{"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\/vi\/example\/application-of-the-fourier-transform-to-the-analysis-of-vibration-signals\/","title":{"rendered":"\u1ee8ng d\u1ee5ng bi\u1ebfn \u0111\u1ed5i Fourier \u0111\u1ec3 ph\u00e2n t\u00edch t\u00edn hi\u1ec7u rung."},"content":{"rendered":"

\u1ee8ng d\u1ee5ng Bi\u1ebfn \u0111\u1ed5i Fourier \u0111\u1ec3 Ph\u00e2n t\u00edch T\u00edn hi\u1ec7u Rung \u0111\u1ed9ng<\/h1>\n

Andrei Shelkovenko. M\u1ed9t trong nh\u1eefng nh\u00e0 ph\u00e1t tri\u1ec3n v\u00e0 l\u00e0 ng\u01b0\u1eddi s\u00e1ng l\u1eadp Vibromera.
\nB\u1ea3n d\u1ecbch b\u00e0i vi\u1ebft n\u00e0y c\u00f3 th\u1ec3 ch\u1ee9a m\u1ed9t s\u1ed1 sai s\u00f3t.<\/p>\n

Bi\u1ebfn \u0111\u1ed5i Fourier v\u00e0 ph\u1ed5 t\u00edn hi\u1ec7u<\/h2>\n

Trong nhi\u1ec1u tr\u01b0\u1eddng h\u1ee3p, nhi\u1ec7m v\u1ee5 l\u1ea5y (t\u00ednh to\u00e1n) ph\u1ed5<\/a> c\u1ee7a m\u1ed9t t\u00edn hi\u1ec7u nh\u01b0 sau. C\u00f3 m\u1ed9t ADC, c\u00e1i m\u00e0 v\u1edbi t\u1ea7n s\u1ed1 l\u1ea5y m\u1eabu T\u00ednh th\u01b0\u1eddng xuy\u00ean<\/a> Fd chuy\u1ec3n \u0111\u1ed5i t\u00edn hi\u1ec7u li\u00ean t\u1ee5c, t\u00edn hi\u1ec7u n\u00e0y \u0111\u1ebfn \u0111\u1ea7u v\u00e0o c\u1ee7a n\u00f3 trong kho\u1ea3ng th\u1eddi gian T, th\u00e0nh c\u00e1c m\u1eabu k\u1ef9 thu\u1eadt s\u1ed1 – N m\u1eabu. Sau \u0111\u00f3, m\u1ea3ng m\u1eabu n\u00e0y \u0111\u01b0\u1ee3c \u0111\u01b0a v\u00e0o m\u1ed9t ch\u01b0\u01a1ng tr\u00ecnh n\u00e0o \u0111\u00f3 (v\u00ed d\u1ee5 nh\u01b0 FourierScope<\/a><\/span>) tr\u1ea3 v\u1ec1 N\/2 gi\u00e1 tr\u1ecb s\u1ed1.<\/p>\n

\u0110\u1ec3 ki\u1ec3m tra xem ch\u01b0\u01a1ng tr\u00ecnh c\u00f3 ho\u1ea1t \u0111\u1ed9ng ch\u00ednh x\u00e1c hay kh\u00f4ng, ch\u00fang ta t\u1ea1o m\u1ed9t m\u1ea3ng c\u00e1c m\u1eabu d\u1eef li\u1ec7u d\u01b0\u1edbi d\u1ea1ng t\u1ed5ng c\u1ee7a hai bi\u1ec3u th\u1ee9c sin(10*2*\u03c0*x) + 0,5*sin(5*2*\u03c0*x) v\u00e0 \u0111\u01b0a v\u00e0o ch\u01b0\u01a1ng tr\u00ecnh. Ch\u01b0\u01a1ng tr\u00ecnh \u0111\u00e3 v\u1ebd ra \u0111\u1ed3 th\u1ecb nh\u01b0 sau:
\n

\"Bi\u1ebfn

H\u00ecnh 1. \u0110\u1ed3 th\u1ecb h\u00e0m th\u1eddi gian c\u1ee7a t\u00edn hi\u1ec7u<\/span><\/i><\/em><\/p><\/div><\/p>\n

 <\/p>\n

\"H\u00ecnh

H\u00ecnh 2. Bi\u1ec3u \u0111\u1ed3 ph\u1ed5 t\u00edn hi\u1ec7u<\/p><\/div>\n

 <\/p>\n

C\u00f3 hai s\u00f3ng h\u00e0i<\/a> tr\u00ean bi\u1ec3u \u0111\u1ed3 ph\u1ed5 – 5 Hz v\u1edbi bi\u00ean \u0111\u1ed9 0,5 V v\u00e0 10 Hz v\u1edbi bi\u00ean \u0111\u1ed9 1 V, t\u1ea5t c\u1ea3 \u0111\u1ec1u gi\u1ed1ng nh\u01b0 trong c\u00f4ng th\u1ee9c c\u1ee7a t\u00edn hi\u1ec7u g\u1ed1c. T\u1ea5t c\u1ea3 \u0111\u1ec1u \u1ed5n, ch\u01b0\u01a1ng tr\u00ecnh ho\u1ea1t \u0111\u1ed9ng ch\u00ednh x\u00e1c.<\/p>\n

\u0110i\u1ec1u n\u00e0y c\u00f3 ngh\u0129a l\u00e0 n\u1ebfu ch\u00fang ta \u0111\u01b0a m\u1ed9t t\u00edn hi\u1ec7u th\u1ef1c \u0111\u01b0\u1ee3c t\u1ea1o th\u00e0nh t\u1eeb s\u1ef1 k\u1ebft h\u1ee3p c\u1ee7a hai s\u00f3ng sin v\u00e0o \u0111\u1ea7u v\u00e0o c\u1ee7a b\u1ed9 chuy\u1ec3n \u0111\u1ed5i t\u01b0\u01a1ng t\u1ef1-s\u1ed1 (ADC), ch\u00fang ta s\u1ebd thu \u0111\u01b0\u1ee3c ph\u1ed5 t\u1ea7n s\u1ed1 t\u01b0\u01a1ng t\u1ef1, bao g\u1ed3m hai th\u00e0nh ph\u1ea7n h\u00e0i.<\/p>\n

V\u1eady n\u00ean, ch\u00fang t\u00f4i th\u1ef1c s\u1ef1 <\/span><\/b><\/strong>t\u00edn hi\u1ec7u \u0111o \u0111\u01b0\u1ee3c k\u00e9o d\u00e0i 5 gi\u00e2y<\/span><\/b><\/strong>, \u0111\u01b0\u1ee3c s\u1ed1 h\u00f3a b\u1eb1ng ADC, t\u1ee9c l\u00e0 \u0111\u01b0\u1ee3c th\u1ec3 hi\u1ec7n theo t\u1eebng ph\u1ea7n <\/span><\/b><\/strong>m\u1eabu, c\u00f3 m\u1ed9t t\u00e1ch bi\u1ec7t, kh\u00f4ng tu\u1ea7n ho\u00e0n <\/span><\/b><\/strong>Ph\u1ed5.
\nT\u1eeb g\u00f3c \u0111\u1ed9 to\u00e1n h\u1ecdc \u2013 c\u00e2u n\u00e0y c\u00f3 bao nhi\u00eau l\u1ed7i? <\/span><\/i><\/em><\/p>\n

B\u00e2y gi\u1edd ch\u00fang ta h\u00e3y th\u1eed \u0111o t\u00edn hi\u1ec7u n\u00e0y trong 0,5 gi\u00e2y.<\/p>\n

\"H\u00ecnh

H\u00ecnh 3. \u0110\u1ed3 th\u1ecb c\u1ee7a h\u00e0m sin(10*2*\u03c0*x)+0,5*sin(5*2*\u03c0*x) trong kho\u1ea3ng th\u1eddi gian \u0111o l\u00e0 0,5 gi\u00e2y<\/p><\/div>\n

