Biomedical Sensors

NOISE IN ELECTRONIC DEVIES AND CIRCUITS

Considering a system in which the input signal

has been switched o , we will see that at the

output of the ampli er will capture a signal like

the one that has been reported on the image.

The plot considering the behavior of the out

voltage in the time domain is representative of

a signal that changes randomly. This is the

noise which be de ned as a disturbance that

is generated inside the circuit due to

fundamental physics mechanisms.

noise

The consists in spontaneous uctuations that occur in some active and passive

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circuit components and which appear as voltages or currents whose evolution over time

continuous stochastic

is regulated by statistical laws. So it is possible to say that it is a

process with a continuous parameter, like time, that originates from some fundamental

physical phenomena such as the thermal agitation of free charge carriers in conductors or

the granular structure of the electric charge. Its existence cannot be denied without

denying fundamental laws of physics as the second law of thermodynamics or the

discrete nature of electric charge.

noise should not be confused with the interferences

It is important to remember that

induces on the circuit by the surrounding environment, like disturbances introduced by

the power supplies or disturbances due to electromagnetic induction. This because,

ideally, interferences can cue removed with the use of ltering and shielding techniques.

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noise limits the precision

In addition, it is possible to say that with which the

instantaneous amplitude of a signal can be measured and, as an extreme situation, can

prevent the detection of a signal. Moreover, noise limits the maximum value of the usable

ampli cation, before saturation occurs. random phenomenon

At the end it is known that noise is a purely so the

instantaneous value of its waveform cannot be predicted at any time. Therefore noise

variables can be described quantitatively through statistical properties such as the mean

- this last one gives the opportunity to

squared value and the power spectral density

study noise’s behaviour in the time of frequency.

stochastic process

In a will be de ned some statistical parameter that permit the

description of noises in electronic circuit. So the properties that can

be de ne for a sigle systems can be related to a certain time

interval (time average) or of many identical system at a certain time

(ensemble average).

An explanation of the behavior of the noise can be the plot on the

left. It is possible to see the trends of output voltage of a set of

identical circuit. We can se di erent behavior od the noise during

the time and the cause is the randomly conduct of the noise.

to occur: veri carsi

1 to shield: proteggere, schermare;

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It can be possible talk about:

- Stationary process: like mean value, mean squared value, etc,

statistical properties, so they do not say with time

can be considered as quantities time-invariant,

- Ergodic process: statistical properties of the process can be determinate by a single

like time average=ensemble

function representing a possible realization of the process,

average. time averages.

On the table below it is possible to the the

QUANTITY DEFINITION OF THE FORMULA

QUANTITY

Mean value It’s just a value which gives not

any information about the

amount of noise property.

Mean squared It gives interesting information

value about the amplitude of the

noise uctuation around the

mean value.

Autocorrelation It gives information about the

function speed of the noise uctuation

around the mean value of

noise.

Considering the plot on the right, it represent the

▶ comparison of two autocorrelation function of

two di erent noise processes. The curves

represent both slowly and rapidly uctuating

random process. It is possible to se that the

maximum pain of the autocorrelation function of

both plots are when τ=0. The di erences are in

the speed of the decrees of the curves but both

will tend to zero when τ goes to in nity.

Remember: this is only an example, it is not a standard behavior.

The crucial parameter that permits to de ne the performance of an electronic system is

signal-to-noise ratio

called which is de ned as a ratio between a quantity that is

associated to the signal and to the noise. In other therms it can also be the ration

between the value of the maximum output voltage signal, in which

there will be both signal and noise contribution, and the mean value of

the noise. These mathematical relations can be seen in the right

rectangular. The nal result has to be a number ≥1; if the ratio is equal

to 1, it means that the signal have the same amplitude of the noise so

the measurement will have a very large uctuation.

It important to remember that the noise v (t) cannot be describe by its behavior as

n

a function of time, because it cannot be predicted. So the noise can be characterized by

the power spectral density in the frequency domain, considering the properties of Laplace

Transform. In this way, the noise can be seen as a sinusoidal signal with speci c

properties. 22 di 34

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Considering the plot on the left in which axis x= V (mV)

out,max

and axis y=numebr of measurments.

