8.2 Common Forms
of Noise
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8.2 : 1/19
Johnson or thermal noise
shot or Poisson noise
1/f noise or drift
interference noise
impulse noise
real noise
Johnson Noise
Johnson noise characteristics
• produced by the thermal motion of electrons in resistors
• the amplitude has a normal pdf with a mean of zero
• the noise magnitude is the standard deviation, σ, of the pdf
• the phase has a uniform pdf over -π/2 to +π/2 radians
• noise can be averaged to zero
The noise voltage is given by, eJ = 4kTRΔf , where k is Boltzmann's
constant, T is the temperature in Kelvin, R is the resistance across
which the voltage is measured, and Δf is the measurement
bandwidth.
Johnson Noise Voltage
5 .10
voltage
The graph shows Johnson
noise for measurements taken
one per second: T = 298 K,
R = 1 kΩ, and Δf = 1 kHz.
7
0
The blue lines are ±2.5σ.
5 .10
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0
50
100
time
150
200
Johnson Noise Spectrum
Noise amplitude versus frequency (the noise spectrum) looks much
the same as noise amplitude versus time.
Noise phase is also random in time and frequency.
Noise Phase vs. Frequenc y
Noise Amplitude vs. Frequency
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1
phase
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voltage
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frequency
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frequency
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Johnson Noise Power
In electronics, power can be computed as: P = ei = e2/R. Using the
equation for Johnson noise voltage, the Johnson noise power is
given by PJ = 4kTΔf. Note that resistance has dropped out of the
equation.
Below is the noise power for T = 298 K and Δf = 1 kHz. The solid
blue line in time is theory from the above equation. The area under
the blue line in frequency equals the temporal amplitude theory
value.
Noise Power vs. Frequency
1.5 .10
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8.2 : 4/19
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50
100
time
150
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power per bandwidth (W/Delta f)
power (W)
Johnson Noise Power
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200
400
600
frequency
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1000
Reducing Johnson Noise
Johnson noise power can be reduced by changing the temperature.
Not much is gained unless liquid helium is used as the coolant.
That is, for liquid nitrogen, 77 K/298 K = 0.26; for liquid helium,
4 K/298 K = 0.013.
Johnson noise power can be reduced by using a smaller bandwidth.
Values down to 0.01 Hz are practical. However, Fourier transforms
tell you that a 0.01 Hz bandwidth will require at least a 100 s
measurement time!
It doesn't matter where in frequency the bandwidth is located. A
1 kHz bandwidth from 0 to 1 kHz has the same noise power as that
from 1.000 MHz to 1.001 MHz.
Johnson noise is simulated by adding normally distributed random
voltages to the true value. Set the mean of the normal pdf to zero
and the standard deviation to the noise voltage, eJ.
8.2 : 5/19
Electronic Shot Noise
Electronic shot noise characteristics
• produced by Poisson fluctuations in the flow of electrons when
discrete charges move across a junction such as those found in
semiconductors or a cathode and anode in vacuum tubes
• the noise is proportional to the square root of the average current
• for average currents above ~1 pA the noise spectrum looks like
Johnson noise, except for a spike at f = 0 due to the average current
• the phase has a uniform pdf over -π/2 to π/2 radians
The noise current is given by, ishot = 2qiavg Δf , where q is the charge
on an electron, iavg is the average current, Δf is the measurement
bandwidth where 1/(2Δf ) = t, the measurement time. (The factor
of two comes from the Nyquist theorem which will be explained
when analog-to-digital converters are covered.)
For currents down to ~1 pA, shot noise is less than 1% of the
average. Circuits composed of non-semiconductor components have
far less shot noise than predicted by the above equation.
8.2 : 6/19
Photon and Ion Shot Noise
When detecting individual photons or ions, the current output comes
in discrete, countable packets of charge, Q. The average current
depends upon the count rate, NR,
iavg = NRQ
The noise follows Poisson statistics for the counted particle, thus
depends upon the square root of the total counts, N1/2 = (NRt)1/2.
