white noise and power spectral

7/30/2019 White Noise and Power Spectral

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Lecture 8

White Noise and Power SpectralDensity

8.1 White NoiseWhite noise is a basic concept underlying the modeling of random disturbances, such as

sensor noise environmental disturbances

P

GS

GA

dA

dS

In contrast to continuous time, white noise is straightforward to characterize in discretetime:

Denition (White noise) A noise signal d[n ] is white if it has zero mean

E (d[n ]) = 0 for all n,with E (d[n ]) being the expected value of the random variable d[n ]. We further dene white noise to have unit variance

E (d[n ]d[n ]) = 1 , for all n,

which may be scaled as appropriate, see the following example with a uniform distribution.Lastly, a noise signal d[n ] is white if it is independent from sample to sample, i.e. not correlated in time

E (d[n ]d[n k]) = 0 , for k = 0 and for all n.1


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By the law of large numbers, the above expected value can be expressed as follows:

E (d[n ]d[n k]) = limT 12T + 1

T

n = T d[n ]d[n k]

So far, we have said nothing about the underlying probability distribution. The above isgeneral and applies to many probability distributions.

Example (Uniform Distribution)

p(x) =1

ba a x b0 otherwise

x

p(x)

a b

Matlab: The command rand draws uncorrelated pseudorandom numbers from a uniform distribution with a = 0 , b = 1 .

Zero mean assumption: a =

b.

Unit variance assumption:1

b a b

ax2 dx = 1

12b bb x2 dx = x

3

6b

b

b=

b2

3= 1

= b = 3

Therefore, the Matlab command d = 2*sqrt(3)*rand-sqrt(3) draws the pseudo-random number d from a uniform distribution with expected value zero and unit vari-ance.

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Example (Normal Distribution)

p(x) =1

2 e x 2

2

4 3 2 1 1 2 3 4

0.1

0.2

0.3

0.4

0.24

x

p(x)

Matlab: The command randn draws uncorrelated pseudorandom numbers from a nor-mal distribution with mean 0 and variance 1.Both probability distributions may be used to generate white noise. We often only careabout mean and variance, so the underlying distribution usually does not matter so much.See the following gure for a time domain example of white noise.

0 5 10 15 20 25 30 35 40 45 50 55 60 3

-1

0

1

3

n

d [ n ]

Figure 8.1: Discrete time representation of white noise with uniform distribution.

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The analysis of a white noise signal in the frequency domain poses several problems. Whitenoise

is not periodic. Fourier Series excluded. has no nite extent. Discrete Fourier Transform excluded. has innite energy. Fourier Transform excluded.

Therefore a rigorous way to handle these types of signals in the frequency domain is needed.A solution is the power spectral density.

8.2 Power Spectral Density Function

For the remainder of this chapter, we assume that all time signals are real . We rstdene the following:

Denition (Auto-Correlation Function) The auto-correlation function is dened as

R xx [k] = E ( x[n ]x[n k]) ,where x[n ] is assumed to be wide sense stationary: R xx [k] does not depend on n (Similar to time invariance). This is a general denition, and not just for white noise.

When x[n ] is white noise,R xx [k] = [k],

the unit impulse. We now dene the power spectral density function of a signal x[n ]:Denition (Power spectral density function) The power spectral density function is the Fourier Transform of the auto-correlation function:

S xx ()F T R xx [k]

S xx () =

k= R xx [k]e j k ,

with on any 2 interval, where < is typical.When x[n ] is white noise,S xx () =

k=

[k]e j k = 1 .

In Fig. 8.2, we show numerical approximations using nite-length ( N = 8192) signals of the auto-correlation function and power spectral density function of a white noise signal.The plotted functions were generated using the Matlab script white_noise.m , which youmay nd on the course website.

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0 5 10 15 20 25 30 35 40 45 50 55 60

0

0.5

1

k

R x x

[ k ]

2 1.5 1 0.5 0 0.5 1 1.5 2102

0

12

3

[rad]

S x x

[

]

RawFiltered

Figure 8.2: Approximated auto-correlation function and power spectral density functionof white noise generated by a normal distribution. For better readability, only 65 of the

total 8192 samples of the respective approximations are plotted. The smoothed powerspectral density function was obtained using a non-causal moving average lter, which willbe covered in future lectures about ltering.

