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Pseudorandom noise

Pseudorandom noise (PRN), also known as pseudo-noise (PN) sequences, refers to a class of deterministic digital signals that mimic the statistical properties of true random noise while being generated by repeatable algorithms, such as linear feedback shift registers (LFSRs).^[1]^[2] These sequences typically consist of binary values (0s and 1s) that appear random, passing standard statistical tests for randomness, and exhibit a flat power spectrum across their bandwidth with no nulls.^[2]^[3] Key properties of PRN include the balance property, where the number of 1s and 0s in a full period differs by at most one; the run property, ensuring a specific distribution of consecutive 1s or 0s; and ideal autocorrelation, characterized by a sharp central peak and low sidelobes for efficient signal detection.^[1] Common generation methods produce maximum-length sequences (m-sequences) with a period of $2^n - 1 for an LFSR of degree n, or more advanced codes like Gold sequences for better cross-correlation, and even chaotic mappings for enhanced randomness in modern implementations.^[1]^[4] PRN has been pivotal in applications requiring noise-like signals for robustness and security, including radar systems for ranging and low-probability-of-intercept operations, satellite navigation like GPS for precise distance measurement and anti-jamming, spread-spectrum communications to improve signal-to-noise ratios, and data scrambling in digital transmission to avoid long runs of identical bits.^[1]^[3]^[4] Historically, PRN sequences were employed in early space missions such as Mariner and Apollo for telemetry, and their use has expanded with digital advancements since the 2000s in fields like noise radar and code-division multiple access (CDMA) systems.^[1]^[2]

Overview

Definition

Pseudorandom noise (PRN), also known as pseudo-noise, is a deterministic signal generated by algorithms or hardware that mimics the statistical properties of true random noise, such as uniform distribution and lack of discernible patterns over short observation intervals.^[5] Unlike genuinely random processes, PRN is produced through repeatable computational methods, ensuring identical outputs when initialized with the same starting conditions, often referred to as a seed value.^[1] The core principle of PRN is its ability to appear statistically random within finite periods of analysis while being fundamentally periodic and predictable over its full cycle, allowing it to pass many tests for randomness without possessing true unpredictability.^[6] This reproducibility makes PRN suitable for applications requiring consistent simulation of noise, as the sequence can be regenerated exactly for verification or synchronization purposes.^[5] A basic example of PRN is a binary sequence consisting of 0s and 1s, such as those produced by shift register-based generators, which can approximate the flat power spectrum of white noise across a defined bandwidth.^[7] These sequences exhibit noise-like behavior in their autocorrelation and power spectral density, providing a practical substitute for random signals in controlled environments.^[6] In contrast to true random noise, which arises from inherently unpredictable physical phenomena like thermal fluctuations in electronic components or atmospheric disturbances, PRN lacks any intrinsic entropy and is entirely algorithmically derived, rendering it pseudo-random by design.^[8]^[9] This distinction ensures PRN's outputs are fully reproducible but limits its use in scenarios demanding absolute unpredictability, such as certain cryptographic primitives.^[10]

Historical Context

The concept of pseudorandom noise emerged in the 1940s amid World War II efforts to develop secure communication systems, where early pseudo-random sequences were employed to generate codes resistant to jamming and interception. A notable milestone was the 1942 patent by actress Hedy Lamarr and composer George Antheil, which proposed frequency-hopping spread spectrum using a pseudo-random sequence recorded on a phonograph disk to synchronize signal changes between transmitter and receiver, laying foundational ideas for pseudorandom code generation in military applications. This analog approach marked the initial shift toward deterministic yet noise-like signals for wartime secrecy, influencing subsequent digital advancements.^[11] In the 1950s and 1960s, research accelerated with the development of linear feedback shift registers (LFSRs) for radar, error-correcting codes, and spread-spectrum systems, transitioning pseudorandom noise from analog mechanisms to digital implementations. Pioneering work by Solomon W. Golomb at the Jet Propulsion Laboratory explored shift register sequences, demonstrating their pseudorandom properties for communication and synchronization tasks.^[12] Researchers like W. Wesley Peterson contributed to integrating LFSRs into coding theory, while E.J. Weldon advanced their application in reliable data transmission over noisy channels.^[13] Golomb's seminal 1967 book, Shift Register Sequences, formalized the theory of these sequences, establishing them as a cornerstone for generating pseudorandom noise with desirable statistical behaviors in digital systems.^[14] From the 1970s onward, pseudorandom noise became integral to digital communications standards, particularly in code-division multiple access (CDMA) for mobile networks, where pseudo-noise sequences enabled multiple users to share bandwidth efficiently. Building on Shannon's 1949 theoretical foundations for interference averaging in spread-spectrum techniques, practical CDMA systems in the late 1980s and 1990s adopted LFSR-generated sequences to mitigate multipath fading and enhance capacity. Qualcomm demonstrated a CDMA system in 1989, leading to the first commercial deployments in 1995.^[15]^[16] This era solidified the evolution from analog noise generators—reliant on physical phenomena like thermal noise—to fully digital pseudorandom noise, driven by computing advances and the need for reproducible, high-period sequences in telecommunications.^[17]