The shot noise charge is given by the noise counts times Q.
Qshot = (N1/2)Q =(NRt)1/2Q = (Q2NRt)1/2 = (Qiavgt)1/2
Finally, Qshot is converted into a noise current by dividing by time.
ishot = Qshot/t = (Qiavg/t)1/2 = (2QiavgΔf)1/2
Note that this is the same equation as electronic shot noise, except
for the use of a charge packet, Q, instead of the charge on one
electron, q.
8.2 : 7/19
Example of Photon Detection
A typical photon counting photomultiplier has a gain of 108, that is,
each photon that ejects an electron from the photocathode
produces a pulse containing 108 electrons. A typical impulse
response of such a detector would be ~5 ns FWHM Gaussian pulse.
Consider an optical signal having 1,000 photons per second.
photon counting noise
Each charge packet has a peak current given by gain, charge on the
electron, and impulse FWHM: 108×1.6×10-19/5×10-9 = ~3.2 mA.
Such a pulse is very easy to detect, thus the counts in one second
would be 103. The Poisson noise would be (103)1/2 = ~32.
current noise
If a picoammeter is used to measure the detector output, it will see
an average current: 103×108×1.6×10-19/1 = 1.6×10-8 A. The noise
current can be computed using the equation on the last slide.
ishot =
8.2 : 8/19
Qiavg
t
108 × 1.6 × 10−19 × 1.6 × 10−8
=
= 5.1× 10−10 A
1
Shot Noise Spectrum
Noise Amplitude vs. Time
Except for f = 0 (the mean), the
amplitude and phase spectra look like
Johnson noise. In the left graph
below the amplitude at f = 0 is
1.6×10-8 A.
current
Shot noise differs from Johnson noise
by having a non-zero mean. This is
shown at the left.
1.8 .10
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1.6 .10
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phase
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current
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Noise Phase vs. Frequenc y
Noise Amplitude vs. Frequenc y
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time
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frequency
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frequency
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Reducing Shot Noise
• the reduction of shot noise is generally not of interest, this is
because a reduction in noise implies a reduction in the mean
• signal-to-noise enhancement is the goal with shot noise
SNR =
iavg
ishot
=
NQ t
= N = NRt
NQ t
• for a fixed counting period, the count rate can be increased by
improving the measurement efficiency, e.g. lens f/# with
fluorescence
• the counting time can be increased (the bandwidth decreased)
With small counts shot noise is simulated by using a Poisson random
number generator. With large counts shot noise can be simulated
with a normal random number generator having the standard
deviation set equal to the square root of the mean.
8.2 : 10/19
One-over-f Noise (1/f)
1/f noise characteristics
• appears as drift in a measurement
• it can be introduced by long term power supply fluctuations,
changes in component values, temperature drifts, etc.
• the longer the time required for a measurement, the more effort
needs to expended to keep everything under control.
• the presence of 1/f noise makes measurements near zero
frequency very difficult
• this noise is generally unimportant above ~1 kHz
Measured 1/f noise can be loosely approximated by
P1/f =
a
b+ f
where a and b are adjustable parameters. Note that noise
amplitude will follow the square root of the power.
8.2 : 11/19
Example 1/f Noise Spectrum
1/f Noise vs . Frequenc y
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power
At the right is the 1/f power spectrum
with a = 1 and b = 0.1. Below left is the
temporal noise corresponding to the
noise spectrum. The low noise
frequencies cause drift. By comparing
1/f noise to temporal Johnson noise (at
the right) it is possible to see that the
high frequencies are attenuated.
0
1/f Amplitude Noise
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voltage
voltage
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time
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frequency
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Comparison of1/f and JohnsonNoise
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Reducing 1/f Noise
• 1/f noise can be reduced if the measurement can be divided by a
reference signal. For example, laser excited fluorescence intensity
can be divided by the average laser power.
