CHAPTER 4: SIGNAL TO NOISE RATIO - Webpage Pace

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Chemistry

CHAPTER 5: SIGNALS AND NOISE


The signal to noise ratio: (S/N)

The fundamental concepts of noise and signal
-
to
-
noise ratio are central to the discipline of
analytical chemistry. Since chemical instrumentation dominates the modern laboratory, it i
s
critical that students understand the role noise plays in limiting the precision of a
measurement. It is equally important that they understand how to implement active and
passive means to mitigate noise. The purpose of this Mathcad document is to allow
the student
to gain a familiarity with the concepts of signal
-
to
-
noise ratios and to explore the advantages
of ensemble averaging and digital filtering analytical signals. Simulated noisy signals are used
to guide the student through a series of individual

exercises (Fig. 1). These exercises culminate
in a capstone experience where they simultaneously apply all of the concepts.


Figure 1. Ensemble average
d signal and original signal.



Every analytical measurement is made up of two components. One component, the signal,
carries information about the analyte that is of interest to the chemist. The second, called noise,
is made up of extraneous informati
on that is unwanted because it degrades the accuracy and
precision of an analysis and also places a lower limit on the amount of analyte that can be
detected.

Noise free data can never be realized in the laboratory because some types of noise arise from
thermodynamic and quantum effects that are impossible to avoid in measurement.

The signal to noise ratio is a representative marker it that is used in describing the quality of an
analytical method or the performance of an instrument.

-
For a dc signal, S
/N = mean / standard deviation = x/s where s is the standard deviation of the
measured signal strength and x is the mean of the measurement
-
x/s is the reciprocal of the
relative standard deviation (RSD)

S/N=I/RSD
(section alB
-
1, appendix 1)

The
average strength of the noise, N, is constant and independent of the magnitude of the signal,
S. The effect of noise on the relative error of a measurement becomes greater and greater as the
quantity being measured decreases in magnitude. The magnitude o
f signal to noise is
conveniently defined as the standard deviation, s, of numerous measurements of the signal
strength, and the signal is given by the mean, X, of the measurements. Thus, S/N is given by:

S/N = Mean/Standard Deviation = X/s

Signal Smoothi
ng Algorithms


Theory:

The signal
-
to
-
noise ratio (SNR) of a signal can be enhanced by either hardware or software
techniques. The wide use of personal computers in chemical instrumentation and their inherent
programming flexibility make software signal smo
othing (or filtering) techniques especially
attractive. Some of the more common signal smoothing algorithms described below.

Moving average algorithm

The simpler software technique for smoothing signals consisting of equidistant points is the
moving averag
e. An array of raw (noisy) data [y
1
, y
2
, …, y
N
] can be converted to a new array of
smoothed data. The "smoothed point" (y
k
)
s

is the average of an odd number of consecutive 2n+1
(n=1, 2, 3, ..) points of the raw data y
k
-
n
, y
k
-
n+1
, …, y
k
-
1
, y
k
, y
k+1
, …, y
k+n
-
1
, y
k+n
, i.e.





The odd number 2n+1 is usually named filter width. The greater the filter width the more intense
is the smoothing effect. This operation is
depicted in the animated picture below.




In this example the filter width is 5. The first five raw data (black squares) within the red
rectangle (moving wi
ndow) are averaged and their average value is plotted as smoothed (green
squares) data point 3. The rectangle is then moved one point to the right and points 2 through 6
are averaged, and the average is plotted as smoothed data point 4, and so on. This pro
cedure is
called a 5
-
point unweighted smooth.

The signal
-
to
-
noise ratio may be further enhanced by increasing the filter width or by smoothing
the data multiple times. Obviously after each filter pass the first n and the last n points are lost.

The results

of this technique are deceptively impressive because of excessive filtering. Actually,
information is lost and/or distorted because too much statistical weight is given to points that are
well removed from the central point. Moving average algorithm is pa
rticularly damaging when
the filter passes through peaks that are narrow compared to the filter width.



Savitzky
-
Golay algorithm

A much better procedure than simply averaging points is to perform a least squares fit of a small
set of consecutive data poin
ts to a polynomial and take the calculated central point of the fitted
polynomial curve as the new smoothed data point.

Savitzky and Golay (see A. Savitzky and M. J. E. Golay, Anal. Chem., 1964, 36, 1627) showed
that a set of integers (A
-
n
, A
-
(n
-
1)

…, A
n
-
1
, A
n
) could be derived and used as weighting coefficients
to carry out the smoothing operation. The use of these weighting coefficients, known as
convolution integers, turns out to be exactly equivalent to fitting the data to a polynomial, as just
describe
d and it is computationally more effective and much faster. Therefore, the smoothed data
point (y
k
)
s

by the Savitzky
-
Golay algorithm is given by the following equation:




Many sets of convolution integers can be used depending on the filter width and the
polynomial degree. Typical sets of these integers for “quadratic smooth” are shown in the table
below:





Filter width (2n+1)

i

11

9

7

5

-
5

-
36







-
4

9

-
21





-
3

44

14

-
2



-
2

69

39

3

-
3

-
1

84

54

6

12

0

89

59

7

17

1

84

54

6

12

2

69

3

3

-
3

3

44

14

-
2



4

9

-
21





5

-
36







Sets of convolution integers can be used to obtain directly, instead of the smoothed signal, its 1st,
2nd, …, mth order d
erivative, therefore Savitzky
-
Golay algorithm is very useful for calculating
the derivatives of noisy signals consisting of descrete and equidistant points.

