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Fundamentals of Noise


V.Vasudevan,Department of Electrical Engineering,Indian Institute of Technology Madras
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Noise in resistors

Random voltage fluctuations across a resistor

Mean square value in a frequency range f proportional to Rand T

Independent of the material, size and shape of the conductor

Contains equal (mean square) amplitudes at all frequencies(upto a very high frequency) Power contained in afrequency range f is the same at all frequencies or thepower spectral density is a constant.
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Questions

Why do these fluctuations occur?

Electrons have thermal energy a finite velocity At random points in time, they experience collisions with

lattice ions velocity changes

Velocity fluctuations lead to current fluctuations

What does the power spectral density of a random signal mean?
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Background

Background on random variables

A sample space of outcomes that occur with certain

probability A function that maps elements of the sample space onto the

real line - random variable

Discrete random variables - example is the outcome of a cointossing experiment

Continuous random variables - Voltage across a resistor
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Background

Continuous Random variables

Continuous random variables are characterized by probabilitydensity functions (pdfs). Examples are

-6 -4 -2 0 2 4 6x

0

0.1

0.2

0.3

0.4

N(0,1

)

Gaussian

-1 -0.5 0 0.5 1x

0

0.2

0.4

0.6

0.8

1

1.2

1.4

f(x)

Uniform

0 2 4 6 8 10x

0

0.2

0.4

0.6

0.8

1

f(x)

Exponential

f(x)dx= 1

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Background

Moments of random variablesMean

x = E{X} =

xf(x)dx

Variance

2

= E{X2

}

2

x

Correlation

RXY = E{XY} =

xyfXY(x, y)dxdy

CovarianceKXY = E{(X X)(Y Y)}

IfKXY = 0, X and Y are uncorrelated. The correlation

coefficient is c= KXYXY .
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Random Processes

Random Processes

Random variable as a function of time

The value at any point in time determined by the pdf at that

time The pdf and hence the moments could be a function of time

If {X1, . . . . .Xn} and {X1 + h, . . . .Xn + h} have the same jointdistributions for all t1, . . . tn and h > 0, it is n

th orderstationary

We will work with wide sense stationary (WSS) processes -The mean is constant and the autocorrelation,Rx(t, t+ ) = Rx()
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Random Processes

Spectrum

Finite energy signals - eg. y(t) = eat. The Fourier transformexists and energy spectral density is |Y(f)|2

Finite power signals - eg. y(t) = sin(2fot) + sin(6fot). In thiscase, can find the spectrum of the average power - eg. for y(t) itis 1

2at frequencies fo and 3fo.

Random signals are finite power signals, but are not periodic andthe Fourier transform does not exist. The question is how do wefind the frequency distribution of average power?
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Random Processes

Limit the extent of the signal

xT(t) = x(t), T/2 t T/2

Find the Fourier transform XT(f) of this signal. The energyspectral density is defined as

ESDT = E{|XT(f)|2}

The power spectral density (PSD) is defined as

Sx(f) = limT

E{|XT(f)|2}

T

The total power in the signal is

Px(f) =

Sx(f)df

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Random Processes

Weiner-Khinchin Theorem

The inverse Fourier transform of |XT(f)|2 is xT(t) xT(t) i.e.

F1 E{XT(f)|2}

T

=

1

TT/2T/2 E{xT(t)xT(t+ )}dt

But for WSS signals,

RX() = E{xT(t)xT(t+ )}

Therefore,F[Rx()] = Sx(f)

F d l f N i
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Random Processes

Noise in resistors

Simple model

Collisions occur randomly - Poisson process

Velocities before and after a collision are uncorrelated Average energy per degree of freedom is kT

2

With these assumptions it is possible to show that SI(f) =4kTR

the autocorrelation is a delta function (white noise). Also, usingcentral limit theorem, it has a Gaussian distribution. However,continuous time white noise has infinite power and is therefore anidealization.

