Week 8: Random Noise Flashcards
What did Robert Brown study? What did he notice?
By the late 19th/early 20th century, there was an idea that microscopic fluctuations could become visible and have observable macroscopic effects.
Robert Brown studied this effect through the irregular movement of pollen powders (London, 1827):
• He put fine pollen powder in liquid, and observe how they mix
• Noticed they jittered around in seemingly random ways in the fluid
o This is what became known as “Brownian Motion”
Botanists and biologists tried to explain this motion.
What were the existing explanations for Brownian motion?
Two primary theories:
- Organic/biological explanation
a. Perhaps pollen has some unknown biological mechanism which drives this motion?
b. Was the dominant theory for ~50 years - Physical explanation:
a. Arose from physicists in the second half of 19th century
b. The idea of “thermal motions”; it’s simply the effect of agitated molecules
c. A Kinetic Theory – each pollen particle is getting
What was the issue with Brownian motion’s Physical explanation? Who resolved it, how? What theory?
While the physical explanation gained traction through the 1800’s, it had a problem: How can a huge object be moved by such tiny particles?
• Like saying asteroid impacts can visibly move a planet
Einstein reasons this problem, and publishes his work in 1905:
• Suggests motion of the pollen is indeed due to collisions with the water
o These collisions vary randomly over time
o The paths of the particles are independent of each other and time independent
Like a bunch of randomly stumbling drunkards; a drunkard’s next step is completely unrelated to their last step, and has no relation to what other drunkards are doing
• Equipartition Theory:
o If system is in thermal equilibrium, particles all have a shared energy level
Every atom has some energy (E = kT)
o If the Pollen is at the same energy level as water, it moves much slower (due to its size)
o So this is still Thermal motion
How did Einstein model Brownian motion? What result did he arrive at?
Einstein modelled the problem by considering how many particles would be within a region, [x, x+dx] in some time [0, t]
• Given a certain time and space window, he considered how many particles would be kicked in to that window
• Arrives at the diffusion equation
o Makes assumption that at time 0, there is no displacement (all particles are together in the same spot)
o This is like adding milk to coffee: Initially, the milk is all concentrated in one spot. But over time, the milk particles diffuse all over randomly, until they are everywhere
• Solving the equation for the particle’s density in space and time is the normal distribution!
o Variance in the distribution increases with time (as particles diffuse further away)
Einstein’s solution for the diffusion equation was then used to determine Avogadro’s Number
• Einstein’s considerations of Brownian Motion introduced “D”; the mean free path
• By knowing a liquid’s properties and using Einstein’s formulation, you could determine NA
o Properties include viscosity, temperature…
• Why was this important?
o Gives a physical method without relying on any chemical reactions
o Thus if it’s close to the physical result, we have a second determine it without assuming atoms, molecules, or their masses
How/Who was Einstein’s estimate for Na tested?
The estimate for Avogadro’s Number was tested in 1909 by Jean Perrin
• A French physicist, who constructed an experiment to measure the mean free path of Brownian Motion
o Idea: If suspended in fluid, the particles won’t simply fall due to gravity
Brownian motion will cause some to randomly move up
o Measuring the density of the fluid at each height can give an estimate of this motion
• Measures 7.05E23
o A reasonable number (today: 6.022E23)
What are the implications of Brownian motion?
- Reality of Atoms
a. Due to cross referencing of Avogadro’s Number - Microscopic Randomness can cause macroscopic fluctuations
a. Before, scientists though all the microscopic randomness would cancel each other out (large numbers lead to macroscopic stability)
b. Here, we saw the fluctuations were the direct result of all the random thermal motion
When was Electronic noise first studied? Why did it matter?
1907: Einstein studies thermal fluctuations of the voltage across capacitors
• There are many free electrons in the metal, so their motion should be detectable by very sensitive measuring tools
Einstein’s ideas weren’t taken serious at the time – but 1910 saw the mobilization of telecommunications
• Vacuum tubes enabled amplification and filtering of signals
• Accelerated the deployment of cross continental radio, phone technology in 1910, 1920’s
Two types of Electronic Noise?
- Shot Noise (Walter Schottky)
a. Schottky, a German physicist, first proposed the possibility of noise in high gain amplifiers
b. Idea: Current is not a constant continuous level, since it is composed of electrons shooting over one-by-one (like pulse width modulation)
i. “Shot noise” since you see noise as electrons are shot across - Thermal Noise (Henry Nyquist)
a. Electrons have random thermal fluctuations (like Einstein proposed with Brownian motion)
b. Thermal agitations will cause random noise
c. Likened a loop of wire to a black body radiator
How did electronic noise impact EE in the 20th C? Why?
• In both cases, noise is “white”
o No preferred frequency, like white light it is just everything everywhere
• So ordinary idea of “frequency components” doesn’t apply; can’t analyse spectrum
o A stochastic process, with random, non-deterministic frequencies
o Need to instead consider the spectral correlation
o Would lead to the development of more rigorous spectral analysis techniques
E.g. Wiener-Khinchin Theorem (1930-1934)
What was the earliest use of Pseudo RNG? Who?
The scrambling signal > “spread spectrum communicator”
• Developed in 1941 by Hedy Lamarr & George Antheil for Allied torpedo guidance systems
o A Hollywood star and musician respectively
• The problem with torpedo communication was that if the guidance system’s frequency was known, the enemies could simply jam that frequency and render the torpedoes useless
• Idea: Periodically swap the frequency of communications for the torpedo
o At a pre-set random cadence, perform a frequency hop
Thus enemies can’t jam a particular frequency to disable the guidance system
• This was used further in the 1980’s with the development of CDMA
