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=1T∫T/2−T/2˜x(t)e−jkω0tdtwhere 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=1T∫T/2−T/2x(t)e−jkω0tdt=1T∫+∞−∞x(t)e−jkω0tdt
- 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)
- And then we have: Tak=2sin(ωT1)ω|ω=kω0That 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 "
c) I want to know that how T0ak will be drawn in continuous?
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