Introduction to Signals and Noise

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Introduction to Signals and Noise
Module Description Goals and Objectives
Signals and Noise Sources of Noise
Signal-to-Noise Enhancement Analog
Filtering Digital Filtering References
Acknowledgements

• Module Description
• This e-module provides an introduction to the
  analytical chemist on the following topics
• The significance of signal and noise in chemical
  measurements
• The origin of noise in chemical measurements
• How noise degrades useful chemical information
• The statistical treatment of noise and the
  definition of a signal-to-noise ratio
• Methods used to improve the reliability of
  chemical measurements by enhancing the
  signal-to-noise ratio
• Steven C. Petrovic
• Department of Chemistry, Southern Oregon
  University, Ashland, OR 97520. Email
  petrovis_at_sou.edu
• This work is licensed under a
• Creative Commons Attribution Noncommercial-Share
  Alike 2.5 License

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Introduction to Signals and Noise
Module Description Goals and Objectives
Signals and Noise Sources of Noise
Signal-to-Noise Enhancement Analog
Filtering Digital Filtering References
Acknowledgements

• Goals and Objectives
• Goal 1 This module will frame the roles of
  signal and noise in chemical measurements.
• Objective 1 Define analytical signals and
  estimate signal parameters that correlate to
  analyte concentrations
• Objective 2 Define noise, estimate the magnitude
  of noise, and investigate how the presence of
  noise interferes with the measurement of
  analytical signals
• Objective 3 Define signal-to-noise ratio (S/N)
  as it relates to method performance and
  investigate how S/N is used to determine the
  detection limit of an analytical method
• Goal 2 This module will describe how to improve
  the signal-to-noise ratio of analytical signals
• Objective 1 Provide an introduction to the
  behavior of passive electronic circuits and show
  how they are used to improve the S/N of an
  analytical measurement
• Objective 2 Provide an introduction to
  software-based methods and show how they are used
  to improve the S/N of an analytical measurement

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Introduction to Signals and Noise
Module Description Goals and Objectives
Signals and Noise Sources of Noise
Signal-to-Noise Enhancement Analog
Filtering Digital Filtering References
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• Signals and Noise
• Defining Signal and Noise
• All analytical data sets contain two components
  signal and noise
• Signal
• This is the part of the data that contains
  information about the chemical species of
  interest (i.e. analyte).
• Signals are often proportional to the analyte
  mass or analyte concentration
• Beer-Lambert Law in spectroscopy where the
  absorbance, A, is proportional to concentration,
  C.

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Introduction to Signals and Noise
Module Description Goals and Objectives
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Signal-to-Noise Enhancement Analog
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• Signals and Noise
• There are other significant relationships
  between signal and analyte concentration
• The Nernst equation where a measured potential
  (E) is logarithmically related to the activity of
  an analyte (ax)
• Competitive immunoassays (e.g. ELISA) where
  labeled (analyte spike) and unlabeled analyte
  molecules (unknown analyte) compete for antibody
  binding sites

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Introduction to Signals and Noise
Module Description Goals and Objectives
Signals and Noise Sources of Noise
Signal-to-Noise Enhancement Analog
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• Signals and Noise
• Noise
• This is the part of the data that contains
  extraneous information.
• Noise originates from various sources in a
  analytical measurement system, such as
• Detectors
• Photon Sources
• Environmental Factors
• Therefore, characterizing the magnitude of the
  noise (N) is often a difficult task and may or
  may not be independent of signal strength (S).
• A more detailed discussion on specific
  relationships between signal and noise may be
  obtained by clicking here and reading Section 3.

