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 is

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 averaged 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 information 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 of 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 Smoothing 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 smoothing (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 average. An array of raw (noisy) data [y1, y2, …, yN] can be converted to a new array of

smoothed data. The "smoothed point" (yk)s is the average of an odd number of consecutive 2n+1

(n=1, 2, 3, ..) points of the raw data yk-n, yk-n+1, …, yk-1, yk, yk+1, …, yk+n-1, yk+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 window) 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 procedure 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 particularly 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 points 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) …, An-1, An) 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

described and it is computationally more effective and much faster. Therefore, the smoothed data

point (yk)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 derivative, 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

moving 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 from 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 characteristics 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 very

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 even 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 usually

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, fundamental 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 the

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 thermal 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 measurement 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 an 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 lowering 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

Although 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 narrowing 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 si2nals 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 simplest 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:

Types 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 instrumentation 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 instrumental 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 resistive 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: vrms = (4kTRf)1/2

where vrms = 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 in input)

f = 1/3tr

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 junction. 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 photocells and vacuum tubes.

shot noises are random and their rate of occurrence is subject to statistical fluctuations

which are defined as follows:

irms. = (2If)1/2

where irms = root mean sq. current fluctuation associated with average 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 ("bandwidth" 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 ampere 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 (amperes2) 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 metallic 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'Energie 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 sources 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 occurs 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 frequency, 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 and 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 recent 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 scattering length, lee, and the electron-phonon scattering length lep. Up until a few years ago, experiments were only possible in the macroscopic regime, where L> lep. The experiments by Steinbach, Martinis and Devoret have confirmed the theory for lengths down to lee. Typical values for a

metal at 50 mK are: l = 50 nm, lee = 1-10 micron, lep = 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 noise 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 is 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-electron regime, the shot noise is slightly enhanced to S = (3/4)1/2eI = 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 is 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 sub-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 in 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 eliminate 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 processes 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 wires 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, and 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 first 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 experiments 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 verification.

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 amplifiers, 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 with 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 flicker 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.

Environmental 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 conductor 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 stations, 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

noise 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 signal 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 detectors

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 reduce 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 radiation 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 dependent 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-frequency 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 breaks 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 symmetrical) 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 (theoretically) 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 "balanced" 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 distance, 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 wires 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 transmission 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 sending 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 amplifier 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, drive 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 impedance 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 horribly 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 manufacturers 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 the 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

Med Dynamic Range

High Dynamic Range

(< 60 dB) (60 to 80 dB) (> 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 circuit 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 amplifier 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 methods 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 signals 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 order 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 what 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 attenuation 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) switch 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 amplifiers:

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 absorption 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 slowly 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 change 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. here 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 derive 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 frequency 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 domain 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/length(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 signal 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 dig1 = 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

http://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/jcewww/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/