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ELEG3410 Random Process & DSP Tutorial # 2 Outline: 1. Probability Density Function (p.d.f) &  Continuous R.V & Discrete R.V. 2. Probability Distribution Function (PDF) 3.Expected Value & Moment 4. Error Function of Gaussian Distribution 5.Error Rate in PCM
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1. Probability Density Function (pdf) Continuous Variable: Gaussian / Normal Laplacian xµ2 1(xµ)1 (x)=expp(x)=exp⎜ ⎟p2 σ2b b 2π σ2 ⎝ ⎠⎝ ⎠
Uniform 1 p(x)=ba
Discrete Variable: Assume the discrete random variablextakes only the values x1,x2, ,xj, ,xn… … with probability  P(x1), P(x2), …, P(xj), …, P(xn). Thep.d.f.ofx is n p(x)=P(x)δ(xx) j j j=1
2
P(x1) P(x2)
P(x3)
P(x4)
P(xn)
…… x1x2x3x4xn Example: Throw a dice: {1,2,3,4,5,6} 1 1 1 1 1 1 p(x)=δ(x1)+δ(x2)+δ(x3)+δ(x4)+δ(x5)+δ(x6) 6 6 6 6 6 6 2. Probability Distribution Function (PDF) Continuous Variable: The distribution function of random variablexx 1 ( ) (x x) W x1=p(x)dx=P1−∞ Thep.d.f.p(xnot a probability but a rate of change of the probability) is dW(x) dx W(x) : Gaussian / Normal 2 ( )1xµ p(x)=exp2 2π σ2σ ⎝ ⎠ probability density function and probability distribution funtion
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Discrete Variable: W(x)=p(x)=P(xx)1 1 xx 1
Probability distribution for discrete random variable Jumps in the distribution function correspond to impulses in the density function. W(x) monotonically nondecreasing inx. 3. Expected Value & Moment The statistical average (or mean, centroid, center of gravity, first moment about the origin, expected value) is defined as: x=xp(x)dx=E[x] − ∞ Ifg(x) is an arbitrary function ofx, the expected value ofg(x) is E[g(x)]=g(x)p(x)dx. −∞
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Moment th zabout the origin is defined as:n moment n n E[x]=x p(x)dx −∞ µ th th zn moment about the mean (n central moment) is defined as: n n E[(xµ)]=(xµ)p(x)dx −∞ st zabout the origin is the expected value of the R.V.1 moment Mean Value of R.V. nd zabout the origin is the average power of the R.V.2 moment 2 2 2 E[x]=x p(x)dx=x −∞ Mean Square Value of R.V. nd zabout the mean is the variance of the R.V.2 moment 2 2 2 2 2 2 σ=E[(xµ)]=E[x](E[x])=xx It measures the “spread” or “dispersion” of the R.V. Uniform 1b forx[a,a+b] p(x)= 0 otherwise p(x)
1/b
a
a + b
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a+b 2 1 1xb a+b E[x]=x=xp(x)dx=xdx= =a+ −∞a ⎢ ⎥ b b2 2 ⎣ ⎦a a+b 3 2 a+b 1xb 2 2 2 [ ]( ) Ex=xp x dx= =a+ab+ a ⎢ ⎥ b3 3 ⎣ ⎦a 2 2 2 2 2 2b2b b 2 σ=xx=a+ab+ −a− −ab= 3 4 12 4. Error function of Gaussian Distribution: 2 (xx) 1 2 2σ p(x)=e σ2π
2 ( ) kσx 122 2σ erf(k)P(kσ<xkσ)=e dx= σ2ππ kσ er( )P(xσ<xx+σ) 2 2 +() ( ) x kσkx x σx1 x=xx 11 1 2 2 2σ2σ =e dx=e dx ∫ ∫ 1 σ2π σ2π xkσkσ
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k 2 2 x e dx 0
k= 1 :erf( 1 ) = 0.683
k= 2 :erf( 2 ) = 0.955
x 2σ2 x Note that in Matlab, the error function is defined differently: k 2x 2 erf(k)=e dx Matlab π 0 You can also check outerfc(k), the complement of error function, or1-erf(k);
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5.Error Rate in PCM 1 =Avolt signal 0 = 0volt t Zero mean noisew Gaussianp.d.f. 2t Varianceσ p(w))( ) ( ) ( ) ( ) P e=P0P e0+P1P e1 1 P(1)=P(0)= p(e/0) p(e/1) 2 AP(e0)=Pw> ⎟ 2A/2A/2 AP(e)P(e1)=Pw≤ − 21 P(e)=[P(e0)+P(e1)] 2 2 10 1A=1erf⎢ ⎥ 4 22σ10 6 10 8 10 10 10 A [dB] 10 14 18 22σ 20 log107.4 = 17.3dB 21dB = 20 log1011.2 For transmission rate = 20 k bit/sec 104 1 error in every 0.5 sec 108 1 error in every 1 hr 23 minutes on average
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How to determine the pdf of a random variable?
t 1
t 2
t 3
T
p(x)
t 4
x=xx 2 1
t
x
x (1) choose a value forx (resolution improves asx) (2) choose a valueTimproves as (accuracy T) x 2 p(x)dx=P(xxx) 1 2 x1 P(x)(xx)P(xxx),x[x,x] 2 1 1 2 1 2 p(x)xx will be approximately constant over that interval, for1 2 (3)P(xxx)1 2 p(x),x[x,x] 1 2 xx 2 1 j t i 1i=1 p(x)≅ • x T  9
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