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Noise / Chaos / Random Numbers
Linear Feedback Shift Registers 

by Bob Paddock.

"Like diseases, noise is never eliminated, just prevented, cured, or endured, depending on its nature, seriousness, and the cost/difficulty of treating it." from Analog-Digital Conversion Handbook, by D.H. Sheingold, Analog Devices.

Noise? I hate noise, I'm always trying to get rid of noise. Why should you take time to read a page about noise?

For starters, did you know that there are five fundamental types of noise and eight "colors" of noise?

Did you ever consider that adding noise to your signal may enhance your system by getting rid of the noise you don't want?

Quick links:

Why would adding noise to my system be good? 

If you're still not convinced that adding noise is helpful, then check out Enrico Simonotto Visual perception of stochastic resonance, that shows (via a Java Applet) what adding the right kind of noise can do to improve your system.

As part of their High Speed Design Techniques series, Analog Devices covers adding noise in the form of dither as a way to enhance high-speed ADC systems.  The section on dither can be found in section 5.45 " Achieving Wide Dynamic Range in High Speed ADCs Using Dither".

National Semiconductor has a similar Application Note: AN-804: Improving A/D Converter Performance Using Dither.

Noise can also show up in places such as cryptology (Architectural considerations for cryptanalytic hardware), Built-In-Self-Test, and other places such as circuit simulation.

Another example of noise is where Texas Instruments uses a noise source for Artifact Mitigation in their white paper on their Andromeda ASIC all-digital approach to projection display.

The last example can be found in a paper (Mind9), that I wrote several years ago, describing a technique related to audio perception via the addition of noise.

Before we can talk about noise and get to some simple demonstration circuits, we need some definitions for noise. The following is taken from sections of:

Federal Standard 1037C 

Telecommunications: Glossary of Telecommunication Terms

Date of Publication: August 7, 1996

Help is available for FS1037C.


a. This glossary provides standard definitions for the fields subsumed by the umbrella discipline of telecommunications...

(188) Terms and definitions in direct support of the MIL-STD-188 series of standards and their associated military handbooks. This is not a source citation.

Black noise: Noise that has a frequency spectrum of predominately zero- power level over all frequencies except for a few narrow bands or spikes. Note: An example of black noise in a facsimile transmission system is the spectrum that might be obtained when scanning a black area in which there are a few random white spots. Thus, in the time domain, a few random pulses occur while scanning.

Blue noise: In a spectrum of frequencies, a region in which the spectral density (i.e., power per hertz) is proportional to the frequency.

Pink noise: In acoustics, noise in which there is equal power per octave.

Pseudorandom noise: Noise that satisfies one or more of the standard tests for statistical randomness. (188) Note 1: Although it seems to lack any definite pattern, pseudorandom noise contains a sequence of pulses that repeat themselves, albeit after a long time or a long sequence of pulses. Note 2: For example, in spread-spectrum systems, modulated carrier transmissions appear as pseudorandom noise to a receiver (a) that is not locked on the transmitter frequencies or (b) that is incapable of correlating a locally generated pseudorandom code with the received signal.

[The following two are relevant as they can be used to make White/Pink noise. For example, see the National Semiconductor MM5437 (which was obsoleted).]

Pseudorandom number generator: 1. A device that produces a stream of unpredictable, unbiased, and usually independent bits. 2. In cryptosystems, a random bit generator used for key generation or to start all the crypto-equipment at the same point in the key stream.

Pseudorandom number sequence: 1. An ordered set of numbers that has been determined by some defined arithmetic process but is effectively a random number sequence for the purpose for which it is required. 2. A sequence of numbers that satisfies one or more of the standard tests for statistical randomness. (188) Note: Although a pseudorandom number sequence appears to lack any definite pattern, it will repeat after a long time interval or after a long sequence of numbers.

White noise: Noise having a frequency spectrum that is continuous and uniform over a specified frequency band. (188) Note: White noise has equal power per hertz over the specified frequency band. Synonym additive white gaussian noise.

The following comes from "Noise and Operational Amplifier Circuits" by Lewis Smith, D.H. Sheingold in Analog Dialogue 3-1, reprinted in the Best of Analog Dialogue page 19 & 31. It has all of the math that you would ever care to apply to noise calculations for the mathematically inclined.

Johnson noise: Thermal agitation of electrons in the resistive portions of impedances results in the random movement of charge through those resistances, causing a voltage to appear corresponding to the instantaneous rate of charge of charge (i.e., current) multiplied by the appropriate resistance. Ideally-pure reactances are free from Johnson noise.