The smoothing effect of the Savitzky
-
Golay algorithm is not so aggressive as in the case of the
mov
ing average and the loss and/or distortion of vital information is comparatively limited.
However, it should be stressed that both algorithms are "lossy", i.e. part of the original
information is lost or distorded. This type of smoothing has only cosmetic
value.



Ensemble Average

In ensemble average successive sets of data are collected and summed point by point. Therefore,
a prerequisite for the application of this method is the ability to reproduce the signal as many
times as possible starting always fro
m the same data point, contrary to the previous two
algorithms which operate exclusively on a single data set.

Typical application of ensemble average is found in NMR and FT
-
IR spectroscopy, where the
final spectrum is the result of averaging thousands of
individual spectra. This is the only way to
obtain a meaningful signal, when a single scan generates a practically unreadable signal heavily
contaminated with random noise.

Repetitive additions of noisy signals tend to emphasize their systematic characteri
stics and to
cancel out any zero
-
mean random noise. If (SNR)
o

is the original signal
-
to
-
noise ratio of the
signal, the final (SNR)
f

after N repetitions (scans) is given by the following equation:




Therefore, by averaging 100 (or 1000) data sets a 10
-
fold (or a 100
-
fold) reduction of noise level
is achieved.


It is of interest to observe the difference between moving average and Savitzky
-
Golay
algorithms using
the signal F3. This signal consists of seven Gaussian peaks of equal height with
progressively decreasing width. By successive applications (passings) of the moving average the
peak
-
height of the more narrow peaks decreases, i.e. a crucial information (the

height) is
distorted. In sharp contrast to moving average, the Savitzky
-
Golay algorithm “respects” this
information and the degradation of the signal is limited.

The ensemble average procedure is best demonstrated with the signal F2, that simulates the ve
ry
narrow peaks encountered in NMR spectra. By adding high level noise, this signal becomes
almost unreadable and only the higher peaks are discernible. By applying ensemble average and
by increasing the number of scans, the signal gradually emerges and ev
en the smaller peaks can
be safely recognized.

SOURCES OF NOISE

It is important for the analyst who uses a particular instrumental method to be aware of the
sources of noise and the instrument components used to minimize this noise because noise
determines

both the accuracy and detection limits of any measurement. Noise enters a
measurement system from environmental sources external to the measurement system (Figure
2.4), or it appears as a result of fundamental, intrinsic properties of the system. It is u
sually
possible to identify the sources of environmental noise and to either reduce or avoid their effects
on the measurement. Such is not the case with fundamental noise because it arises from the
discontinuous nature of matter and energy. Thus, fundame
ntal noise ultimately limits accuracy,
precision. and detection limits in every measurement.


The major kinds of noise associated with solid
-
state electronic devices are thermal, shot,
and flicker.


Fundamental Noise


Thermal Noise
. Noise that originates
from the thermally induced motions in charge carriers is
known as thermal noise. It exists even in the absence of current flow and is represented by the
formula



where V., is the average voltage due to thermal noise, k is the Boltzmann constant, T is th
e
absolute temperature. R is the resistance of the electronic device, and A
f

is the bandwidth of
measurement frequencies, Since thermal noise is independent of the absolute values of
frequencies, it is also known as "white noise." Methods for reducing th
ermal noise are suggested
by Equation 2.3. Sensitive radiation detectors are often cooled to minimize this noise.
Narrowing the frequency bandwidth of the detector is another way to reduce thermal noise,
provided the frequencies important to the measureme
nt of interest are not excluded.



HARDWARE TECHNIQUES FOR SIGNAL
-

TO
-
NOISE ENHANCEMENT


To avoid losing data, the signal from the input transducer (see Section 1.4) should be sampled at
a rate twice that of the highest frequency component of the signal
according to the Nyquist
sampling theorem (see Section 2.6). Adherence to this theorem is important to obtain reliable
results from either hardware or software
S

enhancement methods.



The rise of time of an instrument is its response time in seconds to a
n abrupt change in input and
normally is taken as the time required for the output to increase from 10% to 90% of its final
value. If the rise time is 0.01s, the bandwidth is 33Hz. It is important to note that thermal noise
can also be reduced by lowerin
g the electrical resistance of instrument circuits and by lowering
the temperature of instrument components. Cooling often reduces the thermal noise in
transducers. It is important to note that thermal noise, while dependent on the frequency
bandwidth is

independent of frequency itself, thus it is sometimes termed white noise by analogy
to white light, which contains all visible frequencies. Thermal noise in resistive circuit elements
is independent of the physical size of the resistor.


