Lecture 8

White Noise and Power SpectralDensity

8.1 White Noise

White 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:

Definition (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 define whitenoise 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. notcorrelated in time

E (d[n]d[n k]) = 0, for k 6= 0 and for all n.

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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

baa x b

0 otherwise

x

p(x)

a b

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

Zero mean assumption: a = b. Unit variance assumption:

1

b a ba

x2 dx = 1

1

2b

bb

x2 dx =x3

6b

bb

=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) =12pi

ex

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 figure 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 finite extent. Discrete Fourier Transform excluded. has infinite 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

We first define the following:

Definition (Auto-Correlation Function) The auto-correlation function is defined as

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

When x[n] is white noise,Rxx[k] = [k],

the unit impulse. We now define the power spectral density function of a signal x[n]:

Definition (Power spectral density function) The power spectral density function isthe Fourier Transform of the auto-correlation function:

Sxx()FT Rxx[k]

Sxx() =

k=

Rxx[k]ejk,

with on any 2pi interval, where pi < pi is typical.When x[n] is white noise,

Sxx() =

k=

[k]ejk = 1.

In Fig. 8.2, we show numerical approximations using finite-length (N = 8192) signals ofthe 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 find 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

Rxx[k]

2 1.5 1 0.5 0 0.5 1 1.5 2102

0

1

2

3

[rad]

Sxx[]

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

Engineering Examples and Background

The 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 000V with 1 000 000V coming soon.

PS PD

DestinationSource

vs vdi

R

For a fixed PD = vDi follows vs = vD + iR and

PS = vSi = vDi+ i2R = PD + i

2R,

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

2. Hydraulics

PS PD

fs fd

b

x

Similar PS = PD+x2b. We want really high forces (pressure) and small displacements.

When delivering power, motion is bad.

Power in Frequency Domain

The integral of Sxx() is the expected power of the signal x[n]:

1

2pi

pipi

Sxx() d =1

2pi

pipi

(

k=

Rxx[k]ejk

)d

=

k=

Rxx[k]

(1

2pi

pipi

ejk d

)

[k]

=

k=

Rxx[k][k] = Rxx[0] = E(x2[n]

)

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

1

2pi

(1

2

Sxx() d +

21

Sxx() d

)0 1 < 2 pi.

2112 pipi

Some properties of the power spectral density:

Symmetry Sxx() = Sxx(). We only need to consider the range 0 pi. Conse-quently, the power in a frequency band is

1

pi

21

Sxx() d

Real Sxx() = S

xx(), where denotes the complex conjugate. That is, if Sxx() = a+bj,

then b = 0. Power is a real quantity.

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

Cross Power Spectral Density Function

The following concept is useful for filtering and system identification, topics that we willcover in the next lectures.

Definition Given wide sense stationary x[n] and y[n], the cross correlation function isdefined as

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

Sxy()FT Rxy[k].

One can show that

Rxy[k] = Ryx[k] Sxy() = Syx() Sxy() is generally complex valued

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Relation to LTI Systems

Let y[n] = T {x[n]}, T being a linear time invariant system. We want to calculate Syy()given Sxx(). Let h[n] be the impulse response of T. Then

y[n] =

l=

h[l]x[n l],

and

Ryy[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

Ryy[k] =

l=

p=

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

Comparing to the definition of the auto-correlation function

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

Ryy[k] =

l=

p=

h[l]h[p]Rxx[k + p l].

But,

Rxy[k] = E (x[n]y[n k]) = E(

p=

x[n]h[p]x[n k p])

=

p=

h[p]Rxx[k + p],

therefore we obtain

Ryy[k] =

l=

h[l]Rxy[k l].

In terms of convolutions

Ryy[k] = h[k] Rxy[k]

Rxy[k] =

p=

h[p]Rxx[k + p]

=

p=

h[p]Rxx[k p]

= h[k] Rxx[k].

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Applying Fourier transforms, we obtain

Syy() = H()Sxy()

Sxy() = H()Sxx(),

from which it follows thatSyy() = |H()|2 Sxx(),

since H() = H() and all signals are real. This is a powerful result for the followingtopics: filtering and system identification.

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