s ( t ) = ∑ k = − ∞ ∞ a k p ( t − k T s ) h e ( t ) = p ( t ) ∗ h c ( t ) ∗ h R ( t ) s ( t ) = ∑ k = − ∞ ∞ a k h e ( t − k T s ) = a 0 h e ( t ) + a 1 h e ( t − T s ) + a 2 h e ( t − 2 T s ) + . . . h e ( t = 0 ) = h e max = 1 , s ( t ) | t = n T s → s ( t ) | t = 0 = s ( 0 ) = ∑ k = − ∞ ∞ a k h e ( − k T s ) = a 0 h e ( 0 ) ⏟ = 1 + a 1 h e ( − T s ) ⏟ = 0 + a 2 = h e ( − 2 T s ) ⏟ = 0 + . . . h e ( m T s ) = 0 , m ≠ 0 h e ( t ) ⋅ ∑ k = − ∞ ∞ δ ( t − k T s ) = h e ( 0 ) δ ( t ) ⇒ ∑ k = − ∞ ∞ δ ( t − k T s ) = ∑ k = − ∞ ∞ 1 T s e j 2 π f s t H e ( f ) ∗ 1 T s ∑ k = − ∞ ∞ δ ( f − k f s ) = h e ( 0 ) 1 T s ∑ k = − ∞ ∞ H e ( f − k f s ) = h e ( 0 ) → ∑ k = − ∞ ∞ H e ( f − k f s ) = h e ( 0 ) T s = c t e . {\displaystyle {\begin{aligned}&s(t)=\sum \limits _{k=-\infty }^{\infty }{a_{k}p\left(t-kT_{s}\right)}\\&h_{e}(t)=p(t)*h_{c}(t)*h_{R}(t)\\&s(t)=\sum \limits _{k=-\infty }^{\infty }{a_{k}h_{e}\left(t-kT_{s}\right)=a_{0}h_{e}(t)}+a_{1}h_{e}\left(t-T_{s}\right)+a_{2}h_{e}\left(t-2T_{s}\right)+...\\&h_{e}(t=0)=h_{e\max }=1,\left.s(t)\right|_{t=nT_{s}}\to \\&\left.s\left(t\right)\right|_{t=0}=s\left(0\right)=\sum \limits _{k=-\infty }^{\infty }{a_{k}h_{e}\left(-kT_{s}\right)}=a_{0}\underbrace {h_{e}(0)} _{=1}+a_{1}\underbrace {h_{e}\left(-T_{s}\right)} _{=0}+a_{2}=\underbrace {h_{e}\left(-2T_{s}\right)} _{=0}+...\\&h_{e}\left(mT_{s}\right)=0,m\neq 0\\&h_{e}(t)\cdot \sum \limits _{k=-\infty }^{\infty }{\delta \left(t-kT_{s}\right)=}h_{e}(0)\delta \left(t\right)\Rightarrow \sum \limits _{k=-\infty }^{\infty }{\delta \left(t-kT_{s}\right)=}\sum \limits _{k=-\infty }^{\infty }{{\frac {1}{T_{s}}}e^{j2\pi f_{s}t}}\\&H_{e}(f)*{\frac {1}{T_{s}}}\sum \limits _{k=-\infty }^{\infty }{\delta \left(f-kf_{s}\right)=}h_{e}(0)\\&{\frac {1}{T_{s}}}\sum \limits _{k=-\infty }^{\infty }{H_{e}\left(f-kf_{s}\right)=}h_{e}(0)\to \sum \limits _{k=-\infty }^{\infty }{H_{e}\left(f-kf_{s}\right)=}h_{e}(0)T_{s}=cte.\\\end{aligned}}}
E j . H e ( f ) = ∏ ( f 2 B ) → h e ( t ) = 2 B sinc ( 2 B t ) , f s = R s h e ( t ) ⋅ ∑ k = − ∞ ∞ δ ( t − k T s ) = h e ( 0 ) δ ( t ) ⇒ ∑ k = − ∞ ∞ H e ( f − k f s ) = h e ( 0 ) T s = c t e . T s = 1 2 B , 2 2 B , 3 2 B , . . . → R s = 2 B n ( s i m b o l o s ╱ s ) {\displaystyle {\begin{aligned}&Ej.\\&H_{e}(f)=\prod {\left({\frac {f}{2B}}\right)}\to h_{e}(t)=2B\operatorname {sinc} \left(2Bt\right),f_{s}=R_{s}\\&h_{e}(t)\cdot \sum \limits _{k=-\infty }^{\infty }{\delta \left(t-kT_{s}\right)=}h_{e}(0)\delta \left(t\right)\Rightarrow \sum \limits _{k=-\infty }^{\infty }{H_{e}\left(f-kf_{s}\right)=}h_{e}(0)T_{s}=cte.\\&T_{s}={\frac {1}{2B}},{\frac {2}{2B}},{\frac {3}{2B}},...\to R_{s}={\frac {2B}{n}}\left({}^{simbolos}\!\!\diagup \!\!{}_{s}\;\right)\\\end{aligned}}}