Filtering


Al
though amplitude and the phase relationship of input and output signals can be used to
discriminate between meaningful signals and noise, frequency is the property most commonly
used. As discussed in the previous section, white noise can be reduced by nar
rowing the range
of measured frequencies, environmental noise can be eliminated by selecting the proper
frequency. Three kinds of electronic filters are used to select the band of measured frequencies:
low
-
pass filters that allow the passage of all si2nal
s below a predetermined cutoff frequency,
high
-
pass filters that transmit all frequencies above a given cutoff point, and band pass filters that
combine the properties of the other two filters to pass only a narrow band of frequencies (Figure
2.5). The sim
plest filters are composed of passive circuit elements (resistors, R, capacitors, C.
and inductance coils, L) with the transmitted frequencies determined by values of the individual
circuit components.




Sources of noise in instrumental analysis:


Ty
pes of noise:


Chemical:

This noise arises from uncontrollable variables in the chemistry of the
system such as variation in temperature, pressure, humidity, light and
chemical fumes present in the room.


Instrumental : Noise that arises due to the instrum
entation itself. It could come from
any of the following components
-

source, input transducer all signal
processing elements, and the output transducer. This noise has many
types and can arise from several sources. There are four main categories
of inst
rumental noise: Thermal or Johnson, Shot, Flicker, and
Environmental.




Thermal noise or Johnson noise:



arises from thermal agitation of electrons or other charged carriers
in
resistors, capacitors,
radiation detectors, electrochemical cells and other re
sistive elements in the instrument.



this agitation is random and can create charge variations that create voltage fluctuations
that we view as noise



the magnitude of thermal noise is given by:
v
rms

= (4kTR

f)
1/2

where
v
rms

= root mean square noise voltage,


f = frequency bandwidth,

k = l.38 x 10
-
27

J/K (Boltzman Constant), T = temperature in Kelvin,

R = resistance in ohms of the resistive element




the bandwidth is inversely proportional to the rise time (t),



(response time in seconds to an abrupt change i
n input)


f = 1/3t
r



rise time is taken as the time required for the output to increase from 10% to 90 of the
final value



narrowing bandwidth can decrease thermal noise but it will also slow the rate of the
machine therefor increasing time required to make

a reliable measurement



can also be lowered by lowering temperature or electrical resistance



dependent upon frequency bandwidth but independent of frequency

Shot Noise:

Shot noise is encountered whenever electrons and other charged particles cross a junct
ion. In
typical electronic circuits, these junctions are found at
pn

interfaces. The shot noise in



arises when current involves the movement of electrons or charged particles across a
junction



these junctions are typically found at
pn

interfaces; in ph
otocells and vacuum tubes.



shot noises are random and their rate of occurrence is subject to statistical fluctuations
which are defined as follows:


i
rms
.

= (2I


f)
1/2



where
i
rms

= root mean sq. current fluctuation associated with avera
ge direct



current (I), e = l.60 x 10
-
19

C, f = bandwidth of frequencies



shot noise can be minimized only by reducing bandwidth

Shot noise

refers to the random fluctuations of the electric current in an electrical conductor,
which are caused by the fact
that the current is carried by discrete charges (electrons). The
strength of this noise increases for growing magnitude of the average current flowing through the
conductor. Shot noise is to be distinguished from current fluctuations in equilibrium, which
happen without any applied voltage and without any average current flowing. These equilibrium
current fluctuations are known as Johnson
-
Nyquist noise.

Shot noise is important in electronics, telecommunication, and for fundamental physics.

The strength of

the current fluctuations can be expressed by giving the variance of the current,
<(
I
-
<
I
>)>
2
, where <
I
> is the average ("macroscopic") current. However, the value measured in
this way depends on the frequency range of fluctuations which is measured ("bandw
idth" of the
measurement): The measured variance of the current grows linearly with bandwidth. Therefore, a
more fundamental quantity is the noise power, which is essentially obtained by dividing through
the bandwidth (and, therefore, has the dimension amp
ere squared divided by Hertz). It may be
defined as the zero
-
frequency Fourier transform of the current
-
current correlation function:

<math>S=
\
int_{
-
\
infty}^{+
\
infty} (
\
left
-
\
left<I
\
right>^2) dt </math></center>

Note: This is the total noise power, which

includes the equilibrium fluctuations (Johnson
-
Nyquist noise). Some other commonly employed definitions may differ by a constant pre
-
factor.

Note:

There is often a minor inconsistency in referring to shot noise in an optical system: many
authors refer to

shot noise loosely when speaking of the mean square shot noise current
(amperes
2
) rather than noise power (watts).

Shot noise, the time
-
dependent fluctuations in electrical current caused by the
discreteness of the electron charge, is well known to occur

in solid
-
state devices, such as
tunnel junctions, Schottky barrier diodes and p
-
n junctions. Most textbooks on electronic
devices will tell you that there is no shot noise in metallic resistors, just thermal noise and
a frequency
-
dependent noise known as
1/f
noise. However, our basic knowledge of
electrical conduction in small devices has advanced to the stage where it is clear that this
notion does not hold.

