intTypePromotion=1
zunia.vn Tuyển sinh 2024 dành cho Gen-Z zunia.vn zunia.vn
ADSENSE

Table of Fourier Transform Pairs

Chia sẻ: NGO NAM | Ngày: | Loại File: PDF | Số trang:5

149
lượt xem
15
download
 
  Download Vui lòng tải xuống để xem tài liệu đầy đủ

Tài liệu tham khảo về các công thức toán (tài liệu bằng tiếng anh)

Chủ đề:
Lưu

Nội dung Text: Table of Fourier Transform Pairs

  1. Table of Fourier Transform Pairs Function, f(t) Fourier Transform, F(w) Definition of Inverse Fourier Transform Definition of Fourier Transform ¥ ¥ 1 jwt - jwt f (t ) = 2p ò F (w )e dw F (w ) = ò f (t )e dt -¥ -¥ f (t - t 0 ) F (w )e - jwt0 f (t )e jw 0t F (w - w 0 ) f (at ) 1 w F( ) a a F (t ) 2pf (-w ) d n f (t ) ( jw ) n F (w ) dt n (- jt ) n f (t ) d n F (w) dw n t F (w ) + pF (0)d (w ) ò f (t )dt jw -¥ d (t ) 1 e jw 0 t 2pd (w - w 0 ) sgn (t) 2 jw Signals & Systems - Reference Tables 1
  2. 1 sgn(w ) j pt u (t ) 1 pd (w ) + jw ¥ ¥ å Fn e jnw 0t 2p å Fnd (w - nw 0 ) n = -¥ n = -¥ t wt rect ( ) tSa( ) t 2 B Bt w Sa( ) rect ( ) 2p 2 B tri (t ) w Sa 2 ( ) 2 pt t Ap cos(wt ) A cos( )rect ( ) 2t 2t t (p ) 2 - w 2 2t cos(w 0 t ) p [d (w - w 0 ) + d (w + w 0 )] sin(w 0 t ) p [d (w - w 0 ) - d (w + w 0 )] j u (t ) cos(w 0 t ) p [d (w - w 0 ) + d (w + w 0 )] + 2 jw 2 2 w0 - w u (t ) sin(w 0 t ) p 2 [d (w - w 0 ) - d (w + w 0 )] + 2w 2 2j w0 - w u (t )e -at cos(w 0 t ) (a + jw ) w 0 + (a + jw ) 2 2 Signals & Systems - Reference Tables 2
  3. u (t )e -at sin(w 0 t ) w0 w 0 + (a + jw ) 2 2 e -a t 2a a2 +w2 2 /( 2s 2 ) 2 w2 / 2 e -t s 2p e -s u (t )e -at 1 a + jw u (t )te -at 1 (a + jw ) 2 Ø Trigonometric Fourier Series ¥ f (t ) = a 0 + å (a n cos(w 0 nt ) + bn sin(w 0 nt ) ) n =1 where 1 T 2T a0 = T ò0 f (t )dt , a n = ò f (t ) cos(w 0 nt )dt , and T0 2T bn = ò f (t ) sin(w 0 nt )dt T 0 Ø Complex Exponential Fourier Series ¥ 1T f (t ) = å Fn e jwnt , where Fn = ò f (t )e - jw 0 nt dt T 0 n = -¥ Signals & Systems - Reference Tables 3
  4. Some Useful Mathematical Relationships e jx + e - jx cos( x) = 2 e jx - e - jx sin( x) = 2j cos( x ± y ) = cos( x) cos( y ) m sin( x) sin( y ) sin( x ± y ) = sin( x) cos( y ) ± cos( x) sin( y ) cos(2 x) = cos 2 ( x) - sin 2 ( x) sin( 2 x) = 2 sin( x) cos( x) 2 cos2 ( x) = 1 + cos(2 x) 2 sin 2 ( x) = 1 - cos(2 x) cos 2 ( x) + sin 2 ( x) = 1 2 cos( x) cos( y ) = cos( x - y ) + cos( x + y ) 2 sin( x) sin( y ) = cos( x - y ) - cos( x + y ) 2 sin( x) cos( y ) = sin( x - y ) + sin( x + y ) Signals & Systems - Reference Tables 4
  5. Useful Integrals sin(x) ò cos( x)dx - cos(x) ò sin( x)dx cos( x) + x sin( x) ò x cos( x)dx sin( x) - x cos( x) ò x sin( x)dx òx 2 cos( x)dx 2 x cos( x) + ( x 2 - 2) sin( x) òx 2 sin( x)dx 2 x sin( x) - ( x 2 - 2) cos( x) ax òe dx e ax a ax ò xe dx éx 1 ù e ax ê - 2 ú ëa a û 2 ax òx e dx é x 2 2x 2 ù e ax ê - 2 - 3 ú ëa a a û dx 1 ò a + bx b ln a + bx dx 1 bx ò a 2 + b 2x2 tan -1 ( ) ab a Signals & Systems - Reference Tables 5
ADSENSE

CÓ THỂ BẠN MUỐN DOWNLOAD

 

Đồng bộ tài khoản
2=>2