8.2 Common Forms of Noise

Transcription

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 8.2 : 1/19

2 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. The graph shows Johnson noise for measurements taken one per second: T = 298 K, R = 1 kω, and Δf = 1 khz. The blue lines are ±2.5σ. voltage Johnson Noise Voltage 8.2 : 2/ time

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 Noise Amplitude vs. Frequency Noise Phase vs. Frequenc y voltage phase frequency frequency 8.2 : 3/19

4 Johnson Noise Power In electronics, power can be computed as: P = ei = e 2 /R. Using the equation for Johnson noise voltage, the Johnson noise power is given by P J = 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. Johnson Noise Power Noise Power vs. Frequency power (W) power per bandwidth (W/Delta f) time frequency 8.2 : 4/19

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 =.26; for liquid helium, 4 K/298 K =.13. Johnson noise power can be reduced by using a smaller bandwidth. Values down to.1 Hz are practical. However, Fourier transforms tell you that a.1 Hz bandwidth will require at least a 1 s measurement time! It doesn't matter where in frequency the bandwidth is located. A 1 khz bandwidth from to 1 khz has the same noise power as that from 1. MHz to 1.1 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, e J. 8.2 : 5/19

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 = 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 = 2qiavgΔf, where q is the charge on an electron, i avg 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

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, N R, i avg = N R Q The noise follows Poisson statistics for the counted particle, thus depends upon the square root of the total counts, N 1/2 = (N R t) 1/2. The shot noise charge is given by the noise counts times Q. Q shot = (N 1/2 )Q =(N R t) 1/2 Q = (Q 2 N R t) 1/2 = (Qi avg t) 1/2 Finally, Q shot is converted into a noise current by dividing by time. i shot = Q shot /t = (Qi avg /t) 1/2 = (2Qi avg Δf) 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

8 Example of Photon Detection A typical photon counting photomultiplier has a gain of 1 8, that is, each photon that ejects an electron from the photocathode produces a pulse containing 1 8 electrons. A typical impulse response of such a detector would be ~5 ns FWHM Gaussian pulse. Consider an optical signal having 1, photons per second. photon counting noise Each charge packet has a peak current given by gain, charge on the electron, and impulse FWHM: /5 1-9 = ~3.2 ma. Such a pulse is very easy to detect, thus the counts in one second would be 1 3. The Poisson noise would be (1 3 ) 1/2 = ~32. current noise If a picoammeter is used to measure the detector output, it will see an average current: /1 = A. The noise current can be computed using the equation on the last slide. 8.2 : 8/19 i shot Qiavg = = = A t 1

9 Shot Noise Spectrum Noise Amplitude vs. Time Shot noise differs from Johnson noise by having a non-zero mean. This is shown at the left. Except for f = (the mean), the amplitude and phase spectra look like Johnson noise. In the left graph below the amplitude at f = is A. current time Noise Amplitude vs. Frequency Noise Phase vs. Frequenc y current phase : 9/19 frequency frequency

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 iavg NQ t SNR = = = N = NRt i NQ t shot 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 : 1/19

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 P 1/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

12 Example 1/f Noise Spectrum At the right is the 1/f power spectrum with a = 1 and b =.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. power /f Noise vs. Frequency frequency /f Amplitude Noise Comparison of 1/f and JohnsonNoise voltage voltage : 12/19 time time

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 =, 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

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 11 V power, 12 Hz frequency of fluorescent lights, etc. Shown below is a temporal Gaussian signal with superimposed interference noise at.5 Hz. Temporal Signal Plus Interfering Cosine Spectrum of Signal Plus Interfering Cosine 1.2 voltage.5 voltage : 14/ time frequency

15 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 11 V power line can be reduced or eliminated by the use of 6 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

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. Temporal Signal Plus Impulse Noise Spectrum of Signal Plus Impulse Noise 1 voltage.5 voltage : 16/19 time frequency

17 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

18 Noise Summary 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 = random except at f = 1/ f random drift 1/ f random interference cosine impulse fixed impulse impulse all frequencies non-random 8.2 : 18/19

19 Real Noise 8.2 : 19/19 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