 <\/p>\n

\"H\u00ecnh

H\u00ecnh 4. \u0110\u1ed3 th\u1ecb ph\u1ed5 c\u1ee7a h\u00e0m<\/p><\/div>\n

 <\/p>\n

C\u00f3 g\u00ec \u0111\u00f3 kh\u00f4ng \u1ed5n \u1edf \u0111\u00e2y! S\u00f3ng h\u00e0i \u1edf t\u1ea7n s\u1ed1 10 Hz \u0111\u01b0\u1ee3c v\u1ebd b\u00ecnh th\u01b0\u1eddng, nh\u01b0ng thay v\u00ec s\u00f3ng h\u00e0i \u1edf t\u1ea7n s\u1ed1 5 Hz th\u00ec l\u1ea1i xu\u1ea5t hi\u1ec7n m\u1ed9t s\u1ed1 s\u00f3ng h\u00e0i kh\u00f4ng r\u00f5 r\u00e0ng.<\/p>\n

Tr\u00ean m\u1ea1ng ng\u01b0\u1eddi ta n\u00f3i r\u1eb1ng c\u1ea7n ph\u1ea3i th\u00eam c\u00e1c s\u1ed1 0 v\u00e0o cu\u1ed1i m\u1eabu th\u00ec \u0111\u1ed3 th\u1ecb ph\u1ed5 m\u1edbi \u0111\u01b0\u1ee3c v\u1ebd b\u00ecnh th\u01b0\u1eddng.<\/p>\n

\"H\u00ecnh

H\u00ecnh 5. Ch\u00fang t\u00f4i \u0111\u00e3 th\u00eam c\u00e1c gi\u00e1 tr\u1ecb b\u1eb1ng 0 v\u00e0o m\u1eabu d\u1eef li\u1ec7u cho \u0111\u1ebfn gi\u00e2y th\u1ee9 5<\/p><\/div>\n

 <\/p>\n

\"H\u00ecnh

H\u00ecnh 6. Ph\u1ed5 thu \u0111\u01b0\u1ee3c.<\/p><\/div>\n

 <\/p>\n

Kh\u00f4ng ph\u1ea3i v\u1eady \u0111\u00e2u. T\u00f4i s\u1ebd ph\u1ea3i t\u00ecm hi\u1ec3u v\u1ec1 l\u00fd thuy\u1ebft n\u00e0y. Ch\u00fang ta h\u00e3y chuy\u1ec3n sang Wikipedia<\/b><\/strong><\/a><\/span>\u00a0\u2013 ngu\u1ed3n tri th\u1ee9c.<\/p>\n

H\u00e0m li\u00ean t\u1ee5c v\u00e0 bi\u1ec3u di\u1ec5n d\u01b0\u1edbi d\u1ea1ng d\u00e3y Fourier c\u1ee7a n\u00f3<\/h2>\n

V\u1ec1 m\u1eb7t to\u00e1n h\u1ecdc, t\u00edn hi\u1ec7u c\u1ee7a ch\u00fang ta c\u00f3 \u0111\u1ed9 d\u00e0i T gi\u00e2y l\u00e0 m\u1ed9t h\u00e0m f(x) \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a tr\u00ean kho\u1ea3ng {0, T} (trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y, X l\u00e0 th\u1eddi gian). H\u00e0m nh\u01b0 v\u1eady lu\u00f4n c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n d\u01b0\u1edbi d\u1ea1ng t\u1ed5ng c\u1ee7a c\u00e1c h\u00e0m h\u00e0i (sin ho\u1eb7c cos) theo d\u1ea1ng:<\/p>\n

\"H\u00e0m

\u00a0(1), trong \u0111\u00f3:<\/p>\n

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

k l\u00e0 s\u1ed1 th\u1ee9 t\u1ef1 c\u1ee7a h\u00e0m l\u01b0\u1ee3ng gi\u00e1c (s\u1ed1 th\u1ee9 t\u1ef1 c\u1ee7a th\u00e0nh ph\u1ea7n h\u00e0i, s\u1ed1 th\u1ee9 t\u1ef1 c\u1ee7a s\u00f3ng h\u00e0i)
\nT \u2013 \u0111o\u1ea1n m\u00e0 h\u00e0m \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh (th\u1eddi gian t\u1ed3n t\u1ea1i c\u1ee7a t\u00edn hi\u1ec7u)
\nAk \u2013 bi\u00ean \u0111\u1ed9 c\u1ee7a th\u00e0nh ph\u1ea7n h\u00e0i th\u1ee9 k,
\n\u03b8k \u2013 pha ban \u0111\u1ea7u c\u1ee7a th\u00e0nh ph\u1ea7n h\u00e0i th\u1ee9 k
\n\u201cBi\u1ec3u di\u1ec5n h\u00e0m s\u1ed1 d\u01b0\u1edbi d\u1ea1ng t\u1ed5ng c\u1ee7a d\u00e3y Fourier\u201d c\u00f3 ngh\u0129a l\u00e0 g\u00ec? \u0110i\u1ec1u \u0111\u00f3 c\u00f3 ngh\u0129a l\u00e0 b\u1eb1ng c\u00e1ch c\u1ed9ng c\u00e1c gi\u00e1 tr\u1ecb c\u1ee7a c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i trong d\u00e3y Fourier t\u1ea1i m\u1ed7i \u0111i\u1ec3m, ta s\u1ebd thu \u0111\u01b0\u1ee3c gi\u00e1 tr\u1ecb c\u1ee7a h\u00e0m s\u1ed1 t\u1ea1i \u0111i\u1ec3m \u0111\u00f3.
\n(N\u00f3i ch\u00ednh x\u00e1c h\u01a1n, sai s\u1ed1 b\u00ecnh ph\u01b0\u01a1ng trung b\u00ecnh c\u1ee7a d\u00e3y so v\u1edbi h\u00e0m f(x) s\u1ebd ti\u1ebfn t\u1edbi 0, nh\u01b0ng m\u1eb7c d\u00f9 c\u00f3 s\u1ef1 h\u1ed9i t\u1ee5 b\u00ecnh ph\u01b0\u01a1ng trung b\u00ecnh, d\u00e3y Fourier c\u1ee7a m\u1ed9t h\u00e0m n\u00f3i chung kh\u00f4ng nh\u1ea5t thi\u1ebft ph\u1ea3i h\u1ed9i t\u1ee5 t\u1edbi h\u00e0m \u0111\u00f3 theo t\u1eebng \u0111i\u1ec3m.)
\nD\u00e3y s\u1ed1 n\u00e0y c\u0169ng c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c vi\u1ebft d\u01b0\u1edbi d\u1ea1ng:<\/p>\n

\"(2),\"

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

 <\/p>\n

 <\/p>\n

 <\/p>\n

\u1ede \u0111\u00e2u \"Ph\u01b0\u01a1ng , bi\u00ean \u0111\u1ed9 ph\u1ee9c th\u1ee9 k.<\/p>\n

 <\/p>\n

ho\u1eb7c<\/p>\n

\"

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

 <\/p>\n

 <\/p>\n

 <\/p>\n

M\u1ed1i quan h\u1ec7 gi\u1eefa c\u00e1c h\u1ec7 s\u1ed1 (1) v\u00e0 (3) \u0111\u01b0\u1ee3c th\u1ec3 hi\u1ec7n qua c\u00e1c c\u00f4ng th\u1ee9c sau:<\/p>\n

