Non-Deterministic Random-Number-Generator-RNG (Monte Carlo Estimation of π Using Zener Diode Noise)#

Overview#

This project implements Monte Carlo estimation of π using random bits extracted from analog voltage measurements of a Zener diode noise source. The voltage noise is digitally sampled, converted into a binary sequence via statistical thresholding, and then transformed into 2D coordinate pairs in [0,1)² using bit pairing strategies. Two distinct algorithmic approaches are compared:

• Version 0 (v0.mlx): Sequential non-overlapping bit consumption with MSB/LSB interpretation variants
• Version 1 (v1.mlx): Overlapping sliding-window strategy achieving approximately 40× more usable Monte Carlo sample points

Both methods enable high-accuracy π estimation from a limited set of physical random bits.

Project Structure#

.
├── Data_Post_Processing_v0.mlx
├── Data_Post_Processing_v1.mlx
├── LICENSE
├── README.md
└── Data_from_Scope/
    └── scope_6_2.csv

File Descriptions

• Data_Post_Processing_v0.mlx: MATLAB Live Script implementing sequential non-overlapping bit grouping with MSB-first, LSB-first, and combined (BOTH) interpretation methods for coordinate generation and Monte Carlo π estimation.

• Data_Post_Processing_v1.mlx: MATLAB Live Script implementing overlapping sliding-window approach with 1-bit step size, maximizing bit utilization and generating 5,376 coordinate pairs from 5,415 extracted bits.

• Data_from_Scope/scope_6_2.csv: Raw voltage measurements from oscilloscope containing 1,000,001 samples acquired at 20 kHz over 50 seconds.

Methodology Overview#

Data Acquisition

Raw voltage samples are acquired and stored in CSV format:

• File:

  Data_from_Scope/scope_6_2.csv

• Total Samples: 1,000,001
• Sampling Frequency: 20 kHz
• Measurement Duration: 50 seconds
• Voltage Range: Approximately 0.247–0.255 V

Signal Statistical Analysis

The voltage signal is characterized through computation of:

• Mean: μ_V = 0.251235 V
• Standard Deviation: σ_V = 0.001946 V (≈ 2 mV)
• Upper Threshold: V_max = μ_V + 2σ_V = 0.255126 V
• Lower Threshold: V_min = μ_V - 2σ_V = 0.247344 V

Bit Extraction

Voltage samples are converted to binary bits using threshold detection:

• V(k) > V_max → Bit value 1
• V(k) < V_min → Bit value 0
• V_min ≤ V(k) ≤ V_max -> Discarded

Extraction Statistics:

• Valid bits extracted: 5,415
• Discarded samples: 994,586
• Extraction efficiency: 0.54%

Algorithm Version 0 (v0.mlx): Sequential Non-Overlapping Approach#

Bit Organization

The 5,415 extracted bits are partitioned into consecutive, non-overlapping 40-bit windows:

• Bits per coordinate pair: 40 bits total (20 for x-coordinate, 20 for y-coordinate)
• Sequential grouping: Each pair consumes unique bits with no overlap
• Coordinate pair capacity: ⌊5,415 / 40⌋ = 135 coordinate pairs
• Unused bits: 15 bits (0.28% waste)

Binary Fraction Conversion Methods

Each 20-bit sequence is converted to a decimal value in [0, 1) using one of two interpretation methods:

MSB Method (Most Significant Bit First)

• Bits processed left-to-right with first bit as 2⁻¹, second as 2⁻²
• Formula: x = Σ(b_i × 2^(-i)) for i=1 to 20
• Example [1,0,1,1]: x = 0.5 + 0 + 0.125 + 0.0625 = 0.6875

LSB Method (Least Significant Bit First)

• Bits reversed before conversion, last extracted bit as 2⁻¹
• Formula: x = Σ(b_(21-i) × 2^(-i)) for i=1 to 20
• Example [1,0,1,1] → Reversed [1,1,0,1]: x = 0.5 + 0.25 + 0 + 0.0625 = 0.8125

Version 0 Results

Method 1: MSB Only

• Coordinate pairs: 135
• Points inside circle: 105
• Circle/Total ratio: 0.777778
• Estimated π: 3.11111111
• Absolute error: 0.03048154
• Relative error: 0.9703%