Schottky noise: Shottky noise arises whenever current is passed through a transistor junction. The noise is normally expressed as a current, which will produce voltage drops in in impedance, such as transistor emitter resistance....

Flicker noise (1/f noise): In the frequency range below 100 Hz, most amplifiers exhibit another noise component that dominates over Johnson and Schottky components and becomes the chief source of error at these frequencies. Flicker noise is thought to be a result of imperfect surface conditions on the transistors...

Flicker noise does not have an equal contribution at each frequency. The spectral noise density of this type of noise typically exhibits a –3 dB per octave slope.

[259 and counting, papers on 1/f noise can be found at the Bibliography on 1/f Noise page.]

White noise: In a white noise spectrum, e-sub-n [Spectral noise density] is constant as a function of frequency....

Pink noise: A generic term applied to ideal 1/f noise, for which e-sub-n is exactly proportional to SQR(1/f), is Pink noise. [snip]

1. Pink noise contributes equal increments of RMS noise over each octave or each decade of the spectrum. Each increment with be 1.52k per decade, or 0.83k per octave, where k = e-sub-n or i-sub-n at 1 Hz.

2. Bandwidth for white noise is substantially equal to the higher frequency, if one is considering bandwidths greater than 1 decade.

The following comes from "D-C Amplifier Noise Revisited Understanding, Measuring, and Testing for Random Noise A New Op-Amp Noise Fixture for Automatic Benchtop Tests with LTS-2010" by Al Ryan and Tim Scarnton in Analog Dialogue 18-1, reprinted in the Best of Analog Dialogue page 151 & 159.

..."White" noise, which may be thermal (Johnsons), or shot (Schottky) has a constant distribution across the frequency spectrum but looks as if it is heavily oriented to the higher frequencies....

..."Pink", or "1/f", or "flicker" noise is dominated by low frequencies.... "Flicker noise" or "1/f" noise, has a noise power spectral density varying inversely with frequency.

The energy spectrum of noise is classified in colors following the idea of the light spectrum as you get from shining white light through a prism.

Colors of noise pseudo FAQ, V.1.3
by Joseph S. Wisniewski
as posted on alt.sci.physics.acoustics in Oct 1996.

That e-mail just keeps coming in. So, here's the latest rev. Thanks to the many people who pointed out the flaws in my pink and blue definitions. Thanks Kev for the pointer to FS-1037C. Due to popular demand, I am reversing my previous stand and adding the definition of orange noise.

The noises are now in spectral order (artistic license has been taken over where white, black, gray, and brown fit into a spectrum). Anyone is welcome to help fill in the gaps. We're up to three definitions of black noise. Keep them coming!


White noise (common definition:) power density is constant over a finite frequency range. AKA Johnson noise.

Pink noise (common definition): power density decreases 3 dB per octave with increasing frequency (density proportional to 1/f) over a finite frequency range which does not include DC. Each octave contains the same amount of power. Many point out that this is not a trivial filtering problem. AKA flicker noise.

Red noise (common definition within the oceanographic field, contributed by P.J. "Josh" Rovero) (Anyone have the spectrum?): oceanic ambient noise (ie, noise distant from the sources) is often described as "red" due to the selective absorption of higher frequencies.

Orange noise (anonymous contribution) (Anyone foolish enough to want the spectrum?): quasi-stationary noise with a finite power spectrum with a finite number of small bands of zero energy dispersed throughout a continuous spectrum. These bands of zero energy are centered about the frequencies of musical notes in whatever system of music is of interest. Since all in-tune musical notes are eliminated, the remaining spectrum could be said to consist of sour, citrus, or "orange" notes. Orange noise is most easily generated by a room full of primary school students equipped with plastic soprano recorders.

Green noise (defined by some folks producing relaxation tapes, Mystic Moods, I believe): supposedly the background noise of the world. A really long term-power spectrum averaged over several outdoor sites. Rather like pink noise with a hump added around 500 Hz. (Anyone have the spectrum?)

Blue noise (FS-1037C): power density increases 3 dB per octave with increasing frequency (density proportional to f) over a finite frequency range. This can be good noise for dithering.

Purple noise (original definition, contributed by Jon Risch): power density increases 6 dB per octave with increasing frequency (density proportional to f^2) over a finite frequency range. Differentiated white noise. AKA violet noise.

Gray noise (heard this one a couple of times, but can't put my finger on a source): noise subjected to a psycho acoustic equal loudness curve (such as an inverted a-weight curve) over a given range of frequencies, so that it sounds like it is equally loud at all frequencies. This would be a better definition of "white noise" than the "equal power at all frequencies" definition, since real "white light" has the power spectrum of a 5400K black body, not an equal power spectrum.

