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

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

3. 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.

4. Gaussian RV

5. General Gaussian RV

6. PDF of Gaussian Distribution
Standard Norma Distribution

7. CDF of Gaussian Distribution

8. 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.

9. 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)

10. 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

11. Logic 0 and 1 probability distributions

12. 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-

13. The Q Function
Fx(x) = 1 – Q(X)

19. Error Probability of On-Off Signaling

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

22. Part-ii Matched filter detection