vs. Signal Noise

Signal vs. Noise
Every measurement is affected by processes not related
to the measurement of interest. The magnitude of this
noise, compared to the magnitude of the signal, directly
determines one’s ability to make an accurate
measurement.
Signal
In all experiments, there is:
Sample Response: the instrument’s response when the analyte is
present.
Blank Response: the instrument’s response when the analyte is
absent.
output voltage
The Signal: the difference between the sample and the blank
response.
sample
sample
signal
blank
blank
time
Background or Baseline
output voltage
Ideally, the blank response of an instrument would be exactly 0.
Then the sample response would be equal to the signal. This is
never the case, though it can often be adjusted to be close to 0.
There is always a residual signal associated with an instrument’s
blank response. This is called the background or the baseline.
sample
sample
signal
blank
blank
baseline
time
The baseline is subtracted from both the blank and the sample
response.
Drift
Ideally, the baseline response is constant in time. In such a case,
a constant correction factor is easily subtracted from the blank
and sample to correct the signal. Invariably, however, the
baseline changes slowly with time. This is called drift.
Sometimes the drift is linear in time, but often it is more complex
and difficult to predict.
output voltage
sample
sample
signal
blank
blank
baseline
time
We need to know the value of the baseline at the time we make a
measurement.
Noise
Noise is a random (or almost random) time-dependent change in
the instrument’s output signal that is unrelated to the analyte
response. These variations will tend to make the accurate
measurement of sample, blank, and baseline response less
certain.
Noise arises from many sources (to be discussed soon). The
frequency response can span the entire spectrum. We can treat
noise as if it were a sine wave, or at least the sum of many
(infinite?) sine waves.
Measuring the intensity of the noise and comparing it to the signal
is the key to determining the accuracy of a measurement and in
specifying the smallest signal level one is able to measure
(detection limit).
Peak-to-peak Noise
One measure of the amplitude of a sine wave is the peak-to-peak
amplitude (this is twice the amplitude which appears in the defining
equation for a sine wave).
4
3
2
1
V(peak-to-peak)
0
-1
-2
0
5
10
15
20
or
Vp-p
-3
-4
Noise is usually specified by measuring the peak-to-peak maximum
over a reasonable length of time (“reasonable” depends upon
length of time needed to make a measurement).
Peak-to-peak Noise
Even though the noise is clearly not a perfect sine wave, we know it
can be decomposed into a collection of sine waves and we can
treat it mathematically as a sine wave.
5
4
3
2
1
0
-1 0
-2
-3
-4
-5
2
4
6
8
10
12
14
16
18
20
Vp-p
Average Noise
Another way of measuring the intensity of noise might be the
average noise.
Naverage = 0 if noise is truly random. (Excursions above zero should
balance excursions below zero over time).
If not 0, then another signal must be present and we need to
account for it.
Naverage is not a useful measure of noise.
Root-Mean-Square Noise
It was the plus and minus excursions averaging out that made
Naverage not useful.
Squaring the signal, makes everything positive. This can then be
averaged meaningfully. Take the square root of that result to get
back a value that can be related to the original signal.
For a perfect sine wave, we can calculate its rms value. A
theoretical analysis gives us that
N RMS 
1
2 2
N p p
A quick estimate of the RMS noise is NRMS = 0.35 Np-p.
Signal-to-Noise Ratio
Total signal level nor noise level determine an experiment’s ability
to accurately detect an analyte. Rather it is the ratio of the two that
is critical. The S/N Ratio.
1.2
Np-p = 0.10
1
0.8
S = 0.75
0.6
0.4
Baseline = 0.25
0.2
0
0
2
4
NRMS = 0.354 Np-p = 0.354 x 0.10 = 0.035
6
8
S/N = 0.75/0.035 = 21.4
Signal-to-Noise Ratio
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Same signal level. Same baseline. S/N = 3.
6
7
8
Signal-to-Noise Ratio
3.5
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
In this experiment, the signal-to-noise is 1. Note how you could
not make a reasonable measurement of the signal under these
conditions.
Sources of Electrical
Noise
When sample is abundant, signal is high, background (baseline) is
low, we hardly worry about noise. But at some point, every
experiment needs to account for noise. Electrical noise can be
divided into four principal sources:
• Thermal Noise
• Shot Noise
• Flicker Noise
• Interference
Thermal Noise
Also known as white noise, Johnson noise, or Nyquist noise.
Arises because the atoms of a solid state conductor are vibrating
at all temperatures and they bump into conductors (electrons).
This imposes a new, random motion on those conductors which
generates noise.
Vnoise,rms  4 kB T R B
Vnoise, rms is the RMS voltage of the noise
kB is Boltzmann’s constant = 1.38 x 10-23 J K-1 (V2 s W-1 K-1)
T is the temperature in kelvin
R is the resistance in ohms
B is the bandwidth response of the instrument in Hz (s-1)
Bandwidth
Every instrument responds to rapid or slow signal changes
differently. We specify the bandwidth or bandpass by referring to
the range of frequencies over which it can effectively measure
signals. Usually the bandwidth of an instrument can be adjusted
by changing electronic filters.
