ECE 5221 Personal Communication Systems Prepared by Dr

ECE 5221 Personal Communication Systems Prepared by: Dr. Ivica Kostanic Lecture 14: Frequency allocation and channelization Spring 2011 Florida Institute of technologies

Outline ØOrthogonal codes o Walsh / OVSF codes o M-codes (PN codes) ØUse of codes in CDMA systems Important note: Slides present summary of the results. Detailed derivations are given in notes. Florida Institute of technologies Page 2

Walsh codes Generation of the Walsh code matrices Ø Orthogonal codes Ø Length – power of 2 (1, 2, 4, 8, …) Ø Orthogonality maintained under perfect synchronization Ø Used for user channelization when the synchronization between the users can be maintained o On the DL of the cellular system Example of WC sequence generation: Florida Institute of technologies Page 3

Walsh code orthogonality Ø Code is given as a row in WC matrix Ø To generate a code o “ 0” -> “ 1” o “ 1” -> “-1” Ø Example: Codes W 4, 2 and W 4, 3 o W 8, 2 : (0, 0, 1, 1, 0, 0, 1, 1) -> (1, 1, -1, 1, 1, -1) o W 8, 3 : (0, 1, 1, 0, 0, 1, 1, 0) -> (1, -1, 1, 1, -1, 1) When synchronized – codes are orthogonal When out of sync – codes are not orthogonal Florida Institute of technologies Page 4

M codes (PN codes) Ø Have “noise like” auto-correlation properties Ø Generated as output of shift registers that have taps indicated by primitive polynomials o Taps need to be in “special places” o Location of taps for different code lengths: http: //www. newwaveinstruments. com/resources/articles/m_sequence_linear_feedba ck_shift_register_lfsr. htm Shift register for generation of binary sequence Florida Institute of technologies Page 5

M sequences - properties 1. An m-bit register produces an m-sequence of period 2 m-1. 2. An m-sequence contains exactly 2(m-1) ones and 2(m-1)-1 zeros. 3. The modulo-2 sum of an m-sequence and another phase (i. e. time-delayed version) of the same sequence yields yet a third phase of the sequence. 3 a. (A corollary of 3. ) Each stage of an m-sequence generator runs through some phase of the sequence. (While this is obvious with a Fibonacci LFSR, it may not be with a Galois LFSR. ) 4. A sliding window of length m, passed along an m-sequence for 2 m-1 positions, will span every possible mbit number, except all zeros, once and only once. That is, every state of an m-bit state register will be encountered, with the exception of all zeros. 5. Define a run of length r to be a sequence of r consecutive identical numbers, bracketed by non-equal numbers. Then in any m-sequence there are: 1 run of ones of length m. 1 run of zeros of length m-1. 1 run of ones and 1 run of zeros, each of length m-2. 2 runs of ones and 2 runs of zeros, each of length m-3. 4 runs of ones and 4 runs of zeros, each of length m-4. … 2 m-3 runs of ones and 2 m-3 runs of zeros, each of length 1. 6. If an m-sequence is mapped to an analog time-varying waveform, by mapping each binary zero to 1 and each binary one to -1, then the autocorrelation function for the resulting waveform will be unity for zero delay, and -1/(2 m-1) for any delay greater that one bit, either positive or negative in time. The shape of the autocorrelation function between -1 bit and +1 bit will be triangular, centered around time 0. That is, the function will rise linearly from time = -(one-bit) to time 0, and then declinearly from time 0 to time = +(one-bit). Florida Institute of technologies Page 6

Circular autocorrelation of PN sequence of length N: Circular autocorrelation: Note: PN sequences are practically orthogonal to their delayed versions For PN sequences Consider N=15 sequence in the attached spreadsheet Florida Institute of technologies Page 7

Use of CDMA in cellular systems Ø Coding different on UL and DL Ø DL – 2 levels of coding o Within each cell – users separated by Walsh codes o Cells – separated by different PN codes CDMA DL Ø On the UL o Users separated by PN codes Ø Additional level of coding usually added for encryption purposes Note: WC used to separate synchronized transmissions, PN used to separate asynchronous transmission CDMA UL Florida Institute of technologies Page 8