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

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Part I Gaussian distribution

• ELE 745 AWGN Channel

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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, s2) are the defining
  parameters
• The standard normal distribution is the normal
  distribution with zero mean (µ0) and unity
  variance (s2 1)
• Many measurements, from psychological to thermal
  noise can be approximated by the Gaussian
  distribution.

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Gaussian RV
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General Gaussian RV
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PDF of Gaussian Distribution
Standard Norma Distribution
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CDF of Gaussian Distribution
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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(?) distribution is approximately
  normal N(?,??) for large values of ?.
• The chi-squared distribution approaches normal
  for large k.
• The Students t-distribution t(?) approaches
  normal N(0,?1) when ? is large.

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Area under Gaussian PDF
The area within /- s is 68 (dark blue) The
area within /- 2s is 95 (medium and dark
blue) The area within /- 2s is 99.7 (light,
medium, and dark blue)
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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 lt 10-6 while Optical BER lt
  10-12
• Voice ? Low BER while Data ? High BER

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Logic 0 and 1 probability distributions
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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-
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The Q Function
Fx(x) 1 Q(X)
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Error Probability of On-Off Signaling
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BER (Pe) versus Q factor in a Typical Digital
Communication Link
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Part-iiMatched filter detection
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