Brown noise (Jon M. Risch, rbmccammon): power density decreases 6 dB per octave with increasing frequency (density proportional to 1/f^2) over a frequency range which does not include DC. Is not named for a power spectrum that suggests the color brown, rather, the name is a corruption of Brownian motion. If we were going to pick a color, red might be good since pink noise lies between this noise and white noise. Unfortunately, red is already taken. AKA "random walk" or "drunkard's walk" noise.

Three different definitions of black (silent) noise:

Black noise (contributed by Jeff Mercure, his own definition) whatever comes out of an active noise control system and cancels an existing noise, leaving the world noise-free. (The comic book character "Iron Man" used to have a "black light beam" that could darken a room like this, and popular SCI-FI has an annoying tendency to portray active noise control in this light.)

Black noise (seen in the sales literature for an ultrasonic vermin repeller) power density is constant for a finite frequency range above 20kHz. Ultrasonic white noise. This black noise is like the so-called "black light" with frequencies too high to be perceived as sound, but still capable of affecting you or your surroundings.

Black noise (Manfred Schroeder, "fractals, chaos, power laws," contributed by Mike Arnao) has an f ^ -beta spectrum, with beta > 2, and is characteristic of " natural and unnatural catastrophes like floods, droughts, bear markets, and various outrageous outages, such as those of electrical power." further, "Because of their black spectra, such disasters often come in clusters."

The most common way to create Pseudo-Noise is the 

Linear Feedback Shift Register. 

A Linear Feedback Shift Register generally consists of  two or more D-type flip-flops and one or more exclusive-OR (XOR) gate.  The flip-flops form a shift register.  Whether it shifts right or left does not really matter and is usually determined by the requirements of the circuit that the LFSR is driving or the method that it is being constructed by.

In applications such as cryptology, the shift register can be preset to a known initial condition. but in general they are either set to all zeros except for one bit, or set to all ones except for one bit.

If XORs are used to generate the feedback input to the shift register, then the state of all zeros is not allowed as the system would never leave the all zero state.  If XNORs are used, then the state of all ones is not allowed for the same reasons.  The choice largely depends on how the LFSR is being implemented. For example, some CPLD's have no Preset option and only a Reset option where other devices are the opposite.

LFSR are not truly random devices because after a certain number of cycles, the cycle out of the LFSR will repeat, hence they are termed "pseudo-random devices."  The maximum number of cycles before the cycle repeats can be determined by the formula: (2^n)-1.   Where n represents the number of flip-flops.  The term –1 comes from the fact that either the all zero or all one state is disallowed.

The placement of the XOR/XNOR taps determine the bit sequence of the noise generated.  A poor selection of taps can lead to a LFSR that has a cycle length much less than the (2^n)-1 maximum.  A good book on the subject is The Art of Electronics by Paul Horowitz, Winfield Hill;  About the Authors.

In the past Best Proto gave away a free LFSR design program.  They have discontinued  the support of this program, but you can down load the older version here.

[I've used this program on several machines but for some reason it always crashes on my Gateway-2000 machine, as do several other programs. If anyone knows why or how to fix it, please let me know.  I've really begun to dislike cows and milk.]

The numbers below show the value out of a 17-bit LFSR initialized to a value of 10000h shifting right for the first 64 cycles:

10000, 08000, 04000, 02000, 01000, 00800, 00400, 00200, 00100, 00080, 00040, 00020, 10010, 08008, 04004, 02002, 01001, 10800, 08400, 04200, 02100, 01080, 00840, 00420, 10210, 08108, 04084, 02042, 01021, 00810, 00408, 00204, 00102, 00081, 10040, 08020, 14010, 0A008, 05004, 02802, 01401, 10A00, 08500, 04280, 02140, 010A0, 10850, 08428, 14214, 0A10A, 05085, 12842, 09421, 04A10, 02508, 01284, 00942, 004A1, 00250, 00128, 10094, 0804A, 04025, 02012 ...

An often overlooked use for an LFSR is in implementing counters and state machines. An LFSR takes less resources and frequently runs much faster than a conventional counter. For example, sequential states are not always needed, such as in a FIFO head and tail pointer, the states only need to be unique.

There are also two fundamental types of LFSR, the difference being whether the next stage of the shift register is fed by one of the XOR/XNOR gates, the Galosi type, or whether the XOR/XNOR gates are solely involved in the feedback path, the Fibonacii type. The paper Architectural considerations for cryptanalytic hardware goes into more details if you are interested.

