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Additive White Gaussian Noise (AWGN) Channel and Matched Filter Detection. ELE 745 – Digital Communications Xavier Fernando. ELE 745 – AWGN Channel. Part I – Gaussian distribution. Gaussian (Normal) Distribution.
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Additive White Gaussian Noise (AWGN) Channel and Matched Filter Detection ELE 745 – Digital Communications Xavier Fernando
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.
PDF of Gaussian Distribution Standard Norma 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
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 Vththat is decided to minimize the Pe; Often Vth = V+ + V-
The Q Function Fx(x) = 1 – Q(X)
BER (Pe) versus Q factor in a Typical Digital Communication Link