ELE 745 – Digital Communications Xavier Fernando Additive White Gaussian Noise (AWGN) Channel and Matched Filter Detection ELE 745 – Digital Communications Xavier Fernando

Part I – Gaussian distribution ELE 745 – AWGN Channel Part I – Gaussian distribution

Gaussian (Normal) Distribution The Normal or Gaussian distribution, is an important family of continuous probability distributions The mean ("average", μ) and variance (standard deviation squared, σ2) are the defining parameters The standard normal distribution is the normal distribution with zero mean (μ=0) and unity variance (σ2 =1) Many measurements, from psychological to thermal noise can be approximated by the Gaussian distribution.

Gaussian RV

General Gaussian RV

PDF of Gaussian Distribution Standard Norma Distribution

CDF of Gaussian Distribution

The Central Limit Theorem The sum of independent, identically distributed large number of random variables with finite variance is approximately normally distributed under certain conditions Ex: Binomial distribution B(n, p) approaches normal for large n and p The Poisson(λ) distribution is approximately normal N(λ, λ) for large values of λ. The chi-squared distribution approaches normal for large k. The Student’s t-distribution t(ν) approaches normal N(0, 1) when ν is large.

Area under Gaussian PDF The area within +/- σ is ≈ 68% (dark blue) The area within +/- 2σ is ≈ 95% (medium and dark blue) The area within +/- 2σ is ≈ 99.7% (light, medium, and dark blue)

Bit Error Rate (BER) BER is the ratio of erroneous bits to correct bits BER is an important quality measure of digital communication link BER depends on the signal and noise power (Signal to Noise Ratio) BER requirement is different for different services and systems Wireless link BER < 10-6 while Optical BER < 10-12 Voice  Low BER while Data  High BER

Logic 0 and 1 probability distributions

Digital Receiver Performance Probability of error assuming Equal ones and zeros Where, Depends on the noise variance at on/off levels and the Threshold voltage Vth that is decided to minimize the Pe; Often Vth = V+ + V-

The Q Function Fx(x) = 1 – Q(X)

Error Probability of On-Off Signaling

BER (Pe) versus Q factor in a Typical Digital Communication Link

Part-ii Matched filter detection