# Gaussian Distribution

Document Sample

```					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.
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
• 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
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