\"C\u00f4ng<\/p>\n

 <\/p>\n

 <\/p>\n

\"C\u00f4ng<\/p>\n

 <\/p>\n

 <\/p>\n

L\u01b0u \u00fd r\u1eb1ng c\u1ea3 ba c\u00e1ch bi\u1ec3u di\u1ec5n chu\u1ed7i Fourier n\u00e0y ho\u00e0n to\u00e0n t\u01b0\u01a1ng \u0111\u01b0\u01a1ng. \u0110\u00f4i khi khi l\u00e0m vi\u1ec7c v\u1edbi chu\u1ed7i Fourier, s\u1ebd thu\u1eadn ti\u1ec7n h\u01a1n n\u1ebfu d\u00f9ng s\u1ed1 m\u0169 v\u1edbi \u0111\u1ed1i s\u1ed1 \u1ea3o thay v\u00ec sin v\u00e0 cos, t\u1ee9c l\u00e0 d\u00f9ng bi\u1ebfn \u0111\u1ed5i Fourier \u1edf d\u1ea1ng ph\u1ee9c. Nh\u01b0ng v\u1edbi ch\u00fang ta, s\u1ebd thu\u1eadn ti\u1ec7n khi d\u00f9ng c\u00f4ng th\u1ee9c (1), trong \u0111\u00f3 chu\u1ed7i Fourier \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n nh\u01b0 t\u1ed5ng c\u00e1c cosin v\u1edbi c\u00e1c bi\u00ean \u0111\u1ed9 v\u00e0 pha t\u01b0\u01a1ng \u1ee9ng. N\u00f3i ch\u00ednh x\u00e1c, bi\u1ebfn \u0111\u1ed5i Fourier c\u1ee7a m\u1ed9t t\u00edn hi\u1ec7u th\u1ef1c th\u1ef1c s\u1ef1 t\u1ea1o ra c\u00e1c h\u1ec7 s\u1ed1 ph\u1ee9c (d\u1ea1ng (3)): m\u1ed7i h\u1ec7 s\u1ed1 mang c\u1ea3 bi\u00ean \u0111\u1ed9 l\u1eabn pha c\u1ee7a h\u00e0i t\u01b0\u01a1ng \u1ee9ng. \u0110\u1ed1i v\u1edbi t\u00edn hi\u1ec7u th\u1ef1c, c\u00e1c h\u1ec7 s\u1ed1 ph\u1ee9c n\u00e0y c\u00f3 t\u00ednh \u0111\u1ed1i x\u1ee9ng li\u00ean h\u1ee3p (Hermitian) \u2014 n\u1eeda t\u1ea7n s\u1ed1 \u00e2m ch\u1ec9 \u0111\u01a1n gi\u1ea3n ph\u1ea3n chi\u1ebfu n\u1eeda t\u1ea7n s\u1ed1 d\u01b0\u01a1ng v\u00e0 kh\u00f4ng mang th\u00eam th\u00f4ng tin n\u00e0o. \u0110\u00f3 l\u00e0 l\u00fd do t\u1eeb c\u00e1c h\u1ec7 s\u1ed1 ph\u1ee9c, ta lu\u00f4n c\u00f3 th\u1ec3 chuy\u1ec3n sang c\u00e1c bi\u00ean \u0111\u1ed9 kh\u00f4ng \u00e2m th\u1ef1c Ak v\u00e0 c\u00e1c pha \u03b8k c\u1ee7a c\u00f4ng th\u1ee9c (1) \u2014 v\u00e0 ch\u00ednh ph\u1ed5 bi\u00ean \u0111\u1ed9 n\u00e0y \u0111\u01b0\u1ee3c c\u00e1c ch\u01b0\u01a1ng tr\u00ecnh ph\u00e2n t\u00edch hi\u1ec3n th\u1ecb.<\/p>\n

 <\/p>\n

T\u00f3m l\u1ea1i:<\/span><\/b><\/u><\/strong>
\n<\/span><\/b><\/strong>C\u01a1 s\u1edf to\u00e1n h\u1ecdc cho ph\u00e2n t\u00edch ph\u1ed5 t\u00edn hi\u1ec7u l\u00e0 bi\u1ebfn \u0111\u1ed5i Fourier.<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>
\n<\/span><\/b><\/strong>Bi\u1ebfn \u0111\u1ed5i Fourier cho ph\u00e9p bi\u1ec3u di\u1ec5n m\u1ed9t h\u00e0m li\u00ean t\u1ee5c f(x) (t\u00edn hi\u1ec7u) \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a tr\u00ean kho\u1ea3ng {0, T} d\u01b0\u1edbi d\u1ea1ng t\u1ed5ng c\u1ee7a m\u1ed9t d\u00e3y v\u00f4 h\u1ea1n (d\u00e3y Fourier) c\u00e1c h\u00e0m l\u01b0\u1ee3ng gi\u00e1c (sin v\u00e0\/ho\u1eb7c cos) c\u00f3 bi\u00ean \u0111\u1ed9 v\u00e0 pha x\u00e1c \u0111\u1ecbnh, c\u0169ng \u0111\u01b0\u1ee3c xem x\u00e9t tr\u00ean kho\u1ea3ng {0, T}. D\u00e3y nh\u01b0 v\u1eady \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 d\u00e3y Fourier.<\/span><\/b><\/strong><\/p>\n

H\u00e3y l\u01b0u \u00fd th\u00eam m\u1ed9t s\u1ed1 \u0111i\u1ec3m sau \u0111\u00e2y, vi\u1ec7c n\u1eafm v\u1eefng nh\u1eefng \u0111i\u1ec3m n\u00e0y l\u00e0 \u0111i\u1ec1u c\u1ea7n thi\u1ebft \u0111\u1ec3 \u00e1p d\u1ee5ng bi\u1ebfn \u0111\u1ed5i Fourier m\u1ed9t c\u00e1ch ch\u00ednh x\u00e1c trong ph\u00e2n t\u00edch t\u00edn hi\u1ec7u. N\u1ebfu xem x\u00e9t d\u00e3y Fourier (t\u1ed5ng c\u1ee7a c\u00e1c h\u00e0m sin) tr\u00ean to\u00e0n b\u1ed9 tr\u1ee5c X, ta s\u1ebd th\u1ea5y r\u1eb1ng ngo\u00e0i kho\u1ea3ng {0, T}, h\u00e0m d\u00e3y Fourier s\u1ebd l\u1eb7p l\u1ea1i h\u00e0m ban \u0111\u1ea7u theo chu k\u1ef3.<\/p>\n

V\u00ed d\u1ee5, trong \u0111\u1ed3 th\u1ecb \u1edf H\u00ecnh 7, h\u00e0m ban \u0111\u1ea7u \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a tr\u00ean kho\u1ea3ng {-T\/2, +T\/2}, c\u00f2n d\u00e3y Fourier bi\u1ec3u di\u1ec5n m\u1ed9t h\u00e0m tu\u1ea7n ho\u00e0n \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a tr\u00ean to\u00e0n b\u1ed9 tr\u1ee5c x.<\/p>\n

\u0110i\u1ec1u n\u00e0y l\u00e0 do c\u00e1c h\u00e0m sin t\u1ef1 th\u00e2n l\u00e0 c\u00e1c h\u00e0m tu\u1ea7n ho\u00e0n, n\u00ean t\u1ed5ng c\u1ee7a ch\u00fang c\u0169ng s\u1ebd l\u00e0 m\u1ed9t h\u00e0m tu\u1ea7n ho\u00e0n.<\/p>\n

\"H\u00ecnh

H\u00ecnh 7: Bi\u1ec3u di\u1ec5n h\u00e0m ngu\u1ed3n kh\u00f4ng tu\u1ea7n ho\u00e0n b\u1eb1ng d\u00e3y Fourier<\/p><\/div>\n

Do \u0111\u00f3:<\/p>\n

H\u00e0m ban \u0111\u1ea7u c\u1ee7a ch\u00fang ta l\u00e0 m\u1ed9t h\u00e0m li\u00ean t\u1ee5c, kh\u00f4ng tu\u1ea7n ho\u00e0n, \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a tr\u00ean m\u1ed9t \u0111o\u1ea1n c\u00f3 \u0111\u1ed9 d\u00e0i T.
\nPh\u1ea1m vi c\u1ee7a h\u00e0m n\u00e0y l\u00e0 r\u1eddi r\u1ea1c, t\u1ee9c l\u00e0 n\u00f3 \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n d\u01b0\u1edbi d\u1ea1ng m\u1ed9t d\u00e3y v\u00f4 h\u1ea1n c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i \u2013 m\u1ed9t d\u00e3y Fourier.
\nTh\u1ef1c ra, d\u00e3y Fourier x\u00e1c \u0111\u1ecbnh m\u1ed9t h\u00e0m tu\u1ea7n ho\u00e0n, tr\u00f9ng kh\u1edbp v\u1edbi h\u00e0m c\u1ee7a ch\u00fang ta tr\u00ean kho\u1ea3ng {0, T}, nh\u01b0ng \u0111\u1ed1i v\u1edbi ch\u00fang ta, t\u00ednh tu\u1ea7n ho\u00e0n n\u00e0y kh\u00f4ng ph\u1ea3i l\u00e0 y\u1ebfu t\u1ed1 thi\u1ebft y\u1ebfu.<\/p>\n

