Pseudo-Random Noise (PN) Sequence

A PN sequence is generated by a Linear Feedback Shift Register (LFSR) defined by a primitive polynomial. Given the same initial seed state, the generator will produce the exact same sequence every time. This determinism is critical because it allows a remote receiver with knowledge of the polynomial and initial state to generate an identical local replica for coherent despreading. The sequence is not random; it is pseudo-random, meaning it passes statistical tests for randomness but is fully predictable if the generator architecture is known.

An m-sequence, the most common type of PN code, achieves the maximum possible period for an LFSR of a given length. For an n-stage register, the sequence length is N = 2^n - 1 chips before repeating. This maximal-length property ensures the sequence exhausts every possible non-zero state of the register exactly once per cycle. The long period is essential for spreading signals below the noise floor and for providing a large number of distinct code phases for Code Division Multiple Access (CDMA) ranging and synchronization.

PN sequences satisfy strict statistical randomness postulates. The Balance Property states that in one full period, the number of binary '1's exceeds the number of '0's by exactly one, ensuring a near-zero DC component. The Run Property dictates that runs of consecutive identical digits occur with predictable frequency: one-half of all runs have length 1, one-quarter have length 2, and so on. These properties produce an autocorrelation function that closely approximates ideal white noise, which is fundamental to the sequence's utility in synchronization and multiple access interference rejection.

The defining characteristic of a maximal-length PN sequence is its two-valued, periodic autocorrelation function. When the sequence is perfectly aligned with itself, the correlation is a peak of value N (the sequence length). For any non-zero time shift, the correlation drops immediately to a constant low value of -1. This thumbtack autocorrelation profile enables precise time-of-arrival estimation in ranging systems and allows a Rake Receiver to resolve individual multipath components that are separated by more than one chip duration.

While a single m-sequence has ideal autocorrelation, its cross-correlation with other m-sequences can be high. For multi-user environments, specific code families like Gold Codes and Kasami Sequences are constructed by combining preferred pairs of m-sequences. These families sacrifice some autocorrelation sharpness to guarantee a bounded, low cross-correlation between any two member codes. This property is the physical layer foundation of CDMA systems, where it minimizes mutual interference between simultaneous users sharing the same frequency band.

Multiplying a narrowband data signal by a high-rate PN sequence spreads its energy over a wide bandwidth, transforming the power spectral density from a distinct sinc function into a noise-like, flat spectrum. The spectral envelope follows a (sin x / x)^2 shape with nulls at integer multiples of the chip rate. This whitening effect is what gives Direct Sequence Spread Spectrum (DSSS) its Low Probability of Intercept (LPI) and anti-jamming capabilities, as the signal becomes statistically indistinguishable from background thermal noise to an interceptor without knowledge of the code.

The fundamental metric quantifying a spread spectrum system's resilience to interference. It is defined as the ratio of the transmitted spread bandwidth to the original information bandwidth, or equivalently, the ratio of the chip rate to the data rate. A high processing gain allows the receiver to compress the desired signal while spreading narrowband jammers, effectively suppressing them below the noise floor after despreading.

A closed-loop control circuit that maintains fine synchronization between the incoming PN code and the receiver's local replica. The DLL correlates the received signal with two versions of the local code: an early replica and a late replica. The difference between these correlation outputs forms an error signal that drives a voltage-controlled oscillator, continuously tracking the code phase to within a fraction of a chip.