s M − A S K ( t ) = A c ∑ k = − ∞ ∞ a I k p ( t − k T s ) cos ( ω c t ) a I k = { 0 , A , + 2 A , . . . } {\displaystyle {\begin{aligned}&s_{M-ASK}(t)=A_{c}\sum \limits _{k=-\infty }^{\infty }{a_{I_{k}}p\left(t-kT_{s}\right)\cos \left(\omega _{c}t\right)}\\&a_{I_{k}}=\left\{0,A,+2A,...\right\}\\\end{aligned}}}
y R ( t ) = A R ∑ k = − ∞ ∞ a I k p ( t − k T s ) + n I ( t ) a I k = { 0 , A , 2 A , . . . M − 1 } m I k = 0 , A T s , 2 A T s , . . . E s = E s c 2 → 1 M ⋅ Q ( | V T − m | σ 2 ) M − 2 → 1 M ⋅ 2 Q ( | V T − m | σ 2 ) {\displaystyle {\begin{aligned}&y_{R}(t)=A_{R}\sum \limits _{k=-\infty }^{\infty }{a_{I_{k}}p\left(t-kT_{s}\right)+n_{I}(t)}\\&a_{I_{k}}=\left\{0,A,2A,...M-1\right\}\\&m_{I_{k}}=0,AT_{s},2AT_{s},...\\&E_{s}=E_{sc}\\&2\to {\frac {1}{M}}\cdot Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)\\&M-2\to {\frac {1}{M}}\cdot 2Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)\\\end{aligned}}}
| V T − m | = A T s 2 σ 2 = η 2 T s P s ≃ 2 × 1 M ⋅ Q ( | V T − m | σ 2 ) + ∑ i = 1 M − 2 1 M ⋅ 2 Q ( | V T − m | σ 2 ) = 2 × 1 M ⋅ Q ( | V T − m | σ 2 ) + ( M − 2 ) 1 M ⋅ 2 Q ( | V T − m | σ 2 ) = 2 ⋅ M − 1 M Q ( | V T − m | σ 2 ) = 2 ⋅ M − 1 M Q ( A T s 2 η 2 T s ) = 2 ⋅ M − 1 M Q ( A 2 T s 2 4 η 2 T s ) = 2 ⋅ M − 1 M Q ( A 2 T s 2 η ) {\displaystyle {\begin{aligned}&\left|V_{T}-m\right|={\frac {AT_{s}}{2}}\\&{\sqrt {\sigma ^{2}}}={\sqrt {{\frac {\eta }{2}}T_{s}}}\\&P_{s}\simeq 2\times {\frac {1}{M}}\cdot Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)+\sum \limits _{i=1}^{M-2}{{\frac {1}{M}}\cdot 2Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)}=\\&2\times {\frac {1}{M}}\cdot Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)+\left(M-2\right){\frac {1}{M}}\cdot 2Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)=2\cdot {\frac {M-1}{M}}Q\left({\frac {\left|V_{T}-m\right|}{\sqrt {\sigma ^{2}}}}\right)=\\&2\cdot {\frac {M-1}{M}}Q\left({\frac {\frac {AT_{s}}{2}}{\sqrt {{\frac {\eta }{2}}T_{s}}}}\right)=2\cdot {\frac {M-1}{M}}Q\left({\sqrt {\frac {\frac {A^{2}T_{s}^{2}}{4}}{{\frac {\eta }{2}}T_{s}}}}\right)=2\cdot {\frac {M-1}{M}}Q\left({\sqrt {\frac {A^{2}T_{s}}{2\eta }}}\right)\\\end{aligned}}}
