Please show working... If a periodic signal is expressed as an exponential Fouri

Question

Please show working...

If a periodic signal is expressed as an exponential Fourier series as given below Show that the exponential Fourier series for f^(t) = f(t-T) is given by In which and This result show that time shifting of a periodic signal by T seconds merely changes the phase spectrum by n omega T. The amplitude spectrum is unchanged. Show that the exponential Fourier series for f(t) = f(alpha t) is given by These result shows that time compression of a periodic signal by a factor alpha expands its Fourier spectra by a factor alpha.

Explanation / Answer

f(t) = summation [Dn*e^(Jnwo*t)

a) Dn^ =fouriere series coffiecint of f(t-T)

   =(1/T) integtation[ f(t-T)*e^(-njwot)] , .........(1) where t = 0 to T

let t-T =p

    then dt = dp   , and   p = - T   to 0

Dn^   = (1/T) integtation[ f(t)e^(-jwot)]  

         = (1/T) integtation[ f(p)*e^(-jnTwo)*e^(-jwot)]

= e^(-jnTwo)*(1/T) * integtation[ f(p)*e^(-njwop)]

     Dn^     =e^(-jnTwo)*Dn

               = Dn<(-wonT)

b) f(t) =f(a*t)

Dn^ =(1/T) integtation[ f(a*t)*e^(-jnwot)]  

           = (1/T) * integtation[ f(p)e^(-jnwoP/*a)]

            =

hence     f(a*t) =summation [Dn^*e^(Jnwo*t)

                         =summation [Dn*[e^(Jnwo*t)]^a

                         =summation [Dn*[e^(J*a*nwo*t)]

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