טכניקות בתקשורת מרחיבת סרט (Spread Spectrum) Chapter 1c

1. מצגת זו תכלול כנראה דיון של הקהל, אשר יביא ליצירת פריטי פעולה. השתמש ב- PowerPointכדי לעקוב אחר פריטי פעולה אלה במהלך המצגת. • בהצגת שקופיות, לחץ באמצעות לחצן העכבר הימני. • בחר באפשרות “מפקח הישיבות”. • בחר בכרטיסיה “פריטי פעולה”. • הקלד את פריטי הפעולה כאשר הם מופיעים. • לחץ על אישור כדי להסיר תיבה זו. פעולה זו תיצור אוטומטית שקופיות לפריטי פעולה בסוף המצגת, והנקודות שהעלית יוזנו בתוכה. טכניקות בתקשורת מרחיבת סרט(Spread Spectrum) Chapter 1c ד"ר משה רן כל הזכויות שמורות לחברת MostlyTek Ltd. אין לצלם, לשכפל או להעתיק בכל צורה שהיא ללא קבלת אישור בכתב מד"ר משה רן Dr. Moshe Ran- Spread Spectrum

2. נושאי לימוד 8 שו"ת 8 שו"ת 8 שו"ת 8 שו"ת 8 שו"ת 8 שו"ת 4 שו"ת Dr. Moshe Ran / Spread Spectrum

3. 3. Pseudo- Noise (PN) Sequence Definition PN Implementation ,, ,, ,, ,, ,, ,, Dr. Moshe Ran / Spread Spectrum

4. 3.1 PN Sequence Definition A PN sequence is a deterministic sequence known to the receiver and transmitter which has features of a random sequence. • Spectrum • Correlation • Frequency of occurrence of subsequences. • WHY PN and not True Random?! • True random = sample of a sequence of independent r.v uniformly distributed on the alphabet • True random SpSp is like one-time pad in cryptographic system. • Generation, recording and distribution of “sample random sequences” at very high rates to provide PG is not feasible. Dr. Moshe Ran / Spread Spectrum

5. 3.2 PN Sequence Implementation Methods implementation of a finite pseudo-noise sequence of length N or a periodic pseudo-noise sequence with a period N . • Memory of N cells. Suitable only for short sequence. • Counter with additional logic. No simple logic can be found. • Linear feedback shift register (LFSR)or equivalently Pseudo- Random Binary Sequence (PRBS ). Most useful method. Dr. Moshe Ran / Spread Spectrum

6. Address generator ROM PN based on Memory of N cells • Example of ROM –based generator Assume – period N=16 is desired, and the specified bn should be 1010,1100,1011,0010. Can we replace the ROM with Boolean function?! That is – address is 4-bits binary counter producing consecutive numbers in the range {0,…, 15} Dr. Moshe Ran / Spread Spectrum

7. ROM based - cont. • Possible bn • NOTES: • This mapping is a “replacement” function: every input an is mapped to bn. • I.E., an address-to-bit mapping – specified by a table. • Need deep understanding of Finite Fields theory to design PN generators. The mapping in the example above – RM(1,m) Dr. Moshe Ran / Spread Spectrum

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9. Counter Based PN generator This solution can be described by + 4-bit counter Dr. Moshe Ran / Spread Spectrum

10. Complexity issues • Number of operation: linear function of ~2k (exponential in k) • Are counters good for implementing PN? Probably NO.Since the sequences do not look “random”. While PN -on the average- changes every other bit The counter sequences are changing “much slower” • Linear recursive relations are much better choice. Dr. Moshe Ran / Spread Spectrum

11. 3.3 Linear Feedback Shift Register Configurations Fibonacci configuration Galois configuration The sequence is detemined by: Number of cells, Feedback, Initial state of the shift register. The all zero state produces a sequence with period one. The order of a sequence is the length of the shortest LFSR which may generate the sequence. Dr. Moshe Ran / Spread Spectrum

12. 3.4 M-sequence An m- sequence or a maximal length sequence is defined as a sequence generated by a linear feedback shift register with m cells and with a period of . While generating the m-sequence, the generator passes through all possible states of the register besides the all zero state. The number of different binary m-sequence of order . Where are the prime number in the decomposition of Dr. Moshe Ran / Spread Spectrum