Ti\u1ebfp theo.<\/p>\n

C\u00e1c chu k\u1ef3 c\u1ee7a c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a kho\u1ea3ng th\u1eddi gian {0, T}, tr\u00ean \u0111\u00f3 h\u00e0m ban \u0111\u1ea7u f(x) \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh. N\u00f3i c\u00e1ch kh\u00e1c, c\u00e1c chu k\u1ef3 c\u1ee7a c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a th\u1eddi gian \u0111o t\u00edn hi\u1ec7u. V\u00ed d\u1ee5, chu k\u1ef3 c\u1ee7a th\u00e0nh ph\u1ea7n h\u00e0i th\u1ee9 nh\u1ea5t trong d\u00e3y Fourier b\u1eb1ng kho\u1ea3ng th\u1eddi gian T m\u00e0 h\u00e0m f(x) \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a. Chu k\u1ef3 c\u1ee7a th\u00e0nh ph\u1ea7n h\u00e0i th\u1ee9 hai trong d\u00e3y Fourier b\u1eb1ng kho\u1ea3ng th\u1eddi gian T\/2. V\u00e0 c\u1ee9 th\u1ebf (xem H\u00ecnh 8).<\/p>\n

\"H\u00ecnh

H\u00ecnh 8: Chu k\u1ef3 (t\u1ea7n s\u1ed1) c\u1ee7a c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i trong d\u00e3y Fourier (\u1edf \u0111\u00e2y T = 2\u03c0)<\/p><\/div>\n

Theo \u0111\u00f3, t\u1ea7n s\u1ed1 c\u1ee7a c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i l\u00e0 b\u1ed9i s\u1ed1 c\u1ee7a 1\/T. Ngh\u0129a l\u00e0, t\u1ea7n s\u1ed1 c\u1ee7a c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i F_k \u0111\u01b0\u1ee3c t\u00ednh theo c\u00f4ng th\u1ee9c F_k = k\/T, trong \u0111\u00f3 k thay \u0111\u1ed5i t\u1eeb 0 \u0111\u1ebfn \u221e; v\u00ed d\u1ee5: k = 0 th\u00ec F\u2080 = 0; k = 1 th\u00ec F\u2081 = 1\/T; k = 2 th\u00ec F\u2082 = 2\/T; k = 3 th\u00ec F\u2083 = 3\/T; \u2026 Fk = k\/T (t\u1ea1i t\u1ea7n s\u1ed1 b\u1eb1ng kh\u00f4ng, th\u00e0nh ph\u1ea7n h\u1eb1ng s\u1ed1).<\/p>\n

Gi\u1ea3 s\u1eed h\u00e0m ban \u0111\u1ea7u c\u1ee7a ch\u00fang ta l\u00e0 m\u1ed9t t\u00edn hi\u1ec7u \u0111\u01b0\u1ee3c ghi l\u1ea1i trong kho\u1ea3ng th\u1eddi gian T = 1 gi\u00e2y. Khi \u0111\u00f3, chu k\u1ef3 c\u1ee7a s\u00f3ng h\u00e0i th\u1ee9 nh\u1ea5t s\u1ebd b\u1eb1ng th\u1eddi gian c\u1ee7a t\u00edn hi\u1ec7u T\u2081 = T = 1 gi\u00e2y v\u00e0 t\u1ea7n s\u1ed1 c\u1ee7a s\u00f3ng h\u00e0i n\u00e0y b\u1eb1ng 1 Hz. Chu k\u1ef3 c\u1ee7a s\u00f3ng h\u00e0i th\u1ee9 hai s\u1ebd b\u1eb1ng th\u1eddi gian c\u1ee7a t\u00edn hi\u1ec7u chia cho 2 (T2 = T\/2 = 0,5 gi\u00e2y) v\u00e0 t\u1ea7n s\u1ed1 b\u1eb1ng 2 Hz. \u0110\u1ed1i v\u1edbi s\u00f3ng h\u00e0i th\u1ee9 ba, T3 = T\/3 gi\u00e2y v\u00e0 t\u1ea7n s\u1ed1 l\u00e0 3 Hz. V\u00e0 c\u1ee9 th\u1ebf ti\u1ebfp t\u1ee5c.<\/p>\n

Trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y, kho\u1ea3ng c\u00e1ch gi\u1eefa c\u00e1c s\u00f3ng h\u00e0i l\u00e0 1 Hz.<\/p>\n

Do \u0111\u00f3, m\u1ed9t t\u00edn hi\u1ec7u c\u00f3 th\u1eddi gian k\u00e9o d\u00e0i 1 gi\u00e2y c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c ph\u00e2n t\u00edch th\u00e0nh c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i (\u0111\u1ec3 thu \u0111\u01b0\u1ee3c ph\u1ed5) v\u1edbi \u0111\u1ed9 ph\u00e2n gi\u1ea3i t\u1ea7n s\u1ed1 l\u00e0 1 Hz.
\n\u0110\u1ec3 t\u0103ng \u0111\u1ed9 ph\u00e2n gi\u1ea3i l\u00ean 2 l\u1ea7n th\u00e0nh 0,5 Hz, c\u1ea7n ph\u1ea3i t\u0103ng th\u1eddi gian \u0111o l\u01b0\u1eddng l\u00ean 2 l\u1ea7n th\u00e0nh 2 gi\u00e2y. T\u00edn hi\u1ec7u 10 gi\u00e2y c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c ph\u00e2n t\u00e1ch th\u00e0nh c\u00e1c th\u00e0nh ph\u1ea7n \u0111i\u1ec1u h\u00f2a (ph\u1ed5) v\u1edbi \u0111\u1ed9 ph\u00e2n gi\u1ea3i t\u1ea7n s\u1ed1 0,1 Hz. Kh\u00f4ng c\u00f3 c\u00e1ch n\u00e0o kh\u00e1c \u0111\u1ec3 t\u0103ng \u0111\u1ed9 ph\u00e2n gi\u1ea3i t\u1ea7n s\u1ed1. B\u1ea1n c\u00f3 th\u1ec3 kh\u00e1m ph\u00e1 m\u1ed1i quan h\u1ec7 n\u00e0y v\u1edbi M\u00e1y t\u00ednh \u0111\u1ed9 ph\u00e2n gi\u1ea3i FFT<\/a>.<\/p>\n

C\u00f3 m\u1ed9t c\u00e1ch \u0111\u1ec3 k\u00e9o d\u00e0i th\u1eddi gian t\u00edn hi\u1ec7u m\u1ed9t c\u00e1ch nh\u00e2n t\u1ea1o b\u1eb1ng c\u00e1ch th\u00eam c\u00e1c s\u1ed1 0 v\u00e0o m\u1ea3ng m\u1eabu. Tuy nhi\u00ean, \u0111i\u1ec1u n\u00e0y kh\u00f4ng l\u00e0m t\u0103ng \u0111\u1ed9 ph\u00e2n gi\u1ea3i t\u1ea7n s\u1ed1 th\u1ef1c.<\/p>\n

T\u00edn hi\u1ec7u r\u1eddi r\u1ea1c v\u00e0 bi\u1ebfn \u0111\u1ed5i Fourier r\u1eddi r\u1ea1c<\/h2>\n

