Download

13.1 INTRODUCTION 393Chipping Codes. These codes must have specific characteristics in order to separateconcurrent transmissions in the same RF band. The first characteristic is thatthe number of 1’s and 0’s in a code should be the same or differ at most by one.The other characteristics are related to functions called cross-correlation andautocorrelation.Correlation is the numerical value that results from multiplying bit-by-bit twobinary sequences of equal length (encoded with bipolar values þ1 and 21) thenadding the bits of the product. The same result can be obtained with standardbinary values (1’s and 0’s) by comparing the two sequences bit-by-bit and then subtractingthe number of mismatches from the number of matches. In mathematicalterms, correlation is the scalar that results from the dot product (or innerproduct) of the vectors that represent the two sequences.Cross-correlation is the correlation between two different sequences; autocorrelationis the correlation between two instances of the same sequence that are phaseshifted(in a circular fashion) by a whole number of chips. For instance, if wecompare the two code sequences of Fig. 13.1-1 (0101 and 0011), we find twomatches (the first and the last bit) and two mismatches (the two middle bits), sothe cross-correlation is 2 2 2 ¼ 0. If we then take the first sequence (0101) andcorrelate it with an instance of itself that has been shifted to the right by one chip(1010), we have zero matches and four mismatches, yielding an autocorrelationof 0 2 4 ¼ 24. Autocorrelation when in-phase is 4 2 0 ¼ 4.A set of equal-length codes that have cross-correlation values of zero (calledorthogonal codes because the corresponding vectors are orthogonal) can be usedto separate communication flows in the same RF band, as long as sender and receiverare in-phase. Walsh codes (see below) are an example of orthogonal codes.With other types of code (PN codes, see below) autocorrelation properties can beused to separate communication flows, by spreading and despreading source signalswith different phase-shifted instances of the same code sequence. The key propertyof these codes is that the autocorrelation value when two instances of the sequenceare in-phase is much higher than any of the out-of-phase values. These code typesalso provide a means of acquiring code synchronization between source andreceiver: the source transmits a known pattern spread with a known chipping code;the receiver then despreads the received transmission with different instances of thesame chipping code, each shifted progressively by a whole number of chips, until theknown pattern is detected. That indicates that the sender and receiver are in-phase.Code Types. CDMA systems use two types of codes: Walsh codes (also calledHadamard codes) and pseudorandom noise (PN) codes.Walsh codes. These codes are relatively short (from 2 to 512 chips) and come inorthogonal sets generated from a square matrix (Hadamard matrix [1,4,5], so thenumber of codes in a set is equal to their length. They provide the best performancein terms of number of channels that can be supported in a cell, but maintain theirorthogonality only when source and receiver are phase-synchronized. If senderand receiver are out of phase by even one chip, orthogonality may be lost. To ensure