Guessing an ideal condition, the plot will be characterized by a

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line but in real case the output will be characterized by the

contribution of both signal and noise voltage; for this reason it

is necessary consider a number of measurement. The result will

be a Gaussian distribution with a variance equal to the mean

squared of the noise voltage.

At this point, the critical thing is to nd a way to evaluate the

means square value of the noise in frequency domain

considering the Fourier transform. Looking to the mathematical

consideration it is possible to consider both Fourier transform,

which is the rst writing, and the inverse Fourier transform, which

is the second. noise sources:

It can be possible considered some

- S (f) - [V /Hz]: spectral density which describe the uctuation of noise

2

V

voltage generated by a noise voltage generator.

- S (f) - [A /Hz]: spectral density which described the current noise generator.

2

I

Guessing: supponendo

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Considering the circuit on righe it is possible to recognize:

- v (t): noise voltage at the input of a linear network with a

IN

transfer function T(s) with a. Power spectral density S V,i

(f);

- v (t): noise voltage at the output of the linear network

UN

with a power spectral density S (t).

V,u

Repeating the steps that led to the de nition on the power spectral density for a noise

variable and applying them to v , it is possible to evaluate the power spectral density at

u,N

the output of the network, considering the following mathematical development.

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fi Parte 2

white noise

A is the de nition of the noise that is generated from a white light, which is

characterized by a constant power spectral density that it means that all the frequency

value are the same.

The rst plot on the left is a plot that derives from the relation between

N (f), where N (f)=A. Considering the kind of plot it is possible to say that

v V

all the frequency components give the same contribution: there is no

correlation between samples of the noise voltage measured at di erent

time. What about the autocorrelation function? It is possible to say that

this relation considers the Dirac’s delta writing. In fact, considering the

second plot, it is possible to say that R (τ9 =A*∂(t).

V

In practical system it will be di cult see a purely white noise because

every electronic system, the frequency never go to in nite; in fact the

circuit will operate till a maximum value and then its perfomence will

degree around that value.

On the slide on the right, it can be seen

a situation that is closer to the reality. In

this case the white noise source touch a

maximum value of frequency called f .,

0

which can be consider as a limit over

which the circuit will not operated.

Considering the bilateral frequency, it is

possible to evaluate the autocorrelation

function. 25 di 34

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NOISE SOURCES IN ELECTRONIC DEVICES AND CIRCUITS

In this section we will talk about three di erent kind of noise sources, that will be:

▶ Shot noise

- Thermal noise

- 1/f noise.

- thermal noise.

The rst noise source that we will analyzed is the

It is known that in electronic circuit it is possible to have resistors, which are usually made

by metal, a kind of material which is characterized by conductive properties that permits a

better owing of electron, whose movement determinate the current. When temperature

T > 0K

( ),

of metal is above the absolute zero the electrons acquire an amount of energy

thermal agitation,

due to the rise of temperature. As consequence, they also present

which means that they acquire random velocity component, starting moving randomly

and the electrons start to collide casually. The result of this situation is a uctuation of a

random current signal that, thanks to the relation of the Ohm’s law, will traslate also in a

randomly voltage signal. evaluate the power spectral

After this consideration it is possible to density, starting

from the basic physic equations, like the secondo principle of thermodynamics and the

Boltzmann’s theorem about the equipartition of energy. The result result that we will

obtain will be random thermal motion of electron.

Norton’s model,

Considering the we see that is

possible to model the noise current source in parallel with

resistor; according to physical laws and the characteristic of

the circuit it is possible to see, through the mathematical

symbols, the evaluation of S (f) where k is the Boltzmann’s

I

constant (k =1,38*10 J/K), T is the temperature and R is the

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B

value of the resistor. It is possible to see that this power

spectral density does not depend on the current which ows across the resistor. This

thermal noise is independent of the average current

because the that is above the

but it depends on the voltage applies outside the resistor.

resistor

Thermal noise is always present: also when isn’t apply any

voltage, we still have thermal noise. Thevenin’s model,

An alternative model is called in which

the resistance and the noise voltage are put in series. In this

case the mathematical symbols show the value of S (f). The

V

voltage (V) across the resistor, according to Ohm’s law, is equal

to the product between R and I: so it is possible de ned a

transfer function between t