Power measured
Center Frequency
A simple, RC circuit acts like a low
pass filter; it smooths (integrates)
rapid changes. It allows slowly
varying signals to pass unimpeded.
The relationship between its time
constant t = RC and its bandwidth
B is just
B ≈ 1/4t
Bandwidth
Other filters have other
relationships.
Frequency
Low Pass Filter
A series RC circuit functions as a low pass filter, when taking the
output voltage across the capacitor. Then ac signals at low
frequency pass unattenuated.
Low Pass Filter
R
C
0.9
Vout
Vout
XC

Vin
R2  XC2
Filter Attentuation Factor
Vin
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
10
100
1000
10000
Frequency (Hz)
100000
1000000
High Pass Filter
A series RC circuit functions as a high pass filter, when taking the
output voltage across the resistor. Then ac signals at high
frequency pass unattenuated.
High Pass Filter
R
C
Vout
R

Vin
R2  XC2
1
0.9
Filter Attenuation Factor
Vin
Vout
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
10
100
1000
Frequency (Hz)
10000
100000
1000000
Band Pass Filter
Many more complex filters can be designed. The frequency
response an be very complex. Here is a simple combination of a
high pass and low pass filter, to produce a band pass filter.
Band Pass Filter
R1 = 70 kW
1
C1 = 100 nF
R2 = 5 kW
C2 = 100 nF
F2 = 318 Hz
Filter Attentuation Factor
F1 = 23 Hz
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
10
100
1000
Frequency (Hz)
10000
100000
1000000
Thermal Noise Reduction by Cooling
A 10 kW resistor is used as a current-to-voltage converter. The
voltage across it is amplified by an amplifier with a bandwidth of 15
kHz. What is the rms noise voltage at 20 ˚C? at liquid nitrogen
temperature (77 K)? at liquid helium temperature (4.2 K)?
Vnoise,rms T  298K   4 k B T R B
 41.38 10 23 29810 4 1.5  10 4 
 2.43 10 12  1.56  10 6 V  1.56V
Vnoise,rms T  77K   0.80V
Vnoise,rms T  4.2K   0.19V
Cooling has dropped the noise originating in the resistor. We have
(incorrectly) ignored noise in the amplifier itself.
Thermal Noise Reduction by
Bandwidth
Consider the previous resistor at room temperature. Pass the
signal through a noiseless RC circuit (impossible, since the R in
this new circuit will introduce noise, but lets pretend, O.K.?) which
has a time constant of 0.1 s. What is the expected rms noise from
this filtered signal?
B = 1/(0.1 x 4) = 2.5
Vnoise,rms T  298K, B  2.5s1 
 41.38 10 23 29810 4 2.5
 2.0  10 8 V  20 nV
Noise reduction by filtering was much greater than by cooling, but we
are now much more limited to the speed with which we can make a
measurement and hence the rates of processes we can monitor.
Shot Noise
Also known as quantum noise or Schottky noise.
Arises because charge and energy are quantized. Electrons and
photons leave sources and arrive at detectors as quanta; while the
average flow rate may be constant, at a given instant there are more
quanta arriving than at another instant. There is a slight fluctuation
because of the quantum nature of things.
Inoise,rms  2 q Idc B
q is the electron charge = 1.602 x 10-19 C
Idc is the dc current flowing across the measurement interface
B is again the measurement bandwidth in Hz
Shot Noise Reduction by Bandwidth
What is the shot noise for a 1 amp dc current for a 15 kHz
measurement bandwidth? What is it when the bandwidth is
reduced to 2.5 Hz?
I noise,rms  B  15kHz
 21.602  10 19 11.5  10 4 
 6.9  10 8 A  69 nA
I noise,rms  B  2.5Hz  8.9 10 10 A  890 pA
Again a lower noise level comes at the expense of only being able to
measure slow enough processes.
Flicker Noise
Also known as 1/f noise or pink noise.
Origins are uncertain. Depends upon material, design, nature of
contacts, etc. Flicker noise is determined for every measurement
device. It is recognized by its 1/f dependence. Most important at
low frequencies (from dc to ~200 Hz).
Long term drift in all instruments comes from flicker noise.
Measurements taken above 1 kHz can neglect flicker noise.
A narrow bandwidth makes flicker noise seem constant over that
bandwidth and so it is indistinguishable from white noise.
Modulation
Flicker noise, because of its 1/f behaviour, is particularly
unforgiving when attempting to amplify dc signals. This is remedied
by modulating the signal to a higher frequency, then amplifying, and
demodulating.
Noise with a frequency characteristic different from that for the
modulation-demodulation process is averaged to zero.
Two important solutions are:
• Chopper Amplifier
• Lock-in Amplifer
Chopper Amplifier
An input dc signal is turned into a square wave by alternately
grounding and connecting the input line. This square wave is
amplified and then synchronously demodulated and filtered to give
an amplified dc signal that avoids flicker noise.
6 mV
6 mV
0
6V
3V
input
Gain = 1500/6 = 250
1000 x
Amplifier
1500 mV
output
Lock-in Amplifier
More modern solution is to employ a lock-in amplifier. Can recover
useful signal even when S/N < 1. Key components are:
• Sine wave reference signal that also perturbs the system under
investigation.
