Transcript Slide 1

Introduction to Signals and Noise
Module Description
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
References
Acknowledgements
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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: [email protected]
This work is licensed under a
Creative Commons Attribution Noncommercial-Share Alike 2.5 License
Introduction to Signals and Noise
Goals and Objectives
Module Description
Goals and Objectives

Goal 1: This module will frame the roles of signal and noise
in chemical measurements.
Signals and Noise
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Sources of Noise
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Signal-to-Noise Enhancement
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Analog Filtering
Digital Filtering
References
Acknowledgements
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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
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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
Introduction to Signals and Noise
Signals and Noise
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Defining Signal and Noise
All analytical data sets contain two components:
signal and noise
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Signal
1.
This is the part of the data that contains
information about the chemical species of interest
(i.e. analyte).
2.
Signals are often proportional to the analyte mass
or analyte concentration
a.
Beer-Lambert Law in spectroscopy where the
absorbance, A, is proportional to concentration, C.
A  bC
Introduction to Signals and Noise
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b.
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The Nernst equation where a measured potential
(E) is logarithmically related to the activity of an
analyte (ax)
RT
E  E 
ln a x
nF
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There are other significant relationships between
signal and analyte concentration
Sources of Noise
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Signals and Noise
c.
Competitive immunoassays (e.g. ELISA) where
labeled (analyte spike) and unlabeled analyte
molecules (unknown analyte) compete for antibody
binding sites
A  kNbindingsites
Clabeled
C(labeledunlabeled)
Introduction to Signals and Noise
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Sources of Noise
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Noise
1.
This is the part of the data that contains
extraneous information.
2.
Noise originates from various sources in a
analytical measurement system, such as:
Signal-to-Noise Enhancement
a.
b.
c.
Analog Filtering
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Signals and Noise
3.
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.
Introduction to Signals and Noise
Module Description
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Sources of Noise
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Figure of Merit
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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
S
mean
x


N s tan dard deviation σ
Introduction to Signals and Noise
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Measuring Signals
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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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Introduction to Signals and Noise
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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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The peak area of a transient signal is the
integrated response, which in this case has units
of (µV*min). 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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Module Description
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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
σ is the standard deviation of the population
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 ( x  μ) 2 

exp 

2σ 2 

y
σ 2π
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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
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Quantifying Noise
Normalization
dividing each member of the data set by the
distribution standard deviation so that σ = 1
The x-axis is now represented by a unitless
quantity, z
z = (x-µ)/σ
Introduction to Signals and Noise
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Normal Error Curve
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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.
Introduction to Signals and Noise
Module Description
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Normal Error Curve
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The area underneath the curve represented by “z”
multiples of the standard deviation are shown in
the table below:
±z
1.00
1.64
1.96
2.58
3.00
Area Represented Under Normal Error Curve
(Confidence Level)
68.3%
90.0%
95.0%
99.0%
99.7%
Introduction to Signals and Noise
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Signal-to-Noise Enhancement
Calculating S/N
BACK
1.
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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Signal Intensity (µV)
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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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Introduction to Signals and Noise
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Estimate root mean square noise
Calculate the standard deviation (VRMS) of the noise. At the 99%
confidence level: VN = ±2.58σ.Therefore:
  VRMS 
Signals and Noise
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V  Vmin   2.50  (2.50)  0.97 V
VN
 max
 2.58
5.16
5.16
Estimate the S/N. The signal is 16.0 µV.
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Introduction to Signals and Noise
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Estimating S/N
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First, calculate the standard deviation (VRMS) of the noise.
At the 99% confidence level: VN = ±2.58σ.Therefore:
V  Vmin   2.50  (2.50)  0.97 V
VN

 max
 2.58
5.16
5.16
Second, calculate the S/N. The signal is 16.0 µV.
S 16.0 μV

 16.5
N 0.97 µV
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Signal Intensity (µV)
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Introduction to Signals and Noise
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Sources of Instrumental Noise
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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:
VRMS  4kTR f
VRMS:Root-mean-square noise voltage with a frequency
bandwidth of Δf (in Hertz).
k:
Boltzmann’s constant (1.38 x 10-23 J/K)
T:
Temperature (K)
R:
Resistance of resistive element (Ω)
Reduction of Johnson Noise is accomplished most easily by:

Cooling the detector (reducing T)

Decreasing the frequency bandwidth of the signal
(reducing Δf)

