H
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f
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π
4
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h
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H
β
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f
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{
cos
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2
π
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t
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⟷
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2
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1
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sinc
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sinc
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sinc
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sin
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sinc
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⏟
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sinc
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sin
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2
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sin
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sin
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sin
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{
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sin
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cos
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π
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{\displaystyle {\begin{aligned}&H(f)=H_{\beta }(f)*{\frac {1}{R}}\prod {\left({\frac {f}{R}}\right)}\\&H_{\beta }(f)={\frac {\pi }{4\beta }}\cos \left({\frac {\pi }{2\beta }}f\right)\prod {\left({\frac {f}{2\beta }}\right)}\\&h_{\beta }(t)=\mathbb {F} ^{-1}\left[H_{\beta }(f)\right]\to \left\{{\begin{aligned}&\cos \left(2\pi f_{0}t\right){\underset {\mathbb {F} }{\longleftrightarrow }}{\frac {1}{2}}\delta \left(f-f_{0}\right)+{\frac {1}{2}}\delta \left(f+f_{0}\right)\\&\prod {\left({\frac {t}{T}}\right){\underset {\mathbb {F} }{\longleftrightarrow }}T\operatorname {sinc} (Tf)}\\\end{aligned}}\right\}\\&\cos \left({\frac {\pi }{2\beta }}f\right)=\cos \left({\frac {\pi }{2\beta }}{\frac {2}{2}}f\right)=\cos \left({\frac {1}{4\beta }}2\pi f\right)\\&h_{\beta }(t)={\frac {\pi }{4\beta }}\left({\frac {1}{2}}\delta \left(t-{\frac {1}{4\beta }}\right)+{\frac {1}{2}}\delta \left(t+{\frac {1}{4\beta }}\right)\right)*2\beta \operatorname {sinc} \left(2\beta t\right)=\\&h_{\beta }(t)={\frac {\pi }{4}}\left(\delta \left(t-{\frac {1}{4\beta }}\right)+\delta \left(t+{\frac {1}{4\beta }}\right)\right)*\operatorname {sinc} \left(2\beta t\right)\\&\operatorname {sinc} (Tt)={\frac {\sin \left(T\pi t\right)}{T\pi t}}\\&h_{\beta }(t)={\frac {\pi }{4}}\left(\underbrace {\operatorname {sinc} \left(2\beta \left(t-{\frac {1}{4\beta }}\right)\right)} _{\operatorname {sinc} \left(2\beta t-{\frac {1}{2}}\right)}+\underbrace {\operatorname {sinc} \left(2\beta \left(t+{\frac {1}{4\beta }}\right)\right)} _{\operatorname {sinc} \left(2\beta t+{\frac {1}{2}}\right)}\right)=\\&{\frac {\pi }{4}}\left({\frac {\sin \left(2\beta \pi \left(t-{\frac {1}{4\beta }}\right)\right)}{2\beta \pi \left(t-{\frac {1}{4\beta }}\right)}}+{\frac {\sin \left(2\beta \pi \left(t+{\frac {1}{4\beta }}\right)\right)}{2\beta \pi \left(t+{\frac {1}{4\beta }}\right)}}\right)={\frac {\pi }{4}}\left({\frac {\sin \left(2\beta \pi t-{\frac {\pi }{2}}\right)}{2\beta \pi t-{\frac {\pi }{2}}}}+{\frac {\sin \left(2\beta \pi t+{\frac {\pi }{2}}\right)}{2\beta \pi t+{\frac {\pi }{2}}}}\right)=\\&\left\{{\begin{aligned}&\sin \left(a\pm b\right)=\sin a\cos b+\sin b\cos a\to \\&\sin \left(a\pm {\frac {\pi }{2}}\right)=\sin a\overbrace {\cos \left(\pm {\frac {\pi }{2}}\right)} ^{0}+\sin \left(\pm {\frac {\pi }{2}}\right)\cos a=\pm \cos a\\\end{aligned}}\right\}\\&{\frac {\pi }{4}}\left({\frac {-\cos \left(2\beta \pi t\right)}{2\beta \pi t-{\frac {\pi }{2}}}}+{\frac {\cos \left(2\beta \pi t\right)}{2\beta \pi t+{\frac {\pi }{2}}}}\right)={\frac {\pi }{4}}\cos \left(2\beta \pi t\right)\left({\frac {1}{2\beta \pi t+{\frac {\pi }{2}}}}-{\frac {1}{2\beta \pi t-{\frac {\pi }{2}}}}\right)=\\&{\frac {\pi }{4}}\cos \left(2\beta \pi t\right)\left({\frac {\left(2\beta \pi t-{\frac {\pi }{2}}\right)-\left(2\beta \pi t+{\frac {\pi }{2}}\right)}{\left(2\beta \pi t\right)^{2}-\left({\frac {\pi }{2}}\right)^{2}}}\right)\\\end{aligned}}}
