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

ELE 745 – Digital CommunicationsXavier Fernando

PART I – GAUSSIAN DISTRIBUTIONELE 745 – AWGN Channel

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

1( ) ( /1)thV

thP V p y dy

0 ( ) ( / 0)th

th VP V p y dy

1

1 02 ( ) ( )e th thP P V P V

Digital Receiver Performance

0 ( ) ( / 0)th

th VP V p y dy

1( ) ( /1)thV

thP V p y dy

11 02 ( ) ( )e th thP P V P V

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-IIMATCHED FILTER DETECTION