Chapter 7 Generating and Processing Random Signals. 第一組 電機四 B93902016 蔡馭理 資工四 B93902076 林宜鴻. Outline. Outline. Stationary and Ergodic Process Uniform Random Number Generator Mapping Uniform RVs to an Arbitrary pdf Generating Uncorrelated Gaussian RV Generating correlated Gaussian RV
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Outline
Gaussian
xi+1=[axi+c]mod(m)
xi+1=[axi+c]mod(m)
 c≠0 and relative prime tom
 a1 is a multiple of p, where p is the
prime factors of m
 a1 is a multiple of 4 ifm is a
multiple of 4
so, a1=4‧2‧5‧k =40k
xi+1=[241xi+ 1323]mod(5000)
xi+1=[axi]mod(m)
 m isprime (usaually large)
 a is a primitive elementmod(m)
am1/m = k =interger
ai1/m ≠ k, i=1, 2, 3,…, m2
xi+1=[axi]mod(2n)
the period is achieved if
 The multiplier a is 3 or 5
 The seed x0is odd

(i)rand(1,2048)
(ii)xi+1=[65xi+1]mod(2048)
(iii)xi+1=[1229xi+1]mod(2048)
Let X = X[n]
& Y = X[n1]
Assume X[n] and X[n1] are correlated and X[n] is an ergodic process
Let
X and Z are uncorrelated and zero mean
D>2 – negative correlation
D=2 –uncorrelation (most desired)
D<2 – positive correlation
xi+1=[16807xi]mod(2311)
X = FX1(U)
∴
Setting FR(R) = U
∵RV 1U is equivalent to U (have same pdf)
∴
Solving for R gives
more accuracy!
1.The sum of uniform method
2.Mapping a Rayleigh to Gaussian RV
3.The polar method
.
represent independent uniform R.V
is a constant that decides the var of
Y converges to a Gaussian R.V.
U is the uniform RV in [0,1]
and their joint pdf
let and
and
R is Rayleigh RV and is uniform RV
and they are all on the interval (0,1)
independent and uniform on (1,1)
else back to step2
X and Z
1.mean of input and output
2.variance of input and output
3.inputoutput crosscorrelation
4.autocorrelation and PSD
the Statistics of
substitute previous equation
The End Modulus
Thanks for listening