Part 2: Some Noise Theory

Noise?

In communication systems, noise is an error or undesired random disturbance of a useful information signal. The noise is a summation of unwanted or disturbing energy from natural and sometimes man-made sources (Wikipedia).

The above is the common definition of noise in electronic circuits and communication systems. Blencowe [1] slightly narrows this definition in that the term noise…

[…] is more specifically reserved for intrinsic circuit noise. […] Intrinsic noise […] is generated ‘inside’ components and is an unavoidable consequence of the physics of materials, and sets the ultimate limit on how quiet a circuit can be.

Moreover noise in the above narrower sense must be distinguished from all sorts of deterministic signals like hum or RF pickup, as true noise is 100% chaotic. This means that the amplitude of the noise signal can only be determined in the form of a probability. Usually the probability follows a Gaussian bell curve, therefore we talk about Gaussian Noise.

The above definitions make a lot of sense when dealing with audio systems, even more when dealing with noise purposely generated for audio measurement and analysis, so for the rest of this series these definitions shall stick.

Noise Colors

She comes in colours everywhere
She combs her hair
She’s like a rainbow

_{(She's like a Rainbow/The Rolling Stones)}

Noise is commonly categorized by its spectral content with designators deriving from optics (i.e. red for lowest and blue for the highest frequencies in the light spectrum). There’s a fairly good (even though having some rough edges) overview (with some explaining figures) here.

White Noise

White Noise denominates a form of noise that, simply said, is made up from infinite different frequencies, evenly distributed (again, think of white light in this respect). More technically this means that noise power per frequency (or rather bandwidth) is always the same, independent of the band within the spectrum. E.g. noise power between 1 Hz and 2 Hz is the same as between 200 Hz and 201 Hz or 10005 Hz and 10006 Hz.

Considering this we can make easily make use of White Noise to determine the frequency response of audio devices (e.g. amplifiers) by simply feeding their inputs with white noise and check the Fourier spectrum of the output signal.

White Noise is common in electronic systems, e.g. in the form of thermal (Johnson) noise (see below Types of Noise).

Pink Noise

In Pink Noise the power distribution over frequency is generally constant in some form of logarithmic scale, e.g. per decade or per octave. It’s level therefore falls with frequency on a linear scale and consequently is also referred to as 1/f noise. I.e. power between e.g. 200 and 400 Hz is the same as between 400 and 800 Hz. Pink Noise particularly falls by 3 dB per octave in level. It’s commonly used for equalizing room acoustics and in speaker and crossover development. More on this in Self [2].

Pink Noise is (in analogue systems) usually derived from White Noise using a so-called pinkening filter, i.e. a low pass filter with a 3 dB per octave slope. Unfortunately the stop-band slope of even a 1^st order filter already shows a 6 dB per octave slope so this must be approximated by using a series of poles and zeroes (more on that when it comes to the practical design of the noise generator).

Red Noise

Red Noise is similar to Pink Noise, except that its power falls with 6 dB per octave (1/f² noise). It’s therefore easily derived from white noise by integrating (or 1^st order low pass filtering) a White Noise signal. As far as I know its only use in audio is for synthesizing sounds for electronic music. Sometimes it’s also called “Brown Noise” or “Brownian Noise”, “Brown” being a name, not a color.

Gray Noise

(White) Noise weighted by a psychoacoustic weighting function, e.g. A-Law or µ-Law, is commonly called Gray Noise. It’s used e.g. to determine the subjective perception of audio systems, rooms or noise itself.

Not strictly Gray Noise but very useful for the design of low noise (particularly tube-based) phono stages is the use of an RIAA slope for the weighting filter (cmp. Blencowe [3]), as will be pointed out in a future series on RIAA stage design.

Violet & Blue Noise

These are sort of the opposite hands of Pink and Red Noise, i.e. their power spectrum raises with frequency (+3 dB per octave for Violet and +6dB per octave for Blue Noise respectively). To my best knowledge there’s no use in audio for any of these. Probably one might use it in synthesizers?

Types of Noise in Electronic Circuits

Noise is an inevitable by-product in every electronic circuit, usually highly unwanted. Knowing about the physics behind the noise sources helps to minimize unwanted noise (or, as in this particular case of designing a noise generator, maximize it).

Thermal / Johnson Noise

Johnson Noise is the unavoidable implication of atoms vibrating in any solid-state body (e.g. a resistor) at temperatures above 0°K (-273.15°C). As atoms are made up of charged particles and electric current is defined as the movement of such charged particles, it becomes clear that this movement of the atoms causes uncorrelated currents, i.e. noise. Or, as Blencowe [3] puts it:

This […] leads to the perhaps unexpected fact that an isolated resistor, completely unconnected from anything, dissipates some power. This can be thought of as the power delivered to the resistor by the universe at large to keep it at ambient temperature.

As expected noise power is proportional to temperature:

with

• k … Boltzman’s constant
• T … absolute temperature in Kelvin
• B … (measurement) bandwidth in Hertz

Considering that \(P = U² / R \) we get the more common result (in V_RMS):

Given that our devices will usually be operated at room temperatures and audio bandwidth can be considered a constant 20 Hz to 20 kHz, one quickly realizes that the amount of Johnson Noise is mostly depending on the resistance of the device (Resistor). Using resistances as low as possible therefore is one of the most important rules to follow in low noise designs.

Flicker Noise, Excess Noise

Flicker or Excess Noise is noise that is basically due to imperfections in electronic components. This makes Excess/Flicker Noise much harder to predict, compared to Johnson noise. Usually the term “Excess Noise” is used related to passive components, particularly (film) resistors, whereas “Flicker Noise” is more often used in conjunction with active components (semiconductors or vacuum tubes).

Flicker as well as Excess Noise are Pink (1/f) Noise variants. This is significant, as most of the noise power lies in low frequency (i.e. audio) regions and this makes 1/f noise often much more significant and troublesome, compared to e.g. Johnson Noise.

Excess Noise in resistors is usually caused by imperfections in the helix cut into the resistive film or loose contact between the resistive film and the (possibly corroded) end caps. It is proportional to current flowing through the device. The only remedy is to use high quality resistors (noise figures are normally provided in the manufacturers' data sheets). Wirewound and foil resistors are virtually free free of Excess Noise.

Flicker Noise Blencowe [4] in vacuum tubes is (similar to Excess Noise) proportional to the (DC) current through the tube and some device-specific parameters. A basic formula to determine the noise current as a function of DC current is given by:

with

• K … material constant
• I_DC … mean value of current
• a … material constant (also determines noise color)
• b … material constant
• f … 1 Hertz wide frequency band

The material constants a and b can be approximated by a = 2 and b = 1, so in the end K is the only remaining material constant (to be determined experimentally by measurement). To finally get the amount of Flicker Noise within a certain frequency band (f_hi - f_lo) the above function gets integrated and we finally end up with:

In short: Flicker Noise is proportional to DC current and depending on the material constant K. Again something we have to keep in mind for future low noise projects.

Shot Noise

Shot Noise is a form of White Noise that originates from the discrete (quantized) nature of electric charge. It’s basically described by Schottky’s Formula:

with

• q … electron charge (1.6 x 10^-19)
• I_DC … mean value of current
• B … noise measurement bandwidth

So, again we have a noise current proportional to (the square root of) DC current through the device. We will get back to Shot Noise in the following posts of this series as Shot Noise is the technical basis for deliberate noise generation, e.g. in vacuum noise diodes.

References