ECE 5221 Personal Communication Systems
Prepared by:
Dr. Ivica Kostanic
Lecture 14: Frequency allocation and
channelization
Spring 2011
Florida Institute of technologies
Outline
Orthogonal codes
o Walsh / OVSF codes
o M-codes (PN codes)
Use of codes in CDMA systems
Important note: Slides present summary of the results. Detailed
derivations are given in notes.
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Walsh codes
 Orthogonal codes
Generation of the Walsh code matrices
 Length – power of 2 (1,2,4,8,…)
0 0 
W2  

0 1 
 Orthogonality maintained under
perfect synchronization
 Used for user channelization when the
synchronization between the users
can be maintained
W2n1 W2n1 
W2n  

W
W
n1
n1
2 
 2
o On the DL of the cellular system
Example of WC sequence generation:
W
W4   2
W2
0
W2  0


W2  0

0
0 0 0
1 0 1
0 1 1

1 1 0
0
0

0

0
W8  
0

0
0

0
0 0 0 0 0 0 0
1 0 1 0 1 0 1
0 1 1 0 0 1 1

1 1 0 0 1 1 0
0 0 0 1 1 1 1

1 0 1 1 0 1 0
0 1 1 1 1 0 0

1 1 0 1 0 0 1
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Walsh code orthogonality
 Code is given as a row in WC matrix
 To generate a code
o “0” -> “1”
o “1” -> “-1”
 Example: Codes W4,2 and W4,3
o W8,2 : (0,0,1,1,0,0,1,1) -> (1,1,-1,-1,1,1,-1,-1)
o W8,3 : (0,1,1,0,0,1,1,0) -> (1,-1,-1,1,1,-1,-1,1)
When synchronized – codes are orthogonal
W8,2  W8,3  (1,1,1,1,1,1,1,1)  (1,1,1,1,1,1,1,1)  0
When out of sync – codes are not orthogonal
W8,2  shift ( W8,3 ,1)  (1,1,1,1,1,1,1,1)  (1,1,1,1,1,1,1,1)  8
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M codes (PN codes)
 Have “noise like” auto-correlation properties
 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
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M sequences - properties
1. An m-bit register produces an m-sequence of period 2m-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.
3a. (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 2m-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.
…
2m-3 runs of ones and 2m-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/(2m-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 decline linearly from time 0 to time
= +(one-bit).
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Circular autocorrelation of PN sequence
PN sequence of length N:
xn 
Circular autocorrelation:
1
R p v  
N
N
 xnxmod( n  v, N )
n 1
For PN sequences
 1
R p v    1

 N
Note: PN sequences are
practically orthogonal to their
delayed versions
,v  0
,v  0
Consider N=15 sequence in the attached spreadsheet
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Use of CDMA in cellular systems
 Coding different on UL and DL
 DL – 2 levels of coding
o Within each cell – users
separated by Walsh codes
o Cells – separated by different
PN codes
CDMA DL
 On the UL
o Users separated by PN codes
 Additional level of coding usually
added for encryption purposes
Note: WC used to separate
synchronized transmissions,
PN used to separate
asynchronous transmission
CDMA UL
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