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Introduction to Signals and Noise
Module Description Goals and Objectives
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Signal-to-Noise Enhancement Analog
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Figure of Merit Why is Noise Unwanted? Noise
degrades the accuracy and precision of a signal,
and therefore our knowledge about how much
analyte is present. Signal-to-Noise Ratio (S/N)
A Figure of Merit The quality of a signal may be
expressed by its signal-to-noise ratio
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Introduction to Signals and Noise
Module Description Goals and Objectives
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Measuring Signals If the signal is at
steady-state, as in the case of flame atomic
absorption spectroscopy (FAAS), S is best
estimated as the average signal magnitude, shown
below by the solid line.
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Module Description Goals and Objectives
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Signal-to-Noise Enhancement Analog
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Measuring Signals If the signal is transient, as
in the case of chromatographic peaks, S is best
estimated as the peak height or peak area. In the
figure below, the peak height is measured from
the midpoint of the baseline fluctuations (bottom
horizontal line) to the top of the peak.
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Measuring Signals The peak area of a transient
signal is the integrated response, which in this
case has units of (µVmin). The peak area of this
response is roughly equivalent to the area of the
shaded triangle superimposed on the
chromatographic peak.
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• Quantifying Noise
• All data contains some level of uncertainty due
  to random fluctuations in the measurement
  process. We will focus on describing random
  fluctuations that may be described mathematically
  using a Gaussian distribution shown below.
• In this relationship
• y is the frequency that a value x will occur
• µ is the population mean
• s is the standard deviation of the population

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Introduction to Signals and Noise
Module Description Goals and Objectives
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• Quantifying Noise
• Of course, there are such a myriad of samples
  and measurement methods that each case yields a
  unique distribution with a unique mean and
  standard deviation.
• In order to generally describe the Gaussian
  distribution, one must represent the Gaussian
  distribution in a standardized format. This can
  be done in two steps
• Mean-Centering
• subtracting the population mean from all the
  members of the data set so that µ 0
• Normalization
• dividing each member of the data set by the
  distribution standard deviation so that s 1
• The x-axis is now represented by a unitless
  quantity, z
• z (x-µ)/s

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Module Description Goals and Objectives
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Normal Error Curve If we look at a standardized
Gaussian distribution the so-called Normal
Error Curve shown below you can see that the
probability of any one measurement being a member
of this particular distribution increases as the
magnitude of z increases.
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Normal Error Curve The area underneath the curve
represented by z multiples of the standard
deviation are shown in the table below
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• Calculating S/N
• Calculating the signal to noise ratio based on
  our brief discussion of Gaussian statistics can
  be achieved as follows
• Find a section of the data that contains a
  representative baseline. Notice that on the
  chart, the representative baseline does not
  contain any signal.

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Estimate peak-to-peak noise (VN) If the data is
on a piece of paper, draw two lines that are
parallel with the baseline and tangential to the
edges of the baseline. See the example on the
left side of the page. If the data is digitized
(e.g. in a spreadsheet or text file), locate the
maximum and minimum values in a representative
section of the dataset that only represents the
noise level.
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Module Description Goals and Objectives
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• Estimate root mean square noise
• Calculate the standard deviation (VRMS) of the
  noise. At the 99 confidence level VN
  2.58s.Therefore
• Estimate the S/N. The signal is 16.0 µV.

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Estimating S/N First, calculate the standard
deviation (VRMS) of the noise. At the 99
confidence level VN 2.58s.Therefore
Second, calculate the S/N. The signal is 16.0 µV.
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• Sources of Instrumental Noise
• Johnson Noise
• Also called thermal noise, this source of noise
  results in random voltage fluctuations produced
  by the thermal agitation of electrons as they
  pass through resistive elements in the
  electronics of an instrument.
• The relationship between Johnson Noise and
  experimental parameters is as follows
• VRMSRoot-mean-square noise voltage with a
  frequency bandwidth of ?f (in Hertz).
• k Boltzmanns constant (1.38 x 10-23 J/K)
• T Temperature (K)
• R Resistance of resistive element (O)
• Reduction of Johnson Noise is accomplished most
  easily by
• Cooling the detector (reducing T)
• Decreasing the frequency bandwidth of the signal
  (reducing ?f)
• Actual measurements of Johnson Noise may be found
  by clicking here

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• Sources of Instrumental Noise
• Shot Noise
• This source of noise results in current
  fluctuations produced by electrons crossing a
  junction in a random fashion, which highlights
  the quantized nature of electron flow
• The relationship between Shot Noise and
  experimental parameters is as follows
• iRMSRoot-mean-square current fluctuation (in
  Amperes)
• I Average direct current (A)
• e electronic charge (1.60 x 10-19 C)
• ?f frequency bandwidth (Hz)
• Reduction of Shot Noise is accomplished most
  easily by
• Decreasing the frequency bandwidth of the signal
  (reducing ?f)
• A good discussion of Shot Noise may be found by
  clicking here

