2-D Random Walk Probability Displayed Using “DIY” Electronic Circuitry

A class-room demonstration "two-dimensional random walk" involves random variable r , equal to the sum of the squares of two independent Gaussian random variables x and y . Each has zero mean with identical variance. "Do-It-Yourself" analog electronic circuitry includes: (a) an avalanche diode broadband white noise generator, (b) an L-R-C narrow-band pass filter of angular center frequency ω_o coupled to an audio amplifier driving a loudspeaker. A receiving microphone signal is amplified and displayed. This BP filtered noise random variable is modeled by n(t)=x(t)cos(ω_ot) - y(t)sin(ω_ot). In-phase and quadrature circuitry isolates x(t) and y(t) random variables, x and y. Multiplier circuits, an op-amp adder and square root circuitry generate the r(t) random variable r characterized by the Rayleigh PDF distribution. An analog PDF circuit with a voltage window comparator output chopped on-off, is counted for 4 seconds. In sequence, {bin#, counts} histogram data is generated and displayed by an LED matrix.