Additive White Gaussian Noise (AWGN) Channel and Matched Filter Detection

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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

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

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