• Phase Sensitive Detector, including a four-quadrant multiplier and
phase shifter.
Experimental
System
Four-Quadrant
Multiplier
Sine wave
Generator
Phase Shifter
Integrator/Filter
Lock-in
Output
Interference
Also known as environmental noise or electrical pickup.
Broadcasting electric and magnetic fields.
Line noise and harmonics (60 Hz, 120 Hz, 180 Hz, etc.)
Electrical devices (elevators, air conditioners, motors)
Broadcasting stations (radio, T.V.)
Microphonics (mechanical vibrations coupled capacitively)
Often observed in a narrow frequency with a large, fixed amplitude.
Remediate by shielding, eliminate ground loops, rigidly fix all
cables and detectors, isolate from temperature variations,
compensating magnetic fields, etc.
Software Methods
Computers have dramatically changed the way with which we deal
with noise. Many of these can help “pull the signal out of the noise”.
• Software “low pass filtering”
• Ensemble averaging
• Fourier Transform filtering
Software Low Pass
Filtering
An X-ray Photoelectron spectrum (XPS) of Au nanocrystals attached
to a silicon surface by 3-mercaptopropyl-trimethoxysilane.
XPS of Au-Nanocrystals on Silicon
200
Si 2p
Counts
150
Au 4f
100
50
0
70
80
90
100
110
Binding Energy (eV)
1 scan; 0.5 eV step size
S/N = 29 on Si peak at 100 eV.
Software Low Pass
Filtering
A 5 point moving average to smooth the data.
XPS - 5 Point Moving Average
200
Counts
150
100
50
0
70
80
90
100
110
Binding Energy (eV)
Noise is decreased but so is
peak amplitude.
S/N = 53 on Si peak at 100 eV.
Software Low Pass
Filtering
A weighted 5 point moving average: weighting factors are 1:2:3:2:1
XPS - 5 point weighted moving average
200
Counts
150
100
50
0
70
75
80
85
90
95
100
105
110
115
Binding Energy (eV)
Noise is decreased but so is
peak amplitude but not by as
much as for the non-weighted
smoothing.
S/N = 57 on Si peak at 100 eV.
Software Low Pass
Filtering
A 5 point weighted moving average using Savitzky-Golay weighting
factors for a quadratic fit. They are -3:12:17:12:-3
5 point Savitzky-Golay
200
Counts
150
100
50
0
70
75
80
85
90
95
100
105
110
115
Binding Energy (eV)
Best noise reduction without
compromising peak intensity.
S/N = 81 on Si peak at 100 eV.
A. Savitzky and M.J.E. Golay, Anal. Chem. 1964, 36, 1627.
Software Low Pass
Filtering
You can also do differentiation by choosing the right integers. Here
is the first derivative using the weighting factors 1:-8:0:8:-1
First derivative by S-G
D(counts)/d(eV)
1000
500
0
70
75
80
85
90
95
100
105
110
-500
-1000
Binding Energy (eV)
This can help to identify the peak position
more accurately.
115
Ensemble Averaging
Noise is randomly distributed but signal is not. If we do an
experiment a second time, the signal appears in the same place, but
the noise will be doing something different. If we add two runs
together, the signal increases, but the noise tends to smooth itself
out. Signal increases as N but noise increases as √N. Hence, the
S/N increases as √N.
XPS of Au nanocrystal on Silicon
4000
Counts
3000
2000
1000
0
70
80
90
100
110
Binding Energy (eV)
16 scans, added together. S/N = 109 (about 4x that for the 1 scan spectrum).
Fourier Smoothing
A Fourier transform can decompose a spectrum into the many
sinusoidal contributions that make it up. Noise is generally high
frequency; drift is low frequency; environmental noise is of specific,
narrow frequencies. A Fourier transform, selective removal of
offending frequency components, followed by an inverse Fourier
transform, can significantly improve a spectrum’s appearance.
Time Domain with Noise
and Interference
The signal here is a 30 Hz sine wave, contaminated by a much
higher frequency sine wave and white noise.
Unfiltered Time Domain
15
Amplitude
10
5
0
-5
-10
-15
0
200
400
Time (ms)
600
800
1000
Unfiltered Spectrum
A Fourier transform of the time domain provides a spectral
decomposition of the frequency components, revealing the
interfering signal (at 430 Hz) and the white noise.
Unfiltered Spectrum
1.4
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
300
Frequency (Hz)
350
400
450
500
Filtered Spectrum
By convoluting the spectrum with a box-like multiplication function
with the low frequency end of the spectrum, we effectively apply a
low-pass filter.
Filtered Spectrum
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
0
100
200
300
Frequency (Hz)
400
500
Filtered Time Domain
An inverse Fourier transform takes the filtered spectrum and
produces the filtered time domain data. It is not perfectly clean yet,
because of the white noise present with in the selected filter
bandwidth.
Filtered Time Domain
6
Amplitude
4
2
0
-2
-4
-6
0
200
400
600
Time (msec)
800
1000