Bitstream Encoding Made Easy¶

A stochastic number is not “random junk” — it is a deliberate encoding of a real value as the statistics of a bit sequence.

Unipolar encoding¶

For \(x \in [0,1]\):

\[P(X_t = 1) = x, \qquad \hat{x} = \frac{1}{N}\sum_{t=1}^{N} X_t\]

Value \(x\)	Example length-8 pattern (one draw)	Mean
0.00	`00000000`	0
0.25	`10001000`	0.25
0.50	`10101010`	0.50
1.00	`11111111`	1

Unipolar multiply (independent streams): \(X \land Y\) has mean \(x\cdot y\).

Bipolar encoding¶

For \(x \in [-1,1]\):

\[P(X_t = 1) = \frac{x+1}{2}, \qquad x = 2\,P(X_t=1) - 1\]

Bipolar multiply (independent streams): \(X \odot Y\) via XNOR yields mean \(\approx x\cdot y\) on the bipolar scale.

graph TD
    U[Unipolar x in 0..1] --> AND[AND = multiply]
    B[Bipolar x in -1..1] --> XNOR[XNOR = multiply]
    U --> MUX[MUX / OR hybrids = scaled add]
    B --> SCALE[Scaled add via multiplexers]

Generating streams (SNG sketch)¶

A common stochastic number generator:

Draw a uniform random integer \(R_t\) each cycle (LFSR, TRNG, Sobol, …).
Compare to a binary register holding \(x\).
Output bit \(X_t = [R_t < x]\).

Deterministic variants replace \(R_t\) with a low-discrepancy sequence so short \(N\) still tracks \(x\) closely. See Number Generators.

Decoding¶

Hardware usually counts 1s over a window (or uses progressive counters) and optionally renormalizes. Window choice sets the precision–latency trade-off covered in Error, Precision & Latency.

Try it¶

The bitstream multiply lab on the “Why Probabilistic Computing?” page samples unipolar streams live in the browser.