hw5 - EE210 Signal Systems (March 4, 2009) Home Work Set 5...

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EE210 Signal Systems (March 4, 2009) Home Work Set 5 1. Let g ( t ) = R t 0 f ( τ ) dτ, 0 t T and f ( t ) is defined over [0 , T ]. Let the Fourier Series(FS) expansion of f ( t ) = n c n e jnw 0 t , w 0 = 2 π T . Find the FS expansion of g ( t ) in terms of { c n } . 2. Let x ( t ) L 2 [0 , T ]. Now, let b x ( t ) = N X n = - N ( α n + n ) e jnw 0 t , w 0 = 2 π T . Show that if { α n , β n } N n = - N are computed such that min { α n n } N n = - N Z T 0 | x ( t ) - b x ( t ) | 2 dt , then α n + n = c n , where c n is the FS coefficient in the FS expansion of x ( t ). 3. Let x p ( t ) = x p ( - t ) be a periodic function with period T . Show that the FS expansion contain only the cosine terms and a dc term.
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