Generation Methods

Linear Feedback Shift Registers

A linear feedback shift register (LFSR) consists of a shift register composed of flip-flops or memory elements, where the input to the register is determined by a linear feedback function applied to selected output bits, known as taps, using exclusive-OR (XOR) operations over the finite field GF(2.^[18] The feedback taps are chosen according to the coefficients of a primitive irreducible polynomial of degree n, where n is the length of the register, ensuring the generation of sequences with desirable pseudorandom properties for noise-like applications.^[19] The generation process begins with an initial non-zero seed state loaded into the register. On each clock cycle, the bits shift right (or left, depending on configuration), the bit shifted out serves as the output bit contributing to the pseudorandom sequence, and the new input bit is computed as the modulo-2 linear combination of the tapped bits.^[18] This process repeats, producing a deterministic yet apparently random binary sequence suitable for pseudorandom noise.^[19] The feedback is mathematically described by a linear recurrence relation defined by the characteristic polynomial. For an n-stage LFSR, the sequence satisfies

s_{n+k} = \sum_{i=1}^{n} c_i s_{k+i-1} \pmod{2},

where s_j are the sequence terms (0 or 1), and c_i \in \{0,1\} are the polynomial coefficients corresponding to the tap positions (with c_n = 1).^[18] When the feedback polynomial is primitive over GF(2), the LFSR achieves a maximum period of $2^n - 1, cycling through all possible non-zero states before repeating, which maximizes the sequence length and enhances its pseudorandom characteristics.^[18] For example, a 4-bit LFSR with taps at positions 4 and 3 (corresponding to the primitive polynomial x^4 + x^3 + 1) and initial state 1000 generates the output sequence 100011110101100... over its period of 15 bits.^[19]