F d t l f N i
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Random Processes

Representation of a noisy resistor

Sv (f) = 4kTR

SI(f) =4kT

R

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LTI systems

Noise in Linear time-invariant systems

If h(t) is the impulse response of an LTI system,

y(t) = h(t) x(t)

Assume x(t) is a WSS process with autocorrelation Rx()

E{y(t)y(t+ )} =

h()h()E{x(t )x(t+ )}dd

=

h()h()Rx(+ )dd

y(t) is also a WSS process and

Ry() = h() h() Rx()

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LTI systems

PSD and Variance

IfH(f) = F{h(t)}, the power spectral density at the output is

Sy(f) = F{Ry()} = |H(f)|2Sx(f)

Noise power at the output is

Ry(0) =

Sy(f)df

In linear systems, if the input has a Gaussian distribution, theoutput is also Gaussian

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LTI systems

Example - RC Circuit

R

C

Vo

S=4kTR

The output PSD is

Sy(f) =4kTR

1 + 42f2R2C2

The total power at the output is

Py

=

0

4kTR

1 + 42f2R2C2df

=kT

C

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LTI Two port

Linear two ports - RLC circuits

Resistors are the noise sources

Assuming N noise sources, the in terms of the y parameters,the output current can be written as

Iin = yiiVin + yioVo +N

j=1

yijVj

Iout = yoiVin + yooVo +

Nj=1

yojVj

Vj is the random noise source due to the jth resistor

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LTI Two port

Noisy RLC two ports - Representation

If Ini = N

j=1 yijVj and Ino = N

j=1 yojVj, the two-port can berepresented as a noiseless network with two additional noisecurrent at the two ports. Can also have a Z network, with noisevoltage sources at the two ports

Ini InoY

vni

vno

Z

+ +

To get Ini and Ino, find the short circuit current at the two ports.For vno and vni, find the open circuit voltage

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LTI Two port

Examples

R1

R2

R3

v2ni = 4kT(R1 + R2)f

v2

no = 4kT(R3 + R2)f

The two noise sources arecorrelated

R1 R2

I2ni =4kT

R1f

I2no =4kT

R2f

The two sources are uncorrelated

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LTI Two port

Generalized Nyquist theorem

Supposing we have an RLC network and we wish to find the noiseat the output port. First remove all noise sources.

Connect a unit current source Io at the output

Voltage across resistor Rj due to Io is Vj = Zjo Power absorbed in Rj is Pj =

|Zjo|2

Rj Total power dissipated

in the network is

P=N

j=1

|Zjo|2

Rj

Power delivered to the network by the source is

P= Re(VoIo) = Re(Zo)

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LTI Two port

Power delivered = Power dissipated

Re(Zo) =

Nj=1

|Zjo|2

Rj

Now include noise current sources due to resistors. Output

voltage due to noise generated by resistors is

Vo2

= 4kTfN

j=1

|Zoj|2

Rj

Since the network is reciprocal Zoj = Zjo

Vo2

= 4kTRe(Zo)f

Called the Generalized Nyquist theorem

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LTI Two port

Vn In Representation

vn

in

+

Replace all noise sources in the network by an equivalent noisevoltage and current source in the input port, so that thecorrect output noise spectral density is obtained

Once again, the two sources could be correlated

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LTI Two port

Comparing with the In In representation,

Vn =Inoyoi

In = Ini yii

yoiIno

The correlation coefficient is

civ(f) =E{InV

n

}

[E{I2n }E{V2n }]

1

2=

Siv(f)

[Sni(f)Snv(f)]1

2

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LTI Two port

Noise figure of a two-port network

Ins In

Vn

+

Ys

F =Total noise power at the output per unit bandwidth

Output noise power per unit bandwidth due to the input

F =

E{|Ins + In + YsVn|2}

E{|Ins|2}

= 1 +Sni(f)

Ss(f)+ |Ys|

2Snv(f)|

2

Ss(f)+ 2Re(civY

s )

[Sni(f)Snv(f)]1

2

Ss(f)

Depending on the correlation coefficient, one can use the right

type of source impedence to minimize noise figure

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LTI Two port

References

1. H.A.Haus et al, Representation of noise in linear twoports, Proc.IRE, vol.48, pp 69-74, 1960

2. H.T.Friss, Noise figure of radio receivers, Proc. IRE, vol.32, 1944,pp 4-9-423

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