Method 2: LSB Only

• Coordinate pairs: 135
• Points inside circle: 111
• Circle/Total ratio: 0.822222
• Estimated π: 3.28888889
• Absolute error: 0.14729624
• Relative error: 4.6886%

Method 3: BOTH (Combined)

• Coordinate pairs: 270 (MSB + LSB interpretations)
• Points inside circle: 216
• Circle/Total ratio: 0.800000
• Estimated π: 3.20000000
• Absolute error: 0.05840735
• Relative error: 1.8592%

Statistical Analysis of v0 Coordinates

Observations:

• Both X and Y coordinates exhibit positive bias (3.6% toward higher values)
• Y-coordinate bias exceeds X-coordinate bias by ~22%, indicating asymmetric noise distribution
• Standard deviations comparable, suggesting similar spread in both dimensions

Algorithm Version 1 (v1.mlx): Overlapping Sliding-Window Approach#

Sliding Window Mechanism

A 40-bit window slides across the 5,415-bit sequence with a step size of 1 bit:

Window position k:
- X-coordinate bits: positions [k, k+1, ..., k+19]
- Y-coordinate bits: positions [k+20, k+21, ..., k+39]
- Next window (k+1): shift by 1 bit, reusing 39 bits

Overlap Structure

Each successive window reuses 39 of 40 bits from the previous window:

• Window 1: bits [0–39] → (x₁, y₁)
• Window 2: bits [1–40] → (x₂, y₂) [39 bits reused]
• Window 3: bits [2–41] → (x₃, y₃) [39 bits reused]
• ... continues until end of bit sequence

Coordinate Pair Capacity

Maximum windows: N_windows = Total_bits − Window_size + 1

N_windows = 5,415 − 40 + 1 = 5,376 coordinate pairs

Efficiency comparison:

• Sequential (v0): 135 pairs, 85% utilization
• Overlapping (v1): 5,376 pairs, 99.997% utilization
• Improvement factor: 40× more sample points

Binary Fraction Conversion

V1 uses the same MSB conversion as v0's Method 1 (left-to-right bit ordering):

For 20-bit sequence [b₁, b₂, ..., b₂₀]:
x = Σ(b_i × 2^(-i)) for i=1 to 20

Version 1 Results

Primary: Overlapping Method

• Coordinate pairs: 5,376
• Points inside circle: 4,217
• Circle/Total ratio: 0.78441
• Estimated π: 3.13765
• Absolute error: 0.00394265
• Relative error: 0.1255%

Extended: Bidirectional Variant

• Coordinate pairs: 10,752 (MSB + LSB interpretations across overlapping windows)
• Points inside circle: 8,399
• Circle/Total ratio: 0.78116
• Estimated π: 3.12463
• Absolute error: 0.01696265
• Relative error: 0.5401%

Comparative Performance Analysis#

Sample Generation Efficiency

π Estimation Accuracy Comparison

Key Findings

1. V1 overlapping achieves 15× better accuracy than v0 BOTH despite 97.5% inter-sample correlation
2. 40× increase in sample count overwhelms correlation effects in Monte Carlo estimation
3. Monte Carlo convergence (O(1/√N)): √5,376/√270 ≈ 4.46× theoretical improvement vs. 14.8× observed improvement
4. Statistical independence trade-off validated: Ensemble size and averaging power dominate independence considerations

Monte Carlo π Estimation Method#

For each coordinate pair (x, y), test whether it lies within the inscribed circle:

Distance condition: (x - 0.5)² + (y - 0.5)² ≤ 0.25

The mathematical relationship:

π ≈ 4 × (N_inside / N_total)

Where:

• N_inside = count of points satisfying the circle condition
• N_total = total number of coordinate pairs
• Factor of 4 accounts for unit circle inscribed in unit square area ratio (π/4)