Engineering Examples and BackgroundThe power spectral density function has its name from the fact that in many engineeringsystems, power is proportional to the square of a signal.

Example Resistor with current i(t)

v(t)+ R

i(t)

v(t) = Ri (t) Power = v(t)i(t ) =v2(t )

R= i2(t)R

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Example Damper with force

b

f (t)x(t)

f (t) = bx(t) Power = x(t)f (t) =f 2(t)

b= x2(t)b

Incidentally, these simple examples illustrate two powerful design decisions:

1. High Voltage Lines: 100 000 V with 1 000 000 V coming soon.

P S P D

DestinationSource

vs vdi

R

For a xed P D = vD i follows vs = vD + iR and

P S = vS i = vD i + i2R = P D + i2R,

so its best to make vD really large and i really small.

2. Hydraulics

P S P D

f s f d

b

x

Similar P S = P D + x2b. We want really high forces (pressure) and small displace-ments. When delivering power, motion is bad.

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Power in Frequency DomainThe integral of S xx () is the expected power of the signal x[n ]:

12 S xx () d = 12

k= R xx [k]e j k d

=

k= R xx [k]

12 e j k d

[k]=

k=

R xx [k] [k] = R xx [0] = E x2[n ]

We may also compute the power contained in a frequency band:

12 12 S xx () d +

2

1S xx ()d 0 1 < 2 .

2 1 21 Some properties of the power spectral density:

Symmetry S xx () = S xx (). We only need to consider the range 0 . Conse-quently, the power in a frequency band is1 21 S xx () d

Real S xx () = S xx (), where denotes the complex conjugate. That is, if S xx () =a + bj , then b = 0. Power is a real quantity.

Non-negative S xx () 0 if x[n ] is real, as the power of a signal must be positive overany frequency band.

Cross Power Spectral Density FunctionThe following concept is useful for ltering and system identication, topics that we willcover in the next lectures.

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Denition Given wide sense stationary x[n ] and y[n ], the cross correlation function is dened as

R xy [k] = E ( x[n ]y[n k]) .The corresponding Fourier transform results in the cross power spectral density function S xy ()

F T R xy [k].

One can show that

R xy [k] = R yx [k] S xy () = S yx ()

S xy () is generally complex valuedRelation to LTI SystemsLet y[n ] = T {x[n ]}, T being a linear time invariant system. We want to calculate S yy ()given S xx (). Let h[n ] be the impulse response of T. Then

y[n ] =

l= h[l]x[n l],

and

R yy [k] = E ( y[n ]y[n k]) = E

l= h[l]x[n l]

p=

h[ p]x[n k p]

=

l=

p=

h[l]h[ p]E (x[n l]x[n k p])

Introducing a new index m = n l, we obtain

R yy [k] = l=

p=

h [l]h[ p]E (x[m ]x[m + l k p]) .

Comparing to the denition of the auto-correlation function

R xx [k] = E ( x[n ]x[n k]) ,we obtain

R yy [k] =

l=

p=

h[l]h[ p]R xx [k + p l].

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But,

R xy [k] = E (x[n ]y[n k]) = E p=

x[n ]h[ p]x[n k p] = p=

h[ p]R xx [k + p],

therefore we obtain

R yy [k] =

l= h [l]R xy [k l].

In terms of convolutions

R yy [k] = h [k] R xy [k]

R xy [k] = p=

h[ p]R xx [k + p]

=

p= h[ p]R xx [k p]

= h [k] R xx [k].Applying Fourier transforms, we obtain

S yy () = H ()S xy ()

S xy () = H ()S xx (),from which it follows that

S yy () = |H ()|2 S xx (),since H () = H () and all signals are real. This is a powerful result for the followingtopics: ltering and system identication.

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white noise and power spectral

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