Nonlinear Techniques

Nonlinear feedback shift registers (NFSRs) represent a key advancement in pseudorandom noise generation by employing nonlinear Boolean functions for feedback, contrasting with the linear XOR operations in traditional shift registers. In an NFSR, the input bit is derived from a nonlinear combination of previous states, often incorporating AND or OR gates to compute the feedback function f(x_0, x_1, \ldots, x_{n-1}) = x_0 + F(x_1, \ldots, x_{n-1}), where F is a nonlinear Boolean polynomial. This structure enables sequences with maximum periods up to $2^n - 1 for an n-stage register, while achieving significantly higher linear complexity—potentially up to $2^n - 2—compared to linear counterparts.^[20] Beyond NFSRs, chaotic maps provide another nonlinear approach to generating pseudorandom bits, leveraging the sensitive dependence on initial conditions inherent in chaotic systems. The logistic map, defined by the iteration x_{n+1} = r x_n (1 - x_n) with parameter r = 4, produces a chaotic trajectory when initialized with a non-rational value in (0,1); bits are extracted from the fractional part of x_n (e.g., by taking multiple decimal digits and converting via modulo operations to binary). This method yields sequences with strong ergodic properties, suitable for pseudorandom noise in simulations and encryption. Similarly, cellular automata (CA) generate pseudorandom sequences through parallel local rules applied to a grid of binary cells. In one-dimensional CA, such as Rule 30—where each cell updates as a_i^{(t+1)} = a_{i-1}^{(t)} \oplus (a_i^{(t)} \lor a_{i+1}^{(t)})—evolution from a simple seed produces aperiodic, noise-like patterns with long cycle lengths exceeding $2^{32} for registers of size 49 or larger.^[21]^[22] These nonlinear techniques offer advantages including superior resistance to cryptanalytic attacks, as their complex state transitions defy linear approximation methods that exploit linear feedback shift registers. In some configurations, they achieve effective periods approaching $2^n without relying on primitive polynomials, enhancing scalability in resource-constrained environments. A prominent example is the shrinking generator, which nonlinearly combines outputs from two independent linear feedback shift registers (LFSRs): the first LFSR provides candidate bits, while the second controls selection—outputting the candidate only if its bit is 1, otherwise skipping—resulting in a decimated sequence with period (2^m - 1) \times 2^{k-1} and high linear complexity on the order of $2^{m + k - 1} for LFSR lengths m (candidate) and k (control), assuming primitive polynomials and ideal selection rate.^[23] This approach mitigates predictability while preserving efficiency.^[24] In hardware and software implementations, nonlinear techniques like NFSRs and CA are favored in embedded systems for pseudorandom noise generation, as they avoid the polynomial irreducibility constraints of linear methods and can be realized with simple gate logic or iterative algorithms on microcontrollers. For instance, NFSRs using AND gates have been prototyped on FPGAs for orders up to 27, delivering high-throughput streams for real-time applications.^[20]

Properties

Statistical Characteristics

Pseudorandom noise sequences are engineered to replicate the statistical behavior of truly random binary sequences, ensuring they appear unpredictable in short segments while maintaining determinism overall. A fundamental characteristic is the balance property, where the number of 1s and 0s in a full period differs by at most one. For maximal-length sequences (m-sequences) of period $2^n - 1, there are exactly $2^{n-1} ones and $2^{n-1} - 1 zeros, yielding a proportion of ones approximately equal to 0.5.^[25] Another key feature is the run-length distribution, which aligns with expectations for independent fair coin flips. In an m-sequence period, approximately half the runs (consecutive identical bits) are of length 1, one-quarter of length 2, one-eighth of length 3, and so on, with one run of n consecutive 1s and one run of n-1 consecutive 0s; moreover, runs of the same length alternate between ones and zeros.^[25] This distribution ensures that the sequence avoids excessive clustering or gaps, contributing to its pseudorandom appearance. To verify these properties, pseudorandom noise sequences are subjected to standardized statistical tests, such as those in the NIST Special Publication 800-22 suite. The frequency test assesses balance by checking if the proportion of ones deviates significantly from 0.5 across the sequence. The runs test evaluates run-length distribution by comparing the total number of runs to the expected value under randomness, while the poker test examines the uniformity of overlapping m-bit patterns (e.g., m=5) to detect biases in higher-order distributions. Well-designed PRN sequences, including m-sequences, typically pass these tests when evaluated over multiple periods or sufficient length.^[26] Entropy measures further quantify the randomness, with Shannon entropy serving as a primary metric for the information content per bit. For an m-sequence, the binary entropy H = -p \log_2 p - (1-p) \log_2 (1-p), where p is the proportion of ones, approaches the maximum value of 1 bit per symbol as the period lengthens, reflecting near-uniform distribution.^[27] Despite these strong statistical traits, pseudorandom noise sequences have inherent limitations due to their deterministic nature: given the initial state and generation algorithm, the entire sequence is fully predictable and periodic, failing true randomness tests over the full period or when the state is known.^[1]

Correlation Functions

The autocorrelation function of a pseudorandom noise (PRN) sequence s = \{s_k\}, typically mapped to \pm 1 for analysis, is defined as