Fibonacci LFSR

Galois LFSR

A example of an LFSR in use in a in a spread spectrum systems can be found in the paper LFSR Signal Spreading.

Altera offers Solution Brief 11 (Linear Feedback shift Register), while Xilinx offers XAPP052: Efficient Shift Registers, LFSR Counters, and Long Pseudo-Random Sequence Generators .

Texas Instruments covers how they use LFSR for pseudo-random pattern generation (PRPG), and a parallel signature analyzer (PSA). An LFSR and PSA are used to test a TI application-specific integrated circuits; see their Application Report Abstract: What's an LFSR?.

A free example high-level language implementation of a Linear Feedback Shift Register (LFSR) can be downloaded from here.

Associated Professional Systems has made their Linear Recursive Sequence Analyzer Software avaliable as Freeware.

The PN Simulation Program allows users and designers of Direct Sequence Spread Spectrum systems to generate, display, correlate, simulate and analyze Linear Recursive Sequences. Two sequences can be set and analyzed at a time. 

Random Numbers

A good starting place to look for any thing mathematical whether it be Random Numbers or CRC code is Numerical Methods Reference Material page.  Another is FAQ: Numerical Analysis & Associated Fields Resource Guide.

Web Sites for Random Number Generators has links to several sites that deal with RNG's.

Are you wondering why you just can't use the RANDOM() function in the language of our choice?  Did any one give any data on just how random that function is?

Creating Analog Noise 

Visit the Linear Technology Corporation web site  for DN70 A Broadband Random Noise Generator.

The company Noise/Com is the only company that I'm aware of dedicated to making noise. Check out their application notes and noise diodes.

Martin Saxon has a paper Noise Measurement Briefing with tips on how to approach some noise measurements.

Micronetics Wireless manufactures RF microwave and noise components as well as network infrastructure test equipment designed for use across a broad range of applications, including CDMA, cellular, PCS, HomePNA, Bluetooth, broadband, radar, satellite, military, and electronic simulation.


Loosely, chaos is defined as aperiodic dynamics in deterministic systems in which there is sensitive dependence to the initial conditions. The typical example cited, translated from Mathematician to English is what is known as "The Butterfly Effect."  The Butterfly Effect is where a butterfly flapping its wings today in China is the initial condition that causes a thunderstorm to form over Kennerdell, Pennsylvania, next Saturday during my picnic.

While mathematicians like to cloak things in mathematical terms, actual circuits can be devilishly simple:

Michael Cross at the California Institute of Technology has put together a simple introduction of  how the circuit works, how to build the circuit, and why you should be interested in Chaos.  There is also a Java Applet for simulating Chua's Circuit.  More info can be found at Michael Cross - Site Map California Institute of Technology. [The schematic shown here shows Michael Cross's single op-amp version rather than the dual op-amp Chua version.]

On a personal note, when I fiddle with such circuits on my breadboard, I prefer to use Texas Instruments' rail-splitter the TLE2426. Two 9-V batteries supply 18 V and the TLE2426 outputs a  midpoint ground rather than using the 9-V batteries directly, to get the +/– 9 V.  Not having to wonder if my two supplies are truly balanced gives me one less thing to worry about in my design debugging.

This class of circuit is what has become known as "Chua's Circuit", named for Professor Leon O. Chua from the Nonlinear Electronics Laboratory, University of California at Berkeley.

The details can be found in:

World Scientific Series on Nonlinear Science Series A: Monographs & Treatises

World Scientific Series on Nonlinear Science Series B: Special Theme Issues and Proceedings

Tao Yang is also an assistant to Prof. L.O. Chua on his course EE129: Neural and Nonlinear Information Processing and has several interesting links at his site:

  • Tao Yang, "Optimizing Stochastic Resonance in Visual System,'' Physics Letters A, 10 Aug. 1998, Vol. 245, (no. 1-2):79–86. Download the C source edtcnnsr.c (Optimizing algorithm) Download the C source dtcnnsr2.c (moving sum). 
The Introduction to Nonlinear and Chaotic Phenomena tutorial provides a short introduction to nonlinear and chaotic phenomena, as well as several Java applets are provided for interactive experimentation.

Simulation software for Chua's oscillator is described by M. P. Kennedy (alternate address), Three steps to chaos, IEEE Trans. CAS part I, Vol. CAS-40, no. 10, pp. 640–674, October 1993, and publicly available via FTP from  More info can be found at the Nonlinear Electronics Group, University College Dublin.