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• Sources of Instrumental Noise
• Flicker Noise
• Flicker noise is also called 1/f noise because
  the magnitude of flicker noise is inversely
  proportional to frequency. The source of flicker
  noise is uncertain and it seems to be significant
  only at low frequencies (lt100 Hz)
• A good summary of flicker noise (and Johnson
  noise) may be found by clicking here
• Environmental Noise
• These are sources of noise that interfere with
  analytical measurements. Examples of such sources
  include
• electrical power lines (e.g. 50 or 60 Hz line
  noise)
• electrical equipment (e.g. motors, fluorescent
  lights, etc.)
• RF sources (e.g. cell phones)
• environmental factors (drift in temperature,
  aging of electronic components)

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Module Description Goals and Objectives
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Signal-to-Noise Enhancement Analog
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Introduction to Signal-to-Noise Enhancement As
the S/N of an analytical signal decreases, so
does the accuracy and precision of that signal.
The pair of plots below illustrate this point.
The plot to the left contains three analyte
peaks with a peak-to-peak noise level of 0.19 µV.
The S/N for each peak is 52, 26, and 10
respectively. Increasing the peak-to-peak noise
level ten-fold (1.9 µV) decreases the S/N of each
peak by a factor of ten. (5.2, 2.6, 1.0
respectively)
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Module Description Goals and Objectives
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Introduction to Signal-to-Noise Enhancement Note
that the signal at 2 minutes, with a S/N ratio of
3, is at a level commonly known as the detection
limit, which is defined as the magnitude at which
the signal is statistically distinguishable from
the noise. The signal at 3 minutes, which has a
S/N equal to 1, is indistinguishable from the
baseline noise. This comparison illustrates the
need to reduce noise to a level at which chemical
information is not compromised. A spreadsheet has
been designed to illustrate the relationship
between signal and noise. Click here to perform
these exercises.
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Module Description Goals and Objectives
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Introduction to Signal-to-Noise Enhancement Can
Noise be Reduced After the Data has been
Recorded? In the examples below, the frequency
of the signal is less than the frequency of the
noise. In all cases, if the signal frequency and
the noise frequency are not equal, then there
should be at least one suitable approach to noise
reduction.
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• Overview of S-N Enhancement Techniques
• This module will describe two general categories
  of noise reduction techniques
• Analog Filtering (Hardware-Based)
• Digital Filtering (Software-Based)
• Most of these S/N enhancement methods, whether
  analog or digital, are based on either
• Bandwidth Reduction (i.e. decreasing ?f).
• Signal Averaging (i.e. decreasing ?f or averaging
  out random noise fluctuations)
• Bandwidth reduction is important --- Remember,
  if fsignal ? fnoise, we have a chance of
  isolating the signal from the noise. This results
  in an enhanced signal-to-noise ratio and more
  reliable information about the chemical sample of
  interest.
• We will see that there are limitations to how
  much bandwidth reduction can be applied before
  distorting the instrumental signal. Nevertheless,
  these can be effective approaches to improving
  the quality of the instrumental signal.

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• Analog Filtering
• Signals and noise are almost always expressed as
  electrical quantities. The electrical quantities
  you should be familiar with are
• Voltage Voltage is a measure of energy
  available when an electron moves from a point of
  higher potential to a point of lower potential.
  The SI Unit for voltage is the Volt (V). 1V 1
  Joule/Coulomb.
• Physicochemical phenomena that generate voltage
  include
• Chemical Reactions, such as those that take place
  in a battery
• Electromagnetic Induction, such as moving a coil
  of wire through a magnetic field (i.e. an
  electric generator)
• Photovoltaic Cells, which convert light energy
  into electrical work
• Current Current is a measure of the amount of
  electronic charge flowing per unit time past a
  given point. The SI Unit for current is the
  Ampere (A). 1A 1 Coulomb/second. Types of
  current include
• Direct Current (DC) Charges are flowing in the
  same direction.
• Heres an applet that demonstrates the production
  of pulsed DC http//micro.magnet.fsu.edu/electrom
  ag/java/generator/dc.html
• Alternating Current (AC) Charges change
  direction periodically.
• Heres an applet that demonstrates the production
  of AC http//micro.magnet.fsu.edu/electromag/java
  /generator/ac.html