C(\tau) = \sum_{k=0}^{N-1} s_k s_{k+\tau},

where N is the period of the sequence and indices are modulo N.^[28] This measures the similarity between the sequence and its shifted version by \tau, with C(0) = N representing perfect alignment. For maximal-length sequences (m-sequences) generated by linear feedback shift registers of degree n, the period is N = 2^n - 1, and the autocorrelation exhibits an ideal two-level behavior: C(\tau) = 2^n - 1 for \tau = 0 (modulo N), and C(\tau) = -1 otherwise.^[28] This sharp peak at zero shift and near-zero values elsewhere approximate the delta function of true white noise, enabling precise synchronization and detection in noisy environments. The cross-correlation function between two distinct PRN sequences s and t is similarly defined as C_{st}(\tau) = \sum_{k=0}^{N-1} s_k t_{k+\tau}. Low cross-correlation values are crucial for multi-user systems, where multiple sequences operate simultaneously without significant mutual interference. For example, Gold sequences, constructed from preferred pairs of m-sequences, achieve bounded cross-correlations with magnitudes at most $2^{(n+2)/2} + 1, ensuring robust performance in code-division multiple access. The power spectral density (PSD) of PRN sequences, particularly m-sequences, features a flat spectrum that closely approximates white noise over the signal bandwidth, with S(f) \approx 1 (normalized) and discrete spectral lines spaced by the fundamental frequency $1/(N \Delta t), where \Delta t is the chip duration.^[29] This uniform energy distribution across frequencies minimizes detectable structure, enhancing noise-like properties for applications requiring broad-spectrum coverage. Phase shifts \tau in the autocorrelation directly influence detection performance; non-zero shifts yield low correlation values, which suppress false detections from timing errors. Low sidelobes (the out-of-phase correlation levels) are particularly vital for interference rejection, as they limit the contribution of delayed or multipath signals, preventing elevation of noise floors in correlation-based receivers.^[28]

Applications

Communications Systems

Pseudorandom noise (PRN) sequences are integral to spread-spectrum techniques in communications systems, enabling robust signal transmission in noisy or adversarial environments. In direct-sequence spread spectrum (DSSS), the original data signal is multiplied by a high-rate PRN code, which spreads the narrowband signal across a much wider bandwidth. This process, known as spreading, imparts noise-like properties to the transmitted signal, making it resistant to jamming and intentional interference. Upon reception, the receiver uses the same PRN code to despread the signal, recovering the original data while rejecting narrowband noise and jammers through the processing gain, which is the ratio of the spread bandwidth to the data bandwidth.^[30]^[31]^[32] In code-division multiple access (CDMA) systems, PRN codes serve as unique orthogonal or near-orthogonal signatures assigned to each user, facilitating simultaneous transmission over the same frequency channel without significant mutual interference. This multiple-access capability is foundational to 3G cellular networks, such as wideband CDMA (WCDMA), where users' signals are distinguished at the receiver via code-specific despreading. The pseudorandom nature of these codes ensures low cross-correlation between different users' signatures, minimizing bit error rates in multi-user scenarios and supporting higher spectral efficiency compared to time- or frequency-division methods.^[33]^[34]^[35] Synchronization in PRN-based systems relies on the acquisition and tracking of these codes to align the receiver's local replica with the incoming signal phase. In global positioning system (GPS) receivers, for instance, the coarse/acquisition (C/A) code—a 1023-chip Gold code variant modulated at 1.023 MHz—enables initial code alignment for satellite signal detection, followed by fine tracking loops to maintain lock amid Doppler shifts and noise. Similarly, in cellular CDMA systems, PRN code synchronization ensures accurate despreading, as misalignment would degrade signal recovery and increase error rates. Their correlation properties, with sharp autocorrelation peaks and low sidelobes, facilitate rapid acquisition even at low signal-to-noise ratios.^[36]^[37]^[38] PRN sequences also contribute to error detection in communications by generating test patterns or enhancing checksum mechanisms, where their statistical randomness helps verify data integrity against transmission errors. For example, in embedded control networks, checksum computations are evaluated for pseudorandom properties to improve undetected error rates in packet-based protocols. In GPS applications, the C/A code not only aids synchronization but also supports civilian positioning by providing a publicly available PRN sequence that resists spoofing through its balanced correlation characteristics.^[39]^[38]