C\u00f9ng v\u1edbi s\u1ef1 ph\u00e1t tri\u1ec3n c\u1ee7a c\u00f4ng ngh\u1ec7 s\u1ed1, c\u00e1c ph\u01b0\u01a1ng th\u1ee9c l\u01b0u tr\u1eef d\u1eef li\u1ec7u \u0111o l\u01b0\u1eddng (t\u00edn hi\u1ec7u) \u0111\u00e3 c\u00f3 nhi\u1ec1u thay \u0111\u1ed5i. Trong khi tr\u01b0\u1edbc \u0111\u00e2y t\u00edn hi\u1ec7u c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c ghi l\u1ea1i tr\u00ean m\u00e1y ghi \u00e2m v\u00e0 l\u01b0u tr\u1eef tr\u00ean b\u0103ng t\u1eeb d\u01b0\u1edbi d\u1ea1ng t\u01b0\u01a1ng t\u1ef1, th\u00ec hi\u1ec7n nay c\u00e1c t\u00edn hi\u1ec7u \u0111\u01b0\u1ee3c s\u1ed1 h\u00f3a v\u00e0 l\u01b0u tr\u1eef d\u01b0\u1edbi d\u1ea1ng c\u00e1c t\u1ec7p tin trong b\u1ed9 nh\u1edb m\u00e1y t\u00ednh d\u01b0\u1edbi d\u1ea1ng m\u1ed9t d\u00e3y s\u1ed1 (gi\u00e1 tr\u1ecb \u0111\u1ebfm).<\/p>\n

Quy tr\u00ecnh th\u00f4ng th\u01b0\u1eddng \u0111\u1ec3 \u0111o l\u01b0\u1eddng v\u00e0 s\u1ed1 h\u00f3a t\u00edn hi\u1ec7u di\u1ec5n ra nh\u01b0 sau.<\/p>\n

C\u1ea3m bi\u1ebfn \u0111o \u2014- B\u1ed9 chu\u1ea9n h\u00f3a t\u00edn hi\u1ec7u \u2014- B\u1ed9 chuy\u1ec3n \u0111\u1ed5i t\u01b0\u01a1ng t\u1ef1-s\u1ed1 (ADC) \u2014\u2013 M\u00e1y t\u00ednh
\n(H\u00ecnh 9. S\u01a1 \u0111\u1ed3 k\u00eanh \u0111o l\u01b0\u1eddng)<\/p>\n

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

T\u00edn hi\u1ec7u t\u1eeb b\u1ed9 chuy\u1ec3n \u0111\u1ed5i \u0111o l\u01b0\u1eddng \u0111\u01b0\u1ee3c truy\u1ec1n \u0111\u1ebfn b\u1ed9 chuy\u1ec3n \u0111\u1ed5i t\u01b0\u01a1ng t\u1ef1-s\u1ed1 (ADC) trong kho\u1ea3ng th\u1eddi gian T. C\u00e1c gi\u00e1 tr\u1ecb t\u00edn hi\u1ec7u (d\u1eef li\u1ec7u l\u1ea5y m\u1eabu) thu \u0111\u01b0\u1ee3c trong kho\u1ea3ng th\u1eddi gian T \u0111\u01b0\u1ee3c truy\u1ec1n \u0111\u1ebfn m\u00e1y t\u00ednh v\u00e0 l\u01b0u tr\u1eef trong b\u1ed9 nh\u1edb.<\/p>\n

\"H\u00ecnh

H\u00ecnh 10. T\u00edn hi\u1ec7u s\u1ed1 h\u00f3a \u2013 N m\u1eabu nh\u1eadn \u0111\u01b0\u1ee3c trong kho\u1ea3ng th\u1eddi gian T<\/p><\/div>\n

C\u00e1c y\u00eau c\u1ea7u \u0111\u1ec3 s\u1ed1 h\u00f3a c\u00e1c th\u00f4ng s\u1ed1 t\u00edn hi\u1ec7u l\u00e0 g\u00ec? M\u1ed9t thi\u1ebft b\u1ecb chuy\u1ec3n \u0111\u1ed5i t\u00edn hi\u1ec7u t\u01b0\u01a1ng t\u1ef1 \u0111\u1ea7u v\u00e0o th\u00e0nh m\u00e3 r\u1eddi r\u1ea1c (t\u00edn hi\u1ec7u s\u1ed1) \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 b\u1ed9 chuy\u1ec3n \u0111\u1ed5i t\u01b0\u01a1ng t\u1ef1-s\u1ed1 (ADC) (\u00a9 Wiki).<\/p>\n

M\u1ed9t trong nh\u1eefng th\u00f4ng s\u1ed1 c\u01a1 b\u1ea3n c\u1ee7a b\u1ed9 chuy\u1ec3n \u0111\u1ed5i t\u01b0\u01a1ng t\u1ef1-s\u1ed1 (ADC) l\u00e0 t\u1ed1c \u0111\u1ed9 l\u1ea5y m\u1eabu t\u1ed1i \u0111a \u2013 t\u1ea7n s\u1ed1 l\u1ea5y m\u1eabu c\u1ee7a m\u1ed9t t\u00edn hi\u1ec7u li\u00ean t\u1ee5c theo th\u1eddi gian. T\u1ed1c \u0111\u1ed9 l\u1ea5y m\u1eabu \u0111\u01b0\u1ee3c \u0111o b\u1eb1ng hertz. ((\u00a9 Wiki))<\/p>\n

Theo \u0111\u1ecbnh l\u00fd Kotelnikov, n\u1ebfu m\u1ed9t t\u00edn hi\u1ec7u li\u00ean t\u1ee5c c\u00f3 ph\u1ed5 gi\u1edbi h\u1ea1n b\u1edfi t\u1ea7n s\u1ed1 Fmax, t\u00edn hi\u1ec7u \u0111\u00f3 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c kh\u00f4i ph\u1ee5c ho\u00e0n to\u00e0n v\u00e0 duy nh\u1ea5t t\u1eeb c\u00e1c m\u1eabu r\u1eddi r\u1ea1c c\u1ee7a n\u00f3 l\u1ea5y t\u1ea1i c\u00e1c kho\u1ea3ng th\u1eddi gian \u0394t \u2264 1\/(2*Fmax), t\u1ee9c l\u00e0 v\u1edbi t\u1ea7n s\u1ed1 l\u1ea5y m\u1eabu Fd \u2265 2*Fmax, trong \u0111\u00f3 Fd – t\u1ea7n s\u1ed1 l\u1ea5y m\u1eabu; Fmax – t\u1ea7n s\u1ed1 cao nh\u1ea5t c\u1ee7a ph\u1ed5 t\u00edn hi\u1ec7u. N\u00f3i c\u00e1ch kh\u00e1c, t\u1ea7n s\u1ed1 s\u1ed1 h\u00f3a t\u00edn hi\u1ec7u (t\u1ea7n s\u1ed1 l\u1ea5y m\u1eabu c\u1ee7a ADC) ph\u1ea3i \u00edt nh\u1ea5t g\u1ea5p \u0111\u00f4i t\u1ea7n s\u1ed1 cao nh\u1ea5t c\u1ee7a t\u00edn hi\u1ec7u m\u00e0 ch\u00fang ta mu\u1ed1n \u0111o.<\/p>\n

V\u00e0 \u0111i\u1ec1u g\u00ec s\u1ebd x\u1ea3y ra n\u1ebfu ch\u00fang ta l\u1ea5y m\u1eabu v\u1edbi t\u1ea7n su\u1ea5t th\u1ea5p h\u01a1n m\u1ee9c y\u00eau c\u1ea7u theo \u0111\u1ecbnh l\u00fd Kotelnikov?<\/p>\n

Trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y c\u00f3 m\u1ed9t “ch\u1ed3ng ph\u1ed5<\/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

\"H\u00ecnh

H\u00ecnh 11. S\u1ef1 xu\u1ea5t hi\u1ec7n c\u1ee7a t\u00edn hi\u1ec7u t\u1ea7n s\u1ed1 th\u1ea5p gi\u1ea3 khi t\u1ed1c \u0111\u1ed9 l\u1ea5y m\u1eabu kh\u00f4ng \u0111\u1ee7 cao<\/p><\/div>\n