Using the Data Post-Processing Scripts#

Version 0 (v0.mlx) Workflow

1. Load Data: Reads

   Data_from_Scope/scope_6_2.csv

2. Statistical Analysis: Computes mean, standard deviation, and thresholds
3. Bit Extraction: Applies 2σ threshold detection to generate 5,415 binary bits
4. Coordinate Generation: Sequential 40-bit grouping with MSB and LSB interpretation
5. Monte Carlo Estimation: Performs circle test on 135 (or 270 with BOTH) coordinate pairs
6. π Calculation: Computes estimates for MSB, LSB, and combined methods
7. Error Analysis: Calculates absolute/relative error for each method
8. Visualization: Generates plots of voltage time series, coordinate scatter plots, and error metrics

Version 1 (v1.mlx) Workflow

1. Load Data: Reads

   Data_from_Scope/scope_6_2.csv

2. Statistical Analysis: Computes mean, standard deviation, and thresholds
3. Bit Extraction: Applies 2σ threshold detection to generate 5,415 binary bits
4. Overlapping Window Generation: Sliding window with 1-bit step creates 5,376 coordinate pairs
5. Binary Fraction Conversion: Converts 20-bit sequences to [0,1) decimals
6. Monte Carlo Estimation: Performs circle test on all 5,376 pairs
7. Optional Bidirectional: Also generates 10,752 pairs using MSB+LSB interpretation
8. π Calculation: Computes overlapping and bidirectional estimates
9. Error Analysis: Calculates absolute/relative error and convergence metrics
10. Visualization: Generates scatter plots, convergence curves, and comparison histograms

Key Variables and Functions

Common to both versions:

• voltage_data

  : Raw 1,000,001 voltage samples

• mu_V

  ,

  sigma_V

  : Signal statistics (mean and standard deviation)

• V_threshold_upper

  ,

  V_threshold_lower

  : Extraction thresholds (2σ band)

• extracted_bits

  : Binary sequence (5,415 bits)

• pi_estimate

  : Calculated π value(s)

• error_absolute

  ,

  error_percent

  : Error metrics

Version 0 specific:

• coordinates_MSB

  ,

  coordinates_LSB

  : Generated coordinate pairs from MSB/LSB methods

• inside_circle_MSB

  ,

  inside_circle_LSB

  : Circle membership arrays

Version 1 specific:

• coordinates_overlap

  : 5,376 coordinate pairs from overlapping windows

• coordinates_bidir

  : 10,752 coordinate pairs from bidirectional method

• inside_circle_overlap

  ,

  inside_circle_bidir

  : Circle membership arrays

Theoretical Foundations#

Independence vs. Ensemble Size Trade-off

Version 0 maintains strict statistical independence between consecutive coordinate pairs at the cost of severe bit waste (85% utilization) and limited sample count (135 pairs). Each pair consumes unique, non-overlapping bits.

Version 1 sacrifices inter-sample independence (97.5% bit overlap between adjacent pairs) to maximize ensemble size (5,376 pairs) and bit utilization (99.997%). The central limit theorem ensures that despite correlation, the accumulated decisions across thousands of samples converge to the true expectation.

Principle: In Monte Carlo estimation, ensemble size and averaging power dominate over individual sample independence. When resources constrain the total entropy available, generating many correlated samples outperforms generating fewer independent samples.

Practical Advantages and Applications#

• Data Efficiency: V1 extracts 40× more information from limited entropy source
• Computational Simplicity: No complex algorithms; straightforward threshold detection and bit processing
• Accuracy: V1 achieves 0.1255% π estimation error with only 5,415 extracted bits
• Reproducibility: Hardware-based randomness enables validation of inherent noise properties
• Educational Value: Demonstrates signal processing challenges (threshold selection, entropy extraction, coordinate generation) and statistical concepts (Monte Carlo convergence, ensemble averaging)

References#

• D. H. Bailey, "Numerical results on the transcendence of constants involving π, e, and Euler's constant," Mathematics of Computation, vol. 50, no. 181, pp. 275–281, Jan. 1988.

• D. E. Knuth, The Art of Computer Programming: Seminumerical Algorithms, vol. 2, 3rd ed. Addison-Wesley, 1997.

• G. S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications. Springer-Verlag, 1996.

• L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, "Stochastic resonance," Reviews of Modern Physics, vol. 70, no. 2, pp. 223–287, Apr. 1998.