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:
$$\widetilde{x}(t)=\sum_{k=-\infty}^{+\infty}a_ke^{jk\omega_0t} \tag{4.3} $$ $$a_k=\frac{1}{T}\int_{-T/2}^{T/2}\widetilde{x}(t)e^{-jk\omega_0t}dt \tag{4.4} $$
where $\omega_0=2\pi/T$. Since $\widetilde{x}(t)=x(t)$ for $|t| < T/2$, and also, since $x(t)=0$ outside this interval, eq. $(4.4)$ can be rewritten as
$$a_k=\frac{1}{T}\int_{-T/2}^{T/2}x(t)e^{-jk\omega_0t}dt=\frac{1}{T}\int_{-\infty}^{+\infty}x(t)e^{-jk\omega_0t}dt$$
- My question is In the last equation on the L.H.S, there is no $\widetilde{x}(t)$ but it was mentioned in the eq. $(4.4)$
- And then we have: $$Ta_k= \left.\frac{2 \sin(\omega T_1)}{\omega}\right\rvert_{\omega = k\omega_0}$$ That is, with $\omega$ thought of as a continuous variable, the function ($2\sin\omega T_1/ \omega$) represents envelope of $Ta_k$. Here I want to know what is meant by envelope?I want to add more details from the book exactly " Specifically we have plot $T_0a_k$ rather than $a_k$ and we have also modified the horizontal spacing in each plot. here is the plot "
c) I want to know that how $T_0a_k$ will be drawn in continuous?
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