Although there have been a few intriguing theoretical predictions concerning shot noise
in metall
ic resistors, experimental evidence has been difficult to obtain. Now, a
collaboration between Andrew Steinbach and John Martinis at the US National Institute
of Standards and Technology in Boulder, Colorado, and Michel Devoret at the
Commissariat à l'Ener
gie Atomique in Saclay, France, has performed very accurate noise
measurements for silver thin
-
film resistors. These clearly demonstrate the existence of
several different noise regimes.

In 1918 Schottky reported that in ideal vacuum tubes where all sourc
es of spurious noise
have been eliminated there are two types of noise, described by him as the
Wärmeeffekt

and the
Schroteffekt
. The first of these is now known as Johnson
-
Nyquist or thermal
noise. It is caused by the thermal motion of the electrons and o
ccurs in any conductor that
has a resistance,
R
. The second is the shot noise.

Noise is best characterized by the Fourier transform of the time
-
varying fluctuations in
electric current, which is called the noise spectral density,
S
. For thermal noise, the

spectral density is given by
4kT/R
, where
k

is Boltzmann's constant and
T

is the
temperature. Thermal noise is thus white noise
-

the spectral density is independent of
frequency.

Shot noise results from the fact that the current is not a continuous flow

but the sum of
discrete pulses in time, each corresponding to the transfer of an electron through the
conductor. Its spectral density is proportional to the average current,
I
, and is
characterized by a white noise spectrum up to a certain cut
-
off frequen
cy, which is
related to the time taken for an electron to travel through the conductor. In contrast to
thermal noise, shot noise cannot be eliminated by lowering the temperature.

In devices such as tunnel junctions the electrons are transmitted randomly a
nd
independently of each other. Thus the transfer of electrons can be described by Poisson
statistics, which are used to analyse events that are uncorrelated in time. For these devices
the shot noise has its maximum value at
2eI
, where
e

is the electronic
charge.

However, shot noise is absent in a macroscopic, metallic resistor because the ubiquitous
inelastic electron
-
phonon scattering smoothes out current fluctuations that result from the
discreteness of the electrons, leaving only thermal noise. But rec
ent progress in
nanofabrication technology has revived the interest in shot noise, particularly since
nanostructures and "mesoscopic" resistors allow measurements to be made on length
scales that were previously inaccessible experimentally.

Shot noise in
mesoscopic devices has already proved to be a fruitful playground for
theoretical physicists. Recent calculations show that shot noise should exist in
mesoscopic resistors, although at lower levels than in a tunnel junction. For these devices
the length of

the conductor is short enough for the electron to become correlated, a result
of the Pauli exclusion principle. This means that the electrons are no longer transmitted
randomly, but according to sub
-
Poissonian statistics.

The sub
-
Poissonian shot
-
noise
power, S, of a metallic resistor as
a function of its length, L, as
predicted by theory. Indicated are
the elastic mean
-
free path, l, the
electron
-
electron sc
attering
length, l
ee
, and the electron
-
phonon scattering length l
ep
. Up
until a few years ago,
experiments were only possible in
the macroscopic regime, where
L> l
ep
. The experiments by
Steinbach, Martinis and Devoret
have confirmed the theory for
lengths
down to l
ee
. Typical
values for a metal at 50 mK are: l = 50 nm, l
ee

= 1
-
10 micron, l
ep

= 0.1 mm. The dashed

line gives the shot
-
noise spectral density of a Poisson process as found in, for example,
tunnel junctions.


Theorists have predicted the shot noi
se in a metallic resistor as a function of its length
(see figure). In the so
-
called ballistic regime the resistor is so short that it does not
contain any impurities. The electrons cannot be scattered, so the probability of an
electron being transmitted i
s unity and there is no shot noise.

When the resistor is longer than the mean
-
free path for elastic scattering, the electrons are
scattered by impurities in the metal. The electron motion becomes diffusive but the
energy of each electron remains constant.

It has been calculated by Carlo Beenakker at
the University of Leiden and Marcus Büttiker at the University of Geneva, and by Kirill
Nagaev at the Russian Academy of Sciences in Moscow, that for this elastic regime
S=(2/3)eI
, three times smaller than the
2eI

seen in tunnel junctions.

If the resistor is longer than the electron
-
electron scattering length, the electrons undergo
inelastic collisions. Their total energy is conserved, so they are heated above the lattice
temperature. In this so
-
called hot
-
elec
tron regime, the shot noise is slightly enhanced to
S
= (3/4)
1/2
eI

= 0.87eI
, as predicted by Nagaev and independently by Victor Kozub and
Alexander Rudin at the A.F. Ioffe Institute in St. Petersburg. Intriguingly, in both of these
regimes the shot noise i
s proportional to the full Poissonian shot noise. Moreover, the
constant of proportionality is a simple numerical coefficient that does not depend on the
shape of the resistor nor on the material that it is made of.

The first experimental observation of s
ub
-
Poissonian shot noise in a metallic resistor was
made in a collaboration between the University of Utrecht and Philips Research. We
investigated a gate
-
defined wire in a semiconductor structure, where the wire was 17
micron long. The measurements were i
n agreement with theory but lacked the precision
needed to discriminate between the predicted values.

