Pseudo-Random Binary Excitation

• Pseudo-Random Binary Excitation is the deterministic generation of binary sequences that mimic true randomness using chaotic maps and algebraic integer methods.
• It employs approaches from cubic algebraic chaotic mappings to all-optical circuit designs, achieving high-speed operation and robust modulation.
• Comprehensive statistical validations demonstrate superior independence and aperiodicity, making it pivotal for secure communications, radar detection, and simulations.

Pseudo-random binary excitation refers to the generation and utilization of deterministic, yet unpredictable, binary sequences that closely approximate statistical properties of true randomness. These sequences underpin applications in computational modeling, secure communications, radar detection, and ultrafast photonic logic. The rigorous generation of pseudo-random binary sequences (PRBS) encompasses both classical algorithmic constructs as well as physical implementations, with focus areas ranging from cubic algebraic integer-based chaotic mappings to all-optical circuit designs. This entry addresses key theoretical frameworks, state-of-the-art generation methodologies, statistical validation regimes, implementation platforms, and domain-specific deployment scenarios.

1. Mathematical Foundations of Pseudo-Random Binary Sequences

Pseudo-random binary sequences are defined as deterministic bit streams whose statistical characteristics (e.g., autocorrelation, distribution, independence) resemble those of ideal coin-tossing or white noise. In the context of nonlinear dynamics, the Bernoulli (doubling) map $M_B : x \mapsto 2x \bmod 1$ on the unit interval serves as an archetype for fair-coin sequence generation. For typical real inputs $x_0 \in [0, 1)$, the binary indicators $\epsilon_n = 1$ if $x_n > 1/2$, $\epsilon_n = 0$ otherwise, yield independent and identically distributed bits.

Standard finite-precision arithmetic implementations, however, introduce periodicities and structural artifacts. The algebraic approach—representing states $x_n$ as roots of monic irreducible cubics $f(x) = x^3 + b x^2 + c x + d$—enables exact orbit computation and circumvents premature convergence to cycles or fixed points by maintaining the algebraic integer structure throughout iteration (Saito et al., 2017).

2. Generation via Chaotic Maps and Algebraic Integers

The generator presented by Saito & Yamaguchi constructs PRBS through exact computation of chaotic orbits of the Bernoulli map on cubic algebraic integers. The state space is organized as triples $(b, c, d)$ satisfying:

• $b^2 - 3c \leq 0$ (strict monotonicity),
• $d < 0$,
• $1 + b + c + d > 0$ (unique real root $\alpha \in (0, 1)$).

Each update operates entirely over integers:

• If $\alpha < 1/2$, the update rule is linear scaling:

$$\begin{pmatrix} b' \ c' \ d' \end{pmatrix} = \begin{pmatrix} 2 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 8 \end{pmatrix} \begin{pmatrix} b \ c \ d \end{pmatrix}$$

• If $\alpha > 1/2$, matrices and constant-terms yield new coefficients:

$$\begin{pmatrix} b' \ c' \ d' \end{pmatrix} = \begin{pmatrix} 2 & 0 & 0 \ 4 & 4 & 0 \ 2 & 4 & 8 \end{pmatrix} \begin{pmatrix} b \ c \ d \end{pmatrix} + \begin{pmatrix} 3 \ 3 \ 1 \end{pmatrix}$$

Binary output at each step is derived by evaluating $s_n = 1 + 2b_n + 4c_n + 8d_n$: output $0$ when $s_n > 0$, output $1$ otherwise. This scheme ensures maximal entropy, exact aperiodicity, and deterministic independence under arbitrarily long bit shifts (Saito et al., 2017).

3. Seed Selection, Independence, and Nonmerging Orbits

High-quality pseudo-random binary excitation requires both statistical uniformity and stream independence. The seed set construction utilizes large $c$ and appropriate parity of $b,c$, forming seeds $(b, c, d)$ with $d \in \{-1, -2, \dots, -(b+c)\}$:

• Roots $\alpha(d)$ are nearly equispaced in $[0,1)$, guaranteeing no bias.
• Careful selection ensures each seed is a "source point," with no preimage under $\bar{M}_B$, prohibiting orbit merging.
• Each sequence remains statistically noncorrelated from others even under arbitrary bit shifting (Saito et al., 2017).