Actual measurements of Johnson Noise may be found by
clicking here
Introduction to Signals and Noise
Module Description
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2.
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
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Acknowledgements
Sources of Instrumental Noise
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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:
iRMS  2Ie f
iRMS:Root-mean-square current fluctuation (in Amperes)
I: Average direct current (A)
e: electronic charge (1.60 x 10-19 C)
Δf: frequency bandwidth (Hz)
Reduction of Shot Noise is accomplished most easily by:
Decreasing the frequency bandwidth of the signal
(reducing Δf)
A good discussion of Shot Noise may be found by clicking here
Introduction to Signals and Noise
Module Description
BACK
3.
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 (<100 Hz)
A good summary of flicker noise (and Johnson noise) may
be found by clicking here
4.
Environmental Noise
These are sources of noise that interfere with analytical
measurements. Examples of such sources include:
Goals and Objectives
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Signal-to-Noise Enhancement
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Sources of Instrumental Noise
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)
Introduction to Signals and Noise
Introduction to Signal-to-Noise Enhancement
Module Description
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-topeak 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)
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
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SignalIntensity
Intensity(µV)
(µV)
Signal
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Introduction to Signal-to-Noise Enhancement
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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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Introduction to Signals and Noise
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Introduction to Signal-to-Noise Enhancement
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Can Noise be Reduced After the Data has been Recorded?
Signals and Noise
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.
Sources of Noise
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Introduction to Signals and Noise
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Module Description
Goals and Objectives
Signals and Noise
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2.
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
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Acknowledgements
1.
2.
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 Δf).
Signal Averaging (i.e. decreasing Δf 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.
Introduction to Signals and Noise
Analog Filtering
Module Description
Signals and noise are almost always expressed as electrical
quantities. The electrical quantities you should be familiar with are:
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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.
o
Acknowledgements

Here’s an applet that demonstrates the production of pulsed DC:
http://micro.magnet.fsu.edu/electromag/java/generator/dc.html
Alternating Current (AC): Charges change direction periodically.
o
Here’s an applet that demonstrates the production of AC:
http://micro.magnet.fsu.edu/electromag/java/generator/ac.html
Introduction to Signals and Noise
Module Description
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Analog Filtering - Ohm’s Law
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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 Ohm’s Law, which is
mathematically expressed as:
V
R
I
V = Voltage across the conductor (in Volts, V)
I = Current through the conductor (in Amperes, A)
R = Resistance of the conductor (in Ohms, Ω)
Simple applets to test out Ohm’s Law:
http://micro.magnet.fsu.edu/electromag/java/ohmslaw/
http://phet.colorado.edu/simulations/veqir/VeqIRColored.swf
http://www.walter-fendt.de/ph14e/ohmslaw.htm
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Passive Electronic Components
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Resistor
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A resistor is a component that resists electron flow.

The unit of resistance is called an ohm (Ω). 1Ω = 1V/A

In an electronic circuit schematic, a resistor is represented by:
1.0 K
Sources of Noise
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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:
C
References
Acknowledgements
A simple applet used to illustrate the principle of resistance
http://micro.magnet.fsu.edu/electromag/java/filamentresistance/index.html
Simple applets used to illustrate the principle of capacitance
http://micro.magnet.fsu.edu/electromag/java/capacitance/index.html
http://micro.magnet.fsu.edu/electromag/java/capacitor/
Introduction to Signals and Noise
Module Description
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Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Passive Electronic Circuits
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BACK
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, let’s 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 Ohm’s Law.
Analog Filtering
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Acknowledgements
Circuit #1
V 1.0 Volt
According to Ohm’s Law I  
 0.050 Amperes
R
20
Ohms
o
V = 1.0 Volt
o
R = 20 Ohms
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Passive Electronic Circuits
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Resistors in Series
Practically speaking, we are not limited to a single resistor. Circuit
#1 could also be represented by Circuit #2 below:
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


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
RT  (R1  R2 )  10   10   20 
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Passive Electronic Circuits
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Resistors in Parallel
Resistors placed in a “side-to-side” configuration are in parallel.
Signals and Noise
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Acknowledgements
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:
RT 
1
R1
Applying this to Circuit #3:
R1R2
1

 R1 R1  R2
2
RT 
1
40 * 40

 20 ohms
1
1
 40 40  40
40
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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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
Signals and Noise
Sources of Noise
10 ohms
R1
Signal-to-Noise Enhancement
Analog Filtering
V(in) = 1.0 V
V1
V(out) = ?
Digital Filtering
10 ohms
R2
References
Acknowledgements