Indeed, noise experiments are notoriously difficult. The combination of high frequencies
-

typically from 10 kHz up to 10 GHz, which is enough to elimina
te the spurious
1/f

noise
-

and cryogenic temperatures is problematic, since the radiation from the measuring leads
tends to heat up the device. In addition the high currents needed to obtain a significant
noise signal increase the rate of inelastic proces
ses inside the sample.

Therefore Steinbach and co
-
workers had to develop a dedicated measurement system to
pinpoint the characteristic numerical coefficient of the shot noise. Their low
-
temperature
circuit is not directly connected to the outside world
-

a coil produces a flux that is
detected by a low
-
noise, high
-
bandwidth SQUID (superconducting quantum interference
device) preamplifier. This set
-
up allows for an exceptional measuring accuracy of 2%.

The experiments were performed on thin
-
film silver wir
es of lengths between 1 micron
and 7 mm and at temperatures between 50 and 400 mK. The results are in excellent
agreement with theory. They confirm that the shot noise is close to the value of
0.87eI

for
lengths corresponding to the hot
-
electron regime, an
d a transition to the macroscopic
regime is clearly observed in the longest resistors. Moreover, the shot noise in the 1
micron resistor is substantially less than the hot
-
electron value and close to the value
expected in the elastic regime. This is the fi
rst time that the shot noise has been measued
with sufficient accuracy to distinguish between these two regimes.

The exploration of shot noise in other mesoscopic systems has also seen impressive
progress in the last two years. For example, recent noise e
xperiments have looked into the
effects of Coulomb blockade in a double tunnel junction, a structure that was formed by a
scanning tunneling microscope positioned above a nanoparticle. The effects of
conductance quantization in quantum point contacts have
also been explored. For the
future, several theoretical predictions
-

including shot noise effects in metal
-
superconductor junctions and how two
-
electron quantum interference effects may affect
the shot noise in ring
-
shaped devices
-

await experimental ver
ification.


Flicker noise:



its magnitude is inversely proportional to frequency of signal



causes of flicker noise not understood but recognized by frequency dependence



can be significant at frequencies lower than 100 Hz



causes long term drift in de amplifi
ers, meters, and galvanometers



can be reduced significantly by using wire
-
wound or metallic film resistors rather than
composition type


Flicker Noise is associated with crystal surface defects in semiconductors and is also found in
vacuum tubes. The noise

power is proportional to the bias current, and, unlike thermal and shot
noise, flicker noise decreases with frequency.


An exact mathematical model does not exist for flicker noise because it is so device
-
specific.
However, the inverse proportionality wit
h frequency is almost exactly 1/f for low frequencies,
whereas for frequencies above a few kilohertz, the noise power is weak but essentially flat.
Flicker noise is essentially random, but because its frequency spectrum is not flat, it is not a
white noise
. It is often referred to as pink noise because most of the power is concentrated at the
lower end of the frequency spectrum.


The objection to carbon resistors mentioned earlier for critical low noise applications is due to
their tendency to produce flick
er noise when carrying a direct current. In this connection, metal
film resistors are a better choice for low frequency, low noise applications.

Change in electrical flicker noise power in hot carrier semiconductors can be explained by
fluctuations in the
intensity of impurity scattering, which contradicts the Hooge
-
Kleinpenning
-
Vandamme hypothesis, which relates flicker conduction noise to lattice scattering. It has been
shown that such noise can be caused by fluctuations in the effective number of neutral

scattering
centers within the semiconductor volume. This source modulates carrier mobility, i.e., mobility
fluctuations are a secondary effect. We offer herein an explanation of known experimental data
on 1/f noise in silicon and gallium arsenide.



Envir
onmental noise:


Environmental noise is due to a composite of noises from different sources in the
environment surrounding the instrument. Figure 5
-
3 shows some common sources of
environmental noise.


Much environmental noise occurs because each conduct
or in an instrument is potentially an
antenna capable of picking up electromagnetic radiation and converting it to an electrical signal.
There are numerous sources of electromagnetic radiation in the environment including ac power
lines, radio and TV stat
ions, gasoline engine ignition systems, arcing switches, brushes in
electrical motors, lightening, and ionospheric disturbances.





Signal to noise enhancement:




for some measurements only minimal efforts are required for maintaining a good signal to
no
ise ratio because the signals are relatively strong and the requirements for precision
and accuracy are low

o

Examples:

o

weight determinations made in synthesis and color comparisons made in chemical
content determinations

o

when precision is important the sig
nal to noise ratio can become the limiting
factor and must be improved, there are two ways to approach this:



Hardware:



noise is reduced by incorporating into the instrument components such as filters,
choppers, shields, modulators, and synchronous detect
ors



these will remove or attenuate noise without affecting the analytical signal significantly

Types of Hardware:

Grounding and shielding:

Shielding, grounding, and minimizing the lengths of conductors with the instrumental system can
often substantially r
educe noise that arises from environmentally generated electromagnetic
radiation.



shielding consists of surrounding a circuit, or some of the wires in a circuit with a
conducting material that is attached to earth ground



this allows electromagnetic radia
tion to be absorbed by the shield thus avoiding noise
generation in

the instrument circuit
-
important when using high
-
impedance transducers
(i.e. glass electrodes)

This is the first line of defense against outside noise caused by high
-
frequency electrical
field and magnetic field coupling as well as electromagnetic radiation. The theory is that the
shielding wire, foil or conduit will prevent the bulk of the noise coming in from the outside.

