SYSC4602 Review of Probability Concepts

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SYSC-4602Review of Probability Concepts
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Review of Probability Concepts

• Probability Space
• Consists of a number of events Ai. Each event is
  associated with a probability measure P(Ai)

S
A1
A2
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Conditional Probability
S
A1
A2
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Random Variables

• A random variable is a function that maps events
  in a given probability space to the real line.
• A random variable follows a mathematical law as
  defined by the probability density function (pdf)
  ? fX(x)
• Discrete r.v. has discrete pdf. Examples include
  Bernoulli trials, binomial distribution, Poisson
  distribution
• Continuous r.v. has continuous pdf. Examples
  include Uniform distribution, Exponential
  distribution, and Gaussian (Normal) distribution.

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Statistical Averages
Mean (Expected) Value of r.v. X
nth Moment of r.v. X
Central Moments of r.v. X
Variance
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Examples
fX(x)
Uniform r.v.
x
b
a
Normal r.v.
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Two Random Variables, X and Y
Joint Moments
Independence
Covariance
Correlation Coefficient
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Random Processes

• Observing the sample functions at time tk we find
  that the resulting collection of numbers
  xj(tk), j1,2,,n forms a random variable.
• Observing the sample functions at t1, t2, , tk
  forms a vector of random variables X(t).

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Stationarity

• A random process is said to be stationary in the
  strict sense (strictly stationary) if the joint
  probability function is invariant under shifts in
  time

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Mean, Correlation, and Covariance Functions
Mean of the process X(t)
Autocorrelation Function
Autocovariance Function
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Wide-Sense Stationary

• A random process is said to be wide-sense
  stationary (wss) when

• Power Spectral Density

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Properties of Power Spectral Density of a Random
Process

• Mean Square Value

• Non-negativity

• Symmetry

• Filtered Random Process

Y(t)
H(f)
X(t)
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Example
Y(t)
X(t)
H(f)
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Gaussian Process

• A random process X(t) is said to be a Gaussian
  process if the joint distribution of the random
  variables X(t1), X(t2),,X(tk) is Gaussian
  distributed

m is the kth dimensional vector of the means L is
the k x k covariance matrix

• If a Gaussian process is wide sense stationary
  then it is also stationary in the strict sense
• If a Gaussian process is applied to a stable
  linear filter, then the random Y(t) produced at
  the output of the filter is Gaussian

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White Noise

• White noise is an idealized noise process with a
  power spectral density that independent of
  frequency

SN(f)
N0/2 d(t)
N0/2
f
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Example

• Ideal Low-Pass Filtered White Noise

SN(f)
H(f)
N0/2
1
f
B
-B
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Gaussian White Noise in BandPass Filter
SN(f)
N0/2
f
H(f)
1
fc
fc
f
t
SY(f)
N0/2
fc
fc
f
nI(t) and nQ(t) are two baseband Gaussian
processes
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Thermal Noise

• Thermal Noise
• Due to the thermally induced motion of electrons
  in conducting media. Power spectral density of
  thermal noise is given by

h Plancks constant 6.63x10-34 k
Boltzmanns constant 1.38x10-23 T absolute
temperature 290 at 17c