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
Introduction to Signals and Noise
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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 Ohm’s Law
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10 ohms
R1
Signal-to-Noise Enhancement
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V(in) = 1.0 V
V1
V(out) = ?
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Vin  IR1  R2 
10 ohms
R2
References
Acknowledgements
Vin = I(R1 + R2)
For a discussion of loading errors, click here.
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Voltage Divider (Page 3)
If the readout device (i.e. a meter) is placed across R2, than the
voltage read by the meter is
Signals and Noise
Sources of Noise
10 ohms
R1
Signal-to-Noise Enhancement
Analog Filtering
V(in) = 1.0 V
V1
V(out) = ?
Digital Filtering
References
Acknowledgements
10 ohms
R2
Vout
IR 2
R2


Vin
IR1  R 2  R1  R 2
Or in other words, the divider output equals the instrument output
multiplied by R2 over the total resistance
 R2 

Vout  Vin 
R

R
 1
2
Introduction to Signals and Noise
Module Description
Goals and Objectives
BACK
Passive Electronic Circuits
Voltage Divider (Page 4)
In this case, the divider output is:
Signals and Noise
10 ohms
R1
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
V(in) = 1.0 V
V1
V(out) = ?
Digital Filtering
10 ohms
R2
References
Acknowledgements


10 
  0.5 V
Vout  1.0 V
10


10



NEXT
Introduction to Signals and Noise
BACK
Module Description
Goals and Objectives
Signals and Noise
However, the impedance of a capacitor is frequency dependent,
as shown by the following equation:
Signal-to-Noise Enhancement
1
XC 
2πfC
Analog Filtering
References
Acknowledgements
NEXT
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.
Sources of Noise
Digital Filtering
Passive Electronic Circuits
•
•
•
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!
Introduction to Signals and Noise
Goals and Objectives
•
•
Signals and Noise
•
Sources of Noise
Passive Electronic Circuits
BACK
Module Description
NEXT
Low-Pass Filters
Used when the signal frequency < 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)
Signal-to-Noise Enhancement
10 ohms
R1
Analog Filtering
Digital Filtering
V(in) = 1.0 V
V1
References
V(out) = ?
10 µF
C1
Acknowledgements
Vout
 12πf C 
 XC 
1

  Vin 
  Vin 
 Vin 


1
R

X
R

2
π
fRC

1


c 

2πf C

Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
Passive Electronic Circuits
NEXT
BACK
High-Pass Filters
•
Used when the signal frequency > 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)
Signal-to-Noise Enhancement
V(out) = ?
10 ohms
R1
Analog Filtering
Digital Filtering
V(in) = 1.0 V
V1
References
Acknowledgements
10 µF
C1


 R 
R
2πfRC 
  Vin 
  Vin 
Vout  Vin 


1
R

2
π
fRC

1


 R  Xc 
2πf C

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
•
Decibel Scale
NEXT
BACK
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.
Introduction to Signals and Noise
Goals and Objectives
Bode Plots
BACK
Module Description
•
NEXT
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.
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
0.1
0
-5
-10
Analog Filtering
Digital Filtering
dB
-15
References
Acknowledgements
-20
-25
-30
-35
-40
1
10
Frequency (Hz)
100
1000
10000
100000
Introduction to Signals and Noise
Goals and Objectives
Bode Plots
BACK
Module Description
•
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/(2πRC)
The cutoff frequency, which is 1592 Hz for this particular circuit,
corresponds to a 3 dB attenuation, and can be used as a figure-ofmerit for the response of the filter.
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
0.1
Digital Filtering
0
-5
-10
References
dB
-15
Acknowledgements
NEXT
-20
-25
-30
-35
-40
1
10
Frequency (Hz)
100
1000
10000
100000
Introduction to Signals and Noise
Goals and Objectives
Bode Plots
BACK
Module Description
•
NEXT
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.
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
0.1
Analog Filtering
References
Acknowledgements
dB
Digital Filtering
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
1
10
Frequency (Hz)
100
1000
10000
100000
Introduction to Signals and Noise
BACK
Module Description
Passive Electronic Circuits
Goals and Objectives
Analog Filter Demo
Signals and Noise
Sources of Noise
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.
Signal-to-Noise Enhancement
Bode Plot Exercise
An exercise on interpreting the frequency response of RC filters
using a Bode plot can be accessed by clicking here.
Analog Filtering
Digital Filtering
Analog Filter Exercises
References
A couple of exercises have been included to reinforce your
understanding about the design and application of analog filters.
Acknowledgements
•
•
Click here to access Exercise #1
Click here to access Exercise #2
Introduction to Signals and Noise
Digital Filtering
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
References
Acknowledgements
NEXT
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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Module Description
Digital Filtering
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Ensemble Averaging
Goals and Objectives
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:
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement


Nuclear Magnetic Resonance Spectroscopy (NMR)
Fourier Transform Infrared Spectroscopy (FTIR)
Analog Filtering
Ensemble averaging also works well with multiple datasets once
data acquisition is complete. In either case, this method of S/N
enhancement requires that:
Digital Filtering
References
Acknowledgements


The analyte signal must be stable
The source of noise is random
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
0.012
0.01
Analog Filtering
Acknowledgements
0.008
Absorbance
References
NEXT
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.
Signal-to-Noise Enhancement
Digital Filtering
Digital Filtering
BACK
0.006
0.004
0.002
Sum the data
points at l 1, divide
by the number of
spectra, and plot
the average
0
350
370
390
Repeat the averaging process at each subsequent wavelength
(data acquired at l 2, l 3,…, ln are highlighted in dashed
rectangles). The ensemble average at each wavelength for
these five spectra is plotted here as a black line.
410
430
450
Wavelength (nm)
470
490
510
530
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Sources of Noise
BACK
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NEXT
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/N<1

Ensemble averaging is straightforward to implement

Improvement in S/N is proportional to:
Signal-to-Noise Enhancement
# of datasets averaged together
Analog Filtering
Digital Filtering
References
Acknowledgements
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)
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
References
Acknowledgements
Signal Intensity (µV)
Digital Filtering
NEXT
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.
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
BACK
40
30
20
10
0
-10
-20
0
1
2
3
Time (min)
4
5
Introduction to Signals and Noise
Module Description
BACK
Digital Filtering
NEXT
Boxcar Averaging
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
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 softwarebased low-pass filter. Boxcar averaging can be done both in real
time and after data acquisition is complete.
Analog Filtering
Digital Filtering
References
Acknowledgements
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 xcoordinate.
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
BACK
Digital Filtering
NEXT
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
0.011
0.009
Analog Filtering
Digital Filtering
References
Acknowledgements
0.007
l boxcar1 l boxcar2
0.005
l1 l2 l3
0.003
Sum the data points within the boxcar, divide by the number of points in
the boxcar, and plot the average at the middle boxcar point (i.e. l 2 for
the first 3-point boxcar, l 5 for the second 3-point boxcar, etc.)
0.001
-0.001
570 550 530 510 490 470 450 430 410 390 370
Wavelength (nm)
Absorbance (Raw Data)
Signal-to-Noise Enhancement
Introduction to Signals and Noise
BACK
Module Description
Goals and Objectives
Signals and Noise
Analog Filtering
Digital Filtering
References
Acknowledgements
NEXT
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:
# of data point s in boxcar
Sources of Noise
Signal-to-Noise Enhancement
Digital Filtering


(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.
Introduction to Signals and Noise
Goals and Objectives
Signals and Noise




Analog Filtering

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.

Signal (µV)
Acknowledgements
Peak at 1.00 minutes with a width of 0.04 minutes
Peak at 2.00 minutes with a width of 0.40 minutes



Signal-to-Noise Enhancement
References
NEXT
Example of Boxcar Averaging:

There are two 5 µV signals below
Sources of Noise
Digital Filtering
Digital Filtering
BACK
Module Description
20
15
10
5
0
-5
0
1
2
3
Time (min)
4
5
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
BACK
Digital Filtering
NEXT
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 lowpass, 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
1960’s.
Introduction to Signals and Noise
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Goals and Objectives
Signals and Noise
Sources of Noise
Digital Filtering
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NEXT
Convolution
Before we explore the differences in the meaning and construction
of unweighted versus weighted filters, the concept of convolution
needs to be addressed. Let’s 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.
8
Signal-to-Noise Enhancement
7
6
Digital Filtering
References
Response (µV)
Analog Filtering
5
4
3
2
Acknowledgements
1
0
-1
0
2
4
6
Time (s)
8
10
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
BACK
Digital Filtering
Convolution (cont’d)
For the moment, let’s 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=[00136763100]
and a three-point unweighted filter to convolve the raw data
f=[111]
The result will be a smoothed data matrix, x’
The convolution process involves the following steps:
1.
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.
a.
This operation is called the dot product.
n
References
f  x   fi xi
(n  length of f ilter)
i 1
Acknowledgements
NEXT
b.
So for the first set of three raw data points:
f  x  f1 f2
 x1 
 