It works by relying on two properties

1. Reflection back to the
outside world where it can't do any harm... (and, to a small extent,
re
-
reflection within the shield, but this is a VERY small extent)

2. Absorption
-

where the energy is absorbed by the shield and sent to ground.

The effectiveness of the shield is depen
dent on its:

1. Thickness
-

the thinner the shield the less effective. This is particularly true of low
-
frequency noise... Aluminum foil shield works well at rejecting up to 90 dB at frequencies
above 30 MHz, but it's inadequate at fending off low
-
frequen
cy magnetic fields (in fact it's
practically transparent below 1 kHz), We rely on balancing and differential amplifiers to get
rid of these.

2. Conductivity
-

the shield must be able to sink all stray currents to the ground plane more
easily than anything

else.

3. Continuity
-

we cannot break the shield. It must be continuous around the signal paths,
otherwise the noise will leak in like water into a hole in a boat. Don't forget that the holes in
your equipment for cooling, potentiometers and so on are br
eaks in the continuity. General
guideline: keep the diameter of your holes at less than 1/20 of the wavelength of the highest
frequency you're worried about to ensure at least 20 dB of attenuation. Most high
-
frequency
noise problems are caused by openings
in the shield material.


Balanced transmission lines

It is commonly believed even at the highest levels of the audio world that a balanced signal
and a
differential

or
symmetrical

signal are the same thing. This is not the case. A
differential (or symmetr
ical) signal is one where one channel of audio is sent as two
voltages on two wires, which are usually twisted together. These two signals are identical in
ever respect with the exception that they are opposite in polarity. These signals are known
by such
names as "inverting and non
-
inverting" or "Live and Return"
-

the "L" and "R" in
XLR (the X is for eXternal
-

the ground). They are received by a differential amplifier
which subtracts the return from the live and produces a single signal with a gain of 6
dB
(since a signal minus its negative self is the same as 2 times the signal and therefore 6 dB
louder). The theoretical benefit of using this system is that any noise that is received on the
transmission cables between the source and the receiver is (theo
retically) identical on both
wires. When these two versions of the noise arrive at the receiver's differential amplifier,
they are theoretically eliminated since we are subtracting the signal from a copy of itself.
This is what is known as the Common Mode
Rejection done by the differential input. The
ability of the amplifier to reject the common signals (or mode) is measured as a ratio
between the output and one input leg of the differential amplifier and is therefore called the
Common Mode Rejection Ratio
(CMRR).

Having said all that, I want to come back to the fact that I used the word "theoretical" a little
too often in the last paragraph. The amount and quality of the noise on those two
transmission lines (the live and the return) in the so
-
called "bala
nced" wire is dependent on a
number of things.

1. The proximity to the noise source. This is what is causing the noise to wind up on the two
wires in the first place. If the source of the noise is quite near to the receiving wire (for
example, in the case

of a high
-
voltage/current AC cable sitting next to a microphone cable)
then the closer wire within our "balanced" pair will receive a higher level of noise than the
more distant wire. Remember that this is inversely proportional to the square of the dista
nce,
so it can cause a major problem if the AC and mic cables are sitting side by side. The
simplest way to avoid this difference in the noise on the two wires is to wrap them together.
This ensures that, over the length of the cable, the two internal wire
s average out to being
equally close to the AC cable and therefore we pick up the same amount of noise
-

therefore
the differential amplifier will cancel it.

2. The termination impedance of the two wires. In fact, technically speaking, a balanced
transmis
sion line is one where the impedance between each of the two wires and ground is
identical for each end of the transmission. Therefore the impedance between live and ground
is identical to the impedance between return and ground at the output of the sendin
g device
and at the input of the receiving device. This is not to say that the input and output
impedances are matched. They are not. If the termination impedances are mismatched, then
the noise on each of the wires will be different and the differential a
mplifier will not be
subtracting a signal from a copy of itself
-

therefore the noise will get through. Some
manufacturers are aware of this and save themselves some money while still providing you
with a balanced output. Mackie consoles, for example, driv
e the signal on the tip of their
1/4" balanced outputs, but only put a resistor between the ring and ground (the sleeve) on
the same output connector. This is still a balanced output despite the fact that there is no
signal on the ring because the impedanc
e between the tip and ground matches the
impedance between the ring and ground (they're careful about what resistor they put in
there...)

Grounding

The grounding of audio equipment is there for one primary purpose: to keep you alive. If
something goes hor
ribly wrong inside one of those devices and winds up connecting the 120
V AC from the wall to the box (chassis) itself, and you come along and touch the front panel
while standing in a pool of water, YOU are the path to ground. This is bad. So, the
manufac
turers put a third pin on their AC cables which is connected to the chassis on the
equipment end, and the third pin in the wall socket. This third pin in the wall socket is called
the ground bus and is connected to the electrical breaker box somewhere in t
he facility. All
of the ground busses connect to a primary ground point somewhere in the building. This is
the point at which the building makes contact with the earth through a spike or piling called
the grounding electrode. The wires which connect these
grounds together MUST be heavy
-
gauge (and therefore very low impedance) in order to ensure that they have a MUCH lower
impedance than you when you and it are a parallel connection to ground. The lower this
impedance, the less current will flow through you
if something goes wrong.