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Analog Filtering - Ohms Law In 1827, Georg Ohm
published his work Die galvanische Kette
mathematisch bearbeitet, indicating that the
current flowing through a conductor is
proportional to the voltage across the conductor.
All conductors of electricity obey Ohms Law,
which is mathematically expressed as V
Voltage across the conductor (in Volts, V) I
Current through the conductor (in Amperes, A) R
Resistance of the conductor (in Ohms,
O) Simple applets to test out Ohms
Law http//micro.magnet.fsu.edu/electromag/java/o
hmslaw/ http//phet.colorado.edu/simulations/veqir
/VeqIRColored.swf http//www.walter-fendt.de/ph14e
/ohmslaw.htm
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• Passive Electronic Components
• Resistor
• A resistor is a component that resists electron
  flow.
• The unit of resistance is called an ohm (O). 1O
  1V/A
• In an electronic circuit schematic, a resistor is
  represented by
• Capacitor
• A capacitor is an electronic component that
  stores charge
• It consists of two conductive plates separated by
  an insulating medium
• The unit of capacitance is called a farad (F). 1F
  1C/V
• In an electronic circuit schematic, a capacitor
  is represented by
• A simple applet used to illustrate the principle
  of resistance
• http//micro.magnet.fsu.edu/electromag/java/filame
  ntresistance/index.html
• Simple applets used to illustrate the principle
  of capacitance
• http//micro.magnet.fsu.edu/electromag/java/capaci
  tance/index.html
• http//micro.magnet.fsu.edu/electromag/java/capaci
  tor/

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• Passive Electronic Circuits
• Remember that signal-to-noise ratios can be
  enhanced if the signal frequency is different
  than the noise frequency. You will be introduced
  to these frequency-dependent analog filters at
  the end of this section. For now, lets start
  very simply
• Resistor Fundamentals
• The simplest circuit involving a resistor and a
  voltage source is shown below. The dotted lines
  are just there to represent where a high-quality
  voltmeter would be connected if we wished to
  measure the voltage across the resistor.
  Calculating the current flowing through this
  resistor requires the use of Ohms Law.
• Circuit 1
• According to Ohms Law
• V 1.0 Volt
• R 20 Ohms

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• Passive Electronic Circuits
• Resistors in Series
• Practically speaking, we are not limited to a
  single resistor. Circuit 1 could also be
  represented by Circuit 2 below
• Resistors placed in a head-to-tail
  configuration are in series.
• The total resistance is the sum of all the
  individual resistances
• Putting resistors together in series gives a
  larger total resistance

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Passive Electronic Circuits Resistors in
Parallel Resistors placed in a side-to-side
configuration are in parallel. The total
resistance is the reciprocal of the sum of each
reciprocal resistance. So for a pair of resistors
as shown in Circuit 3 above Applying this
to Circuit 3 Putting resistors together in
parallel always gives a smaller total resistance.
Note that Circuit 3 has the same current as
Circuits 1 and 2.
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• Passive Electronic Circuits
• Voltage Divider
• Sometimes, the output of an instrument is too
  large for a readout device. One circuit used to
  reduce a voltage is a voltage divider
• Note that
• A representation for a voltmeter has been added
  to the schematic
• The voltage is only being accessed across one of
  the two resistors

