tutorial 2

Rqzhang

Description

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 11. Probability Density Function (pdf) Continuous Variable: Gaussian / Normal Laplacian 2 ⎛ x − µ ⎞1⎛ ⎞1 ()x − µ ⎜ ⎟ p()x = exp − ⎜ ⎟p()x = exp − ⎜ ⎟2⎜ ⎟ 2b b2 σ2 π σ ⎝ ⎠ ⎝ ⎠ Uniform 1p()x = b − a Discrete Variable: Assume the discrete random variable x takes only the values x1, x2, …, xj, …, xn with probability P(x1), P(x2), …, P(xj), …, P(xn). The p.d.f. of x is n p(x) = P(x ) ⋅ δ (x − x )∑ j jj =1 2P(x ) 3P(x ) P(x ) 4 nP(x ) P (x ) 1 2……x x x x x1 2 3 4 n Example: Throw a dice: {1,2,3,4,5,6} 1 1 1 1 1 1p(x) = δ (x −1) + δ (x − 2) + δ (x − 3) + δ (x − 4) + δ (x − 5) + δ (x − 6) 6 6 6 6 6 6 2. Probability Distribution Function (PDF) Continuous Variable: The distribution function of random variable x x1W()x = p(x)dx = P(x ≤ x) 1 1∫− ∞The p.d.f. p(x) is not a probability but a rate of change of the probability dW(x)dxW(x) : Gaussian / Normal 2⎛ ⎞1 ()x − µ ⎜ ⎟p()x = exp −⎜ 2 ⎟2 σ2 π σ ⎝ ⎠probability density function and probability distribution funtion 3Discrete Variable: ...

Informations

• Publié par: Muchang
• Nombre de lectures: 21
• Langue: English

Extrait

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

1

1. Probability Density Function (pdf) Continuous Variable: Gaussian / Normal Laplacian ⎛ x−µ ⎞ 2 1 ⎛ ( x−µ ) ⎞ 1 ( x ) =exp ⎜ − ⎟ p ( x ) =exp ⎜ ⎟ p − 2 σ 2b b 2π σ 2 ⎝ ⎠ ⎝ ⎠

Uniform 1 p ( x ) = b−a

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(x−x) ∑ j j j=1

2

P(x 1 ) P(x 2 )

P(x 3 )

P(x 4 )

P(x n )

…… x 1 x 2 x 3 x 4 x n Example: Throw a dice: {1,2,3,4,5,6} 1 1 1 1 1 1 p(x)=δ(x−1)+δ(x−2)+δ(x−3)+δ(x−4)+δ(x−5)+δ(x−6) 6 6 6 6 6 6 2. Probability Distribution Function (PDF) Continuous Variable: The distribution function of random variablex x 1 ( ) ( x x ) W x 1 =p(x)dx=P≤ 1 ∫ −∞ Thep.d.f.p(xnot a probability but a rate of change of the probability) is dW(x) dx W(x) : Gaussian / Normal 2 ⎛ ( ) ⎞ 1 x−µ p ( x ) =exp ⎜ − ⎟ 2 2π σ 2σ ⎝ ⎠ probability density function and probability distribution funtion

3

Discrete Variable: W ( x ) =p(x)=P ( x≤x ) ∑ 1 1 x≤x 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. ∫ −∞

4

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[ ( x−µ ) ]= ( x−µ ) 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 [( x−µ )] =E [ x ] − ( E [ x ] ) =x−x 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

5

a+b 2 1 1 ⎡ x ⎤ b ∞a+b E [ x ] =x=x⋅p ( x ) dx=xdx= =a+ ∫ −∞ ∫ a ⎢ ⎥ b b2 2 ⎣ ⎦ a a+b 3 2 a+b 1 ⎡ x ⎤ b 2 2 2 [ ] ( ) Ex=x⋅p x dx= =a+ab+ ∫ a ⎢ ⎥ b3 3 ⎣ ⎦ a 2 2 2 2 2 2 b 2 b b 2 σ=x−x=a+ab+ −a− −ab= 3 4 12 4. Error function of Gaussian Distribution: 2 ( x−x ) − 1 2 2σ p(x)=e σ2π

2 ( ) kσ x − 1 2 2 2σ erf(k)≡P(−kσ<x≤kσ)=e dx = ∫ σ2π π −kσ er( )≡P(x−σ<x≤x+σ) 2 2 + ( − ) ( ) x kσk x x σ x 1 x=x−x − 1 − 1 1 2 2 2σ2σ =e dx=e dx ∫ ∫ 1 σ2π σ2π x−kσ−kσ

6

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 2 −x 2 erf(k)=e dx ∫ Matlab π 0 You can also check outerfc(k), the complement of error function, or1-erf(k);

7

5.Error Rate in PCM 1 =Avolt signal 0 = 0volt t Zero mean noisew Gaussianp.d.f. 2 t Varianceσ p(w) )( ) ( ) ( ) ( ) P e=P 0 ⋅P e 0 +P 1 ⋅P e 1 1 P ( 1 ) =P ( 0 ) = p(e/0) p(e/1) 2 ⎛ A ⎞ P ( e0 ) =P ⎜ w> ⎟ ⎝ 2 ⎠ A/2 A/2 ⎛ A ⎞ P(e) P ( e1 ) = P ⎜ w ≤ − ⎟ ⎝ 2 ⎠ 1 P ( e ) = [ P ( e0 ) + P ( e1 ) ] 2 2 10 1 ⎡ A ⎤ = 1 − erf ⎢ ⎥ 4 2 ⎣ 2σ ⎦ 10 6 10 8 10 10 10 A [dB] 10 14 18 22 σ 20 log 10 7.4 = 17.3dB 21dB = 20 log 10 11.2 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

8

How to determine the pdf of a random variable?

∆t 1

∆t 2

∆t 3

T

p(x)

∆t 4

∆x=x−x 2 1

t

x

∆x (1) choose a value for∆x (resolution improves as∆x↓) (2) choose a valueTimproves as (accuracy T↑) x 2 p(x)dx=P(x≤x≤x) ∫ 1 2 x 1 P(x)(x−x)≅P(x≤x≤x),x∈[x,x] 2 1 1 2 1 2 p(x) x≅x will be approximately constant over that interval, for 1 2 (3) P(x≤x≤x) 1 2 p(x)≅,x∈[x,x] 1 2 x−x 2 1 j ∆t ∑ i 1 i=1 p(x)≅ • ∆x T  9