Noise Fundamentals

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Newsletter – Noise Fundamentals - Where Noise IS the Signal

Experiments

Noise Quantification
Johnson Noise
Shot noise
Calibrations
Projects

Noise Quantification

Noise describes an electronic signal which has zero mean value and random time variation, yet with stable statistical properties. But how is noise actually measured? How can we turn a signal that is ac and contains all frequencies into a single, measurable, number?

The noise voltages which students will manipulate in this apparatus are systematically quantified by an actual all-analog re-creation of the theoretical mean-square definition of noise. Students configure the apparatus so that a tiny noise voltage V(t) is:

• pre-amplified, by a gain factor G1
• filtered, in frequency space, by a filter function G(f), typically used to define lower and upper bounds to the spectrum
• further amplified by a user-chosen gain factor G2
• squared by a real-time analog multiplier
• and, finally, time-averaged over a user-chosen averaging time.

The whole process results in a nearly steady d.c. voltage which is easily measured by a multi-meter attached to the output connector of the high-level electronics box. It is this voltage that is related, by known coefficients, to the crucial ‘mean square noise’ <V2(t)>. Here, writing the quantity as <brackets> indicates that it represents a time average. The filter function, G(f), defines the ‘effective noise bandwidth’ Δf, which appears in theoretical predictions of the statistical properties of noise.

Johnson Noise

Our unit is shipped to allow immediate out-of-the-box measurement of Johnson noise. This noise source arises as a spontaneously-appearing and fluctuating emf, VJ(t), across any resistor or dissipative electrical element. A famous theory of Nyquist claims that the size of this noise is given by

< [VJ(t)]2> = 4 kB T R Df.

In our apparatus, students can measure the left-hand-side via the actual mean-square definition, and test the predicted dependence on

• source resistance R, from 10Ω to 100 MΩ
• source temperature T, from 77 K to 400 K
• bandwidth Df, variable in width and location in frequency space, up to frequency 100 kHz

With these three tests performed, students can then extract a measured value of kB, Boltzmann’s constant.

Shot Noise

Another very important form of electronic noise is shot noise, which arises because (some) electric currents are subject to fluctuations due to the quantization of charge. In fact, for any current consisting of the statistically-independent arrivals of electrons, each of charge –e, constituting collectively an average current of idc, the instantaneous current i(t) = idc + δi(t) has fluctuations, of a size first predicted by Schottky as:

< [δi(t)]2 > = 2 e idc Df.

In our apparatus, the simplest shot-noise source is the photocurrent produced when a p-n junction photodiode is illuminated by an incandescent bulb. Installing this assembly within the pre-amp, and re-configuring the front end to be a current-to-voltage converter, students can translate the current fluctuations into a measured mean-square voltage noise. They can test the predicted variation of current fluctuations

• over a range of dc currents idc from <10 nA to >100 μA
• and for bandwidths Df varying in width and location in frequency space.

With those idc and Df dependencies verified, students can harvest an all-electronic measurement of the fundamental charge e. Precisions of order 1%, and accuracies of a few per cent, are available with the apparatus as-is, and the accuracy can be improved to 1-2% by attention to various calibration tasks.

What’s more, students can verify that not all currents are subject to shot noise of this form, by showing that alternative sources of dc current display fluctuations much smaller than those predicted by the formula above. In fact, the reconfigurable front end allows the exploration of shot-noise-limited, and sub-shot-noise, currents, from sources dependent on, and independent of, the photoelectric effect.

Calibrations

We have deliberately structured the low-level and high-level electronics to allow full user calibration of every factor involved in experimental noise measurements.

In the as-supplied condition, the pre-amp and main-amp gains G1 and G2 can be trusted to better than 1% over the relevant bandwidths. The filter sections give bandwidths defined to better than 2% as supplied, and simple calibrations can reduce this to <1% uncertainty. What’s more, we’ve chosen circuits with good high-frequency behavior, so their performance in the relevant dc-to-100 kHz band can be trusted in detail. The result is that interested students can pursue statistical, and systematic, errors in this apparatus, at a level of 1%, with confidence.

The built-in Noise Calibrator supplies pseudo-random noise, uniform in density in the

0-32 kHz hand, with output of rms measure near 212 milliVolts. Students will learn how this gives a ‘noise power density’ near 1.40 x 10-6 V2/Hz, and will also learn why this entails a ‘voltage noise density’ near 1.19 mV/ √Hz, milliVolts per root Hertz, in this band.

The Manual also has an Appendix giving the details of how an amplified (but not squared) noise signal can be digitized (say, with a digital ‘scope) and processed (on a computer) to give a Fourier –spectrum view of noise, and noise density. Students can learn to do what ‘scopes’ FFT functions typically will not do: create spectral-density graphs whose vertical axes are properly calibrated in V2/Hz or V/√Hz units.