Signals &systems 10EE52 0  Example 4 Prove that T sin(kw0t) sin(  0 gw0t)dt 0 for 2f w0  1 f  T , integers  k g Solution T  sin(kw0 sin( Let (22)  t) gw0 t)dt D 0 Since cos( cos() cos()  )  sin()sin() or sin()sin()  cos() cos()  cos(  ) Thus, T T cos(kw0 cos(gw0 t) t)dt    (23)  g)w0tdt D 0 0 From Equation (1) then T cos(kw0 cos(gw0 t) t)dt  0  (24) D  0 Adding Equations (23), (26) T T sin(kw0t) sin( gw0  cos(gw0 2D   t) t) t)dt 0 0 T coskw0  t gw0  (25)  tdt 0 T   g)w0 tdt
Signals &systems 10EE52  0 Example 4 Prove that T sin(kw0t) sin( gw0t)dt  0 0 for w0  2f f  1 T k , g  integers Solution T Let D   sin(kw0 t) sin( gw0 t)dt (22) 0 Since cos(  )  cos() cos() sin()sin() or sin()sin()  cos() cos()  cos(  ) Thus, T T D   cos(kw0 t) cos(gw0 t)dt   g)w0tdt (23) 0 0 From Equation (1) then T D   cos(kw0 t) cos(gw0 t)dt  0 (24) 0 Adding Equations (23), (26) T T 2D   sin(kw0t) sin( gw0 t)  t) cos(gw0 t)dt 0 0 T   coskw0 t  gw0 tdt (25) 0 T   g)w0 tdt
U4-L18-_-Fourier-Series-and-examples.docx