Autocorrelation Functionand

Power Spectrum

(based on chapter 9)

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1lim ( )

T

y y t dt⟨ ⟩ = ∫

Consider signal y(t) with the following properties:

1. time average of the signal fluctuations: lim ( )2T

T

y y t dtT→∞

−

⟨ ⟩ = ∫1. time average of the signal fluctuations:

the average fluctuation about the mean is zero: 0y⟨ ⟩ =

y(t)

0

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2 21lim ( )

T

y y t dt⟨ ⟩ = ∫

2. the mean of the square of the signal fluctuations is:

lim ( )2T

T

y y t dtT→∞

−

⟨ ⟩ ∫

since the square of the fluctuation is always positive, the mean square i t 2is not zero: 2 0y⟨ ⟩ ≠

y(t)

0

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Autocorrelation function,

defines how rapidly the signal fluctuates

( )R τ

1( ) ( ) ( ) lim ( ) ( ) 0

T

R y t y t y t y t dtτ τ τ= ⟨ + ⟩ = + ≠∫( ) ( ) ( ) lim ( ) ( ) 02T

T

R y t y t y t y t dtT

τ τ τ→∞

−

⟨ + ⟩ + ≠∫

The idea is to compare the fluctuation at any one point t

with its value at a later point t+τ. a u a a a po τ

Here, τ is a time interval between two points.

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A rapidly fluctuating signal has an autocorrelation function that decaysquickly with respect to τ:

R(τ)y(t) 2 ( )y t

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A slowly fluctuating signal has an autocorrelation function that decaysslowly with respect to τ:

y(t)

R(τ)2 ( )y t( )y

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By definition, the autocorrelation function at τ=0 is equal to the mean square field:

2( 0) ( )R y tτ = = ⟨ ⟩

In general, the autocorrelation R(τ) tends to be large

for small values of τ, and tends to zero for large values of τ.

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If τ is small compared to the timescale of fluctuations,

then the value of the signal at the two time points tend to be similar.

y(t)

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If τ is long compared to the timescale of fluctuations, then

there is no longer any consistent relationship between the signal values.

y(t)

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often, one assumes a simple exponential form for R(τ):

τ2( ) corrR y e

τττ

−

= ⟨ ⟩

parameter τcorr is called correlation time of the signal fluctuations;

rapid fluctuations have a small value of τcorr, while slow fluctuations

have a large value of τcorr.

τcorr depends on system parameters, such as temperature.

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Power spectrum - is the Fourier transform

of the autocorrelation function.

∞ 1 ∞

( ) ( ) iG R e dωτω τ τ∞

−

−∞

= ∫1

( ) ( )2

iR G e dωττ ω ωπ −∞

= ∫F. T.

Parameter τ referred to as ‘lag’. g

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autocorrelation function is symmetric ( ) ( )R Rτ τ− =which means that:

1 1( ) ( ) ( )i iR G d G dωτ ωτ

∞ ∞

∫ ∫0

1 1( ) ( ) ( )

2i iR G e d G e dωτ ωττ ω ω ω ω

π π−∞

= =∫ ∫

this condition leads to simplification:

1( ) ( ) ( )R G d

∞

∫0

( ) ( ) cos( )R G dτ ω ωτ ωπ

= ∫

consideration is then restricted to both 0 0d≥ ≥

Recall: in general the transformed function contains an antisymmetric

i i t ( ) ( )Y Y

consideration is then restricted to both . 0 0andω τ≥ ≥

12

imaginary component ( ) ( )imaginary imaginaryY Yω ω− = −

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Consider the power spectrum:

22 2

( ) 21

corr

corr

G yτωω τ

= ⟨ ⟩+

if the signal fluctuates rapidly, τcorr is short and the power spectrum is broad:

y(t) G(ω)

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if the signal fluctuates slowly τ is long and the power spectrum is narrow:if the signal fluctuates slowly, τcorr is long and the power spectrum is narrow:

y(t)G(ω)G(ω)

The area under the spectrum is independent of τcorr and is given

in the present model by (twice the mean square amplitude

of the signal y(t) )

The area under the spectrum is independent of τcorr and is given

in the present model by (twice the mean square amplitude

of the signal y(t) )

22 ( )y t⟨ ⟩

14

of the signal y(t) ).of the signal y(t) ).

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Electronic noise

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Electronic noise is an unwanted signal characteristic of all electronic circuits.

Example: all electronic amplifiers generate noise.This noise originates from the random thermalThis noise originates from the random thermal motion of carriers (thermal noise) and the discreteness of charge (shot noise).

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thermal noise and shot noise are inherent to all devices

depending on the circuit, noise level can vary greatly

noise signals are random and must be treated by

statistical means

even though we cannot predict the actual noise waveform,

we can predict the statistics such aswe can predict the statistics such as

the mean <y(t)> and

variance < 2(t)> (average of the square)

17

variance <y2(t)> (average of the square).

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Noise Power

The mean value of the noise fluctuations is zero:

Noise Power

1( ) ( ) 0

T

V t V t dtT

⟨ ⟩ = =∫0T ∫

The mean is also zero if we freeze time and take an

infinite number of samples from identical amplifiers

(this is called the ergodic hypothesis) .

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• the variance is non-zero (the signal power is non-zero):

2 21( ) ( ) 0

T

V t V t dtT

⟨ ⟩ = ≠∫

•the rms (root-mean-square) voltage is given by:

0T ∫

2 ( )rmsV V t= ⟨ ⟩

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Power Spectrum of NoisePower Spectrum of Noise

The power spectrum S(ω)=2G(ω) of the noise shows theof the noise shows the distribution of noise power as a function of frequency.

i hiMany noise sources are “white” in that the spectrum is flat (up to extremely high frequencies)( p y g q )

noise waveform is unpredictable h i l i b h i f ithere is no correlation between the noise waveform at time t

and t+τ, no matter how small is τ.