 <\/p>\n

\u0110\u1ec3 tr\u00e1nh hi\u1ec7u \u1ee9ng aliasing, m\u1ed9t b\u1ed9 l\u1ecdc ch\u1ed1ng aliasing \u0111\u1eb7c bi\u1ec7t (b\u1ed9 l\u1ecdc th\u00f4ng th\u1ea5p<\/a>) \u0111\u01b0\u1ee3c \u0111\u1eb7t tr\u01b0\u1edbc ADC. N\u00f3 cho ph\u00e9p c\u00e1c t\u1ea7n s\u1ed1 th\u1ea5p h\u01a1n m\u1ed9t n\u1eeda t\u1ea7n s\u1ed1 l\u1ea5y m\u1eabu ADC v\u00e0 ch\u1eb7n c\u00e1c t\u1ea7n s\u1ed1 cao h\u01a1n.<\/p>\n

\u0110\u1ec3 t\u00ednh ph\u1ed5 t\u00edn hi\u1ec7u t\u1eeb c\u00e1c m\u1eabu r\u1eddi r\u1ea1c c\u1ee7a n\u00f3, ph\u01b0\u01a1ng ph\u00e1p r\u1eddi r\u1ea1c bi\u1ebfn \u0111\u1ed5i Fourier (DFT)<\/a> \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng. L\u01b0u \u00fd l\u1ea1i r\u1eb1ng ph\u1ed5 c\u1ee7a m\u1ed9t t\u00edn hi\u1ec7u r\u1eddi r\u1ea1c “theo \u0111\u1ecbnh ngh\u0129a” b\u1ecb gi\u1edbi h\u1ea1n \u1edf t\u1ea7n s\u1ed1 Fmax nh\u1ecf h\u01a1n m\u1ed9t n\u1eeda t\u1ea7n s\u1ed1 l\u1ea5y m\u1eabu Fd. Do \u0111\u00f3, ph\u1ed5 c\u1ee7a m\u1ed9t t\u00edn hi\u1ec7u r\u1eddi r\u1ea1c c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n b\u1eb1ng t\u1ed5ng c\u1ee7a m\u1ed9t s\u1ed1 h\u1eefu h\u1ea1n <\/u>s\u1ed1 l\u01b0\u1ee3ng th\u00e0nh ph\u1ea7n h\u00e0i, tr\u00e1i ng\u01b0\u1ee3c v\u1edbi t\u1ed5ng v\u00f4 h\u1ea1n c\u1ee7a d\u00e3y Fourier c\u1ee7a m\u1ed9t t\u00edn hi\u1ec7u li\u00ean t\u1ee5c, c\u00f3 ph\u1ed5 c\u00f3 th\u1ec3 l\u00e0 v\u00f4 h\u1ea1n. Theo \u0111\u1ecbnh l\u00fd Kotelnikov, t\u1ea7n s\u1ed1 t\u1ed1i \u0111a c\u1ee7a m\u1ed9t th\u00e0nh ph\u1ea7n h\u00e0i ph\u1ea3i \u0111\u1ea3m b\u1ea3o bao ph\u1ee7 \u00edt nh\u1ea5t hai m\u1eabu, do \u0111\u00f3 s\u1ed1 l\u01b0\u1ee3ng th\u00e0nh ph\u1ea7n h\u00e0i b\u1eb1ng m\u1ed9t n\u1eeda s\u1ed1 m\u1eabu c\u1ee7a t\u00edn hi\u1ec7u r\u1eddi r\u1ea1c. Ngh\u0129a l\u00e0, n\u1ebfu c\u00f3 N m\u1eabu trong d\u00e3y m\u1eabu, s\u1ed1 l\u01b0\u1ee3ng th\u00e0nh ph\u1ea7n h\u00e0i trong ph\u1ed5 s\u1ebd l\u00e0 N\/2.<\/p>\n

B\u00e2y gi\u1edd h\u00e3y xem x\u00e9t ph\u00e9p bi\u1ebfn \u0111\u1ed5i Fourier r\u1eddi r\u1ea1c (DFT).<\/p>\n

\"Ph\u01b0\u01a1ng<\/p>\n

So s\u00e1nh v\u1edbi d\u00e3y Fourier<\/p>\n

 <\/p>\n

\"C\u00f4ng<\/p>\n

Nh\u01b0 ch\u00fang ta c\u00f3 th\u1ec3 th\u1ea5y, hai k\u1ebft qu\u1ea3 n\u00e0y tr\u00f9ng kh\u1edbp v\u1edbi nhau, ngo\u1ea1i tr\u1eeb vi\u1ec7c th\u1eddi gian trong FFT l\u00e0 r\u1eddi r\u1ea1c v\u00e0 s\u1ed1 l\u01b0\u1ee3ng s\u00f3ng h\u00e0i b\u1ecb gi\u1edbi h\u1ea1n \u1edf N\/2, t\u1ee9c l\u00e0 b\u1eb1ng m\u1ed9t n\u1eeda s\u1ed1 m\u1eabu.<\/p>\n

C\u00e1c c\u00f4ng th\u1ee9c DFT \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n b\u1eb1ng c\u00e1c bi\u1ebfn s\u1ed1 nguy\u00ean kh\u00f4ng c\u00f3 \u0111\u01a1n v\u1ecb k, s, trong \u0111\u00f3 k l\u00e0 s\u1ed1 m\u1eabu t\u00edn hi\u1ec7u, s l\u00e0 s\u1ed1 th\u00e0nh ph\u1ea7n ph\u1ed5.
\nGi\u00e1 tr\u1ecb s bi\u1ec3u th\u1ecb s\u1ed1 dao \u0111\u1ed9ng h\u00e0i to\u00e0n ph\u1ea7n trong m\u1ed7i chu k\u1ef3 T (th\u1eddi gian \u0111o t\u00edn hi\u1ec7u). Bi\u1ebfn \u0111\u1ed5i Fourier r\u1eddi r\u1ea1c \u0111\u01b0\u1ee3c s\u1eed d\u1ee5ng \u0111\u1ec3 t\u00ednh to\u00e1n bi\u00ean \u0111\u1ed9 v\u00e0 pha c\u1ee7a c\u00e1c dao \u0111\u1ed9ng h\u00e0i m\u1ed9t c\u00e1ch s\u1ed1, t\u1ee9c l\u00e0 \u201ctr\u00ean m\u00e1y t\u00ednh\u201d.<\/p>\n

Nh\u01b0 \u0111\u00e3 \u0111\u1ec1 c\u1eadp \u1edf tr\u00ean, khi ph\u00e2n t\u00edch m\u1ed9t h\u00e0m kh\u00f4ng tu\u1ea7n ho\u00e0n (t\u00edn hi\u1ec7u c\u1ee7a ch\u00fang ta) th\u00e0nh d\u00e3y Fourier, d\u00e3y Fourier thu \u0111\u01b0\u1ee3c th\u1ef1c ch\u1ea5t t\u01b0\u01a1ng \u1ee9ng v\u1edbi m\u1ed9t h\u00e0m tu\u1ea7n ho\u00e0n c\u00f3 chu k\u1ef3 T (H\u00ecnh 12).<\/p>\n

 <\/p>\n

\"H\u00ecnh

H\u00ecnh 12. H\u00e0m tu\u1ea7n ho\u00e0n f(x) c\u00f3 chu k\u1ef3 T\u2080, v\u1edbi chu k\u1ef3 T > T\u2080<\/p><\/div>\n

