random process examples - binghamton universityws2.binghamton.edu/fowler/fowler personal...
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![Page 1: Random Process Examples - Binghamton Universityws2.binghamton.edu/fowler/fowler personal page/EE521_files/V-3 RP Examples_2007.pdf3/23 Ex. #1: D-T White Noise Also, let x[k] be Gaussian](https://reader034.vdocuments.net/reader034/viewer/2022042111/5e8c0b695e76293fb049ee9a/html5/thumbnails/1.jpg)
1/23
Random ProcessRandom ProcessExamplesExamples
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2/23
Ex. #1: D-T White NoiseLet x[k] be a sequence of RV’s where…
each RV x[k] in the sequence is uncorrelated with all the others:
E{ x[k] x[m] } = 0 for k≠m
Physically, uncorrelated means that knowing x[k] provides no insight into what value x[m] (for m≠k) will be likely to take (roll a die; the value you get provides no insight into what you expect to get on any future roll)
This DEFINES a DT White NoiseAlso called “Uncorrelated Process”
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3/23
Ex. #1: D-T White Noise
Also, let x[k] be Gaussian with Zero mean and Variance σ²
Define RP’s PDF; also called “Normal”
Note: Does not depend on kat least the 1st order PDF is time invariant
X[k] ~ N (0,σ²)
Mean Variance
⎟⎟⎠
⎞⎜⎜⎝
⎛ −
=2
2
2
21);( σ
πσ
x
ekxpPDF :
Recall: Gaussian process are very common, so…
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4/23
Ex. #1: D-T White NoiseTASK : We have a model…. Find the mean, ACF, and check if WSS (also find variance of process)
MEAN of Process :
ACF:
E { x[k] } = 0 CONSTANTBy definition !
RX(k1,k2)=E { x[k1] . x[k2] }
By our definition of white noise, ….this is 0 if k1≠ k2
![Page 5: Random Process Examples - Binghamton Universityws2.binghamton.edu/fowler/fowler personal page/EE521_files/V-3 RP Examples_2007.pdf3/23 Ex. #1: D-T White Noise Also, let x[k] be Gaussian](https://reader034.vdocuments.net/reader034/viewer/2022042111/5e8c0b695e76293fb049ee9a/html5/thumbnails/5.jpg)
5/23
Ex. #1: D-T White Noise
{ } 2221 ][),(
21σ==== kxEkkR kkkx
By definition of variance for zero-mean case
Now for k1=k2=k:
][
,0,),(
212
21
212
21
kk
kkkkkkRx
−=
⎪⎩
⎪⎨⎧
≠==
δσ
σThus,
][)( 2 mmRx δσ=⇒
= m(like τ for cont-time ACF)
ACF for DT White RP
![Page 6: Random Process Examples - Binghamton Universityws2.binghamton.edu/fowler/fowler personal page/EE521_files/V-3 RP Examples_2007.pdf3/23 Ex. #1: D-T White Noise Also, let x[k] be Gaussian](https://reader034.vdocuments.net/reader034/viewer/2022042111/5e8c0b695e76293fb049ee9a/html5/thumbnails/6.jpg)
6/23
Ex. #1: D-T White NoiseACF displays lack of correlation between any pair of any time instants:
Now since we have constant mean and ACF depends only on m = k2-k1 ⇒ WSS
RX[m] = σ²δ[m]
m
σ²
-3 -2 -1 1 2 3
……
![Page 7: Random Process Examples - Binghamton Universityws2.binghamton.edu/fowler/fowler personal page/EE521_files/V-3 RP Examples_2007.pdf3/23 Ex. #1: D-T White Noise Also, let x[k] be Gaussian](https://reader034.vdocuments.net/reader034/viewer/2022042111/5e8c0b695e76293fb049ee9a/html5/thumbnails/7.jpg)
7/23
Ex. #1: D-T White NoiseVariance
2
22
]0[]0[
σ
σ
=
=
−=
x
xx
RxR
=0
For this case:Variance of the Process = Variance of the RV
![Page 8: Random Process Examples - Binghamton Universityws2.binghamton.edu/fowler/fowler personal page/EE521_files/V-3 RP Examples_2007.pdf3/23 Ex. #1: D-T White Noise Also, let x[k] be Gaussian](https://reader034.vdocuments.net/reader034/viewer/2022042111/5e8c0b695e76293fb049ee9a/html5/thumbnails/8.jpg)
8/23
Ex. #2: Filtered D-T RPStart with White RP x[k] in previous example
Recall : Zero Mean ProcessRX [m] = σ² δ[m]⇒ WSS
x[k] D-T Filterh[n] = [1 1]
y[k] = x[k] + x [k-1]
Two–Tap FIR filterTaps = [1 1]
![Page 9: Random Process Examples - Binghamton Universityws2.binghamton.edu/fowler/fowler personal page/EE521_files/V-3 RP Examples_2007.pdf3/23 Ex. #1: D-T White Noise Also, let x[k] be Gaussian](https://reader034.vdocuments.net/reader034/viewer/2022042111/5e8c0b695e76293fb049ee9a/html5/thumbnails/9.jpg)
9/23
Ex. #2: Filtered D-T RPTASK: Is y[k] WSS?