Testing and Simulation

Pseudorandom noise (PRN) plays a critical role in engineering validation by simulating realistic noise environments in controlled settings, particularly through noise injection techniques. In hardware-in-the-loop (HIL) testing for receivers, such as those in GNSS systems, PRN is injected to mimic Gaussian or impulsive noise, allowing evaluation of receiver performance under jamming or interference conditions without real-world risks. This approach enables precise assessment of signal acquisition and tracking robustness by generating pseudo-random noise codes that replicate environmental disturbances.^[40]^[41] Bit error rate (BER) testing relies on PRN to establish controlled noise floors, facilitating measurement of system performance in digital communications. Pseudo-random binary sequences (PRBS), a common form of PRN, are transmitted through the system under test, with errors counted at the receiver to quantify BER under varying signal-to-noise ratios. This method ensures repeatable and statistically valid results, as PRBS patterns approximate random data while allowing deterministic verification. Standards like IEEE 802.3 incorporate PRBS for such evaluations in Ethernet interfaces, ensuring compliance with error rate thresholds.^[42]^[43] In Monte Carlo simulations, PRN serves as seeds for pseudorandom number generators, providing repeatable yet random-like inputs for algorithm verification in engineering models. These seeds enable consistent replication of simulation runs, essential for statistical analysis of system reliability, such as in radio planning where uniform, Gaussian, or Rayleigh distributions are derived from PRN to model propagation and noise effects. The ITU-R SM.2028 methodology outlines pseudorandom generation techniques, including linear congruential methods, to ensure unbiased sampling in large-scale trials.^[44] PRN finds application in audio and video processing testing for tasks like dithering and watermarking. In dithering, subtractive PRN, often generated as pseudo-random uniform noise, is added to signals before quantization to minimize distortion and improve linearity in analog-to-digital conversion during media playback tests. For watermarking, PRN sequences embed imperceptible markers into audio or video streams, verifying integrity and authenticity in conformance checks; for instance, biometric-derived PRN keys are modulated into frequency or wavelet domains to resist compression or filtering attacks.^[45]^[46]^[47] International standards guide the use of PRN in conformance testing to ensure interoperability and performance. The ITU-T P.311 recommendation specifies PRN sequences, such as 4096-point patterns from ITU-T P.501, for evaluating transmission characteristics in telephone-band equipment, including noise injection for distortion measurement. Similarly, ITU-T O.131 defines PRN test signals for quantizing distortion assessment in digital systems, promoting standardized noise simulation across audio and communications hardware. IEEE standards, such as those for serial interfaces, further endorse PRBS patterns for BER and stress testing in compliance procedures.

Types of Sequences

Maximal-Length Sequences

Maximal-length sequences, also known as m-sequences, are binary pseudorandom sequences produced by linear feedback shift registers (LFSRs) initialized with a nonzero state and governed by a primitive polynomial of degree n over the finite field GF(2, yielding a maximum period of $2^n - 1.^[48] These sequences cycle through all possible $2^n - 1 nonzero states of the register, excluding the all-zero state, which would cause the LFSR to lock indefinitely.^[48] To generate an m-sequence, the feedback taps of the LFSR are determined by the coefficients of a primitive polynomial, ensuring the maximum period. For example, the primitive polynomial x^3 + x^2 + 1 for n=3 produces a sequence of length 7, such as 0010111 (depending on initial state), by connecting the feedback from the second and third stages. Primitive polynomials are irreducible polynomials whose roots generate the multiplicative group of GF($2^n), and tables of such polynomials exist for various degrees to facilitate implementation.^[48] A key property of m-sequences, when mapped to \pm 1 values, is their two-level autocorrelation function: N at zero lag and -1 for all nonzero lags modulo the period N = 2^n - 1, where N is the sequence length.^[49] This sharp peak with minimal sidelobes makes m-sequences particularly suitable for ranging and synchronization tasks, as the low out-of-phase correlation suppresses interference and multipath effects effectively.^[49] In radar systems, m-sequences enable pulse compression by modulating the transmit waveform, achieving high range resolution through matched filtering at the receiver, with sidelobe levels approaching -1/N relative to the main lobe.^[50] For frequency-hopping communications, m-sequences provide pseudorandom hopping patterns that enhance anti-jam performance and code-division multiple access, distributing energy across frequencies to minimize detectability.^[51] Despite their strengths, m-sequences are inherently binary, limiting their use in multilevel signaling schemes without additional modulation.^[48] Additionally, their linear structure renders them vulnerable to attacks if the initial state (seed) is known or partially compromised, allowing reconstruction of the entire sequence via the Berlekamp-Massey algorithm after observing $2n consecutive bits.^[52]