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14. 3.5 Statistical Properties of Binary M-sequences • Balance Property The number of “ones” in one period of the m-sequence exceeds the number of “zeros” by 1. b. Events counting Every times except the all zero J-tuple which occurs times. c. shift-and-Add Property The sum of the sequence and a shifted version of the sequence is another shifted version of the sequence. d. Periodic autocorrelation We refer to the sequence Is replaced by 1 and 1 is replaced by –1. The periodic autocorrelation of is defined Dr. Moshe Ran / Spread Spectrum

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16. 3.6 Statistical Properties of Random Binary Sequence • Balance Property The probability of one equals probability of zero. b. Events counting the probability of any c. Autocorrelation we refer to the sequence I.e. 0 us replaced by 1 and 1 is replaced by –1. The autocorrelation of a sequence is defined And is equal to Dr. Moshe Ran / Spread Spectrum

17. 3.7 Autocorrelation of Continuous Sequence The periodic autocorrelation of the continuous waveform is defined by The continuous autocorrelation of can be obtained by connecting the discrete autocorrelation of by straight lines. Dr. Moshe Ran / Spread Spectrum

18. 3.8 Spectrum of an M-sequence The spectrum of a continuous M-Sequence is the Fourier transform of the periodic autocorrelation of . The power spectrum of direct sequence spread spectrum signal is continuous and has deviations from the sinc form. Dr. Moshe Ran / Spread Spectrum

19. 3.9 Selected M-sequences The reversed order sequence is called the complementary sequence and is another m-sequence. If the original sequence is generated by a LFSR of order with the taps and , the complementary sequence is generated by a LFSR of order with the taps and . Dr. Moshe Ran / Spread Spectrum

20. 3.10 Linear Span of a Sequence The linear span of a sequence is defined as the length of the shortest LFSR which generates the sequence. If there are known consecutive bits of a sequence with a leaner span of , all the sequence can be calculated. In the binary case only known consecutive bits are required. For example, suppose that a binary sequence has a linear span of 5 and a portion of the sequence contains …1011101100011… Where the sequence index is increasing from left to right. The binary sequence satisfy the linear recursion Dr. Moshe Ran / Spread Spectrum

21. 3.11 Linear Span Of A Sequence (Cont.) …1011101100011… However we know that , then The solution is Dr. Moshe Ran / Spread Spectrum

22. 3.12 Sequence With Large Linear Span There are sequence with a period of length with a linear span larger then The most common approach to obtain a large linear span is the LFSR with feedforward logic. NON-LINEAR FUNCTION MEMORY LINEAR FUNCTION Dr. Moshe Ran / Spread Spectrum

23. Analytical derived sequence with a large linear span are known such as: • GMW • Bent Dr. Moshe Ran / Spread Spectrum

24. 3.13 Sequence for CDMA Systems In CDMA system many users share the same frequency band with different sequence with a small crosscorrelation. A popular family of sequence is the Gold sequences. In a family of Gold sequences of length there are sequences, and maximum crosscorrelation is approximately The maximum crosscorrelation is + + + Dr. Moshe Ran / Spread Spectrum

25. 3.14 Sequences for FH/CDMA Systems In a FH/CDMA system many users share the same frequency band with different sequences with a minimal probability of hit. FREQUENCY SYNTHESIZER (one-to-one mapping) SEQUENCE SELECTOR Dr. Moshe Ran / Spread Spectrum

26. 3.15 Aperiodic And Odd Autocorrelation The Aperiodic autocorrelation of a sequence cn with a period N is defined as Applications: where only one waveform is transmitted such as radar. The odd autocorrelation of a sequence cn with a period N is defined as The odd autocorrelation is useful in data communication where the one symbol time is equal to one period. It represent the effect of changes in data. The odd and the Aperiodic autocorrelation depend on the initial state of the sequence generator. Dr. Moshe Ran / Spread Spectrum

27. 7.18 Best Odd Autocorrelation M- Sequences The optimization was performed over all initial states of all m-sequences of period Dr. Moshe Ran / Spread Spectrum