signals and noise introduction to signals and noise module description this e-module provides an...

Introduction to Signals and NoiseSignals and Noise

Module DescriptionThis 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

Module Description Goals and Objectives Signals and Noise Sources of Noise Signal-to-Noise Enhancement

Analog Filtering

Digital Filtering References

Acknowledgements

mailto:[email protected]

http://creativecommons.org/licenses/by-nc-sa/2.5/


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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Signals and Noise

Defining Signal and Noise

All analytical data sets contain two components: signal and noise

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.


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Signals and Noise

There are other significant relationships between signal and analyte concentration

b. The Nernst equation where a measured potential (E) is logarithmically related to the activity of an analyte (ax)

c. Competitive immunoassays (e.g. ELISA) where labeled (analyte spike) and unlabeled analyte molecules (unknown analyte) compete for antibody binding sites


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Signals and Noise

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:

a. Detectors

b. Photon Sources

c. 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).

3. 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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http://www.wam.umd.edu/~toh/spectrum/TOC.html

http://www.wam.umd.edu/~toh/spectrum/SignalsAndNoise.html


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

σ

x

deviationdardtans

mean

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Measuring SignalsIf 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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Measuring SignalsIf 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 SignalsThe 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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Quantifying NoiseAll 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

π2σ

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Quantifying NoiseOf 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:

1. Mean-Centering subtracting the population mean from all the members of the data set so that µ = 0

2. 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


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Normal Error CurveIf 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 CurveThe area underneath the curve represented by “z” multiples of the standard deviation are shown in the table below:

±z Area Represented Under Normal Error Curve (Confidence Level)

1.00 68.3%

1.64 90.0%

1.96 95.0%

2.58 99.0%

3.00 99.7%


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Calculating S/NCalculating the signal to noise ratio based on our brief discussion of Gaussian statistics can be achieved as follows:

1. 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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Estimate root mean square noise1. Calculate the standard deviation (VRMS) of the noise. At the 99%

confidence level: VN = ±2.58σ.Therefore:

2. Estimate the S/N. The signal is 16.0 µV.

V97.0

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Estimating S/N

First, calculate the standard deviation (VRMS) of the noise.

At the 99% confidence level: VN = ±2.58σ.Therefore:

Second, calculate the S/N. The signal is 16.0 µV.

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Sources of Instrumental Noise1. 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: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

fkTR4VRMS

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http://yalelab.wikidot.com/start



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Sources of Instrumental Noise

2. Shot NoiseThis source of noise results in current fluctuations produced by electrons crossing a junction in a random fashion, which highlights the quantized nature of electron flowThe relationship between Shot Noise and experimental parameters is as follows:

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

fIe2iRMS

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http://www.st-andrews.ac.uk/~jcgl/Scots_Guide/iandm/part3/page1.html



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Sources of Instrumental Noise

3. Flicker NoiseFlicker 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 NoiseThese 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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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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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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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:

1. Analog Filtering (Hardware-Based)

2. Digital Filtering (Software-Based)

Most of these S/N enhancement methods, whether analog or digital, are based on either:

1. Bandwidth Reduction (i.e. decreasing Δf).

2. 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.

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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. o 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

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http://micro.magnet.fsu.edu/electromag/java/generator/dc.html

http://micro.magnet.fsu.edu/electromag/java/generator/dc.html

http://micro.magnet.fsu.edu/electromag/java/generator/ac.html

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Analog Filtering - Ohm’s 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 Ohm’s 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, 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

Resistor 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:

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 resistancehttp://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/

1.0 K

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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, 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.

Circuit #1

According to Ohm’s Lawo V = 1.0 Volto R = 20 Ohms

V11.0 V

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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:

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

R210 ohms

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201010)RR(R 21T

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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.

V11.0 V

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R240 ohms

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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

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

V1V(in) = 1.0 V

R110 ohms

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V(out) = ?

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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

Vin = I(R1 + R2)

For a discussion of loading errors, click here.

V1V(in) = 1.0 V

R110 ohms

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V(out) = ?

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Assuming that the meter resistance is much larger than R2 (i.e. no loading error occurs), then according to Ohm’s Law

Vin = I(R1 + R2)

For a discussion of loading errors, click here.



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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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In this case, the divider output is:

V1V(in) = 1.0 V

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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!

fCπ2

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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)

C110 µF

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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)

V1V(in) = 1.0 V

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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/(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-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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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/N<1 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)

togetheraverageddatasetsof#

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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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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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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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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.

boxcarinsintpodataof#

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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 1960’s.

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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. 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.

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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 = [ 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:

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.

b. So for the first set of three raw data points:

)filteroflengthn(xfxfn

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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.

a. So in this case, x’2 has the same time as x2 (i.e. time = 2 s).

3. 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,

a. x’3 has the same time as x3 (i.e. time = 3 s).

4. 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


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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x'4 =(x3+x4+x5)/3

x'5 =(x4+x5+x6)/3

x'10 =(x9+x10+x11)/3

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

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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.

