PN Sequences

Pseudorandom noise (PN) sequences are deterministic binary sequences that exhibit statistical properties closely resembling those of truly random noise. They are the backbone of spread spectrum communication, CDMA systems, and synchronization mechanisms. Understanding their generation and correlation properties is essential for any study of modern digital communications.

Core Concept Explanation

A **Linear Feedback Shift Register (LFSR)** is the standard hardware circuit for generating PN sequences. It consists of n flip-flops (stages) connected in series, with a feedback path formed by XOR-ing selected stages (called taps) and feeding the result back to the first stage. The selection of feedback taps is governed by a primitive polynomial over GF(2). With n stages and a primitive polynomial, the LFSR produces a sequence of period 2^n minus 1 before repeating. This is called a **maximal length sequence** or **m-sequence**.

For example, a 4-stage LFSR with a primitive feedback polynomial produces a sequence of period 15. A 10-stage LFSR produces a period of 1023. The sequence looks random — it has nearly equal numbers of ones and zeros (balance property), no long repeating pattern to the eye, and excellent autocorrelation — but it is entirely deterministic and reproducible given the same initial state (seed).

The three key randomness properties of m-sequences are the balance property (number of 1s exceeds number of 0s by exactly one per period), the run property (runs of consecutive identical chips follow a specific distribution), and the shift-and-add property (XOR of a sequence with any shifted version of itself gives another shifted version of the same sequence). These make m-sequences ideal for spreading and synchronization.

Mathematical Expression

The **autocorrelation function** of a PN sequence is the most critical property for spread spectrum. For an m-sequence of period N chips, the discrete periodic autocorrelation R(k) equals N when the shift k equals zero (or a multiple of N), and equals negative one for all other shifts. This two-valued property means the sequence correlates perfectly with itself only when aligned, and produces a very small value otherwise. In continuous normalized form, the peak value is 1 and the off-peak value is negative 1/N.

The **cross-correlation** between two different m-sequences of the same length is not always small, which is a limitation. **Gold codes** and **Kasami sequences** are designed families that offer bounded and low cross-correlation in addition to good autocorrelation. CDMA systems (IS-95, WCDMA) use such code families so that different users' signals minimally interfere with each other after despreading.

Given:
LFSR with n = 5 stages
Primitive feedback polynomial used

Why this formula applies:
m-sequence period depends only on number of stages.

Formula:
Period N = 2^n - 1
Autocorrelation peak = N (unnormalized) or 1 (normalized)
Off-peak autocorrelation = -1 (normalized) or -1/N

Substitution:
N = 2^5 - 1 = 31 chips
Off-peak R(k) = -1/31 ≈ -0.032

Calculation:
Period = 31 chips
Peak autocorrelation value = 1.0
Off-peak value = -0.032

Final Answer:
m-sequence of period 31 chips; autocorrelation is +1 at zero shift and -1/31 at all other shifts.

Practical Understanding

In GPS, each satellite transmits a unique 1023-chip Gold code (C/A code) at 1.023 Mcps. The GPS receiver computes the cross-correlation between the incoming signal and locally generated replicas for each satellite. Because Gold codes have low cross-correlation, the receiver can identify and track multiple satellites simultaneously. The sharp autocorrelation peak also enables precise ranging by measuring the time offset at which the correlation peaks.

In CDMA cellular systems, each user is assigned a unique PN spreading code. At the base station, a matched filter correlates the received composite signal with each user's code. Owing to the near-orthogonality of well-designed code sets, each user's signal is recovered while other users' signals are suppressed as interference. The quality of the code set directly determines the system capacity.

Exam Tip: For a GATE question on m-sequences, remember period = 2^n - 1, normalized autocorrelation is +1 at zero and -1/N elsewhere, and the number of stages n equals the degree of the primitive polynomial. These are frequently tested numerical facts.

Mechanism: Autocorrelation and Synchronization

• LFSR generates m-sequence by XOR-ing selected tap stages and feeding back to stage 1. Primitive polynomial choice determines tap positions.
• Receiver slides its local PN code replica one chip at a time and computes correlation with received signal.
• When local code aligns with received code (zero offset), autocorrelation hits its maximum value of N (or 1 normalized).
• At all other offsets, correlation is close to zero (-1/N), allowing the receiver to distinguish aligned from misaligned state.
• After acquisition (peak found), tracking loops maintain alignment continuously as the signal varies.

Quick Revision

• m-sequence period: N = 2^n - 1, where n is the number of LFSR stages.
• Normalized autocorrelation: +1 at τ=0, -1/N at all other shifts. This two-valued property is unique to m-sequences.
• Balance property: number of 1s equals (N+1)/2, number of 0s equals (N-1)/2 per period.
• Gold codes have bounded three-valued cross-correlation — preferred in CDMA and GPS for multi-user separation.
• PN sequences are deterministic but appear random; they need a nonzero initial seed (all-zeros state is forbidden in LFSR).
• Trap: Do not say autocorrelation is zero off-peak — it is -1/N, not zero. This is a common exam mistake.
• Trap: m-sequence period is 2^n - 1, not 2^n. The all-zeros state is excluded.