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.

SignalIn 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.

The Signal: the difference between the sample and the blank response.

time

ou

tpu

t vo

lta

ge

blank

sample sample

blank

signal

Background or BaselineIdeally, 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.

time

ou

tpu

t vo

lta

ge

blank

sample sample

blank

signal

baseline

The baseline is subtracted from both the blank and the sample response.

DriftIdeally, 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.

time

ou

tpu

t vo

lta

ge

blank

sample

sample

blank

signal

baseline

We need to know the value of the baseline at the time we make a measurement.

NoiseNoise 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 NoiseOne 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).

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).

-4

-3

-2

-1

0

1

2

3

4

0 5 10 15 20

V(peak-to-peak)

or

Vp-p

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

2

3

4

5

0 2 4 6 8 10 12 14 16 18 20

Vp-p

Peak-to-peak Noise

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 NoiseIt 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

A quick estimate of the RMS noise is NRMS = 0.35 Np-p.

NRMS = 12 2

Np− p

Signal-to-Noise RatioTotal 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.

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8

S = 0.75

Baseline = 0.25

Np-p = 0.10

NRMS = 0.354 Np-p = 0.354 x 0.10 = 0.035S/N = 0.75/0.035 = 21.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6 7 8

Same signal level. Same baseline. S/N = 3.

Signal-to-Noise Ratio

0

0.5

1

1.5

2

2.5

3

3.5

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.

Signal-to-Noise Ratio

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 NoiseAlso 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 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)

Vnoise,rms = 4 kB T R B

BandwidthEvery 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.

Center Frequency

Bandwidth

Frequency

Pow

er m

easu

red

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

Other filters have other relationships.

Low Pass Filter

Low Pass Filter

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000 100000 1000000

Frequency (Hz)

Filter

Att

entu

ation F

act

or

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.

R

C

Vin

Vout

Vout

Vin

= XC

R2 + XC2

High Pass Filter

High Pass Filter

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000 100000 1000000

Frequency (Hz)

Filter

Att

enuat

ion F

acto

r

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.

R

C

Vin

Vout

Vout

Vin

= RR2 + XC

2

Band Pass Filter

Band Pass Filter

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000 100000 1000000

Frequency (Hz)

Filter

Att

entu

atio

n F

acto

r

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.

R1 = 70 kΩ

C1 = 100 nF

F1 = 23 Hz

R2 = 5 kΩ

C2 = 100 nF

F2 = 318 Hz

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)?

Cooling has dropped the noise originating in the resistor. We have (incorrectly) ignored noise in the amplifier itself.

Vnoise,rms T = 298K( ) = 4kB T RB

= 4 1.38 ×10−23( ) 298( ) 104( )1.5 ×104( )= 2.43 ×10−12 = 1.56 ×10−6 V = 1.56µV

Vnoise,rms T = 77K( ) = 0.80V

Vnoise,rms T = 4.2K( ) = 0.19V

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?

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.

Vnoise,rms T = 298K, B = 2.5s−1( )= 4 1.38 ×10−23( ) 298( ) 104( ) 2.5( )

= 2.0 ×10−8 V = 20nV

B = 1/(0.1 x 4) = 2.5

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.

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

Inoise,rms = 2 q Idc B

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?

Again a lower noise level comes at the expense of only being able to measure slow enough processes.

Inoise,rms B = 15kHz( )= 2 1.602 ×10−19( )1( ) 1.5 ×104( )= 6.9 ×10−8 A = 69nA

Inoise,rms B = 2.5Hz( ) = 8.9 ×10−10 A = 890 pA

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 AmplifierAn 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.

0

6 mV6 mV 6 V

3 V 1500 mV

1000 x Amplifier

input

output

Gain = 1500/6 = 250

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.

Sine wave Generator

Experimental System

Four-Quadrant Multiplier

Phase Shifter

Integrator/FilterLock-in Output

InterferenceAlso 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

XPS of Au-Nanocrystals on Silicon

0

50

100

150

200

70 80 90 100 110

Binding Energy (eV)

Counts

An X-ray Photoelectron spectrum (XPS) of Au nanocrystals attached to a silicon surface by 3-mercaptopropyl-trimethoxysilane.

1 scan; 0.5 eV step size

Au 4f

Si 2p

S/N = 29 on Si peak at 100 eV.

XPS - 5 Point Moving Average

0

50

100

150

200

70 80 90 100 110

Binding Energy (eV)

Counts

A 5 point moving average to smooth the data.

S/N = 53 on Si peak at 100 eV.Noise is decreased but so is peak amplitude.

Software Low Pass Filtering

XPS - 5 point weighted moving average

0

50

100

150

200

70 75 80 85 90 95 100 105 110 115

Binding Energy (eV)

Counts

A weighted 5 point moving average: weighting factors are 1:2:3:2:1

S/N = 57 on Si peak at 100 eV.Noise is decreased but so is peak amplitude but not by as much as for the non-weighted smoothing.

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

0

50

100

150

200

70 75 80 85 90 95 100 105 110 115

Binding Energy (eV)

Counts

S/N = 81 on Si peak at 100 eV.Best noise reduction without compromising peak intensity.

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

-1000

-500

0

500

1000

70 75 80 85 90 95 100 105 110 115

Binding Energy (eV)

D(c

ounts

)/d(e

V)

This can help to identify the peak position more accurately.

Software Low Pass Filtering

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

0

1000

2000

3000

4000

70 80 90 100 110

Binding Energy (eV)

Counts

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 Noiseand 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

-10

-5

0

5

10

15

0 200 400 600 800 1000

Time (ms)

Am

plitude

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

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200 250 300 350 400 450 500

Frequency (Hz)

Am

pli

tud

e

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

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500

Frequency (Hz)

Am

plit

ude

Filtered Time DomainAn 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

-4

-2

0

2

4

6

0 200 400 600 800 1000

Time (msec)

Am

plit

ude