4. Statistical Validation and Comparison to Classical Generators

To rigorously demonstrate randomness, generated PRBS are subjected to comprehensive test batteries such as DIEHARD, NIST STS, and TestU01:

Test Battery	Number of Sub-Tests	Pass Rate for Bernoulli-Map PRBS
DIEHARD	234 first-level, 6 second-level	232/234, 6/6
NIST STS	188 across $10^3$ sequences	187/188, p-value OK
TestU01	SmallCrush (15), Crush (144), BigCrush (160)	all passed

The Bernoulli-map PRBS exhibits stronger independence properties and resistance to structural correlations compared to MT19937 (Mersenne Twister), notably filling out the unit square in $Y_n$ vs. $Y_{n-227}$ plots where MT19937 points concentrate along diagonals (Saito et al., 2017).

5. Physical Implementations: Ultrafast All-Optical PRBS Generation

Advances in silicon photonic hardware have enabled the design and realization of all-optical PRBS generators utilizing cascaded 2×2 silicon microring resonators (SiMRRs). PRBS generation leverages nonlinear and thermal carrier dynamics:

• Two-photon absorption injects free carriers ($N_{fc}(t)$).
• Free-carrier dispersion and absorption modulate resonance blue-shifts and losses.
• Thermo-optic effect induces slower resonance red-shifts.

A linear-feedback shift register (LFSR) architecture is emulated using all-optical D-flip-flops and XOR gates:

• 3-bit PRBS: Feedback polynomial $x^3 + x^2 + 1$, period $7$.
• 4-bit PRBS: Feedback polynomial $x^4 + x + 1$, period $15$.

This configuration achieves sequence bit-rates exceeding $45\,\text{Gbps}$, $>80\%$ modulation depth, and low pump power thresholds ($\sim28\,\text{mW}$) (Ghosh et al., 2022).

6. Applied Domains: Radar Detection and Information Processing

Pseudo-random binary excitation is central to digital radar detection, where PR sequences of length $N$ maximize channel separation and detection fidelity. The pseudo-random method entails correlating received echoes with time–frequency shifted copies of the transmitted PR sequence, evaluating the ambiguity function $\mathcal{A}(S, R)[\tau, \omega]$:

• For sparsity $r \leq N^{1-\delta}$, probability of detecting all true targets approaches unity, and the expected number of false alarms approaches zero as $N \to \infty$.
• Fast Fourier transforms render the $N \times N$ computational task feasible ($O(N^2 \log N)$ complexity).
• Asymptotic reliability is obtained under channel uniformity, square-root cancellation in noise, and independence of channel and noise processes (Fish et al., 2014).

7. Theoretical Implications and Limitations

Periodicity is inherently suppressed in algebraic-irrational PRBS, yielding aperiodic streams. The normal-number conjecture suggests uniform pattern frequencies in binary expansions of algebraic irrationals. Absolute unpredictability is maintained, as simple automata cannot reproduce such sequences (Adamczewski–Bugeaud theorem). The measure-theoretic isomorphism between the Bernoulli map and coin-flipping idealizes the realization of pseudo-randomness. While slower than linear-recurrence generators (e.g., MT19937), exact-chaotic generators afford provable stream independence, infinite period per seed, and no hidden linear structure (Saito et al., 2017).

Summary Table

Generator Type	Key Mechanism	Unique Attributes	Principal Application
Bernoulli-map on algebraic integers (Saito et al., 2017)	Chaotic map + algebraic integer arithmetic	Infinite period, exact independence, no orbit merging	Cryptography, simulation, tests
SiMRR all-optical (Ghosh et al., 2022)	Nonlinear carrier dynamics, LFSR circuit	45 Gb/s bit-rate, >80% modulation depth, low power	Ultrafast logic, photonics
PR sequences in radar (Fish et al., 2014)	Discrete correlation, ambiguity function	Polynomial reliability, vanishing false alarms	Radar sparse target detection

Pseudo-random binary excitation, realized through deep mathematical constructs or optical device physics, continues to drive reliability, unpredictability, and high-performance across computational, communication, and sensing systems.