⇒ need to find mean & ACF
MEAN: Using filter output expressions gives
{ } { }{ } { }]1[][
]1[][][−+=
−+=kxEkxE
kxkxEkyE
⇒ E {y[k]} = 0
![Page 10: Random Process Examples - Binghamton Universityws2.binghamton.edu/fowler/fowler personal page/EE521_files/V-3 RP Examples_2007.pdf3/23 Ex. #1: D-T White Noise Also, let x[k] be Gaussian](https://reader034.vdocuments.net/reader034/viewer/2022042111/5e8c0b695e76293fb049ee9a/html5/thumbnails/10.jpg)
10/23
Ex. #2: Filtered D-T RPACF :
{ }{ }
{ } { }
{ } { }))1()1((
21)1(
21
)1(21
)(21
2211
2121
1212
1212
]1[]1[][]1[
]1[][][][
])1[][])(1[][(
][][),(
−−−+−
−−−
−−+−+
−+=
−+−+=
=
kkRkkR
kkRkkR
y
xx
xx
kxkxEkxkxE
kxkxEkxkxE
kxkxkxkxE
kykyEkkR
Plug in Eq. for output
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11/23
Ex. #2: Filtered D-T RP
⇒ RY(m) = σ²[2δ[m] + δ[m-1] + δ[m+1]
( )][
12
]1[
12
]1[
12
][
1221
22
22
)1()1()1(
)1()(),(
m
x
m
x
m
x
m
xy
kkRkkR
kkRkkRkkR
δσδσ
δσδσ
−−−++−+
−−+−=
+
−
12 where kkm −=
ACF for 2-Tap Filtered White RP
y[k] is WSS
![Page 12: Random Process Examples - Binghamton Universityws2.binghamton.edu/fowler/fowler personal page/EE521_files/V-3 RP Examples_2007.pdf3/23 Ex. #1: D-T White Noise Also, let x[k] be Gaussian](https://reader034.vdocuments.net/reader034/viewer/2022042111/5e8c0b695e76293fb049ee9a/html5/thumbnails/12.jpg)
12/23
Ex. #2: Filtered D-T RP
Note : Filter introduces correlation between adjacent samples - but still no correlation for samples 2 or more samples apart (for this filter)
Ry[m]
m
2σ²
-3 -2 -1 1 2 3
…… σ²
![Page 13: Random Process Examples - Binghamton Universityws2.binghamton.edu/fowler/fowler personal page/EE521_files/V-3 RP Examples_2007.pdf3/23 Ex. #1: D-T White Noise Also, let x[k] be Gaussian](https://reader034.vdocuments.net/reader034/viewer/2022042111/5e8c0b695e76293fb049ee9a/html5/thumbnails/13.jpg)
13/23
Big Picture: Filtered RPFilters can be used to change the correlation structure of aRP:
x[k] D-T Filterh[n] = [1 1]
y[k] = x[k] + x [k-1]
0 50 100 150 200 250 300
0
White Noise
-20 -10 0 10 200
0.5
1
ACF Rx[m] of Input Input RP (One Sample Function)
km
0 50 100 150 200 250 300
0
-20 -10 0 10 200
0.5
1
ACF Ry[m] of Output Output RP (One Sample Function)
km
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14/23
Big Picture: Filtered RP (cont)
km-20 -10 0 10 20 0 50 100 150 200 250 300
0Filter: ones(1,15)
0
0.5
1
ACF Ry[m] of Output Output RP (One Sample Function)
0 50 100 150 200 250 300
0
Filter: [1 1 1 1]
-20 -10 0 10 200
0.5
1
ACF Ry[m] of Output Output RP (One Sample Function)
km
0 50 100 150 200 250 300