Gold and Kasami Sequences

Gold sequences, introduced by Robert Gold in 1967, form a large family of binary pseudorandom noise (PRN) sequences designed to provide good cross-correlation properties for applications requiring multiple synchronized signals. These sequences are generated by the modulo-2 addition (XOR) of two preferred maximal-length sequences (m-sequences) produced by linear feedback shift registers (LFSRs) of the same degree n, resulting in a set of $2^n + 1 sequences, each with period $2^n - 1. The "preferred pair" of m-sequences is selected such that their cross-correlation takes on only three possible values: -1, -1 ± $2^{(n+1)/2}, which is optimal in the sense of Welch's bound for binary sequences. This three-level cross-correlation ensures low interference in multi-user environments compared to random binary sequences. The explicit construction of a Gold sequence c from two m-sequences s and t is given by

c_k = s_k \oplus t_k, \quad k = 0, 1, \dots, 2^n - 2.

By shifting one of the m-sequences relative to the other, all $2^n + 1 distinct sequences in the family are obtained, including the original m-sequences themselves. These properties make Gold sequences particularly suitable for spread-spectrum systems where a large number of codes with balanced correlation is needed. Kasami sequences, developed by Tadao Kasami in 1966, represent another important family of PRN sequences optimized for low cross-correlation within large sets, often outperforming Gold sequences in terms of set size versus correlation bounds. They are constructed using quadratic trace functions over finite fields or by decimating a single m-sequence of degree $2m with specific ratios, yielding two main variants: the small set and the large set. The small set comprises $2^m sequences of length $2^{2m} - 1, achieving ideal two-level periodic autocorrelation and cross-correlation values bounded by -1 \pm 2^m, which meets the Welch lower bound for family size. The large set, in contrast, provides $2^{3m} + 2^m - 1 sequences of the same length but with cross-correlation up to magnitude $2^{m+1} + 1, offering a trade-off for even larger families. Both Gold and Kasami sequences excel in code-division multiple-access (CDMA) communications, where they enable efficient user separation by minimizing multi-user interference, performing better than purely random codes due to their controlled correlation spectra. For instance, the GPS P(Y)-code employs a Kasami-like structure, combining outputs from LFSRs of lengths 10 and 12 via modulo-2 addition with phase offsets unique to each satellite, to provide secure, high-precision ranging for military applications.

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Audio Testing With Noise - audioXpress
May 13, 2015 · Noise generators come in two basic flavors: random and pseudo-random. Figure 1: Comparison of a swept frequency response (upper trace) with a ...
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[PDF] Linear Feedback Shift Registers (LFSRs)
• The characteristic polynomial of an LFSR generating a maximum-length sequence is a primitive polynomial. • A maximum-length sequence is pseudo-random: – ...
[47]
[PDF] E303: Communication Systems - Imperial College London
Balanced Gold codes are generated by appropriately selecting the relative phases of the two original m-sequences. SUMMARY: By selecting any preferred pair of ...Missing: Optimal | Show results with:Optimal<|control11|><|separator|>
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A novel non-coherent radar pulse compression technique based on ...
This paper proposes a novel technique for implementation of NCPC that is based on well-known m-sequences along with a modified coding scheme. Theoretical ...
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Auto- and Cross-Ambiguity Surface Performance for Frequency-Hop Coded Waveforms
**Summary of m-sequences in frequency hopping from https://ieeexplore.ieee.org/document/657739:**
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[PDF] An Analysis of Linear Feedback Shift Registers in Stream Ciphers
Due to the inherent linearity, LFSR based stream ciphers are susceptible to several general attacks including known plaintext attack [9], algebraic attack [10],.<|control11|><|separator|>