1. 8.2 Common Forms of Noise Johnson or thermal noise
shot or Poisson noise
1/f noise or drift
interference noise
impulse noise
real noise
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2. Johnson Noise Johnson noise characteristics
produced by the thermal motion of electrons in resistors
the random amplitude has a normal pdf with a mean of zero
the noise magnitude is the standard deviation, s, of the pdf
the phase has a uniform pdf over -p/2 to +p/2 radians
Johnson noise can be averaged to zero
The noise voltage is given by, , where k is Boltzmann's constant, T is the temperature in Kelvin, R is the resistance across which the voltage is measured, and Df is the measurement bandwidth (f3dB).
The graph shows Johnson noise for measurements taken one per second: T = 298 K,
R = 1 k, and Df = 1 kHz.
The blue lines are 2.5s. 5s is called the tangential noise.
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3. 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.
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4. 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 = 4kTDf. Note that resistance has dropped out of the equation.
Below is the noise power for T = 298 K and Df = 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 value.
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5. 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 =
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 MHz to 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.
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6. 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 -p/2 to p/2 radians
The noise current is given by, , where q is the charge on an electron, iavg is the average current, Df is the measurement bandwidth where 1/(2Df ) = 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.
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7. 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 = (2QiavgDf)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.
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8. 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 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.
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9. Shot Noise Spectrum
Shot noise differs from Johnson noise by having a non-zero mean. This is shown at the left.
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.
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10. 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
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.
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11. 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
where a and b are adjustable parameters. Note that noise amplitude will follow the square root of the power.
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12. Example 1/f Noise Spectrum
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.
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13. 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, and retain the real part.
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14. 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.
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15. 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
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16. 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.
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17. 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
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18. uniform except impulse at f = 0
Noise Summary
type
time domain
frequency domain
phase
Johnson
random, uniform
mean of zero
random
shot
non-zero mean
uniform except impulse at f = 0
except at f = 0
1/ f
drift
interference
cosine
impulse
fixed
all frequencies
non-random
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19. 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 or lock-in amplifier
impulse noise + Johnson noise:
use a boxcar integrator to eliminate the impulse noise and average the boxcar output to reduce the Johnson noise
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