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Passive Electronic Circuits Voltage Divider (Page
2) Assuming that the meter resistance is much
larger than R2 (i.e. no loading error occurs),
then according to Ohms Law Vin
I(R1 R2) For a discussion of loading errors,
click here.
Passive Electronic Circuits Voltage Divider (Page
2) Assuming that the meter resistance is much
larger than R2 (i.e. no loading error occurs),
then according to Ohms Law Vin
I(R1 R2) For a discussion of loading errors,
click here.
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Passive Electronic Circuits Voltage Divider (Page
3) If the readout device (i.e. a meter) is
placed across R2, than the voltage read by the
meter is Or in other words, the
divider output equals the instrument output
multiplied by R2 over the total resistance
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Passive Electronic Circuits Voltage Divider (Page
4) In this case, the divider output
is
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• Passive Electronic Circuits
• RC Voltage Dividers (Analog Filters)
• Although voltage dividers are extremely useful,
  they are unable to selectively filter signal
  voltages from noise voltages. That is
• Voltage dividers are frequency independent.
• However, the impedance of a capacitor is
  frequency dependent, as shown by the following
  equation
• XC is the impedance of the capacitor (impedance
  is the generalized form of resistance that
  applies to AC signals)
• f is the frequency of the voltage source in Hertz
• C is the capacitance in Farads
• As the frequency increases,
• the impedance of a capacitor decreases!

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• Passive Electronic Circuits
• Low-Pass Filters
• Used when the signal frequency lt noise frequency
• The relationship between Vin and Vout is
  analogous to a frequency independent voltage
  divider
• The desired filter output is obtained across the
  frequency dependent component (capacitor)

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• Passive Electronic Circuits
• High-Pass Filters
• Used when the signal frequency gt noise frequency
• The relationship between Vin and Vout is
  analogous to a frequency independent voltage
  divider
• The desired filter output is obtained across the
  frequency independent component (resistor)

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• Decibel Scale
• Expressing Signal Attenuation of RC filters
• Because an ideal analog filter would not
  attenuate the signal but only the noise, the
  decibel scale is used to express the degree of
  electrical attenuation (or gain) attributable to
  an electronic device, such as a RC filter.
• A decibel is defined as
• dB 20 log (Vout/Vin)
• So 0 dB represents no signal attenuation, and -20
  dB represents an order of magnitude decrease in
  the RC filter output compared with the input.
• Remember that S/N enhancement is possible if the
  frequency of the signal and the noise are
  different. We can express the attenuation of the
  RC filter response as a function of frequency
  using a Bode plot.
• Bode plots are log-log plots decibels are a
  logarithmic quantity and frequency is plotted on
  a logarithmic scale. They are quite frequently
  used to illustrate the frequency response of
  electronic circuits.

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• Bode Plots
• Below is a Bode plot of the low-pass RC filter
  frequency response shown a few slides back.
  Notice that low frequencies are unattenuated, but
  attenuation increases with higher frequencies.

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• Bode Plots
• Every Bode plot has two straight lines the
  relatively flat response where little attenuation
  occurs and a linear response of -20 dB/decade at
  higher frequencies. The intersection point of
  these two lines coincides with the rounded
  section of the plot. This is the cutoff
  frequency, fo, of the RC filter, which is
  expressed by the following relationship fo
  1/(2pRC)
• The cutoff frequency, which is 1592 Hz for this
  particular circuit, corresponds to a 3 dB
  attenuation, and can be used as a figure-of-merit
  for the response of the filter.

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• Bode Plots
• Below is a Bode plot of the high-pass RC filter
  frequency response a few slides back. Note that
  because the same resistor and capacitor was used,
  the cutoff frequency has not changed. The filter
  output is simply accessed across the resistor
  instead of the capacitor.

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• Passive Electronic Circuits
• Analog Filter Demo
• A lecture demonstration of how an RC filter
  isolates noise from signal can be obtained as a
  MS Word document by clicking here or as a web
  page by clicking here.
• Bode Plot Exercise
• An exercise on interpreting the frequency
  response of RC filters using a Bode plot can be
  accessed by clicking here.
• Analog Filter Exercises
• A couple of exercises have been included to
  reinforce your understanding about the design and
  application of analog filters.
• Click here to access Exercise 1
• Click here to access Exercise 2

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• Digital Filtering
• What is a digital filter?
• A digital filter is a noise reduction technique
  that is software-based. It is an approach that
  was popularized once personal computers became
  widely available.
• Digitally-based signal-to-noise enhancement
  techniques described in this e-module include
• Ensemble Averaging
• Boxcar Averaging
• Moving Average (Weighted Unweighted)