f3    x 2   f1x1  f2 x 2  f3 x 3 
 x 3 
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
NEXT
Convolution (cont’d)
2.
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.
x '2 
Sources of Noise
f x
n
 fi

f1x1  f2 x 2  f3 x 3 
f1  f2  f3 
i1
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
BACK
3.
a.
So in this case, x’2 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,
n
Digital Filtering
fi x(i1)

f x  f x  f x 
f  x i1
x '3 

 1 2 2 3 3 4
n
n
f1  f2  f3 
 fi
 fi
References
i1
Acknowledgements
4.
i1
a.
x’3 has the same time as x3 (i.e. time = 3 s).
Repeat step 3 until the leading edge of the filter has the same xvalue as the last point in the raw data matrix. This means that (n1)/2 data points will be lost from each side of x’
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Digital Filtering
BACK
NEXT
Convolution (cont’d)
Because the filter function is unweighted, we call this convolution
process the moving window averaging technique, as shown in the
figure below.
8
Sources of Noise
x'5 =(x4+x5+x6)/3
6
Analog Filtering
Digital Filtering
References
Response (µV)
Signal-to-Noise Enhancement
x'4 =(x3+x4+x5)/3
4
2
x'3 =(x2+x3+x4)/3
x'2 =(x1+x2+x3)/3
0
0
Acknowledgements
x'10 =(x9+x10+x11)/3
-2
5
Slide filter function over one point and apply to raw data until filter is aligned with
last set of n data points, where n is the number of elements of the filter
-4
Time (s)
10
Introduction to Signals and Noise
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Goals and Objectives
Digital Filtering
BACK
NEXT
Convolution (cont’d)
Convolving the filter function with the original response in the
previous figure results in the smoothed response below.
Signals and Noise
7
Sources of Noise
6
5
Analog Filtering
4
Digital Filtering
References
Response (µV)
Signal-to-Noise Enhancement
3
2
1
Acknowledgements
0
0
2
4
6
-1
Time (s)
8
10
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Digital Filtering
BACK
NEXT
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.
7
Analog Filtering
6
Digital Filtering
References
Response (µV)
5
4
3
2
Acknowledgements
1
0
0
2
4
6
-1
Time (s)
8
10
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
Digital Filtering
BACK
NEXT
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.
Filter Points
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Normalizing Factor
13
-11
0
9
16
21
24
25
24
21
16
9
0
-11
143
9
5
-21
14
39
54
59
54
39
14
-21
-3
12
17
12
-3
231
35
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Digital Filtering
BACK
Weighted (Savitzky-Golay) Filters
If we use a five-point filter function, instead of the unweighted
function below
f=[11111]
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
Analog Filtering
Digital Filtering
x '3 
fx
n
 fi
f x  f x  f x  f x  f x 
 1 1 2 2 3 3 4 4 5 5
f1  f2  f3  f4  f5 
i1
References
Acknowledgements
NEXT

( 3 * 0)  (12 * 0)  (17 * 1)  (12 * 3)  ( 3 * 6)
( 3)  12  17  12  ( 3)
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Acknowledgements
8
Response (µV)
References
NEXT
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).
Analog Filtering
Digital Filtering
Digital Filtering
BACK
6
4
2
0
-2 0
2
4
6
Time (s)
8
10
Introduction to Signals and Noise
BACK
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Digital Filtering
NEXT
Points to Consider Using Moving Average Filtering
Goals and Objectives

Signals and Noise


Sources of Noise

Signal-to-Noise Enhancement

Analog Filtering
Digital 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.
References
Click here to work on a moving average exercise
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References
Module Description
Goals and Objectives
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1.
Signals and Noise
Sources of Noise
2.
3.
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
References
Acknowledgements
4.
5.
References
NEXT
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.
Instructor’s Resources
Click here for suggested approaches to solving the exercises.
Acknowledgements
Module Description
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Author
Steven C. Petrovic
Department of Chemistry
Southern Oregon University
1250 Siskiyou Blvd.,Ashland, OR 97520
Email: [email protected]
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering

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
References
Acknowledgements
This work is licensed under a Creative Commons
Attribution Noncommercial-Share Alike 2.5 License
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
Sources of Noise
BACK
Signal-to-Noise Enhancement Exercise #1
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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.
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
References
Acknowledgements
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.
Introduction to Signals and Noise
Module Description
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Signal-to-Noise Enhancement Exercise #1
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Table of Spreadsheet Parameters:
Goals and Objectives
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.
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
Parameter
S/N (signal-tonoise ratio)
Peak Intensity
Mean
Standard Deviation
(Std Dev)
Noise
References
Offset
Acknowledgements
RMS Noise
Peak + Noise
Definition
This figure of merit indicates the magnitude of the signal with respect
to the noise level at the 95% or 99% confidence level.
This value controls the height of the analyte peak. A value of zero
indicates no analyte present.
This value controls the position of the maximum peak intensity.
Values are restricted between 0.5 to 4.5 minutes
This value controls the width of the peak. Values are restricted
between 0.001 and 0.250 minutes
This value controls the amount of noise added to the plot. Since it is
based on Excel’s RAND function, it appears as high-frequency noise
superimposed on the analyte peaks. This is also known as the peakto-peak noise.
This value is used to raise or lower the baseline of this plot. It is most
effective when the plot has large peaks on a noiseless baseline.
This value is the magnitude of the noise (Max Signal – Min Signal)
divided by 5.16 or 3.92 (±zσ), where z = 2.58 or 1.96 (99% or 95%
confidence level). Dividing the RMS noise into the peak intensity
provides the S/N for that analyte.
This cell represents the sum of the peak intensity and the
superimposed noise.
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
BACK
Signal-to-Noise Enhancement Exercise #1
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Part 1: Spreadsheet Orientation
1.
Familiarize yourself with the Signal Noise Exercise spreadsheet.
Observe changes to the plot when:
a.
The peak parameters are adjusted (peak intensity, mean,
standard deviation)
b.
The magnitude of the noise is increased from zero
c.
The magnitude of the offset is increased from zero
2.
Answer the following questions
a.
Which parameter(s) control the signal level?
b.
Which parameter(s) control the noise level?
c.
Which parameter(s) or concept(s) control the character of the
instrumental response?
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
References
Acknowledgements
Introduction to Signals and Noise
Module Description
Goals and Objectives
Signals and Noise
BACK
Signal-to-Noise Enhancement Exercise #1
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Part 2: Evaluating Baseline Noise
1.
Start with a flat baseline by eliminating all traces of signal and
noise.
a.
How would you accomplish this?
b.
Which parameter would you adjust if you can’t see the flat
baseline?
2.
Calculating Noise Magnitude
a.
Based on the discussion of noise in this e-module, if the
peak-to-peak noise (VN) is 1.0 µV, calculate the RMS Noise
at the 99% confidence level.
b.
Increase the noise level to 1.0 µV on the spreadsheet and
look at the RMS Noise. Does your answer agree with the
spreadsheet?
c.
Repeat the RMS noise calculation at the 95% confidence
level. Is the RMS Noise smaller or larger? Explain this
difference based on your knowledge of population
distributions.
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
References
Acknowledgements
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
BACK
Signal-to-Noise Enhancement Exercise #1
NEXT
Part 3: Evaluating Signal-to-Noise Ratios
1.
Enter the following signal parameters into the Signal Noise
Exercise spreadsheet
a.
Peak 1
i.
Peak Intensity = 10 µV
ii.
Mean = 1 minute
iii.
Standard Deviation = 0.1 minute
b.
Peak 2
a.
Peak intensity = 5 µV
b.
Mean = 2 minutes
c.
Standard Deviation = 0.1 minute
c.
Peak 3
a.
Peak Intensity = 1 µV
b.
Mean = 3 minutes
c.
Standard Deviation = 0.1 minute
Introduction to Signals and Noise
Module Description
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Return to S/N Enhancement
Signal-to-Noise Enhancement Exercise #1
Goals and Objectives
Signals and Noise
Part 3: Evaluating Signal-to-Noise Ratios