E s = ∑ i = 0 M − 1 p r ( M ) ⋅ E i = ∑ i = 0 M − 1 1 M ⋅ E i = 1 M ∑ i = 0 M − 1 E i E i = { 0 , A 2 T s , 4 A 2 T s , 9 A 2 T s , . . } E s = 1 M ∑ i = 0 M − 1 E i = 1 M ∑ i = 0 M − 1 i 2 A 2 T s = 1 M A 2 T s ∑ i = 0 M − 1 i 2 ∑ i = 1 n i 2 = n ( n + 1 ) ( 2 n + 1 ) 6 → ∑ i = 0 M − 1 i 2 = ( M − 1 ) M ( 2 ( M − 1 ) + 1 ) 6 = ( M − 1 ) M ( 2 M − 1 ) 6 {\displaystyle {\begin{aligned}&E_{s}=\sum \limits _{i=0}^{M-1}{pr(M)\cdot E_{i}}=\sum \limits _{i=0}^{M-1}{{\frac {1}{M}}\cdot E_{i}}={\frac {1}{M}}\sum \limits _{i=0}^{M-1}{E_{i}}\\&E_{i}=\left\{0,A^{2}T_{s},4A^{2}T_{s},9A^{2}T_{s},..\right\}\\&E_{s}={\frac {1}{M}}\sum \limits _{i=0}^{M-1}{E_{i}}={\frac {1}{M}}\sum \limits _{i=0}^{M-1}{i^{2}A^{2}T_{s}}={\frac {1}{M}}A^{2}T_{s}\sum \limits _{i=0}^{M-1}{i^{2}}\\&\sum \limits _{i=1}^{n}{i^{2}}={\frac {n\left(n+1\right)\left(2n+1\right)}{6}}\to \sum \limits _{i=0}^{M-1}{i^{2}}={\frac {\left(M-1\right)M\left(2\left(M-1\right)+1\right)}{6}}={\frac {\left(M-1\right)M\left(2M-1\right)}{6}}\\\end{aligned}}}
∑ i = 0 M − 1 i 2 = ( M − 1 ) M ( 2 M − 1 ) 6 E s = 1 M A 2 T s ∑ i = 0 M − 1 i 2 = 1 M A 2 T s ⋅ ( M − 1 ) M ( 2 M − 1 ) 6 = A 2 T s 6 ( M − 1 ) ( 2 M − 1 ) E s = A 2 T s 2 ⋅ 3 ( M − 1 ) ( 2 M − 1 ) ↔ A 2 T s 2 = 3 E s ( M − 1 ) ( 2 M − 1 ) P s = 2 ⋅ M − 1 M Q ( A 2 T s 2 η ) ⇒ P s = 2 ⋅ M − 1 M Q ( 3 ( M − 1 ) ( 2 M − 1 ) E s η ) {\displaystyle {\begin{aligned}&\sum \limits _{i=0}^{M-1}{i^{2}}={\frac {\left(M-1\right)M\left(2M-1\right)}{6}}\\&E_{s}={\frac {1}{M}}A^{2}T_{s}\sum \limits _{i=0}^{M-1}{i^{2}}={\frac {1}{M}}A^{2}T_{s}\cdot {\frac {\left(M-1\right)M\left(2M-1\right)}{6}}={\frac {A^{2}T_{s}}{6}}\left(M-1\right)\left(2M-1\right)\\&E_{s}={\frac {A^{2}T_{s}}{2\cdot 3}}\left(M-1\right)\left(2M-1\right)\leftrightarrow {\frac {A^{2}T_{s}}{2}}={\frac {3E_{s}}{\left(M-1\right)\left(2M-1\right)}}\\&P_{s}=2\cdot {\frac {M-1}{M}}Q\left({\sqrt {\frac {A^{2}T_{s}}{2\eta }}}\right)\Rightarrow \\&P_{s}=2\cdot {\frac {M-1}{M}}Q\left({\sqrt {{\frac {3}{\left(M-1\right)\left(2M-1\right)}}{\frac {E_{s}}{\eta }}}}\right)\\\end{aligned}}}
Comprobación:
4 − A S K → P s ≃ 3 2 Q ( E s 7 η ) P s = 2 ⋅ M − 1 M | M = 4 Q ( 3 ( M − 1 ) ( 2 M − 1 ) E s η ) | M = 4 = 3 2 Q ( 3 3 ⋅ 7 E s η ) = 3 2 Q ( 1 7 E s η ) → c . q . d . {\displaystyle {\begin{aligned}&4-ASK\to P_{s}\simeq {\frac {3}{2}}Q\left({\sqrt {\frac {E_{s}}{7\eta }}}\right)\\&P_{s}=\left.2\cdot {\frac {M-1}{M}}\right|_{M=4}\left.Q\left({\sqrt {{\frac {3}{\left(M-1\right)\left(2M-1\right)}}{\frac {E_{s}}{\eta }}}}\right)\right|_{M=4}={\frac {3}{2}}Q\left({\sqrt {{\frac {3}{3\cdot 7}}{\frac {E_{s}}{\eta }}}}\right)={\frac {3}{2}}Q\left({\sqrt {{\frac {1}{7}}{\frac {E_{s}}{\eta }}}}\right)\to c.q.d.\\\end{aligned}}}