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• Digital Filtering
• Ensemble Averaging
• Ensemble averaging is a data acquisition method
  that the enhances the signal-to-noise of an
  analytical signal through repetitive scanning.
  Ensemble averaging can be done in real time,
  which is extremely useful for analytical methods
  such as
• Nuclear Magnetic Resonance Spectroscopy (NMR)
• Fourier Transform Infrared Spectroscopy (FTIR)
• Ensemble averaging also works well with multiple
  datasets once data acquisition is complete. In
  either case, this method of S/N enhancement
  requires that
• The analyte signal must be stable
• The source of noise is random

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• Digital Filtering
• How Ensemble Averaging Works
• Repeated experiments (scans) are performed on the
  chemical system in question. The scans are
  averaged either in real-time or after the data
  acquisition is complete. A visualization of this
  process is shown below for five spectra of 8.8
  µg/mL 1,1-ferrocenedimethanol in water.

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• Digital Filtering
• Pros of Ensemble Averaging
• Ensemble averaging filters out random noise,
  regardless of the noise frequency
• Ensemble averaging is effective, even if the
  original signal has a S/Nlt1
• Ensemble averaging is straightforward to
  implement
• Improvement in S/N is proportional to
• Cons of Ensemble Averaging
• Requirement of a stable signal
• Ensemble averaging will not work if noise is not
  random (e.g. 60 Hz electrical noise)

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Digital Filtering Example of Ensemble
Averaging These simulated 5-µV gaussian signals
illustrate S/N improvement of ensemble averaging.
The bottom dataset represents a S/N of 2 (single
dataset), the middle dataset represents a S/N of
8 (average of 16 datasets), and the top dataset
represents a S/N of 20 (average of 100 datasets).
Click here to work on an ensemble averaging
exercise.
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• Digital Filtering
• Boxcar Averaging
• Boxcar averaging is a data treatment method that
  the enhances the signal-to-noise of an analytical
  signal by replacing a group of consecutive data
  points with its average. This treatment, which is
  called smoothing, filters out rapidly changing
  signals by averaging over a relatively long time
  but has a negligible effect on slowly changing
  signals. Therefore, boxcar averaging mimics a
  software-based low-pass filter. Boxcar averaging
  can be done both in real time and after data
  acquisition is complete.
• How Boxcar Averaging Works
• During Data Acquisition
• The signal is sampled several times.
  Theoretically, any number of points may be used.
• The samples are summed together and an average is
  calculated.
• The average signal (dependent variable) is stored
  in the smoothed data set as the y-coordinate, and
  the average value of the independent variable
  (e.g. time, wavelength) is used as the
  x-coordinate.

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• Digital Filtering
• How Boxcar Averaging Works
• After Data Acquisition (see figure below)
• Sum the data points within the boxcar
• Divide by the number of points in the boxcar
• Plot the average y-value at the central x-value
  of the boxcar
• Repeat with Boxcar 2, etc until the last full
  boxcar is smoothed

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• Digital Filtering
• Main Points about Boxcar Averaging
• Boxcar averaging is equivalent to software-based
  low-pass filtering.
• Boxcar averaging is straightforward to implement.
• Improvement in S/N is proportional to
• (N-1) points are lost from each boxcar in the
  smoothed data set, where N is the boxcar length.
  The data density of the smoothed data set will be
  reduced by (N-1)/N
• Significant loss of information can occur if the
  length of the boxcar is comparable to the data
  acquisition rate. It is best to implement boxcar
  averaging with a sufficient data acquisition
  rate.

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• Digital Filtering
• Example of Boxcar Averaging
• There are two 5 µV signals below
• Peak at 1.00 minutes with a width of 0.04 minutes
• Peak at 2.00 minutes with a width of 0.40 minutes
• Levels of boxcar averaging are as follows
• Bottom dataset Theoretical S/N of 13 (no
  smoothing)
• Middle dataset Theoretical S/N of 29 (Five-point
  boxcar, 0.05 min long)
• Top dataset Theoretical S/N of 39 (Nine-point
  boxcar, 0.09 min long)
• Notice that little distortion occurs if the peak
  width is much larger than the boxcar and
  significant S/N enhancement is possible.
• Signals with frequencies similar to the rate of
  data acquisition are quickly attenuated,
  analogous to a low-pass RC filter.
• Click here to work on a boxcar averaging exercise.

