Monday, 7 September 2015

continuous signals - Confusion in CT Fourier Transform Proof



I am confused trying to understand the Proof of Fourier Transform from Oppenheim book Signals and Systems. I am pasting the equations directly from the book:



˜x(t)=+k=akejkω0t

ak=1TT/2T/2˜x(t)ejkω0tdt


where ω0=2π/T. Since ˜x(t)=x(t) for |t|<T/2, and also, since x(t)=0 outside this interval, eq. (4.4) can be rewritten as


ak=1TT/2T/2x(t)ejkω0tdt=1T+x(t)ejkω0tdt




  1. My question is In the last equation on the L.H.S, there is no ˜x(t) but it was mentioned in the eq. (4.4)

  2. And then we have: Tak=2sin(ωT1)ω|ω=kω0
    That is, with ω thought of as a continuous variable, the function (2sinωT1/ω) represents envelope of Tak. Here I want to know what is meant by envelope?I want to add more details from the book exactly " Specifically we have plot T0ak rather than ak and we have also modified the horizontal spacing in each plot. here is the plot " enter image description here c) I want to know that how T0ak will be drawn in continuous?





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