An added benefit to this ground is that we use it as a huge sink where all of the noise on our
shielding is routed.


Conductor Out From


Low Dynamic
Range

(< 60 dB)


Med Dynamic
Range

(60 to 80 dB)


High Dynamic
Range

(> 80 dB
)


Low
EMI


High
EMI


Low
EMI


High
EMI


Ground Electrode

6

2

00

00

0000

Master Bus

10

8

6

4

0

Local Bus

14

12

12*

12*

10*

Maximum resistance for
any cable (W)

0.5

0.1

0.01

0.001

0.0001

* Do not share ground conductors
-

run

individual branch grounds. In all cases the
ground conductor must not be smaller than the neutral conductor of the panel it services.


Difference amplifiers:



used to attenuate noise generated in the transducer



ac signal induced in the transducer circui
t generally appears in phase at both the inverting
and non
-
inverting terminals; cancellation then occurs at output

Any noise generated in the transducer circuit is particularly critical because it usually appears in
an amplified form in the instrument read

out. To attenuate this type of noise, most instruments
employ a difference amplifier for the first stage of amplification. Common mode noise in the
transducer circuit generally appears in the phase at both the inverting and noninverting inputs of
the am
plifier and is largely subtracted out by the circuit so that the noise at its output is
diminished substantially.

Instrumentation amplifiers are composed of three op amps. Op amp A and op amp B make up
the input stage of the instrumentation amplifier in

which the two op amps are cross coupled
through three resistors. The second stage of the module is the difference amplifier of op amp C.
The overall gain of the circuit is given by:


o

= K(2a + 1)(

2

-


1
)


Analog filtering:



one of the most common met
hods to improve S/N ratio is the use of a low
-
pass analog
filter
-
is effective at removing many high frequency components such as thermal or shot
noise because majority of analyte signals are dc with bandwidths that extend over only a
few Hz
-
high
-
pass sig
nals used in systems where the analyte signal is ac and the filter can
reduce drift and flicker noise
-
to attenuate noise we use narrow band electronic filters
because magnitude of fundamental noise is directly proportional to the sq. rt. of the
frequency
bandwidth signal. significant noise reduction occurs if input signal is restricted
to a narrow band and then an amplifier tuned to this band is used. The band past by the
filter must be wide enough to carry all information carried by the signal.

First or
der analog filters depend on one of two things:


an inductor or a capicitor. Although we did not actually use these particular components, as
much

as Sam wanted to, it is these components that make analog filters work. We decided on
active rather than passive filters because they give better results.








Schematics and Graphs


Lowpass Filter:










Highpass Filter:



To give those not familiar with analog circuits (or maybe its just been a really long time) an idea
of w
hat we wired together we have included some pictures of our breadboard with our first order
lowpass filters.


Ooh, look! An Op
-
Amp.


I think I see some filters.


Lowpass, highpass, yup. There they are.

Analog Second Order Filters


By going to second order filters we were able to shorten the transision band providing greater
at
tenuation of the noise. We were also able to make a bandpass filter using second order design.
Also note that compared to first order filters these circuits only require the addition of one
capacitor.













Schematics and Graphs


Lowpass Filter:





Highpass Filter:







Bandpass Filter:







Once again some pictures for those of you who have not entered the lab
recently:


What a mess!


A bandpass
enters the game.


Close up of our second order filters.





Modulation:



amplifier drift and flicker noise often interfere with the amplification
of a low frequency
or dc signal and 1/f noise is often much larger than noises that predominate at larger
frequencies therefore modulators are used to convert these to a higher frequency where
1/f is less troublesome



the modulated signal is amplified then

filtered with a high
-
pass filter to remove the
amplifier 1/f noise



the signal is then demodulated and filtered with a low
-
pass filter in order to provide an amplified dc
signal to the readout device



noise is a concern because source intensity and detector

sensitivity are low which result
'in a small electrical signal from transducer



IR transducers are heat detectors, this adds environmental noise due to the thermal
radiation of their surroundings
-
a slotted rotating disk placed in the beam path produces a
radiant signal that fluctuates between zero and some max.
-
signal is converted by
transducer to a sq. wave ac electrical signal whose frequency depends upon size of slots
and rate of rotation



in IR, environmental noise is usually dc and can be reduced with

a high
-
pass filter before
amplification
-
(refer to figure 4
-
6 on next page) This is a chopper amplifier that uses a
solid state switch to shot the input or output signal to the ground appearance of signal at
certain stages 0) input a 6
-
mV dc signal a) swi
tch converts to approx. sq. wave signal of
amplitude 6mV b) amplification to ac signal amplitude 6V c) shorted to ground
periodically which results in d) reduced amplitude to 3V e) RC filter serves to smooth
signal and produce 1.5V dc output


Lock
-
in ampli
fiers:



permit recovery of signals even when S/N is unity or less



generally requires a reference signal at same frequency and phase (must have fixed phase
relationship) as signal to be amplified