What is Chaos? A five-part online course for everyone

By Dr. Matthew A. Trump at University of Texas at Austin

Interactive Chaos: The Dynamics of the Standard Map: A "map" in this sense means that it is a pair of mathematical transformation equations that can be plotted on a two-dimensional graph. The Standard Map is a mathematical model that physicists use to understand and describe the phenomenon of chaotic motion. By studying the map, one can can catch a glimpse of the barest conditions under which chaos can arise. A command and important part of the study of  Chaos is " phase space", which is a visual method of representing Chaos, is also covered by the Standard Map tutorial.

Yun Liu  maintains link pages on Chaos Communications and Interesting Sites of Chaos Research.

Nonlinear Dynamics in Optical Systems

Communication with Chaotic Lasers (pdf)

One of the more studied area of Chaos today is that of encryption:

Chaos in the area of biology:
  • Electric Noise can Increase Human Tactile Sensation of mechanical forces, new experiments have shown, opening possibilities for electric devices that can enhance sensitivity to touch in the elderly, stroke patients, and people with diabetes. Researchers in Massachusetts (contact Jim Collins or Kris Richardson, Boston University,( 617) 353-0390) applied a small mechanical force to the finger pads of 11 young, healthy subjects. The force was ordinarily too weak for the subjects to detect, with a magnitude of approximately 0.01 Newtons, roughly equivalent to pressing a pencil tip onto the finger very lightly. However, when the researchers applied this force along with 2 mA worth of randomly fluctuating electrical current through the fingerpad, 9 of the 11 subjects then reported detecting the mechanical stimulus–without feeling the electric current. From the Physics News Update, The American Institute of Physics, Bulletin of Physics News No. 387, August 28, 1998 by Phillip F. Schewe and Ben Stein.
  • Ever have one of those days where your mind felt like it was in Chaos? If so, you might want to take a look at: Chaos in the Brain 
  • Overall objective: To gain a better understanding of the dynamics of the human brain through a study of EEGs, with an emphasis on attempting to characterize the effects of Alzheimer's Disease (and other dementia). In addition to the conventional signal analysis techniques (time-frequency time-scale, like power spectra, autocorrelation, etc.), we examine the data using a variety of techniques from the field of nonlinear dynamics (chaos) -- such as delay-time embedding, dimension estimation, and Lyapunov exponents, etc. 

Journal of the field:

Just like the electronics industry has Circuit Cellar, and the pizza industry has Pizza Today, the field of Chaos has the International Journal of Bifurcations and Chaos.

Other Sources: 

The sci.nonlinear faq compiled by  (James Meiss),  A Microsoft Word (rtf) version is also available.

Nonlinear Sites

A extensive Nonlinear Dynamics Bibliography can be found at the page of the Arbeitsgruppe Nichtlineare Dynamik / Nonlinear Dynamics Group.

A representative patent in the field: 

Kevin Cuomo and Alan V. Oppenheim. "Communication Using Synchronized Chaotic Systems.'' United States Patent # 5291555, issued 1, March, 1994.

Reference books: 

Chaos: Making a New Science by James Gleick.  A good nontechnical introduction and history to the beginnings of the field of Chaos.

The Circuits and Filters Handbook by Wai-Kai Chen: A summary with in-depth explanations.

A couple of interesting pages for their esthetics rather than technical aspects are Chua's Oscillator: Applications of Chaos to Sound and Music and the

Images of Chaos, an Artwork for Television Home Page.

Other related areas of study: 

While fractals are not directly related to the subject of Noise, there is some overlap, so you might find the Mandelbrot Fractal Generator Java Applet by Nick Lilavois of interest. More can be found on Benoit B. Mandelbrot here.

If you wish to dig deeper, some interesting areas that you might want to do a bit of searching for  are Balthazar van der Pol's Neon Bulb Oscillator and the concept of Winding Number by Poincare.

An offline example of Chaos in our own bodies: "Nonlinear dynamics, chaos and complex cardiac arrhythmias" by L. Glass,  A. L. Goldberger,  M. Courtemanche, and A. Shrier.  Dynamic Chaos, proceedings of a Royal Society Discussion Meeting Held on 4 and 5 February 1987, by The London Royal Society. Proc. R. Soc. Lond. A 413, 9-26 (1987).

A paper that I would like to find but have not: L.O. Chua , "The Genesis of Chua's Circuit", AEU 46, 250 (1992).

If you would like to add any information on this topic or request a
specific topic to be covered, contact Bob Paddock

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