What is a Maximum Length Sequence (MLS)?
A Maximum Length Sequence (MLS), also known as an m-sequence, is a special type of binary sequence that exhibits unique properties making it valuable in various applications. These sequences are generated using linear feedback shift registers (LFSRs) and are characterized by their maximum possible period, making them highly useful in fields like communication, cryptography, and testing.
Understanding the Basics
To grasp the concept of MLS, let's break down the key elements:
- Binary Sequence: An MLS consists of a sequence of bits, either 0 or 1.
- Maximum Period: The MLS has the maximum possible length for a given LFSR configuration, ensuring a long and unpredictable pattern.
- Linear Feedback Shift Register (LFSR): An LFSR is a digital circuit used to generate the sequence. It consists of a series of flip-flops and XOR gates, where the output of certain flip-flops is fed back to the input based on a specific polynomial.
How are MLS Generated?
MLS generation involves a carefully designed LFSR and a characteristic polynomial that determines the sequence's length and properties. The process involves:
- Initialization: The LFSR is initialized with a specific starting state (seed).
- Shifting: The flip-flops shift their contents one position to the right in each clock cycle.
- Feedback: The output of the last flip-flop is XORed with specific outputs of other flip-flops, based on the characteristic polynomial. This feedback determines the next bit in the sequence.
Key Properties of MLS
MLS possess several desirable characteristics:
- Maximum Length: As the name suggests, they have the maximum possible period for a given LFSR configuration. This period is 2^n - 1, where n is the number of flip-flops in the LFSR.
- Balanced: An MLS has an almost equal number of 0s and 1s, resulting in a balanced distribution.
- Random-Like: Though generated deterministically, MLS exhibit random-like behavior, making them suitable for applications requiring randomness.
- Autocorrelation: The autocorrelation function of an MLS has a sharp peak at zero lag and is close to zero for all other lags, indicating a high degree of randomness.
Applications of Maximum Length Sequences
MLS find wide applications in various domains due to their unique properties:
- Communication Systems: MLS are used for channel estimation, synchronization, and spread spectrum communication.
- Cryptography: They serve as pseudo-random number generators (PRNG) in cryptographic algorithms, contributing to the security of data transmission.
- Testing and Simulation: MLS are employed for testing digital circuits, simulating noise, and generating random patterns for testing purposes.
- Error Detection and Correction: They are used in coding theory for generating error detection and correction codes.
- Image and Signal Processing: MLS are applied in image processing for watermarking and in signal processing for pattern recognition.
Designing an MLS Generator
To design a generator for MLS, you need to select a characteristic polynomial that meets the desired properties. The following steps guide the process:
- Determine the desired length: The length of the MLS is directly related to the number of flip-flops in the LFSR.
- Choose a primitive polynomial: A primitive polynomial is a polynomial irreducible over the field GF(2) and has a root that generates all non-zero elements of the field. You can use tables or algorithms to find suitable primitive polynomials.
- Implement the LFSR: Construct the LFSR using flip-flops and XOR gates based on the chosen primitive polynomial.
Examples of MLS Generation
Here are examples of generating MLS using different LFSR configurations:
- LFSR with 3 flip-flops:
- Characteristic polynomial: x^3 + x + 1
- Seed: 110
- Generated MLS: 110 011 100 101 010 001 111 ...
- LFSR with 4 flip-flops:
- Characteristic polynomial: x^4 + x + 1
- Seed: 1001
- Generated MLS: 1001 0010 0100 1010 0101 0110 1101 1011 0111 1110 1111 ...
Conclusion
Maximum Length Sequences (MLS) are powerful tools with numerous applications in various fields. Their maximum period, balanced distribution, random-like behavior, and excellent autocorrelation properties make them suitable for tasks requiring randomness, synchronization, and testing. By understanding their generation, properties, and applications, you can leverage the power of MLS in your projects.