Code Sequence and Properties PDF

Code sequence and its properties Code sequence – it is a periodic sequence of logical 1s and 0s following each other in a certain order (often pseudo-random). Transitions in this sequence 0→1, 1→0 are possible in time moments ti: ti = iTt,, Ts = nTt, n >> 1...

Code sequence and its properties Code sequence – it is a periodic sequence of logical 1s and 0s following each other in a certain order (often pseudo-random). Transitions in this sequence 0→1, 1→0 are possible in time moments ti: ti = iTt,, Ts = nTt, n >> 1 Pn(t)=±1 U=1 corresponds to logical “1” U=-1 corresponds to logical “0” Rk=1/Tt - chip rate Clock pulses Tt 1 2 3 4 5 6 7 8 9 Linear feedback shift-register (LFSR) S ( f ) = U 2 T t sin c 2 ( p fT t ) 0 (+) 0=0 0 (+) 1=1 + 1 (+) 0=1 1 (+) 1=0 1110010 D1 D2 D3 c Maximum (longest) sequence (code) is the longest sequence that can be generated by a sequence generator of the n-column shift register type. The length of the binary maximum sequence is 2n-1 clock periods. Linear feedback shift-register (LFSR) + 0 (+) 0=0 0 (+) 1=1 1 (+) 0=1 1110010 1 (+) 1=0 D1 D2 D3 c Maximum (longest) sequence (code) is the longest sequence that can be generated by a sequence generator of the n-column shift register type. The length of the binary maximum sequence is 2n-1 clock periods. Properties of m-sequences The number of units is equal to the number of zeros plus 1. For example, a length-10 generator generates a sequence of 210-1 = 1023 chops. It contains 512 units and 511 zeros. The statistical distribution of “1” and “0” in the maximal sequences is strictly defined. The position of the slots “1” and “0” of a certain length in the separate maximal sequences changes, but the amount does not. The autocorrelation function of the maximal sequence for all shifts is -1, except for the shift of the 0 ±1 chip period, where this function varies linearly in the range of -1...2n-1. The result of summing the sequence according to the modulo-2 with its own displacement is the same sequence, but its shift in time does not coincide with any of the originals. Each possible state of the n-column generator (n-set) exists at some point. That state exists once and only once per code period. The exception is the state of all "0", which must not exist (in this state the generator stops generating). Properties of m-sequences The statistical distribution of “1” and “0” in the maximum sequences is strictly defined. The position of the slots “1” and “0” of a certain length in the separate maximum sequences changes, but the amount does not. Both the logical zero and the logical unit have exactly 2n- (p + 2) of p chip duration in the maximum sequence, except for the n-time slot, which can only consist of “1”, and the n-1-time slot, which can consist of from “0” alone. Slots in 27-1 length m-sequence: Slot duration, Number of Number of Total number p slots (1) slots (0) of slots 1 16 16 32 2 8 8 32 3 4 4 24 4 2 2 16 5 1 1 10 6 0 1 6 7 1 0 7 Code autocorrelation and cross-correlation ¥ Autocorrelation function: For binary bipole ò Y(r) = f (t) f (t - r)dt -¥ sequence [P(t)=±1] and code offset ri=iTt: å Y ( ri ) = Coincidences sutapimai - å nesutapima Discrepancies i 7 chip code ti=iTt Shift (chips) Code/shifted code Coincidences- discrepancies 0 1110100/1110100 7 1 1110100/0111010 -1 Great result 2 1110100/0011101 -1 3 1110100/1001110 -1 4 1110100/0100111 -1 5 1110100/1010011 -1 6 1110100/1101001 -1 7 1110100/1110100 7 m-sequence autocorrelation function Such autocorrelation function is good for: a) Determination of the delay of signal propagation; b) For "catching" signals (suppression of inter-symbol interference) under multipath fading conditions. But bad for: a) Large n (with a long sequence period) the synchronization process takes a long time Shift r = 0, value 2n-1 Offset 0 -1 -1 chip +1 chip  m-sequence cross-correlation function ¥ Cross-correlation function: Y(r) = ò f (t)g(t - r)dt For binary bipole sequence -¥ [P(t)=±1] and code offset ri=iTt: 7 chip codes 1110100 and Y ( ri ) = å Coincidences sutapimai - å nesutapima Discrepancies i 1110010 [ Not enough isolation between codes ] Shift (chips) Code/shifted code Coincidences-discrepancies 0 1110100/1110010 3 1 1110100/0111001 -1 Poor result 2 1110100/1011100 3 3 1110100/0101110 -1 4 1110100/0010111 -1 5 1110100/1001011 -5 6 1110100/1100101 3 7 1110100/1110010 3 Linear summation of codes 1 2 ……..q n 1 2 ……..q n c (clock pulses) a) If both m-sequences are the same but with different phases, the output has the same sequence whose phase does not coincide with any one element. b) If the two m-sequences are different (equal length), the output is a non- maximal sequence of the same length (Gold codes) - they have excellent cross-correlation properties. c) If both m-sequences are of different lengths (2n-1 ir 2p-1 ), then the output code is non-maximal and has (2n-1)(2p-1) length. (JPL codes) Linear summation of codes Summation of 