π
4
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t
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(
−
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(
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cos
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2
β
π
t
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1
1
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2
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h
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t
)
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cos
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2
β
π
t
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1
−
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4
β
t
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2
H
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f
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=
H
β
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f
)
∗
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∏
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f
R
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=
π
4
β
cos
(
π
2
β
f
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∏
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f
2
β
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∗
1
R
∏
(
f
R
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h
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t
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=
h
β
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t
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⋅
sinc
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R
t
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=
cos
(
2
β
π
t
)
1
−
(
4
β
t
)
2
⋅
sinc
(
R
t
)
{\displaystyle {\begin{aligned}&{\frac {\pi }{4}}\cos \left(2\beta \pi t\right)\left({\frac {-\pi }{\left(2\beta \pi t\right)^{2}-\left({\frac {\pi }{2}}\right)^{2}}}\right)={\frac {\pi }{4}}\cos \left(2\beta \pi t\right)\left({\frac {\pi }{{\frac {\pi ^{2}}{4}}-4\beta ^{2}\pi ^{2}t^{2}}}\right)=\\&\cos \left(2\beta \pi t\right)\left({\frac {\pi ^{2}}{\pi ^{2}-16\beta ^{2}\pi ^{2}t^{2}}}\right)=\cos \left(2\beta \pi t\right)\left({\frac {1}{1-16\beta ^{2}t^{2}}}\right)\\&h_{\beta }(t)={\frac {\cos \left(2\beta \pi t\right)}{1-\left(4\beta t\right)^{2}}}\\&\\&H(f)=H_{\beta }(f)*{\frac {1}{R}}\prod {\left({\frac {f}{R}}\right)}={\frac {\pi }{4\beta }}\cos \left({\frac {\pi }{2\beta }}f\right)\prod {\left({\frac {f}{2\beta }}\right)}*{\frac {1}{R}}\prod {\left({\frac {f}{R}}\right)}\\&h(t)=h_{\beta }(t)\cdot \operatorname {sinc} \left(Rt\right)={\frac {\cos \left(2\beta \pi t\right)}{1-\left(4\beta t\right)^{2}}}\cdot \operatorname {sinc} \left(Rt\right)\\\end{aligned}}}
editar
Para enviar datos digitales es común utilizar un pulso rectangular a modo de función “generadora”, sin embargo, esto nos da un problema en frecuencia, la transformada de Fourier de un pulso rectangular es una sinc, que tiene un ancho de banda infinito. Como el espectro es un recurso limitado, tendremos que limitar esa señal de alguna manera, con el consiguiente aumento de la probabilidad de error. Por ello, se usa la función coseno alzado que decrece mas rápidamente que la sinc conforme aumenta la frecuencia (esta mas limitada).
Aquí como se ve, aunque el lóbulo principal del coseno alzado sea mayor que el de la sinc, decrece mas rápidamente haciéndose prácticamente nulo en unos pocos segundos.
Por ejemplo, con el ancho de banda necesario para contener el 98% de la energía en el coseno alzado, contendríamos solamente el 80% de la energía en la función Sinc.