Software:



based on digital computer algorithms that permit
extraction of signal from noisy
environment



requires some hardware to condition output and convert it to digital form



computer and readout system are also needed



common software are generally applicable to non
-
periodic or irregular wave forms, such
as ab
sorption spec. or signals having no synchronizing or reference wave


Ensemble averaging:



successive sets of data (arrays) are collected and summed point by point (often called
coaddition)



then data for each point is averaged (figure 4
-
8)



signal to noise

ratio for signal average:



points must be measured at a frequency at least 2x that of the highest frequency
component of the wave form, much greater frequencies will include more noise but no
additional information
-
wave form reproducibility is important,
generally accomplished
through synchronizing pulse derived from the wave form
-
can result in dramatic
improvements.


Boxcar averaging:



digital procedure for smoothing irregularities, that arise from noise, in a wave form



assumed analog signal varies slowl
y with time and that the average of a small number of
adjacent points is better than any one individual point



usually done by a computer as data is being collected (real time)
-
limitations include loss
of detail, application to only complex signals that c
hange rapidly as a function of time
-
for square wave or repetitive pulsed outputs where only ave. amplitude is important, it is
very important
-
moving window averaging means point I is the ave. of 1, 2 and 3, point 2
is an ave. of points 2, 3 and 4 etc. he
re only the first and last points are lost


Digital Filtering:



moving
-
window boxcar method is a kind of linear filtering where it assumed that there is
an approximate linear relationship among points being sampled
-
more complex
polynomial relationships der
ive a center point for each window



can also be carried out by Fourier transform procedure where the original signal which
varies as a function of time (time domain signal) is converted to a frequency domain
signal where the independent variable is frequenc
y not time
-
accomplished
mathematically on a computer by a Fourier transform procedure



then frequency signal is multiplied by the frequency response of a digital filter which
removes a certain frequency region of the transformed signal



the filtered time d
omain signal is retrieved by an inverse Fourier transform



Digital Filters


Our digital filters were designed and implementated in Matlab. Our code reads our original
signal, creates the noisy signals, creates the filters, filters the signals, and outputs

the recovered
signals. Our Matlab code looks nice and simple but it took us a while to get it to this point.

Matlab Code For Digital Filters



%% Read in original test wav

[y, fs, bits] = wavread('bitch');


%% Noise functions for different frequencies

t = linspace(1,length(y),length(y));

co1 = cos(t/length(y)*2*pi*60);

co2 = cos(t/length(y)*2*pi*65);

co3 = cos(t/length(y)*2*pi*70);

co3a = cos(t/length(y)*2*pi*150);

co3b = cos(t/length(y)*2*pi*80);

co3c = cos(t/length(y)*2*pi*100);

co3d = cos(t/l
ength(y)*2*pi*200);

co4 = cos(t/length(y)*2*pi*10000);

co5 = cos(t/length(y)*2*pi*15000);

co6 = cos(t/length(y)*2*pi*16000);


%% high
-
frequency noise signal

high = y + co5';


%% low
-
frequency noise signal

low = y + co3c';


%% high and low noise sig
nal

both = y + co3c' + co5';


%% Noisy signal to put through crazy elliptical filters

z = .5*y +co1'+co2'+co3'+co3a'+co3b'+co3c'+co3d'+co4'+co5'+co6';


%% Disign 1st and 2nd order butterworth filters

[bhp1,ahp1] = butter(1,212.206/fs*2,'high');

[blp1
,alp1] = butter(1,3120.685/fs*2);

[blp2,alp2] = butter(2, 3000/fs*2);

[bhp2,ahp2] = butter(2,194.14/fs*2,'high');


%% filter

low1 = filter(blp1,alp1,high);

low2 = filter(blp2,alp2,high);

high1 = filter(bhp1,ahp1,low);

high2 = filter(bhp2,ahp2,low);

band2 = filter(bbp3,abp2,both);


%% Design high
-
order elliptical low
-
pass filter

[bl,al] = ellip(10, 1, 100, 2000/(fs/2));


%% Design high
-
order elliptical high
-
pass filter

[bh,ah] = ellip(8, 1, 100, 600/(fs/2),'high');


%% Low
-
pass noisy signal

dig
1 = filter(bl,al,z);

%% then high
-
pass to get result

dig2 = filter(bh,ah,dig1);

















Useful Websites Dealing With Instrumental Analysis

Chemical Abstracts Service:


http://www.cas.org


Chemical Center Home Page:


http://www.chemcenter/org

ht
tp://www.chem.uoa.gr/Applets/appletsmooth/Text_Smooth2.htm


Journal of Chemistry and Spectroscopy:

http://www.kerouac.pharm.uky.edu/asrg/wave/wavehp.html

http://jchemed.chem.wisc.edu/j
cewww/Features/McadInChem/mcad015/


The Analytical Chemistry Springboard at Umea U.

http://www.anachem.umu.se/jumpstation.htm


ZirChrom Separations Home Page:




http://www.zirchrom.com/


Comprehensive Info on HPLC





http://hplc.chem.vt.edu/