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Th l iThermal noise

Thermal noise (Johnson, Nyquist) generats by the fluctuations ( , yq ) g y

of the electric current due to the random thermal motion of the

charge carriers (the electrons)charge carriers (the electrons).

Thermal noise was first measured by John Johnson at Bell Labs in 1928; the results of these measurements were explained by Harry Nyquist (Bell Labs)

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Thermal noise of a resistorThermal noise of a resistor

all resistors generate noise.

noise power of a resistor R can be represented by aseries voltage source with mean square value:

2 4V kTR f⟨ ⟩ = ∆

equivalently, by a current source in shunt:

4V kTR f⟨ ⟩ = ∆

, where G=1/R2 44

kT fi kT fG

R

∆⟨ ⟩ = = ∆

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h k

Resistor Noise Example

Suppose that R = 10kΩT = 20C = 293K4kT = 1.62 × 10−20

2 16

2 8

1.62 10

1 27 10

V f

V V f

−

−

⟨ ⟩ = × × ∆

⟨ ⟩ ∆2 81.27 10rmsV V f= ⟨ ⟩ = × ∆

If the bandwidth is ∆f = 106MHz, then:If the bandwidth is ∆f 106MHz, then:

13rmsV microvolts≈

•this is the limit for the smallest voltage in this bandwidth

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Both equations 2 4V kTR f⟨ ⟩ = ∆

and

2 4kT f∆

come from general thermodynamic consideration

2 4kT fi

R

∆⟨ ⟩ =

come from general thermodynamic consideration, thus explaining the appearance of kT.

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The average motion due to collision : F Vα= −

Consider electron with velocity v(t) at a given time point (v(t) is a random signal);

dVm V

dtα= −The equation of motion is:

dt

with solution: 0 0( ) ,c

tt

mc

mV t V e V e

ατ τ

α

−−= = =

0( ) ( )c c

t

V t V e V t eτ τ

τ ττ+

− −

+ = =At t+τ:

m- is the electron massτc – collision time

τ

Then, the autocorrelation function is: 2( ) ( ) ( ) ( ) cR V t V t V t eτττ τ

−

= ⟨ + ⟩ = ⟨ ⟩

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In thermal equilibrium:2m V kT⟨ ⟩ =q ⟨ ⟩

k - Boltzmann’s constT - temperature

In a resistor of length l, the current produced is ev

il

=

then, autocorrelation for the current is:2

2( ) c

i

e kTR e

ml

τττ

−

ml

Collision time τc is related to the resistance of the material through which the electron moves;

A li d l i fi ld E i d d di d if l i VAn applied electric field E induces steady current, corresponding to a drift velocity Vd :

dVm eE V

dtα= − with the solution: c

d

eEeEV

m

τα

= =

26

dt mα6, 10, 13 -Feb-2009

Thus, current is: (using Vd and E=Vapplied/l): ( g applied )

2 2

2

1d c capplied applied

eV e E eI V V

l ml ml R

τ τ= = = = [Ohm’s Law]

l ml ml RVapplied is applied voltage

2 1ce τ= where R is the resistance

2ml R=

Therefore, autocorrelation for noise current is: ( ) ckT

R eτττ

−

=

, where R is the resistance

( )ic

R eR

ττ

Using the definition one can obtain the power spectrum of noise current:Using the definition one can obtain the power spectrum of noise current:

2

2 2

4 1( ) [ ]

1

d i kTS

df Rω

ω τ⟨ ⟩

= =+

27

1 cdf R ω τ+

6, 10, 13 -Feb-2009

2

Examine the equation: 2

2 2

4 1( ) [ ]

1 c

d i kTS

df Rω

ω τ⟨ ⟩

= =+

For extremely short τc , 2 2

11

1 cω τ=

+

4kT fi

∆=And the noise current is:

df f≡ ∆R

where is the bandwidth in hertz over which the noise is measuredf∆where is the bandwidth in hertz over which the noise is measuredf∆

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thermal noise is an intrinsic property to all resistors;

noise level depends directly upon temperature

if extremely low noise level requires the standard practice is to cool the input stage of the amps

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Shot noise

due to the quantized nature of charge due to the quantized nature of charge

signal is generated from a number of charges, so the charge can

always be written as Q=Ne, where N is the number of charges

(integer) and e is the charge of electron(integer) and e is the charge of electron

when N is low quantization results in noise fluctuations

shot noise is described by a Poisson distribution

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Poisson distribution-is a discrete probability distribution

describes the probability of events occurring with a known rate

in a fixed period of time

If the expected number of events is λ, then the probability h h l k (k i i i k 0 1 2 )that there are exactly k events (k is a non-negative integer, k = 0, 1, 2, ...)

is equal to:

p(k,λ)

( , )!

kep k

k

λλλ−

=

p(k,λ)

!k

32k

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shot noise mean square current fluctuation is:2 2i I f⟨ ⟩ ∆

where is the frequency bandwidth in Hz

2 2i Ie f⟨ ⟩ = ∆f∆

Assume the acquisition time is τm. Then, the mean current is:

Ne

m

NeI

τ=

The current fluctuation about the mean is:The current fluctuation about the mean is:

2 22

2

N e Iei

τ τ⟨ ⟩ = =

here, N is the mean and the variance.

m mτ τ

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( )Ie τ

Autocorrelation function is: ( ) ,2m

m

IeR

ττ ττ

= ≤

Recall, autocorrelation function is symmetrical about its origin

and spans the measurement period.

This gives the shot noise result:

2 2i I f⟨ ⟩ ∆2 2i Ie f⟨ ⟩ = ∆Short noise is independent of temperature !

and

requires the existence of the current flow

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