• the effect of 1/f noise on a measurement can be reduced by
interspersing working standards with the sample
• the effect of 1/f noise can be reduced or eliminated by
modulating the signal to frequencies above ~1 kHz
1/f noise is simulated by the following procedure
• generate a vector of Johnson noise in the time domain
• use Mathcad's CFFT( ) function to generate a Johnson noise
spectrum
• multiply the Johnson noise spectrum by (a/(f+b))1/2, where a/b
controls the noise power at f = 0, and b controls the relative
contributions of low to high frequencies (small b favors low
frequencies)
• use Mathcad's ICFFT( ) function to generate 1/f voltage noise in
the time domain
8.2 : 13/19
Interference Noise
Interference noise characteristics
• interference noise appears within a very narrow range of
frequencies
• often appears as a temporal harmonic wave
• caused by bad shielding of electrical cables, dc power supply
ripple, noise on the 110 V power, 120 Hz frequency of fluorescent
lights, etc.
Shown below is a temporal Gaussian signal with superimposed
interference noise at 0.05 Hz.
Temporal Signal Plus Interfering Cosine
Spectrum of Signal Plus Interfering Cosine
1
voltage
voltage
0.2
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8.2 : 14/19
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time
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frequency
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Reducing Interference Noise
Reducing interference noise
• if interference noise is well-separated spectrally from the
signal, it can usually be reduced or eliminated by electronic band
pass (signal) and/or band reject filters (noise).
• if interference noise is not separated spectrally, it can often be
reduced or eliminated by using modulation to shift the signal
frequencies
• cable pick-up and/or power supply ripple can be eliminated by
proper choice of electronic components
• pick-up off the 110 V power line can be reduced or eliminated
by the use of 60 Hz notch-pass filters (from a company called
CorCom)
• digital filters can easily post-process your data and remove
sinusoidal interference
8.2 : 15/19
Impulse Noise
Impulse noise characteristics
• impulse noise appears as a very sharp spike in the time domain
• the spectrum of impulse noise is very broad
• sources of impulse noise are rapid electrical discharges, such as
lightening and pulsed laser power supplies, or rapid discharge of
capacitors
Shown below is a temporal Gaussian with an impulse interference
at 125 s.
Temporal Signal Plus Impulse Noise
Spectrum of Signal Plus Impulse Noise
0.5
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voltage
voltage
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100
time
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0.005
0
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frequency
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Reducing Impulse Noise
Reducing impulse noise
• frequency-based techniques cannot be used to reduce or
eliminate temporal impulse noise
• If the noise and signal are temporally separated, the
measurement can be terminated during the noise spike (for
example, an impulse followed by an exponentially decaying
signal). This is accomplished with an electronic device called a
boxcar integrator.
• if the noise and signal are temporally overlapped, the data can
be post-processed with an algorithm that only permits small
changes in signal amplitude (this would be premised on the
anticipated maximum rate of change for the signal)
• if the signal can be repeated and the impulse is random in
time, the impulses can be recognized by an algorithm and
removed
8.2 : 17/19
Noise Summary
8.2 : 18/19
type
time domain
frequency
domain
phase
Johnson
random, uniform
mean of zero
random, uniform
mean of zero
random
shot
random, uniform
non-zero mean
random
uniform except
impulse at f = 0
random
except at f = 0
1/ f
random
drift
1/ f
random
interference
cosine
impulse
fixed
impulse
impulse
all frequencies
non-random
Real Noise
Measured noise can consist of all mentioned types. The total noise
is determined by adding together noise power (noise amplitude is
a standard deviation, thus noise power is a variance).
Johnson + 1/f (very common):
modulate the signal to frequencies greater than ~1 kHz to
avoid 1/f noise and use a narrow band pass filter to reduce
Johnson noise
Shot + 1/f (common in counting experiments):
ratio the measurement to reduce 1/f and count for longer
periods of time
Johnson + interference:
modulate the signal to a clean part of the spectrum and use a
narrow band pass filter
impulse noise + Johnson noise:
use a boxcar integrator to eliminate the impulse noise and
average the boxcar output to reduce the Johnson noise
8.2 : 19/19