Una señal FSK puede ser modelada como la suma de dos señales ASK, que son a su vez una codificación unipolar NRZ modulada con un coseno.
s F S K ( t ) = A c ∑ k = − ∞ ∞ p ( t − k T s ) cos ( b k ⋅ ω c t ) = x A S K 1 ( t ) + x A S K 2 ( t ) s F S K ( t ) = A c ∑ k = − ∞ ∞ a k 1 p ( t − k T s ) cos ( ω c 1 t ) + A c ∑ k = − ∞ ∞ a k 2 p ( t − k T s ) cos ( ω c 2 t ) b k = { 1 , 2 } → { 1 → a k 1 = 1 ; a k 2 = 0 2 → a k 1 = 0 ; a k 2 = 1 x I ( t ) = ∑ k = − ∞ ∞ a k p ( t − k T s ) , x Q ( t ) = 0 a ′ 1 ′ = 1 , a ′ 0 ′ = 0 m a k = 1 2 , P a k = 1 2 , σ a k 2 = 1 4 G I ( f ) = σ a k 2 ⋅ R s | P ( f ) | 2 + m a k 2 ⋅ R s 2 ∑ k = − ∞ ∞ | P ( k R s ) | 2 δ ( f − k R s ) G I ( f ) = G N R Z ( f ) = A c 2 4 T s sinc 2 ( T s f ) + A c 2 4 δ ( f − R s ) G x ( f ) = G I ( f − f c ) + G I ( f + f c ) 4 + G Q ( f − f c ) + G Q ( f + f c ) 4 → G x ( f ) = G I ( f − f c ) + G I ( f + f c ) 4 = G x ( f ) = G I ( f ± f c ) 4 G x A S K 1 ( f ) = G x A S K 2 ( f ) = A c 2 4 2 T s sinc 2 ( T s ( f ± f c ) ) + A c 2 4 2 δ ( f ± f c − R s ) → G F S K ( f ) = A c 2 4 2 T s sinc 2 ( T s ( f ± f c 1 ) ) + A c 2 4 2 δ ( f ± f c 1 − R s ) + A c 2 4 2 T s sinc 2 ( T s ( f ± f c 2 ) ) + A c 2 4 2 δ ( f ± f c 2 − R s ) {\displaystyle {\begin{aligned}&s_{FSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{p\left(t-kT_{s}\right)\cos \left(b_{k}\cdot \omega _{c}t\right)}=x_{ASK_{1}}(t)+x_{ASK_{2}}(t)\\&s_{FSK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{k1}p\left(t-kT_{s}\right)\cos \left(\omega _{c1}t\right)}+A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{k2}p\left(t-kT_{s}\right)\cos \left(\omega _{c2}t\right)}\\&b_{k}=\left\{1,2\right\}\to \left\{{\begin{aligned}&1\to a_{k1}=1;a_{k2}=0\\&2\to a_{k1}=0;a_{k2}=1\\\end{aligned}}\right.\\&x_{I}(t)=\sum \limits _{k=-\infty }^{\infty }{a_{k}p\left(t-kT_{s}\right)},x_{Q}(t)=0\\&a_{'1'}=1,a_{'0'}=0\\&m_{a_{k}}={\frac {1}{2}},P_{a_{k}}={\frac {1}{2}},\sigma _{a_{k}}^{2}={\frac {1}{4}}\\&G_{I}(f)=\sigma _{a_{k}}^{2}\cdot R_{s}\left|P(f)\right|^{2}+m_{a_{k}}^{2}\cdot R_{s}^{2}\sum \limits _{k=-\infty }^{\infty }{\left|P(kR_{s})\right|^{2}\delta \left(f-kR_{s}\right)}\\&G_{I}(f)=G_{NRZ}(f)={\frac {A_{c}^{2}}{4}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}f\right)+{\frac {A_{c}^{2}}{4}}\delta \left(f-R_{s}\right)\\&G_{x}(f)={\frac {G_{I}(f-f_{c})+G_{I}(f+f_{c})}{4}}+{\frac {G_{Q}(f-f_{c})+G_{Q}(f+f_{c})}{4}}\to \\&G_{x}(f)={\frac {G_{I}(f-f_{c})+G_{I}(f+f_{c})}{4}}=G_{x}(f)={\frac {G_{I}(f\pm f_{c})}{4}}\\&G_{x_{ASK1}}(f)=G_{x_{ASK2}}(f)={\frac {A_{c}^{2}}{4^{2}}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c}\right)\right)+{\frac {A_{c}^{2}}{4^{2}}}\delta \left(f\pm f_{c}-R_{s}\right)\to \\&G_{FSK}(f)={\frac {A_{c}^{2}}{4^{2}}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c1}\right)\right)+{\frac {A_{c}^{2}}{4^{2}}}\delta \left(f\pm f_{c1}-R_{s}\right)+\\&{\frac {A_{c}^{2}}{4^{2}}}T_{s}\operatorname {sinc} ^{2}\left(T_{s}\left(f\pm f_{c2}\right)\right)+{\frac {A_{c}^{2}}{4^{2}}}\delta \left(f\pm f_{c2}-R_{s}\right)\\\end{aligned}}}
Para la probabilidad de error (BER):
BER de FSK