![{\displaystyle {\begin{aligned}&H(f)=H_{\beta }(f)*{\frac {1}{R}}\prod {\left({\frac {f}{R}}\right)}\\&H_{\beta }(f)={\frac {\pi }{4\beta }}\cos \left({\frac {\pi }{2\beta }}f\right)\prod {\left({\frac {f}{2\beta }}\right)}\\&h_{\beta }(t)=\mathbb {F} ^{-1}\left[H_{\beta }(f)\right]\to \left\{{\begin{aligned}&\cos \left(2\pi f_{0}t\right){\underset {\mathbb {F} }{\longleftrightarrow }}{\frac {1}{2}}\delta \left(f-f_{0}\right)+{\frac {1}{2}}\delta \left(f+f_{0}\right)\\&\prod {\left({\frac {t}{T}}\right){\underset {\mathbb {F} }{\longleftrightarrow }}T\operatorname {sinc} (Tf)}\\\end{aligned}}\right\}\\&\cos \left({\frac {\pi }{2\beta }}f\right)=\cos \left({\frac {\pi }{2\beta }}{\frac {2}{2}}f\right)=\cos \left({\frac {1}{4\beta }}2\pi f\right)\\&h_{\beta }(t)={\frac {\pi }{4\beta }}\left({\frac {1}{2}}\delta \left(t-{\frac {1}{4\beta }}\right)+{\frac {1}{2}}\delta \left(t+{\frac {1}{4\beta }}\right)\right)*2\beta \operatorname {sinc} \left(2\beta t\right)=\\&h_{\beta }(t)={\frac {\pi }{4}}\left(\delta \left(t-{\frac {1}{4\beta }}\right)+\delta \left(t+{\frac {1}{4\beta }}\right)\right)*\operatorname {sinc} \left(2\beta t\right)\\&\operatorname {sinc} (Tt)={\frac {\sin \left(T\pi t\right)}{T\pi t}}\\&h_{\beta }(t)={\frac {\pi }{4}}\left(\underbrace {\operatorname {sinc} \left(2\beta \left(t-{\frac {1}{4\beta }}\right)\right)} _{\operatorname {sinc} \left(2\beta t-{\frac {1}{2}}\right)}+\underbrace {\operatorname {sinc} \left(2\beta \left(t+{\frac {1}{4\beta }}\right)\right)} _{\operatorname {sinc} \left(2\beta t+{\frac {1}{2}}\right)}\right)=\\&{\frac {\pi }{4}}\left({\frac {\sin \left(2\beta \pi \left(t-{\frac {1}{4\beta }}\right)\right)}{2\beta \pi \left(t-{\frac {1}{4\beta }}\right)}}+{\frac {\sin \left(2\beta \pi \left(t+{\frac {1}{4\beta }}\right)\right)}{2\beta \pi \left(t+{\frac {1}{4\beta }}\right)}}\right)={\frac {\pi }{4}}\left({\frac {\sin \left(2\beta \pi t-{\frac {\pi }{2}}\right)}{2\beta \pi t-{\frac {\pi }{2}}}}+{\frac {\sin \left(2\beta \pi t+{\frac {\pi }{2}}\right)}{2\beta \pi t+{\frac {\pi }{2}}}}\right)=\\&\left\{{\begin{aligned}&\sin \left(a\pm b\right)=\sin a\cos b+\sin b\cos a\to \\&\sin \left(a\pm {\frac {\pi }{2}}\right)=\sin a\overbrace {\cos \left(\pm {\frac {\pi }{2}}\right)} ^{0}+\sin \left(\pm {\frac {\pi }{2}}\right)\cos a=\pm \cos a\\\end{aligned}}\right\}\\&{\frac {\pi }{4}}\left({\frac {-\cos \left(2\beta \pi t\right)}{2\beta \pi t-{\frac {\pi }{2}}}}+{\frac {\cos \left(2\beta \pi t\right)}{2\beta \pi t+{\frac {\pi }{2}}}}\right)={\frac {\pi }{4}}\cos \left(2\beta \pi t\right)\left({\frac {1}{2\beta \pi t+{\frac {\pi }{2}}}}-{\frac {1}{2\beta \pi t-{\frac {\pi }{2}}}}\right)=\\&{\frac {\pi }{4}}\cos \left(2\beta \pi t\right)\left({\frac {\left(2\beta \pi t-{\frac {\pi }{2}}\right)-\left(2\beta \pi t+{\frac {\pi }{2}}\right)}{\left(2\beta \pi t\right)^{2}-\left({\frac {\pi }{2}}\right)^{2}}}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf03d993941d2abe7c58bb84937b5e37ab8c96e)
