editar
Recordemos un poco la representación frecuencial de las señales periódicas
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{\displaystyle {\begin{aligned}&x(t+T)=x(t)\to x(t)=\sum \limits _{k=-\infty }^{\infty }{C_{k}e^{j{\frac {2\pi }{T_{s}}}t}}{\text{ }};{\text{ }}D=f_{s}={\frac {1}{T_{s}}}\\&X(f)=\mathbb {F} \left[x(t)\right]=\sum \limits _{k=-\infty }^{\infty }{C_{k}\delta \left(f-kf_{s}\right)}\\&G_{x}(f)={\underset {T\to \infty }{\mathop {\lim } }}\,{\frac {\left|X(f)\right|^{2}}{T}}\\&G_{x}(f)=\sum \limits _{k=-\infty }^{\infty }{\left|C_{k}\right|^{2}\delta \left(f-kf_{s}\right)}\\\end{aligned}}}
Ahora, sabiendo que:
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{\displaystyle {\begin{aligned}&s(t)=\sum \limits _{k=-\infty }^{\infty }{x\left(t-kT_{s}\right)}=x(t)*\sum \limits _{k=-\infty }^{\infty }{\delta \left(t-kT_{s}\right)}=x(t)*{\frac {1}{T_{s}}}\sum \limits _{k=-\infty }^{\infty }{e^{j{\frac {2\pi }{T_{s}}}t}}\\&\mathbb {F} \left[\sum \limits _{k=-\infty }^{\infty }{x\left(t-kT_{s}\right)}\right]=X(f)\cdot {\frac {1}{T_{s}}}\sum \limits _{k=-\infty }^{\infty }{\delta \left(f-kf_{s}\right)}=\sum \limits _{k=-\infty }^{\infty }{\overbrace {\frac {X(kf_{s})}{T_{s}}} ^{C_{k}}}\cdot \delta \left(f-kf_{s}\right)\\\end{aligned}}}
editar
Una señal digital, además de ser también periódica, se caracteriza por tener un conjunto finito de amplitudes/fases posibles. La representamos matemáticamente como:
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{\displaystyle x(t)=\sum \limits _{k=-\infty }^{\infty }{a_{k}p\left(t-kT_{s}\right)}}
Sepamos que :
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{\displaystyle R_{s}\left({}^{baudios}\!\!\diagup \!\!{}_{s}\;\right)={\frac {1}{T_{s}}}=f_{s}}
Ahora, saquemos su DEP:
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lim
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lim
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lim
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{
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↔
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lim
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lim
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{\displaystyle {\begin{aligned}&x(t)=\sum \limits _{k=-N}^{N}{a_{k}p\left(t-kT_{s}\right)}=\sum \limits _{k=-N}^{N}{a_{k}\delta \left(t-kT_{s}\right)}*p(t)\to X(f)=P(f)\cdot \sum \limits _{k=-N}^{N}{a_{k}e^{-j2\pi fkT_{_{s}}}=}\\&{\bar {G}}_{x}(f)={\underset {T\to \infty }{\mathop {\lim } }}\,{\frac {\left|X(f)\right|^{2}}{T}}\to T=\left(2N+1\right)T_{s}\\&{\bar {G}}_{x}(f)=\left|P(f)\right|^{2}{\underset {T\to \infty }{\mathop {\lim } }}\,\cdot {\frac {1}{T}}\left|\sum \limits _{k=-N}^{N}{{\bar {a}}_{k}e^{-j2\pi fkT_{_{s}}}}\right|^{2}=\left|P(f)\right|^{2}{\underset {T\to \infty }{\mathop {\lim } }}\,\cdot {\frac {1}{T}}\sum \limits _{m=-N}^{N}{{\bar {a}}_{m}e^{-j2\pi fmT_{_{s}}}\sum \limits _{n=-N}^{N}{{\bar {a}}_{n}e^{+j2\pi fnT_{_{s}}}=}}\\&{\bar {G}}_{x}(f)=\left|P(f)\right|^{2}{\underset {T\to \infty }{\mathop {\lim } }}\,\cdot {\frac {1}{T}}\sum \limits _{m=-N}^{N}{\sum \limits _{n=-N}^{N}{{\bar {a}}_{n}{\bar {a}}_{m}e^{-j2\pi f\left(m-n\right)T_{_{s}}}}\to \left\{m=n+\tau \leftrightarrow \tau =m-n\right\}}\\&{\bar {G}}_{x}(f)=\left|P(f)\right|^{2}{\underset {T\to \infty }{\mathop {\lim } }}\,\cdot {\frac {1}{T}}\sum \limits _{\tau =-N-n}^{N-n}{\sum \limits _{n=-N}^{N}{{\bar {a}}_{n}{\bar {a}}_{n+\tau }e^{-j2\pi f\tau T_{_{s}}}}}\to R\left(\tau \right)={\bar {a}}_{n}{\bar {a}}_{n+\tau }\\&{\bar {G}}_{x}(f)=\left|P(f)\right|^{2}{\underset {T\to \infty }{\mathop {\lim } }}\,\cdot {\frac {1}{T}}\sum \limits _{\tau =-N-n}^{N-n}{\left(2N+1\right){\bar {a}}_{n}{\bar {a}}_{n+\tau }e^{-j2\pi f\tau T_{_{s}}}}\to {\underset {T\to \infty }{\mathop {\lim } }}\,\cdot {\frac {1}{T}}={\underset {N\to \infty }{\mathop {\lim } }}\,\cdot {\frac {1}{\left(2N+1\right)T_{s}}}\\&{\bar {G}}_{x}(f)=\left|P(f)\right|^{2}{\underset {N\to \infty }{\mathop {\lim } }}\,\cdot {\frac {\left(2N+1\right)}{\left(2N+1\right)T_{s}}}\sum \limits _{\tau =-N-n}^{N-n}{R\left(\tau \right)e^{-j2\pi f\tau T_{_{s}}}}={\frac {\left|P(f)\right|^{2}}{T_{s}}}\sum \limits _{\tau =-\infty }^{\infty }{R\left(\tau \right)e^{-j2\pi f\tau T_{_{s}}}}\\&R\left(\tau \right)={\bar {a}}_{n}{\bar {a}}_{m}=\left\{{\begin{aligned}&P_{x}=\sigma _{x}^{2}+m_{x}^{2},n=m\\&m_{x}^{2},n\neq m\\\end{aligned}}\right.\\&{\bar {G}}_{x}(f)={\frac {\left|P(f)\right|^{2}}{T_{s}}}\left(P_{x}+\underbrace {\sum \limits _{\tau =-\infty }^{\infty }{m_{x}^{2}e^{-j2\pi f\tau T_{_{s}}}}} _{\forall n\neq m}\right)={\frac {\left|P(f)\right|^{2}}{T_{s}}}\left(\sigma _{x}^{2}+\sum \limits _{\tau =-\infty }^{\infty }{m_{x}^{2}e^{-j2\pi f\tau T_{_{s}}}}\right)={\frac {\left|P(f)\right|^{2}}{T_{s}}}\left(\sigma _{x}^{2}+m_{x}^{2}\sum \limits _{\tau =-\infty }^{\infty }{e^{-j2\pi f\tau T_{_{s}}}}\right)\\&\sum \limits _{\tau =-\infty }^{\infty }{e^{\pm j2\pi f\tau T_{_{s}}}}=R_{s}\sum \limits _{\tau =-\infty }^{\infty }{\delta \left(f-\tau R_{s}\right)}=R_{s}\sum \limits _{k=-\infty }^{\infty }{\delta \left(f-kR_{s}\right)}\\&{\bar {G}}_{x}(f)=R_{s}\left|P(f)\right|^{2}\left(\sigma _{x}^{2}+m_{x}^{2}\cdot R_{s}\sum \limits _{k=-\infty }^{\infty }{\delta \left(f-kR_{s}\right)}\right)=\sigma _{x}^{2}\cdot R_{s}\left|P(f)\right|^{2}+m_{x}^{2}\cdot R_{s}^{2}\left|P(f)\right|^{2}\sum \limits _{k=-\infty }^{\infty }{\delta \left(f-kR_{s}\right)}\\&{\bar {G}}_{x}(f)=\sigma _{x}^{2}\cdot R_{s}\left|P(f)\right|^{2}+m_{x}^{2}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}\\\end{aligned}}}
En las modulaciones analógicas la “eficiencia” de una modulación se medía por el ancho de banda que tenía (a menor ancho de banda requerido por la señal, mayor eficiencia de la modulación), también era importante la relación señal a ruido o SNR, para saber la robustez de nuestra señal al ruido.
En las modulaciones digitales hemos visto que el SNR no es una buena elección para ver lo “buena” que es nuestra señal, pues al ser digital un empobrecimiento ( o aumento) de la SNR no tiene que significar un error de bit mayor, en las modulaciones digitales con que superemos cierto umbral basta. Por ello, en vez de SNR se usa el BER (Bit Error Rate), que nos dice el ratio de error del bit, siendo un parámetro de importancia.