Sources of Noise
1.
Signal-to-Noise Enhancement
a.
Analog Filtering
b.
c.
Digital Filtering
d.
References
e.
Acknowledgements
2.
3.
Change the peak-to-peak noise level on the spreadsheet to 1.00
µV.
Look at the S/N ratio for each peak and determine which
peaks are below the detection limit (S/N < 3)
Record the “Peak+Noise” magnitude for each peak.
Obtain replicate data by pressing F9 to refresh the
spreadsheet. Record the new magnitude for each peak.
For each analyte peak, calculate the percent relative error
range and average percent relative error introduced by
adding 1.00 µV peak-to-peak noise to the simulated signal.
As the S/N decreases, comment on any trends regarding the
effect of noise on the accuracy and precision of the signal
Increase the peak-to-peak noise level on the spreadsheet to 5.00
µV and determine which peaks are now below the detection limit.
Based on your results, what role would signal-to-noise
enhancement have in analyte detection?
Introduction to Signals and Noise
Module Description
1.
Goals and Objectives
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Bode Plot Exercise #1
BACK
Using the Bode Plot spreadsheet, answer the following questions
about the circuit below.
a.
Calculate the cutoff frequency and check it against the
spreadsheet.
b.
Determine if this circuit can increase the S/N by a factor of
ten or more with minimal signal attenuation for the following
frequencies:
i.
Signal: 1 Hz, Noise: 5000 Hz
ii.
Signal: 1000 Hz, Noise: 50000 Hz
iii.
Signal: 1 Hz, Noise: 60 Hz
Analog Filtering
4700 ohms
R1
Digital Filtering
References
V(in) = 0.1 V
V1
Acknowledgements
V(out) = ?
0.10 µF
C1
Introduction to Signals and Noise
Analog Filtering Exercise #1
Module Description
1.
Goals and Objectives
a.
Signals and Noise
b.
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
c.
d.
NEXT
The schematic below is an example of a passive RC filter
Is the filter currently configured as a low-pass filter or a highpass filter? Explain how you would convert from one type of
filter to the other.
If the magnitude of the input voltage is 1.00 V, calculate the
output voltage when the input frequency is:
a.
1 Hz
b. 100 Hz
c. 5000 Hz
Which frequency is least affected by the RC filter?
If a 1 Hz signal with a magnitude of 1.00V has 5000 Hz noise
superimposed upon it (also 1.00V)
a.
Calculate the S/N of the unfiltered signal (99% C.L.)
b.
Calculate the S/N after filtering with the circuit below
Digital Filtering
10 ohms
R1
References
Phono Plug to Laptop
Acknowledgements
V(signal generators)
10 µF
C1
Introduction to Signals and Noise
Module Description
2.
Goals and Objectives
Signals and Noise
Sources of Noise
Analog Filtering Exercise #2
BACK
Below is an HPLC peak with 60 Hz noise.
Estimate the frequency of the analyte signal by assuming that the
HPLC peak is one-half of a sine wave. Double the peak width to obtain
the period of the signal, multiply by 60 to convert from minutes to
seconds, then take the inverse to obtain the signal frequency in Hertz.
If only a 47 µF capacitor is available in your lab, design a RC filter
that will improve the S/N of the chromatogram by a factor of three
while having a minimal effect on the signal (<0.5% decrease in
signal)
Signal-to-Noise Enhancement
120
Analog Filtering
References
Acknowledgements
Signal Intensity (µV)
Digital Filtering
100
80
60
40
20
0
-20
0
0.2
0.4
0.6
Time (min)
0.8
1
1.2
Introduction to Signals and Noise
BACK
Module Description
An ensemble averaging spreadsheet similar to the Signal Noise
Exercise spreadsheet can be accessed by clicking here.
Goals and Objectives
1.
Adjust the following parameters in the ensemble averaging
spreadsheet:
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
Ensemble Averaging Exercise
•
Peak #1 Intensity = 2 µV (Peaks 2 & 3 should have zero
intensity)
•
Peak #1 Mean = 1 minute
•
Peak #1 Standard Deviation = 0.01 minute
•
Noise = 6 µV
2.
References
Acknowledgements
Adjust the number of datasets averaged in the spreadsheet
•
Top = 1 dataset
•
Middle = 16 datasets
•
Bottom = 100 datasets
3.
How does the S/N improve with the number of datasets averaged?
4.
Compare the appearance of the original signal with the ensemble
averaged signal.
Introduction to Signals and Noise
BACK
Module Description
1.
Adjust the following parameters in the boxcar averaging
spreadsheet:
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
•
Peak Intensity = 5 µV (All 3 peaks)
•
Peak #1 Mean & Standard Deviation = (1.000 ± 0.005) min
•
Peak #2 Mean & Standard Deviation = (2.000 ± 0.025) min
•
Peak #3 Mean & Standard Deviation = (3.000 ± 0.250) min
•
Noise = 2 µV
2.
References