 <\/p>\n

Nh\u01b0 c\u00f3 th\u1ec3 th\u1ea5y trong H\u00ecnh 12, h\u00e0m s\u1ed1 f(x) c\u00f3 chu k\u1ef3 T0. Tuy nhi\u00ean, do th\u1ef1c t\u1ebf l\u00e0 \u0111\u1ed9 d\u00e0i m\u1eabu \u0111o l\u01b0\u1eddng T kh\u00f4ng b\u1eb1ng chu k\u1ef3 h\u00e0m s\u1ed1 T0, h\u00e0m s\u1ed1 thu \u0111\u01b0\u1ee3c t\u1eeb chu\u1ed7i Fourier c\u00f3 gi\u00e1n \u0111o\u1ea1n t\u1ea1i \u0111i\u1ec3m T. K\u1ebft qu\u1ea3 l\u00e0, ph\u1ed5 c\u1ee7a h\u00e0m s\u1ed1 n\u00e0y s\u1ebd ch\u1ee9a m\u1ed9t s\u1ed1 l\u01b0\u1ee3ng l\u1edbn c\u00e1c \u0111i\u1ec1u h\u00f2a t\u1ea7n s\u1ed1 cao. Hi\u1ec7n t\u01b0\u1ee3ng n\u00e0y \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 r\u00f2 r\u1ec9 ph\u1ed5<\/a>, v\u00e0 trong th\u1ef1c t\u1ebf n\u00f3 \u0111\u01b0\u1ee3c gi\u1ea3m thi\u1ec3u b\u1eb1ng c\u00e1ch c\u1eeda s\u1ed5<\/a> t\u00edn hi\u1ec7u tr\u01b0\u1edbc ph\u00e9p bi\u1ebfn \u0111\u1ed5i. N\u1ebfu th\u1eddi gian l\u1ea5y m\u1eabu \u0111o l\u01b0\u1eddng T tr\u00f9ng v\u1edbi chu k\u1ef3 h\u00e0m s\u1ed1 T0, th\u00ec ph\u1ed5 thu \u0111\u01b0\u1ee3c sau bi\u1ebfn \u0111\u1ed5i Fourier s\u1ebd ch\u1ec9 ch\u1ee9a \u0111i\u1ec1u h\u00f2a \u0111\u1ea7u ti\u00ean (m\u1ed9t sinusoid v\u1edbi chu k\u1ef3 b\u1eb1ng th\u1eddi gian l\u1ea5y m\u1eabu), b\u1edfi v\u00ec h\u00e0m s\u1ed1 f(x) l\u00e0 m\u1ed9t sinusoid.<\/p>\n

N\u00f3i c\u00e1ch kh\u00e1c, ch\u01b0\u01a1ng tr\u00ecnh DFT \u201ckh\u00f4ng bi\u1ebft\u201d r\u1eb1ng t\u00edn hi\u1ec7u c\u1ee7a ch\u00fang ta l\u00e0 m\u1ed9t \u201c\u0111o\u1ea1n c\u1ee7a s\u00f3ng sin\u201d, m\u00e0 c\u1ed1 g\u1eafng bi\u1ec3u di\u1ec5n n\u00f3 d\u01b0\u1edbi d\u1ea1ng m\u1ed9t d\u00e3y h\u00e0m tu\u1ea7n ho\u00e0n c\u00f3 \u0111i\u1ec3m gi\u00e1n \u0111o\u1ea1n do s\u1ef1 gi\u00e1n \u0111o\u1ea1n c\u1ee7a c\u00e1c \u0111o\u1ea1n ri\u00eang bi\u1ec7t c\u1ee7a s\u00f3ng sin.<\/p>\n

Do \u0111\u00f3, c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i xu\u1ea5t hi\u1ec7n trong ph\u1ed5, v\u00e0 t\u1ed5ng th\u1ec3 ch\u00fang ph\u1ea3i ph\u1ea3n \u00e1nh h\u00ecnh d\u1ea1ng c\u1ee7a h\u00e0m s\u1ed1, bao g\u1ed3m c\u1ea3 \u0111i\u1ec3m gi\u00e1n \u0111o\u1ea1n n\u00e0y.<\/p>\n

Do \u0111\u00f3, \u0111\u1ec3 thu \u0111\u01b0\u1ee3c ph\u1ed5 \u201cch\u00ednh x\u00e1c\u201d c\u1ee7a m\u1ed9t t\u00edn hi\u1ec7u l\u00e0 t\u1ed5ng c\u1ee7a nhi\u1ec1u s\u00f3ng sin c\u00f3 chu k\u1ef3 kh\u00e1c nhau, th\u00ec c\u1ea7n ph\u1ea3i s\u1ed1 nguy\u00ean bi\u1ec3u th\u1ecb s\u1ed1 chu k\u1ef3 c\u1ee7a <\/u>M\u1ed7i \u0111\u01b0\u1eddng sin ph\u1ea3i xu\u1ea5t hi\u1ec7n trong kho\u1ea3ng th\u1eddi gian \u0111o c\u1ee7a t\u00edn hi\u1ec7u. Tr\u00ean th\u1ef1c t\u1ebf, \u0111i\u1ec1u ki\u1ec7n n\u00e0y c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c \u0111\u00e1p \u1ee9ng n\u1ebfu th\u1eddi gian \u0111o t\u00edn hi\u1ec7u \u0111\u1ee7 d\u00e0i.<\/p>\n

 <\/p>\n

\"H\u00ecnh

H\u00ecnh 13 V\u00ed d\u1ee5 v\u1ec1 h\u00e0m v\u00e0 ph\u1ed5 t\u00edn hi\u1ec7u l\u1ed7i \u0111\u1ed9ng h\u1ecdc c\u1ee7a h\u1ed9p s\u1ed1<\/p><\/div>\n

 <\/p>\n

\u1ede kho\u1ea3ng th\u1eddi gian ng\u1eafn h\u01a1n, h\u00ecnh \u1ea3nh s\u1ebd tr\u00f4ng \u201cx\u1ea5u h\u01a1n\u201d:<\/p>\n

 <\/p>\n

\"H\u00ecnh

H\u00ecnh 14: V\u00ed d\u1ee5 v\u1ec1 h\u00e0m dao \u0111\u1ed9ng v\u00e0 ph\u1ed5 dao \u0111\u1ed9ng c\u1ee7a r\u00f4to<\/p><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

Trong th\u1ef1c t\u1ebf, vi\u1ec7c ph\u00e2n bi\u1ec7t \u0111\u00e2u l\u00e0 \u201cth\u00e0nh ph\u1ea7n th\u1ef1c\u201d v\u00e0 \u0111\u00e2u l\u00e0 \u201chi\u1ec7n t\u01b0\u1ee3ng nhi\u1ec5u\u201d do s\u1ef1 kh\u00f4ng \u0111\u1ed3ng nh\u1ea5t gi\u1eefa chu k\u1ef3 c\u1ee7a c\u00e1c th\u00e0nh ph\u1ea7n v\u00e0 th\u1eddi gian l\u1ea5y m\u1eabu t\u00edn hi\u1ec7u, hay do c\u00e1c \u201c\u0111i\u1ec3m nh\u1ea3y v\u00e0 \u0111\u1ee9t g\u00e3y\u201d tr\u00ean d\u1ea1ng s\u00f3ng, c\u00f3 th\u1ec3 kh\u00e1 kh\u00f3 kh\u0103n. T\u1ea5t nhi\u00ean, vi\u1ec7c \u0111\u1eb7t c\u00e1c c\u1ee5m t\u1eeb \u201cth\u00e0nh ph\u1ea7n th\u1ef1c\u201d v\u00e0 \u201chi\u1ec7n t\u01b0\u1ee3ng nhi\u1ec5u\u201d trong d\u1ea5u ngo\u1eb7c k\u00e9p l\u00e0 c\u00f3 l\u00fd do. S\u1ef1 hi\u1ec7n di\u1ec7n c\u1ee7a nhi\u1ec1u s\u00f3ng h\u00e0i tr\u00ean bi\u1ec3u \u0111\u1ed3 ph\u1ed5 kh\u00f4ng c\u00f3 ngh\u0129a l\u00e0 t\u00edn hi\u1ec7u c\u1ee7a ch\u00fang ta th\u1ef1c s\u1ef1 bao g\u1ed3m ch\u00fang. \u0110i\u1ec1u n\u00e0y gi\u1ed1ng nh\u01b0 ngh\u0129 r\u1eb1ng s\u1ed1 7 \u201cbao g\u1ed3m\u201d c\u00e1c s\u1ed1 3 v\u00e0 4. S\u1ed1 7 c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c coi l\u00e0 t\u1ed5ng c\u1ee7a 3 v\u00e0 4 \u2013 \u0111i\u1ec1u \u0111\u00f3 l\u00e0 \u0111\u00fang.<\/p>\n

