Signals and Noise
Sections
 5A Signal-to-Noise Ratio √
 5B Noise sources in instruments requires knowledge of
electronics (Skip)
 5C Signal-to-Noise Enhancement
5C-1 Hardware requires knowledge of electronics (Skip)
5C-2 Software methods √
5A Signal-to-Noise Ratio
Remember – every measurement has inherent uncertainty (random
error)
“Noise” is the manifestation of this random error.
 It degrades measurement accuracy/precision
 Determines the detection limit
Every measurement contains 2 components:
1. Signal = response from analyte
2. Noise
Since noise is mostly independent of signal strength, the effect of
noise on the analysis depends on the signal.
A most useful figure of merit: Signal-to-Noise Ratio (S/N)
1
Noise – standard deviation of numerous measurements of signal
strength
Signal – average (mean) of numerous measurements of signal
strength
S/N =
This is nothing new.
Calculate the S/N ratio for the following titrimetric data.
Five titrations of the same amount of material required the
following volumes: 24.38 mL, 24.21 mL, 24.46 mL, 24.30 mL,
24.40 mL.
Why is S/N important? For one thing, it tells whether or not a
signal is detectable.
General Rule: A signal cannot be detected if S/N < 3.
2
To calculate S/N for 2 absorbance spectra: S/N =
Malachite Green Absorbance Spectra
0.45
Signal = 0.39
0.4
0.35
0.3
Concentrated
solution
Absorbance
0.25
0.2
Signal = 0.079
0.15
0.1
0.05
S
0
-0.05
400
Dilute solution
450
500
550
600
650
700
750
800
Wavelength (nm)
Clearly a high S/N is desirable.
Hardware devices to increase S/N (5C-1, electronics/instrument
design)
Software methods to enhance S/N. Routinely done (5C-2)
3
5C-2 Software methods to enhance S/N
1. Ensemble or signal averaging.
Make n repetitive measurements and average the result. (No
different from replicate titrimetric analyses)
In the absence of systematic error…
Find means of replicate measurements, less scatter.
Below is data from an absorbance spectrum of malachite green in a
region where there is no absorbance (i.e. noise or random error).
0.12
0.12
0.09
0.08
0.07
0.08
0.07
0.07
0.08
0.08
0.07
0.06
0.08
0.09
0.08
0.07
0.05
0.07
0.07
0.08
1. Find the mean and standard deviation for this set of 20
measurements.
2. Find the mean and standard deviation from the means
of sets of 5 measurements.
3. Compare the mean and standard deviations.
The standard deviation of the mean is inversely proportional to
where n = number of measurements averaged to generate the
mean. (Note confidence interval equation)
n,
Noise = random error of measurement, measured by standard
deviation.
4
There is a
averaging.
n
dependence on S/N for ensemble or signal
Vertical scale increases as the number
of scans increases.
Random signal fluctuations (noise)
increases with n .
Signal increases with n.
S/N increases
Summary:
5
2. Boxcar Averaging
The average of a small number of adjacent data points is a better
measure of signal than all of the individual data points. (Signal
varies more slowly than noise)
Signal
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2 pt
boxcar
x-axis
Signal
3 pt
boxcar
x-axis
unprocessed
Signal
4.5
0.733
3.5
4
0.2
0.9
1.1
2.2
3.5
4
3.6
3.8
3.3
3.1
2.9
2.4
1.8
1.2
0.8
1.5
0.55
2
3
3.5
1.65
Signal
x-axis
5
5.5
2.5
2
3.233
1.5
3.75
1
7.5
3.7
8
0.5
3.567
0
0
9.5
3.2
11.5
2.65
11
2
4
x-axis
8
6
10
12
14
16
2.8
2 pt boxcar
4
13.5
1.5
14
1.267
3.5
3
2.5
Signal
3 pt boxcar
2
4
1.5
3.5
1
3
0.5
2.5
0
Signal
0
2
4
6
8
10
x-axis
2
1.5
1
0.5
0
0
2
4
6
8
10
12
14
16
x-axis
Summary:
6
12
14
16
3. Digital Filtering
A moving boxcar average – must include an odd number of data
points.
x-axis
Signal
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
3 pt moving
boxcar
x-axis
Signal
0.2
0.9
1.1
2.2
3.5
4
3.6
3.8
3.3
3.1
2.9
2.4
1.8
1.2
0.8
2
3
4
5
6
7
8
9
10
11
12
13
14
5 pt moving
boxcar
x-axis
0.733
1.4
2.267
3.233
3.7
3.8
3.567
3.4
3.1
2.8
2.367
1.8
1.267
Signal
3
4
5
6
7
8
9
10
11
12
13
1.58
2.34
2.88
3.42
3.64
3.56
3.34
3.1
2.7
2.28
1.82
Shown below are the results of moving boxcar averages from real
spectra of many hundreds of data points.
Malachite Green Absorbance Spectra
0.09
0.08
0.07
0.06
Absorbance
0.05
0.04
0.03
0.02
0.01
0
-0.01
400
450
500
550
600
650
700
750
800
Wavelength (nm)
7
Relative Single Beam Signal Intensity
0.5 cm-1 resolution
19 pt. Boxcar averaged
2390.00
2370.00
2350.00
2330.00
2310.00
2290.00
2270.00
2250.00
Wavenumber
Summary:
Polynomial least squares data smoothing: similar to moving boxcar
averaging but with weighted data points.
Reference pp. 121/122 of text. Original ref: Anal. Chem. 1964, 36,
1627.
End of Chapter 5 problems: 6-8, 11-12, 13a,e
(Problem 13 data exists on the website:
www.thomsonedu.com/chemistry/skoog)
8