Filter Points 13 9 5 -6 -11 -5 0 -4 9 -21 -3 16 14 -2 21 39 -3 -1 24 54 12 0 25 59 17 1 24 54 12 2 21 39 -3 3 16 14 4 9 -21 5 0 6 -11

Normalizing Factor 143 231 35

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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

)3(121712)3(

)6*3()3*12()1*17()0*12()0*3(

fffff

xfxfxfxfxf

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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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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

References

1. Adams, M. J. Acquisition and Enhancement of Data. Chemometrics in Analytical Spectroscopy; The Royal Society of Chemistry: Cambridge, 1995; pp 27 – 53.

2. Binkley, D.; Dessy, R. J. Chem. Educ. 1979, 56, 148.

3. 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.)

4. Sharaf, M. A.; Illman, D. L.; Kowalski, B. R. Signal Detection and Manipulation. Chemometrics; John Wiley and Sons: New York, 1986; pp 65 – 117.

5. Skoog, D. A.; Holler, F. J.; Nieman, T. A. Principles of Instrumental Analysis; Harcourt Brace: Philadelphia, 1998.

Instructor’s ResourcesClick here for suggested approaches to solving the exercises.


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Acknowledgements

AuthorSteven C. Petrovic

Department of ChemistrySouthern Oregon University

1250 Siskiyou Blvd.,Ashland, OR 97520Email: [email protected]

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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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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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 zeroc. The magnitude of the offset is increased from zero

2. Answer the following questionsa. 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?

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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.

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Part 3: Evaluating Signal-to-Noise Ratios

1. Enter the following signal parameters into the Signal Noise Exercise spreadsheet

a. Peak 1i. Peak Intensity = 10 µVii. Mean = 1 minuteiii. Standard Deviation = 0.1 minute

b. Peak 2a. Peak intensity = 5 µVb. Mean = 2 minutesc. Standard Deviation = 0.1 minute

c. Peak 3a. Peak Intensity = 1 µVb. Mean = 3 minutesc. Standard Deviation = 0.1 minute

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Part 3: Evaluating Signal-to-Noise Ratios

1. Change the peak-to-peak noise level on the spreadsheet to 1.00 µV.

a. Look at the S/N ratio for each peak and determine which peaks are below the detection limit (S/N < 3)

b. Record the “Peak+Noise” magnitude for each peak.c. Obtain replicate data by pressing F9 to refresh the

spreadsheet. Record the new magnitude for each peak.d. 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.

e. As the S/N decreases, comment on any trends regarding the effect of noise on the accuracy and precision of the signal

2. Increase the peak-to-peak noise level on the spreadsheet to 5.00 µV and determine which peaks are now below the detection limit.

3. Based on your results, what role would signal-to-noise enhancement have in analyte detection?

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Bode Plot Exercise #1

1. 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 Hzii. Signal: 1000 Hz, Noise: 50000 Hziii. Signal: 1 Hz, Noise: 60 Hz

V1V(in) = 0.1 V

R14700 ohms

C10.10 µF

V(out) = ?

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Analog Filtering Exercise #1

1. The schematic below is an example of a passive RC filtera. Is the filter currently configured as a low-pass filter or a high-

pass filter? Explain how you would convert from one type of filter to the other.

b. 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 Hzc. 5000 Hz

c. Which frequency is least affected by the RC filter?d. 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

R110 ohms

C110 µF

V(signal generators)

Phono Plug to Laptop

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Analog Filtering Exercise #2

2. 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)

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Ensemble Averaging Exercise

An ensemble averaging spreadsheet similar to the Signal Noise Exercise spreadsheet can be accessed by clicking here.

1. Adjust the following parameters in the ensemble averaging spreadsheet:

• 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. 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.

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Boxcar Averaging Exercise #1

A boxcar averaging spreadsheet similar to the Ensemble Averaging spreadsheet can be accessed by clicking here.

1. Adjust the following parameters in the boxcar averaging spreadsheet:

• Peak Intensity = 5 µV (All 3 peaks)

• Peak #1 Mean & Standard Deviation = (1.000 ± 0.005) min



• Noise = 2 µV

2. Adjust the number of boxcar elements for each dataset

• Top = 9 data points

• Middle = 3 data points

• Bottom = 1 data point (raw data)

3. 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.

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Boxcar Averaging Exercise #21. Adjust the following parameters in the boxcar averaging

spreadsheet:

• Peak Intensity = 4 µV (Peak #1), 2.5 µV (Peak #2), 1 µV (Peak #3)




• Noise = 2.0 µV

2. Adjust the number of boxcar elements for each dataset

• Top = 9 data points

• Middle = 5 data points

• Bottom = 1 data point (raw data)

3. Discuss the ability of boxcar filters to clearly extract signals from noise at or below the detection limit based on S/N enhancement.

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Moving Average Exercise (Pg. 1)

A moving average spreadsheet similar to the Ensemble Averaging spreadsheet can be accessed by clicking here.

1. Remove all noise from the data, and adjust the following parameters in the moving average spreadsheet:

• Peak Intensity = 5 µV (All 3 peaks)




• 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.

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Moving Average Exercise (Pg. 2)

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 Savitzky-Golay 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?

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Moving Average Exercise (Pg. 3)7. Remove all noise from the data, and adjust the following

parameters in the moving average spreadsheet:

• Peak Intensity = 1 µV (Peak #1), 1 µV (Peak #2), 1 µV (Peak #3).




• Offset = 1 µV

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?

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signals and noise introduction to signals and noise module description this e-module provides an...

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