 or rather not even \u201cour signal\u201d, but a periodic function composed by repeating our signal (sample) can be represented as a sum of harmonics (sine waves) with certain amplitudes and phases. But in many cases important for practice (see figures above) it is indeed possible to relate the harmonics obtained in the spectrum also to real processes having cyclic character and contributing significantly to the form of the signal.<\/p>\n

\u0995\u09bf\u099b\u09c1 \u09ab\u09b2\u09be\u09ab\u09b2<\/h2>\n

1. ADC \u09a6\u09cd\u09ac\u09be\u09b0\u09be digitized, \u0985\u09b0\u09cd\u09a5\u09be\u09ce N\u099f\u09bf discrete sample-\u098f \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09bf\u09a4 T sec. \u09b8\u09cd\u09a5\u09be\u09af\u09bc\u09c0 \u098f\u0995\u099f\u09bf \u09ac\u09be\u09b8\u09cd\u09a4\u09ac measured signal-\u098f\u09b0 spectrum-\u0995\u09c7 N\/2\u099f\u09bf harmonic-\u098f\u09b0 \u09b8\u09ae\u09b7\u09cd\u099f\u09bf \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09af\u09be\u09af\u09bc\u0964<\/p>\n

2. Signal-\u099f\u09bf real value-\u098f\u09b0 \u098f\u0995\u099f\u09bf set \u09b9\u09bf\u09b8\u09c7\u09ac\u09c7 \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09bf\u09a4 \u09b9\u09af\u09bc\u0964 \u098f\u09b0 DFT spectrum \u09b9\u09b2\u09cb conjugate symmetry-\u09b8\u09b9 complex coefficient-\u098f\u09b0 \u098f\u0995\u099f\u09bf set; \u098f\u0997\u09c1\u09b2\u09cb \u09a5\u09c7\u0995\u09c7 amplitude spectrum \u09aa\u09be\u0993\u09af\u09bc\u09be \u09af\u09be\u09af\u09bc \u2014 positive frequency-\u09a4\u09c7 real non-negative amplitude (\u098f\u09ac\u0982 phase)-\u098f\u09b0 set, \u098f\u09ac\u0982 \u09ac\u09be\u09b8\u09cd\u09a4\u09ac\u09c7 plot \u0995\u09b0\u09be \u09b9\u09af\u09bc \u098f\u0987 one-sided amplitude spectrum-\u099f\u09bf\u0987\u0964 Negative frequency-\u09b8\u09b9 two-sided complex form \u098f\u09ac\u0982 one-sided amplitude\/phase form \u098f\u0995\u0987 spectrum-\u098f\u09b0 \u09b8\u09ae\u09a4\u09c1\u09b2\u09cd\u09af \u0989\u09aa\u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u2014 signal analysis-\u098f\u09b0 \u099c\u09a8\u09cd\u09af \u09b8\u09be\u09a7\u09be\u09b0\u09a3\u09a4 one-sided amplitude spectrum-\u098f \u0995\u09be\u099c \u0995\u09b0\u09be\u0987 \u09ac\u09c7\u09b6\u09bf \u09b8\u09c1\u09ac\u09bf\u09a7\u09be\u099c\u09a8\u0995\u0964<\/p>\n

3. \u09b8\u09ae\u09af\u09bc T-\u098f \u09ae\u09be\u09aa\u09be signal \u0995\u09c7\u09ac\u09b2 \u09b8\u09c7\u0987 \u09b8\u09ae\u09af\u09bc\u09b8\u09c0\u09ae\u09be\u09a4\u09c7\u0987 \u09b8\u0982\u099c\u09cd\u099e\u09be\u09af\u09bc\u09bf\u09a4\u0964 measurement \u09b6\u09c1\u09b0\u09c1\u09b0 \u0986\u0997\u09c7 \u0995\u09c0 \u0998\u099f\u09c7\u099b\u09bf\u09b2 \u09ac\u09be \u09aa\u09b0\u09c7 \u0995\u09c0 \u0998\u099f\u09ac\u09c7 \u09a4\u09be \u098f\u0996\u09be\u09a8\u09c7 \u0997\u09c1\u09b0\u09c1\u09a4\u09cd\u09ac\u09aa\u09c2\u09b0\u09cd\u09a3 \u09a8\u09af\u09bc\u0964 time-limited signal-\u098f\u09b0 FFT \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b6\u09b0\u09cd\u09a4\u09c7 \u09a4\u09be\u09b0 component-\u098f\u09b0 amplitude \u0993 frequency \u0997\u09a3\u09a8\u09be \u0995\u09b0\u09a4\u09c7 \u09b8\u0995\u09cd\u09b7\u09ae \u098f\u0995\u099f\u09bf “\u09ac\u09be\u09b8\u09cd\u09a4\u09ac” spectrum \u09a6\u09c7\u09af\u09bc\u0964<\/p>\n

 <\/p>","protected":false},"excerpt":{"rendered":"

Application of the Fourier Transform to the Analysis of Vibration Signals Andrei Shelkovenko. One of the developers and founder of Vibromera. The translation of the article may contain inaccuracies. Fourier transform and signal spectrum In many cases the task of obtaining (calculating) the spectrum of a signal is as follows. […]<\/p>\n","protected":false},"author":2,"featured_media":2161,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ai_generated_summary":"","footnotes":""},"categories":[4],"tags":[],"class_list":["post-2138","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-example"],"_links":{"self":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/posts\/2138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/comments?post=2138"}],"version-history":[{"count":2,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/posts\/2138\/revisions"}],"predecessor-version":[{"id":102137,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/posts\/2138\/revisions\/102137"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/media\/2161"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/media?parent=2138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/categories?post=2138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/bn\/wp-json\/wp\/v2\/tags?post=2138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}