7 symbol length codes 1110100 and 1110010 Shift of second code Code/shifted code Result (chips) 0 1110100/1110010 0000110 1 1110100/0111001 1001101 2 1110100/1011100 0101000 3 1110100/0101110 1011010 4 1110100/0010111 1100011 5 1110100/1001011 0111111 6 1110100/1100101 0010001 7 1110100/1110010 0000110 Autocorrelation function of nonmaximal code Typical non maximal code autocorrelation function Non-maximal codes are often used in practice: a) Gold codes -- low cross-correlation - allow better (than m-sequences) to distinguish signals against the background of other signals, b) JPL codes -- very suitable for ranging -- very long codes are synchronized quickly c) Orthogonal codes -- allow elimination of useless signals when multiple compressed signals are synchronized: T ò p ( t ) g ( t ) dt = 0 0 E.g.: 1 1 1 1, 1 1 -1 -1, 1 -1 1 -1, 1 -1 -1 1  Insertion of information into DSSS signal The most commonly used method of obtaining a DSSS signal is the modification of a code sequence with an information signal followed by a BFSK, QPSK, etc. Such a DSSS signal is obtained in two steps: 1) Code modification is performed - i.e. the information logic signals m(t) and the code sequence pN(t) are summed according to the modulus- 2 (this operation is equivalent to the multiplication of bipolar NRZ signals);. 2) The modified code sequence is executed in BPSK, QPSK, or m-QAM. Such systems are always coherent. The bit and clock starts are synchronized and Tb = KT t For BPSK, DSSS signal is expressed as: Data m(t)=±1 2E s TSPS = m ( t ) p N ( t ) cos 2p f c t Tb Code pN(t)=±1 DSSS signal modulator Synchronous Bipole data m(t) signal m(t) m(t) Data Coder synchronization unipolar- X chain bipolar pN(t) Sequence pN(t) Chip generator generator BPSK Carrier generator (frequency multiplier) X sDSSS  DSSS signal modulator -- multiple sources m1(t) Data Discrete X multilevel synchronization NRZ coding signal OC1(t) m2(t) Data synchronization X Σ X NRZ coding OC2(t) pN(t) m3(t) Data X Scrambling synchronization NRZ coding OC3(t) Spreading Data rates of input signals are equal and Tb= OC (orthogonal code) period, PN period >> OC period, Radio modulator Orthogonal codes - allows eliminating unnecessary Carrier signals, when multiple signals are synchronized. X Code examples: 1 1 1 1, 1 1 -1 -1, 1 -1 1 -1, 1 -1 -1 1 Coherent case: S = [m1(t)OC1(t) + m2(t)OC2(t) + m3 (t)OC3(t)]pN(t)cos(ωt) = sTSPS = m1 (t)OC1(t) pN(t)cos(ωt) + m2(t)OC2(t) pN(t)cos(ωt) + … Efficiency of FDMA system Number of physical channels per single cell Bv BV - total bandwidth, Bc- channel bandwidth m f = Bc N c For hexagonal cell we had S/I ratio: S R -n S 1 æ Dk ö 1 n n = = ç ÷ = (3N c ) 2 I N I Dk-n I 6è R ø 6 Minimal S/I for hexagonal cell: 1 2 æ S ön 1 æ S ön ç 6 ÷ = (3N c) 2 ç 6 ÷ = 3N c è Iø è I ø 3Bv From here the efficiency: m f = 2 æ S ön Bc ç 6 ÷ è I ø Efficiency of FDMA and TDMA systems For path loss exponentν= 4, the efficiency equation: Bv Bv mf = 1 = 1,22 S æ2 Sö 2 Bc Bc ç ÷ I è3 I ø For TDMA case: Bv Here k - the number of m f = 1,22 physical channels within Bc S frequency channel k I Exercise - FDMA/TDMA efficiency Assume mixed time-frequency division multiple access system. The total frequency band Bv = 12.5 MHz is divided into frequency channels of Bk = 200 kHz bandwidth and k = 8 time multiplexed channels per frequency channel. The minimum allowable S/I factor is 18 dB. Find the efficiency of the system by assuming a hexagonal cell configuration with cluster consisting of 7 cells and path loss exponent n = 4 Bv m f = 1,22 Bc S k I CDMA system efficiency Signal-to-noise ratio for K CDMA users S sBv 1 1 = = = N (K -1)sBv +N0 Bv N0 Bv N 0 K -1 + K -1 + s S For digital signals S/N is expressed via Eb/No: S EB R EB 1 Here: M -- Bv = =. processing gain: M = N N0s B N0s M R When only a single user is active: æSö EB R EB 1 EB 1 ç ÷ = = =M è N øV N 0 Bv N 0 M N 0s æ EB ö -1 K - 1 + M çç ÷÷ è N0 ø CDMA system efficiency When only single user is æ EB ö E çç ÷÷ = B active Eb/No ratio is max è N 0s ø max N 0 Let Kmax is the maximum number of transmitters at minimal S/N with sufficient connection quality: æ EB ö 1 çç ÷÷ = M è N 0s -1 ø min æ EB ö K max - 1 + M çç ÷÷ è N0 ø E B N 0 - (E B N 0 s )min K max = 1+ M (E B N 0 )(E B N 0 s )min CDMA system efficiency If thermal noise is lower than interference: E B N 0 - (E B N 0 s )min K max = 1+ M (E B N 0 )(E B N 0 s )min 0 1 K = 1+ M max (E B N 0 s )min Exercise: CDMA system efficiency CDMA system supports data rate Rb=10 kbps with chip rate Rt=10 Mcps. For system operation, energy per bit to noise ratio Eb/No must be >10 dB. For a single active user Eb/No = 16 dB. What is the maximum number of active users? What is the maximum number of active users in ideal conditions (i.e. when there is no thermal noise)? What is the maximum number of active users, when transmit power is reduced by 2 dB? E B N 0 - (E B N 0 s )min K max = 1+ M (E B N 0 )(E B N 0 s )min 0 1 K = 1+ M max (E B N 0 s )min