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Digital Filtering Convolution-Based
Smoothing Overview Digital filtering is a data
treatment method that the enhances the
signal-to-noise ratio of an analytical signal
through the convolution of a data set with an
appropriate filter. This treatment method is
another smoothing technique. If the filter is
unweighted, it will perform in a similar manner
to the boxcar filter. That is, it filters out
rapidly changing signals by averaging over a
relatively long time but has a negligible effect
on slowly changing signals, and it too behaves as
a software-based low-pass filter. However, a
weighted filter may be constructed to mimic a
low-pass, high-pass or even a bandpass filter.
This module will focus on a weighted filter
application based on least-squares quadratic
smoothing that was popularized by Savitzky and
Golay in the 1960s.
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Digital Filtering Convolution Before we explore
the differences in the meaning and construction
of unweighted versus weighted filters, the
concept of convolution needs to be addressed.
Lets start with an analytical signal sampled
every second for ten seconds. The raw data in
this ideal case, which is represented in the
figure below, consists of a slowly changing
peak-shaped function.
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• Digital Filtering
• Convolution (contd)
• For the moment, lets ignore the independent
  variable (i.e. x-axis) and treat this
  instrumental response as a vector. We can
  represent the data above by the following matrix
• x 0 0 1 3 6 7 6 3 1 0 0
• and a three-point unweighted filter to convolve
  the raw data
• f 1 1 1
• The result will be a smoothed data matrix, x
• The convolution process involves the following
  steps
• Matrix multiplication of the first raw data
  segment with the same number of array elements as
  the appropriate filter function, f. The filter
  function has the same sampling rate as the raw
  data.
• This operation is called the dot product.
• So for the first set of three raw data points

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• Digital Filtering
• Convolution (contd)
• Normalizing the dot product with the sum of the
  filter elements and placing the result in the
  smooth data matrix with an x-value equivalent to
  the x-value of the center of the filter function.
• So in this case, x2 has the same time as x2
  (i.e. time 2 s).
• Slide the filter function over one data point and
  repeat the matrix multiplication process, placing
  the next normalized dot product as the next array
  element in the smoothed data matrix. Therefore,
• x3 has the same time as x3 (i.e. time 3 s).
• Repeat step 3 until the leading edge of the
  filter has the same x-value as the last point in
  the raw data matrix. This means that (n-1)/2 data
  points will be lost from each side of x

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Digital Filtering Convolution (contd) Because
the filter function is unweighted, we call this
convolution process the moving window averaging
technique, as shown in the figure below.
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Digital Filtering Convolution (contd) Convolving
the filter function with the original response
in the previous figure results in the smoothed
response below.
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Digital Filtering Effect of Unweighted Filter
Width In the unweighted moving window averaging
approach, we assume that each data point is
equally important in the instrumental response
above. This works well if the peak width is much
larger than the filter width. However, if the
width of the filter is comparable to the peak
width of the signal, applying an unweighted
filter distorts the signal, decreasing the signal
intensity and increasing its width. In the figure
below, the raw data is smoothed by a 3-point,
5-point, and 7-point unweighted filter.
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Digital Filtering Weighted (Savitzky-Golay)
Filters In order to avoid distorting the signal
significantly, one convolves the raw data with a
filter that looks more like the signal itself. A
weighted filter that emphasizes the response at
the central filter element and de-emphasizes the
response at the outer filter elements is used.
This approach, which is called least-squares
polynomial smoothing, was popularized in
analytical chemistry by Savitzky and Golay.
Savitzky and Golay used the least-squares
approach to derive a set of convolution integers
for a given filter width. Below is a list of
Savitzky-Golay coefficients for 5, 9, and
13-point quadratic smoothing of instrumental
responses.
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Digital Filtering Weighted (Savitzky-Golay)
Filters If we use a five-point filter function,
instead of the unweighted function below f 1
1 1 1 1 we would use the Savitzky-Golay
coefficients f -3 12 17 12 -3 using the
original raw data, the normalized dot product for
the first smoothed data point would be
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Digital Filtering Weighted (Savitzky-Golay)
Filters Just like the unweighted moving average
smooth, the raw data would be convolved with the
weighted moving average smooth using the
appropriate Savitzky-Golay coefficients. A
comparison of the 5-point unweighted and weighted
moving average smoothing functions on a noisy
version of the raw data set is shown below.
Notice that the polynomial filter (smoothed
response in black) distorts the signal to a
lesser extent than the unweighted filter
(smoothed response in red).
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• Digital Filtering
• Points to Consider Using Moving Average Filtering
  
