Performance Analysis of TPSK in CDMA
System
Ng Chee Kyun
Department of Computer and Communication Systems
Engineering,
Faculty of Engineering, Universiti Putra Malaysia,
Malaysia.
APAN ‘06, 17 - 21 July, 2006
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CDMA Sequences
• Traditional CDMA sequences
~ m-sequence, Gold code, Walsh-Hadamard code.
~ Binary sequence {+1,-1}.
• Large Area Synchronized (LAS) CDMA sequences
~ Large Area (LA) code and Loosely Synchronous (LS) code.
~ Ternary sequence {+1,0,-1}
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Walsh-Hadamard Code
A Hadamard matrix of order 4
1 1 1 1 
1  1 1  1

H4  
1 1  1  1


1

1

1
1


where each row represents a Walsh-Hadamard sequence
w0  1 1 1 1
w1  1  1 1  1
w2  1 1  1  1
w3  1  1  1 1
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The auto-correlation property of Walsh-Hadamard sequences.
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The cross-correlation property of Walsh-Hadamard sequences.
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LAS Sequences
• The family of Large Area Synchronized (LAS) sequences is
formed by the combination of Large Area (LA) code [Li 99] and
Loosely Synchronous (LS) code [Staňczak 01].
• Li in [Li 99] studied various LA code construction schemes and
their application to CDMA system.
• Staňczak in [Staňczak 01] investigated the construction of
various schemes designed for the generation of various LS codes.
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LA Code
• Ternary codes: {+1,0,-1}
• Three parameters: LAN , K 0 , L 
N is the number of non-zero pulses,
K 0 is the minimum pulse interval, and
L is the length or period of LA code.
• There are certain numbers of zeros padding between the pulses
interval.
 has been
LA16,38,847
• For instant, the construction of
code
proposed in [Li 99]. The 16 pulse positions, are
Pn given by
Pn  {0,38,78,120,164,210,258,308,360,414,470,530,592,660,732,808}
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Pulse positions of LA16,38,847
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The arrangement of 16
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
LA16,38,847
sequences
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The auto-correlation property of LA sequence C2.
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The cross-correlation property between LA sequences C2 and C10.
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LS Code
• LS codes can be denoted as
aPxP

LS M , P,by
W0applying
dimensional Walsh-Hadamard matrix to an orthogonal
complementary set of length M, while inserting
W0 number of zeros padding at the center of the code.
• Then, the total length of the
2MP  W0
code
is given by
LS M , P,W
0
• The four 2 x 2 LS 5,2,4sequences
can be formed as

LS1 ( z)   1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,1
LS 2 ( z)   1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,1
LS 3 ( z )   1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,1
LS 4 ( z)   1,1,1,1,1,1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,1,1,1,1
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Construction of LAS Sequences
LAS N , K 0 , L; M , P,W0 
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The auto-correlation property of LAS sequence.
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The cross-correlation property of LAS sequence.
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Probability of Error over PSK Systems
• Chip Error Rate (CER) is used (instead of BER)
~ Binary Phase Shift Keying (BPSK)  Binary sequence {+1,-1}.
~ Ternary PSK  Ternary sequence {+1,0,-1}.
• Symbol Error Rate (SER) – A symbol  more than one chip.
~ Quadrature PSK (QPSK)  Symbol sequence {+1+1, -1+1, ….}
 d2
SER  Q min
 2N 0





where d min is the Euclidean distance or minimum
distance between signal points in the I-Q
constellation, and Q (x ) is the complementary error
function or simply Q-function which defined by
Q( x )  

x
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  x2 
dx
exp 
2
 2 
1
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Constellation Diagram for BPSK Signals
 2E 

Since d min  2 E then SER  Q

 N0 
In BPSK signalling each symbol in spreading sequence
corresponds to one chip in data pulse, therefore
 2 Ec
CER  SER  Q
 N0
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



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Constellation Diagram for TPSK Signals
 3E
3E
Since d 
and d min  2d  3E then SER  Q
2
 2N0
In TPSK signalling each symbol in spreading sequence
corresponds to one chip in data pulse, therefore
 3 Ec
CER  SER  Q
 2N0
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







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Constellation Diagram for QPSK Signals
 E 

Since d min  2 E then SER  Q

N
0


In QPSK signalling each symbol in spreading sequence
corresponds to two chips in data pulse, therefore
 2 Ec
CER  Q
 N0
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



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LAS Even Ternary (LAS-ET) Sequences
• An additional constraint is imposed to original LA code.
 The pulse interval between two adjacent non-zero
pulses, d n must be even.
 The minimum pulse interval, K 0 must also be even.
 LAS  ET 16,38,818 compared to LA16,38,847
Pulse positions of LA16,38,818
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The arrangement of 16
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sequences

LA16,38,818
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LA Sequences
• For mapping two chips into one symbol, there are nine combinations:
s(t )  [0,0], [0,1], [0,1], [1,.0], [1,1], [1,1], [1,0], [1,1], [1,1]
 9-ary PSK
LAS-ET Sequences
• For mapping two chips into one symbol, there are three combinations:
s(t )  [1,.0], [0,0], [1,0]  TPSK
 3E c
 CER  Q
 N0

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


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SER Performances
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CER Performances
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Concluding Remarks
• Better CER performance in TPSK signalling can be
achieved when two chips in the LAS-ET sequence are
mapped in one symbol period.
• At the same time, the spectrum efficiency of this LA
sequence is also increased.
• The sequence length in LAS-ET sequence is
decreased compared to original LA sequence while
maintaining the original size of IFW.
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Thank You
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