T\u01b0\u01a1ng t\u1ef1 nh\u01b0 v\u1eady, t\u00edn hi\u1ec7u c\u1ee7a ch\u00fang ta\u2026 hay \u0111\u00fang h\u01a1n l\u00e0 kh\u00f4ng h\u1eb3n l\u00e0 \u201ct\u00edn hi\u1ec7u c\u1ee7a ch\u00fang ta\u201d, m\u00e0 l\u00e0 m\u1ed9t h\u00e0m tu\u1ea7n ho\u00e0n \u0111\u01b0\u1ee3c t\u1ea1o th\u00e0nh t\u1eeb vi\u1ec7c l\u1eb7p l\u1ea1i t\u00edn hi\u1ec7u (m\u1eabu) c\u1ee7a ch\u00fang ta, c\u00f3 th\u1ec3 \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n d\u01b0\u1edbi d\u1ea1ng t\u1ed5ng c\u1ee7a c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i (s\u00f3ng sin) v\u1edbi c\u00e1c bi\u00ean \u0111\u1ed9 v\u00e0 pha nh\u1ea5t \u0111\u1ecbnh. Tuy nhi\u00ean, trong nhi\u1ec1u tr\u01b0\u1eddng h\u1ee3p c\u00f3 \u00fd ngh\u0129a th\u1ef1c ti\u1ec5n (xem c\u00e1c h\u00ecnh v\u1ebd \u1edf tr\u00ean), th\u1ef1c s\u1ef1 c\u00f3 th\u1ec3 li\u00ean h\u1ec7 c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i thu \u0111\u01b0\u1ee3c trong ph\u1ed5 v\u1edbi c\u00e1c qu\u00e1 tr\u00ecnh th\u1ef1c t\u1ebf c\u00f3 t\u00ednh ch\u1ea5t tu\u1ea7n ho\u00e0n v\u00e0 \u0111\u00f3ng g\u00f3p \u0111\u00e1ng k\u1ec3 v\u00e0o h\u00ecnh d\u1ea1ng c\u1ee7a t\u00edn hi\u1ec7u.<\/p>\n

M\u1ed9t s\u1ed1 k\u1ebft qu\u1ea3<\/h2>\n

1. M\u1ed9t t\u00edn hi\u1ec7u \u0111o th\u1ef1c c\u00f3 \u0111\u1ed9 d\u00e0i T gi\u00e2y \u0111\u01b0\u1ee3c s\u1ed1 h\u00f3a b\u1eb1ng b\u1ed9 chuy\u1ec3n \u0111\u1ed5i t\u01b0\u01a1ng t\u1ef1-s\u1ed1 (ADC), t\u1ee9c l\u00e0 \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n b\u1eb1ng m\u1ed9t t\u1eadp h\u1ee3p c\u00e1c m\u1eabu r\u1eddi r\u1ea1c (g\u1ed3m N m\u1eabu), c\u00f3 ph\u1ed5 r\u1eddi r\u1ea1c kh\u00f4ng tu\u1ea7n ho\u00e0n \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n b\u1eb1ng m\u1ed9t t\u1eadp h\u1ee3p c\u00e1c th\u00e0nh ph\u1ea7n h\u00e0i (g\u1ed3m N\/2 th\u00e0nh ph\u1ea7n).<\/p>\n

2. T\u00edn hi\u1ec7u \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n b\u1eb1ng m\u1ed9t t\u1eadp c\u00e1c gi\u00e1 tr\u1ecb th\u1ef1c. Ph\u1ed5 DFT c\u1ee7a n\u00f3 l\u00e0 m\u1ed9t t\u1eadp c\u00e1c h\u1ec7 s\u1ed1 ph\u1ee9c c\u00f3 t\u00ednh \u0111\u1ed1i x\u1ee9ng li\u00ean h\u1ee3p; t\u1eeb \u0111\u00f3 thu \u0111\u01b0\u1ee3c ph\u1ed5 bi\u00ean \u0111\u1ed9 \u2014 m\u1ed9t t\u1eadp c\u00e1c bi\u00ean \u0111\u1ed9 kh\u00f4ng \u00e2m th\u1ef1c (v\u00e0 c\u00e1c pha) t\u1ea1i c\u00e1c t\u1ea7n s\u1ed1 d\u01b0\u01a1ng, v\u00e0 tr\u00ean th\u1ef1c t\u1ebf ch\u00ednh ph\u1ed5 bi\u00ean \u0111\u1ed9 m\u1ed9t ph\u00eda n\u00e0y \u0111\u01b0\u1ee3c v\u1ebd. D\u1ea1ng ph\u1ee9c hai ph\u00eda v\u1edbi t\u1ea7n s\u1ed1 \u00e2m v\u00e0 d\u1ea1ng bi\u00ean \u0111\u1ed9\/pha m\u1ed9t ph\u00eda l\u00e0 c\u00e1c bi\u1ec3u di\u1ec5n t\u01b0\u01a1ng \u0111\u01b0\u01a1ng c\u1ee7a c\u00f9ng m\u1ed9t ph\u1ed5 \u2014 \u0111\u1ed1i v\u1edbi ph\u00e2n t\u00edch t\u00edn hi\u1ec7u, th\u01b0\u1eddng thu\u1eadn ti\u1ec7n h\u01a1n khi l\u00e0m vi\u1ec7c v\u1edbi ph\u1ed5 bi\u00ean \u0111\u1ed9 m\u1ed9t ph\u00eda.<\/p>\n

3. D\u1ea5u hi\u1ec7u \u0111\u01b0\u1ee3c \u0111o t\u1ea1i th\u1eddi \u0111i\u1ec3m T ch\u1ec9 \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh t\u1ea1i th\u1eddi \u0111i\u1ec3m T. Nh\u1eefng g\u00ec \u0111\u00e3 x\u1ea3y ra tr\u01b0\u1edbc khi ch\u00fang ta b\u1eaft \u0111\u1ea7u \u0111o d\u1ea5u hi\u1ec7u v\u00e0 nh\u1eefng g\u00ec s\u1ebd x\u1ea3y ra sau \u0111\u00f3 v\u1eabn l\u00e0 \u0111i\u1ec1u ch\u01b0a \u0111\u01b0\u1ee3c khoa h\u1ecdc l\u00e0m s\u00e1ng t\u1ecf. V\u00e0 trong tr\u01b0\u1eddng h\u1ee3p c\u1ee7a ch\u00fang ta, \u0111i\u1ec1u \u0111\u00f3 kh\u00f4ng ph\u1ea3i l\u00e0 v\u1ea5n \u0111\u1ec1 quan tr\u1ecdng. Bi\u1ebfn \u0111\u1ed5i Fourier nhanh (FFT) c\u1ee7a d\u1ea5u hi\u1ec7u c\u00f3 gi\u1edbi h\u1ea1n th\u1eddi gian cho ta ph\u1ed5 \u201cth\u1ef1c\u201d c\u1ee7a n\u00f3, theo ngh\u0129a l\u00e0 trong m\u1ed9t s\u1ed1 \u0111i\u1ec1u ki\u1ec7n nh\u1ea5t \u0111\u1ecbnh, n\u00f3 cho ph\u00e9p t\u00ednh to\u00e1n bi\u00ean \u0111\u1ed9 v\u00e0 t\u1ea7n s\u1ed1 c\u1ee7a c\u00e1c th\u00e0nh ph\u1ea7n c\u1ea5u th\u00e0nh.<\/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\/vi\/wp-json\/wp\/v2\/posts\/2138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/comments?post=2138"}],"version-history":[{"count":2,"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/posts\/2138\/revisions"}],"predecessor-version":[{"id":102137,"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/posts\/2138\/revisions\/102137"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/media\/2161"}],"wp:attachment":[{"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/media?parent=2138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/categories?post=2138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/vibromera.eu\/vi\/wp-json\/wp\/v2\/tags?post=2138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}