Acknowledgements
NEXT
A boxcar averaging spreadsheet similar to the Ensemble
Averaging spreadsheet can be accessed by clicking here.
Goals and Objectives
Digital Filtering
Boxcar Averaging Exercise #1
3.
Adjust the number of boxcar elements for each dataset
•
Top = 9 data points
•
Middle = 3 data points
•
Bottom = 1 data point (raw data)
Which of the peak parameters is the most significant when using
boxcar averaging to increase S/N? Support your choice based on
what you observe in the boxcar averaging spreadsheet.
Introduction to Signals and Noise
BACK
Module Description
1.
Adjust the following parameters in the boxcar averaging
spreadsheet:
Goals and Objectives
•
Peak Intensity = 4 µV (Peak #1), 2.5 µV (Peak #2), 1 µV
(Peak #3)
•
Peak #1 Mean & Standard Deviation = (1.00 ± 0.15) min
•
Peak #2 Mean & Standard Deviation = (2.00 ± 0.15) min
•
Peak #3 Mean & Standard Deviation = (4.00 ± 0.15) min
•
Noise = 2.0 µV
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
2.
Digital Filtering
References
3.
Acknowledgements
Boxcar Averaging Exercise #2
Adjust the number of boxcar elements for each dataset
•
Top = 9 data points
•
Middle = 5 data points
•
Bottom = 1 data point (raw data)
Discuss the ability of boxcar filters to clearly extract signals from
noise at or below the detection limit based on S/N enhancement.
Introduction to Signals and Noise
BACK
Module Description
NEXT
A moving average spreadsheet similar to the Ensemble Averaging
spreadsheet can be accessed by clicking here.
Goals and Objectives
1.
Remove all noise from the data, and adjust the following
parameters in the moving average spreadsheet:
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
Moving Average Exercise (Pg. 1)
•
Peak Intensity = 5 µV (All 3 peaks)
•
Peak #1 Mean & Standard Deviation = (1.00 ± 0.02) min
•
Peak #2 Mean & Standard Deviation = (2.00 ± 0.05) min
•
Peak #3 Mean & Standard Deviation = (3.00 ± 0.10) min
•
Offset = 1 µV
2.
Select unweighted (MA) as the filter type for Dataset 1 and set the
length of the smoothing window to 1. Select MA as the filter type
for Dataset 2 and observe what happens to the smoothed data as
you increase the size of the filter window.
3.
Select weighted (SG) as the filter type for Dataset 2 and observe
what happens to the smoothed data as the size of the filter window
increases.
4.
Compare and contrast unweighted and weighted filters and their
effect on the distortion of the peak width and height of the original
signal.
References
Acknowledgements
Introduction to Signals and Noise
BACK
Module Description
Goals and Objectives
Signal-to-Noise Enhancement
Analog Filtering
Digital Filtering
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Acknowledgements
NEXT
5.
Adjust the number of elements in both moving average filters to 5
points. Make one filter unweighted (MA) and one weighted (SG)
Compare and contrast the effect of unweighted and SavitzkyGolay smoothing filters with equal lengths on peak distortion.
6.
Repeat this exercise with filter lengths of 9, 13, 17 and 21
elements Under what conditions does minimal peak distortion
occur? Will an unweighted filter always distort the original signal?
Will a weighted signal always provide an undistorted signal?
Signals and Noise
Sources of Noise
Moving Average Exercise (Pg. 2)
Introduction to Signals and Noise
BACK
Module Description
7.
Remove all noise from the data, and adjust the following
parameters in the moving average spreadsheet:
Goals and Objectives
•
Peak Intensity = 1 µV (Peak #1), 1 µV (Peak #2), 1 µV (Peak
#3).
•
Peak #1 Mean & Standard Deviation = (1.00 ± 0.02) min
•
Peak #2 Mean & Standard Deviation = (2.00 ± 0.04) min
•
Peak #3 Mean & Standard Deviation = (4.00 ± 0.10) min
•
Offset = 1 µV
Signals and Noise
Sources of Noise
Signal-to-Noise Enhancement
Analog Filtering
Moving Average Exercise (Pg. 3)
8.
Select unweighted (MA) as the filter type for Dataset 1 and set the
length of the smoothing window to 1. Select MA as the filter type
for Dataset 2 and set the length of the smoothing window to 5. The
peak distortion for the widest peak will be ~1%. Gradually increase
the noise until Peak #3 in Dataset #1 has a S/N of ~3. Then select
weighted (SG) as the filter type for Dataset #2 and repeat the
gradual increase in the noise level. Note the appearance of the
raw and smoothed signals in both cases as the noise increases.
9.
Compare and contrast the ability of properly sized unweighted and
weighted filters to clearly extract signals from noise based on S/N
enhancement. Which of the S/N enhancement approaches best
enhances signals at or below the detection limit?
Digital Filtering
References
Acknowledgements