0
Filter: ones(1,10)
-20 -10 0 10 20
0
0.5
1
ACF Ry[m] of Output Output RP (One Sample Function)
km
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15/23
Filtered RPs: Insight
Narrow ACF Rapid FluctuationsBroad ACF Slow Fluctuations
Our study of the ACFs of filtered random processes and the degree of “smoothness” of the sample functions shows the following general result:
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16/23
Ex. #3: Exp. Sig. + White Noise
Define a RP as:
][][ kwxaky k +=
Deterministic Functiona = deter. #
Random Variable(picked once randomly and then fixed)
White Noise(get a new random value at
each k – uncorrelated from sample-to-sample)
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17/23
Ex. #3: Exp. Sig. + WN Further Definition of this RP:
• w[k] is white noise
• Each w[k] is a Gaussian RV with
• X is a Gaussian RV with
• RV X is independent of each RV w[k]⇒ E{ x w[k]} = E{x} E{w[k]} = 0
• The # a is a deterministic number
),0(~ 2XNX σ
),0(~][ 2wNkw σ
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18/23
Ex. #3: Exp. Sig. + WNTASK: Is this WSS?
MEAN: E{ y[k] } = ak E{X} + E{ w[k] } = 0 (Constant)
ACF: Ry(k,k+m) = E{ y[k] y[k+m] }= E {(akX + w[k]) (ak+m X + w[k+m])}
= a2k+m E{X2} + ak E{X w[k+m]} + ak+m E{X w[k]} + E {w[k] w[k+m]}
= 0 = 0
= 0Due toIndep.
][mwδσ= 2
2Xσ
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19/23
Ex. #3: Exp. Sig. + WN
][),( 222 mamkkR wmk
xy δσσ +=+ +
Not WSS !!!
In General: Depends on k not just m
But… If a = ±1 then a2K+M = a2K aM = aM
Then it is WSS
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20/23
Ex. #3: Exp. Sig. + WN
If a = -1
If a = +1RX[m]
m1 2 3-3 -2 -1
2xσ
22wx σσ +
RX[m]
m
12
3-3-2
-1
2xσ
22wx σσ +
2xσ−
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21/23
Ex. #4: Weird Func of WNLet z[k] be white noise with zero mean
& variance of
Define x[n] = Z [int(n/2)]
Then x[0] = z[0]x[1] = z[0]x[2] = z[1]x[3] = z[1]
2zσ
Always appear in pairs
int(D3D2D1.d1d2d3d4)= D3D2D1
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22/23
Ex. #4: Weird Func of WNTASK: Is this RP WSS ?
MEAN: E { x[n] } = E { z[k] } = 0
RX(n, n+m) = E{ x[n] x[n+m] }
If |m| ≥ 2 then x[n] & x[n+m] are two different z[.] values ⇒ uncorrelated
⇒ RX(n,n+m) = 0 for |m| ≥ 2
k = n/2 if n is even= (n-1)/2 if n is odd
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23/23
Ex. #4: Weird Func of WN
n dependence causes it not to be WSS
If m = 0 then RX(n, n ) = E { z2[k] } =
If m = 1 … and n is even: RX(n, n+1) =
… and n is odd: RX(n, n+1) = 0
2zσ
2zσ
m
n