• The moving average technique retains greater data
  density than boxcar averaging.
• The moving average technique is straightforward
  to implement.
• Improvement in S/N is proportional to ( filter
  elements)1/2 if the noise is normally
  distributed.
• (N-1)/2 points are lost on either end of the
  smoothed data set, where N is the filter length.
• Significant distortion and loss of resolution may
  occur if the length of the filter is comparable
  to the peak width. It is best to implement a
  moving average with a filter width much smaller
  than the narrowest peak to be smoothed.
• Optimal filter choices are typically chosen in an
  empirical fashion.
• Click here to work on a moving average exercise

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References
Module Description Goals and Objectives
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• References
• Adams, M. J. Acquisition and Enhancement of Data.
  Chemometrics in Analytical Spectroscopy The
  Royal Society of Chemistry Cambridge, 1995 pp
  27 53.
• Binkley, D. Dessy, R. J. Chem. Educ. 1979, 56,
  148.
• Savitzky, A. Golay, M. J. E. Anal. Chem. 1964,
  36, 1627. (Errors in reported equations corrected
  in Steinier, J. Termonia, Y. Deltour, J. Anal.
  Chem. 1972, 44, 1906.)
• Sharaf, M. A. Illman, D. L. Kowalski, B. R.
  Signal Detection and Manipulation. Chemometrics
  John Wiley and Sons New York, 1986 pp 65 117.
• Skoog, D. A. Holler, F. J. Nieman, T. A.
  Principles of Instrumental Analysis Harcourt
  Brace Philadelphia, 1998.
• Instructors Resources
• Click here for suggested approaches to solving
  the exercises.

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• Author
• Steven C. Petrovic
• Department of Chemistry
• Southern Oregon University
• 1250 Siskiyou Blvd.,Ashland, OR 97520
• Email petrovis_at_sou.edu
• Acknowledgements
• Participants at the following ASDL Curriculum
  Development Workshops
• University of Kansas, Lawrence, KS, June 18-22,
  2007
• University of California at Riverside, Riverside,
  CA, June 9-13, 2008
• This work is licensed under a Creative Commons
  Attribution Noncommercial-Share Alike 2.5 License
  

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Signal-to-Noise Enhancement Exercise
1 Introduction Exercise 1 is designed to
familiarize the student with the effect of noise
on the detectability of a signal. This exercise
is designed to be completed with the Signal Noise
Exercise spreadsheet. This spreadsheet allows the
user to create an ideal separation containing up
to three peaks, which represent three different
compounds. The height of each peak is
proportional to the amount of analyte being
separated. A noise component may be added to the
ideal separation in order to simulate data that
could be acquired in an actual separation.
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Signal-to-Noise Enhancement Exercise 1 Table of
Spreadsheet Parameters The table below
describes all of the parameters on the
spreadsheet needed to complete the exercise
below. Parameters with a light yellow background
may be adjusted. Parameters with a light green
background may not be adjusted.
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• Signal-to-Noise Enhancement Exercise 1
• Part 1 Spreadsheet Orientation
• Familiarize yourself with the Signal Noise
  Exercise spreadsheet. Observe changes to the plot
  when
• The peak parameters are adjusted (peak intensity,
  mean, standard deviation)
• The magnitude of the noise is increased from zero
• The magnitude of the offset is increased from
  zero
• Answer the following questions
• Which parameter(s) control the signal level?
• Which parameter(s) control the noise level?
• Which parameter(s) or concept(s) control the
  character of the instrumental response?

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• Signal-to-Noise Enhancement Exercise 1
• Part 2 Evaluating Baseline Noise
• Start with a flat baseline by eliminating all
  traces of signal and noise.
• How would you accomplish this?
• Which param