Para medir la eficiencia en frecuencia de nuestra modulación se usa el parámetro eficiencia espectral que se define como:
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{\displaystyle e.e.={\frac {R_{s}}{\beta _{T}}}={\frac {\left({}^{bit}\!\!\diagup \!\!{}_{s}\;\right)}{Hz}}}
Un ejemplo de eficiencia espectral, para el filtro coseno alzado:
β
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{\displaystyle {\begin{aligned}&\beta _{T}=W\left(1+\rho \right)\\&W={\frac {1}{T_{s}}}?=R_{s}\\&e.e.={\frac {R_{s}}{\beta _{T}}}={\frac {1}{1+\rho }}\left({}^{bit/s}\!\!\diagup \!\!{}_{Hz}\;\right)\\\end{aligned}}}
s si la señal esta representada en paso banda, en caso de que se quiera usar el valor de W y no confundirlo, debemos referirnos al ancho de banda nominal que es a lo que se refiere W y es efectivamente el ancho de banda si la señal fuese paso banda. en caso de estar en banda base, el ancho de banda de la señal será de 1/2Ts
Para las modulaciones digitales hemos visto que podemos usar mas de dos posibles estados, pudiendo de esa manera enviar mas información en el mismo tiempo. Por ello, se diferencia entre tiempo de bit
(
T
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{\displaystyle \left(T_{b}\right)}
(
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{\displaystyle \left(T_{s}\right)}
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{\displaystyle {\begin{aligned}&T_{b}\left({}^{s}\!\!\diagup \!\!{}_{bit}\;\right),T_{s}\left({}^{s}\!\!\diagup \!\!{}_{simbol}\;\right)\\&R_{b}={\frac {1}{T_{b}}}=\left({}^{bit}\!\!\diagup \!\!{}_{s}\;\right)\\&R_{s}={\frac {1}{T_{s}}}=\left({}^{simbol}\!\!\diagup \!\!{}_{s}\;\right)=\left({}^{baud}\!\!\diagup \!\!{}_{s}\;\right)\\\end{aligned}}}
La relación entre ambos:
T
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→
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log
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{\displaystyle {\begin{aligned}&T_{s}=\log _{2}M\cdot T_{b}\\&{\frac {T_{s}}{T_{b}}}=\log _{2}M\to {\frac {R_{b}}{R_{s}}}=\log _{2}M\\&R_{b}=\log _{2}M\cdot R_{s}\\\end{aligned}}}
![{\displaystyle {\begin{aligned}&x(t+T)=x(t)\to x(t)=\sum \limits _{k=-\infty }^{\infty }{C_{k}e^{j{\frac {2\pi }{T_{s}}}t}}{\text{ }};{\text{ }}D=f_{s}={\frac {1}{T_{s}}}\\&X(f)=\mathbb {F} \left[x(t)\right]=\sum \limits _{k=-\infty }^{\infty }{C_{k}\delta \left(f-kf_{s}\right)}\\&G_{x}(f)={\underset {T\to \infty }{\mathop {\lim } }}\,{\frac {\left|X(f)\right|^{2}}{T}}\\&G_{x}(f)=\sum \limits _{k=-\infty }^{\infty }{\left|C_{k}\right|^{2}\delta \left(f-kf_{s}\right)}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d58fe29a0203700d41ab3050621b08d7f99d83)
![{\displaystyle {\begin{aligned}&s(t)=\sum \limits _{k=-\infty }^{\infty }{x\left(t-kT_{s}\right)}=x(t)*\sum \limits _{k=-\infty }^{\infty }{\delta \left(t-kT_{s}\right)}=x(t)*{\frac {1}{T_{s}}}\sum \limits _{k=-\infty }^{\infty }{e^{j{\frac {2\pi }{T_{s}}}t}}\\&\mathbb {F} \left[\sum \limits _{k=-\infty }^{\infty }{x\left(t-kT_{s}\right)}\right]=X(f)\cdot {\frac {1}{T_{s}}}\sum \limits _{k=-\infty }^{\infty }{\delta \left(f-kf_{s}\right)}=\sum \limits _{k=-\infty }^{\infty }{\overbrace {\frac {X(kf_{s})}{T_{s}}} ^{C_{k}}}\cdot \delta \left(f-kf_{s}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19ff2799fbd99f594e374adb3b09df12c7d5f4ca)
