University:
College
Course:
College Engineering
Academic year:

2023
Views:

498

Pages:

2
Author:

Dipin K.

Expression for Bit error rate and Probability of error in wireless additive white gaussian noise (AWGN) channel. For deriving the expression of probability of error, we are considering BPSK modulation scheme and AWGN channel. The received signal expression for a wireless channel with a transmitted signal ‘x’ is 𝑦 = ℎ𝑥 + 𝑛 (1) Where ℎ is the channel complex fading channel coefficient. The magnitude of ℎ is Rayleigh distributed and angle is uniform distributed. In equation (1) the 1st term in the right hand side represents signal part and second term represents the noise part. Then the signal power is |ℎ𝑥|2 and Noise power is 𝜎 2 . So the SNR is 𝑆𝑁𝑅 = |ℎ𝑥|2 𝜎2 The Bit error rate for wireless AWGN system is 𝐵𝐸𝑅 = 𝑄 (√ |ℎ|2 𝑃 𝜎2 ) (2) Where P is the signal power. The probability of error for wireless AWGN system is 1 𝑆𝑁𝑅 𝑃𝑒 = (1 − √ ) 2 2+𝑆𝑁𝑅 (3) For high value of SNR, 1 𝑆𝑁𝑅 1 2 1 2 𝑃𝑒 = (1 − ( )) 2 2+𝑆𝑁𝑅 (4) −1 2 𝑃𝑒 = (1 − ( + 1) ) 2 𝑆𝑁𝑅 −1 (5) 1 We know (1 + 𝑥) 2 = 1 − 2 𝑥 So from equation (5) 1 1 2 𝑃𝑒 = (1 − (1 − × )) 2 2 𝑆𝑁𝑅 i.e., 𝑃𝑒 = 1 2𝑆𝑁𝑅 (6) Q: Consider a wireless system, find the required SNR to obtain a probability of error of 10−6 𝑃𝑒 = 𝑆𝑁𝑅 = 1 2𝑆𝑁𝑅 1 1 = 2𝑃𝑒 2 × 10−6 𝑆𝑁𝑅 = 0.5 × 106 (𝑖𝑛 𝑛𝑜𝑟𝑚𝑎𝑙 𝑠𝑐𝑎𝑙𝑒) 𝑆𝑁𝑅 = 10 log(0.5 × 106 ) = 56.98𝑑𝐵