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Introduction to M-sequence: Pseudo-random Binary Sequences

Learn about M-sequences, binary sequences that act like random sequences and have applications in telecommunications and computer science. Understand their period and span property.

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Introduction to M-sequence: Pseudo-random Binary Sequences

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  1. Span Property 2001년 6월 2일 정지욱 연세대학교 전기전자공학과 정지욱 <유한체 이론 및 응용> 1/7

  2. Introduction to M-sequence(1) • Definition An m-sequence is a binary sequence that satisfies a linear recurrence whose characteristics polynomial is primitive. (m-sequence is perhaps the best-known family of pseudo noise sequences. It is a linear feedback shift-register sequence having the maximum possible period.) • Usefulness - easily generated binary sequences that behave in many respects as if they were completely random.(pseudo randomness property) - Applications: telecommunications, computer science, etc. 정지욱 <유한체 이론 및 응용> 2/7

  3. Introduction to M-sequence(2) • Period of m-sequence The characteristic polynomial of m-sequence is an irreducible polynomial of degree m which is the minimal polynomial of a primitive root in GF(2m). Since the characteristic polynomial of an m-sequence has period 2m-1, every m-sequence 2m-1.(by Theorem 9.4) • Example 10.1 primitive polynomial: 정지욱 <유한체 이론 및 응용> 3/7

  4. Example of m-sequence Period: 24-1=15 Initial condition: 0001 => 000100110101111 정지욱 <유한체 이론 및 응용> 4/7

  5. Introduction to Span Property • m-gram If (s0,s1, …. , sn-1)is the m-sequence, an m-gram is one of the n subsequences of length m of the form (st, st+1, ….. , st+m-1), for t = 0,1, … , n-1 • Theorem 10.1(Span property) Among the 2m-1 m-grams of an m-sequence {st}, each nonzero binary vector of length m occurs once and only once. 정지욱 <유한체 이론 및 응용> 5/7

  6. Proof of Span Property 정지욱 <유한체 이론 및 응용> 6/7

  7. Examples of Span Property » m=3, , (st) = (0010111) m-grams: (001), (010), (101), (011), (111), (110), (100) » m=4, , (st) = (000100110101111) m-grams: (0001), (0010), (0100), (1001), (0011), (0110), (1101), (1010), (0101), (1011), (0111), (1111), (1110), (1100), (1000) » m=5, , (st) = (0000100101100111110001101110101) m-grams: (00001), (00010), (00100), (01001), (10010), (00101), (01011), (10110), (01100), (11001), (10011), (00111), (01111), (11111), (11110), (11100), (11000), (10001), (00011), (00110), (01101), (11011), (10111), (01110), (11101), (11010), (10101), (01010), (10100), (01000), (10000) 정지욱 <유한체 이론 및 응용> 7/7

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