Bit Error Rate Analysis in Wired AWGN Channel

p {margin: 0; padding: 0;} .ft00{font-size:18px;font-family:Times New Roman,Bold;color:#000000;} .ft01{font-size:18px;font-family:Times New Roman;color:#000000;} .ft02{font-size:18px;font-family:Cambria Math;color:#000000;} .ft03{font-size:16px;font-family:Cambria Math;color:#000000;} .ft04{font-size:12px;font-family:Cambria Math;color:#000000;} .ft05{font-size:16px;font-family:Times New Roman;color:#000000;} .ft06{font-size:14px;font-family:Cambria Math;color:#000000;} .ft07{font-size:21px;font-family:Times New Roman;color:#000000;} .ft08{font-size:21px;font-family:Cambria Math;color:#000000;} .ft09{font-size:16px;font-family:Times New Roman,Bold;color:#000000;} .ft010{font-size:18px;line-height:22px;font-family:Times New Roman;color:#000000;} .ft011{font-size:16px;line-height:23px;font-family:Cambria Math;color:#000000;} .ft012{font-size:21px;line-height:25px;font-family:Cambria Math;color:#000000;} .ft013{font-size:21px;line-height:30px;font-family:Cambria Math;color:#000000;} Expression for bit error rate in wired additive white gaussian noise (AWGN) channel  Wired Communication  Let  x  be  the  transmitted  signal  in  wired  communication,  where  additive  white  gaussian  noise channel is considered, then received signal y is represented as                                                            𝑦 = 𝑥 + 𝑛      Where n is the AWGN signal.  In AWGN channel Bit error rate(BER) is expressed as,   𝐵𝐸𝑅 = 12 𝑒𝑟𝑓𝑐 (√ 𝐸 𝑏 𝑛 0 ) =  𝑄(√𝑆𝑁𝑅)   Where  𝑒𝑟𝑓𝑐(𝑥)   is  the  complementary  error  function,  𝐸 𝑏   is  the  noise  signal  energy  and  𝑛 0   is  the  variance.  This  can also  represent  in  terms  of probability  (Q)  function and  signal  to  noise power  ratio (SNR), expressed as   𝑆𝑁𝑅 =   ( 𝑃 𝜎 2 )   So  BER = 𝑄 (√ 𝑃 𝜎 2 )   Where   𝑃  is the signal power and  𝜎 2  is the noise power   The equation for probability of error in wired system can be formed by applying Chernoff bound,  i.e.,    𝑄(𝑥)   ≤ 1 2 𝑒 −𝑥 2   So   𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟(𝑃𝑒) = 12 𝑒 −𝑆𝑁𝑅 2   Example:  Q:  Calculate  the  SNR  required  to  achieve  a  probability of error   of  10 −6     in  a  wired  AWGN  system.  Solution: We know that for the expression for probability of error in a wired AWGN system is   𝑃 𝑒 = 12 𝑒 −𝑆𝑁𝑅 2   Here  𝑃 𝑒 =   10 −6  ,    So  SNR can be calculated as   𝑆𝑁𝑅 = −2𝑙𝑜𝑔 𝑒 (2 𝑃 𝑒 ) ,   So SNR can be calculated as SNR = 14.19dB.  Conclusion: The system require 14.19dB of SNR to get a bit error rate of  10 −6 .