Wireless Communication Technologies
Lecture 17 & 18
Wireless Communication Technologies
16:332:559 (Advanced Topics in Communications)
Lecture #17 and #18 (April 1, April 3, 2002)
Instructor
Prof. Narayan Mandayam
Summarized by
Sandeepa Mukherjee (smdatta@ece.rutgers.edu)
This lecture note is about multiuser CDMA systems and we study in detail about
the performance of such systems.
Sandeepa Mukherjee
Page 1 of 18
Wireless Communication Technologies
Lecture 17 & 18
1.1Multiuser BPSK CDMA System
Let’s assume that we have a CDMA system with K users indexed as j =1,2…K
and each user transmits BPSK modulated signal waveforms simultaneously.
Then the transmitted signal of one such user j can be represented as,
(1)
where,
c j t  
cj
(n)
(n)
c
n  
(n)
j
pTc (t  nTc )
  1,1
b j t  
bj
n 
n 
b
n  
(n)
j
pT (t  nT )
  1,1
T = NTc
N = Processing Gain
The received signal at the receiver can be represented as,
(2)
where,
 j = relative time offset
j = phase offset
N
(t)  N (0, o )
2
In general signal from the various users are received at various propagation
delays, j  [0, T ] and j  [0,2 ]
Receiver in the multiuser scenario can be modeled as figure below:







Sandeepa Mukherjee
Page 2 of 18
Wireless Communication Technologies

S1(t)
1
S2(t)
2
SK(t)
K
Lecture 17 & 18

Rx
x
(t)
Figure 1: Received signal in multiuser scenario
1.1.2 Single User Reception
Focusing on user 1, we try to look at the demodulation of bit stream of user 1.
Assume the receiver is synchronized to user 1. So, without loss of generality we
can say, 1 = 0 and 1 = 0.
We will consider demodulation of a single bit of user 1, which is bit b1(0).
The matched filter receiver matched to user 1 is illustrated in the following figure,
r(t)
Z1(t)
C1cosct
The output of the matched filter corresponding to user 1 is given as,
(3)
K
T
j 1
0
z1   2 Pj  c1 (t )c j (t   j )b j (t   j ) cos( c t   j ) cos( c t )dt  1
where,
Sandeepa Mukherjee
Page 3 of 18
Wireless Communication Technologies
Lecture 17 & 18
T
1    (t )c1 (t ) cos( c t )dt
0
Neglecting all double frequency terms,


P1 ( 0) K Pj ( 1)
Tb1  
b j R j ,1 ( j )  b (j 0) Rˆ j ,1 ( j ) cos  j  1
2
2
j 2
We introduce continuous time partial cross-correlation functions,
z1 
(4)

R j ,1 ( )   c1 (t )c j (t   )dt
0
(5)
T
Rˆ j ,1 ( )   c1 (t )c j (t   )dt

0  T
Thus, in one bit interval of the desired user i.e. user 1, there would be effectively
two bits of every interferer j, by assuming that the signals of the interferers are
delayed by not more than one bit interval. This is illustrated in the following
figure:
b1(t)
b1(0)
T
bj(t)
bj(-1)
T
bj(0)
Sandeepa Mukherjee
Page 4 of 18
Wireless Communication Technologies
Lecture 17 & 18
Case of Synchronous users
For the case of synchronous system, j = 0 and j = 0 for all j. So there is only
one term from each interferer i.e. one bit of j interferes with one bit of the desired
user. Then from eq. (4) we have,
z1 
P1 ( 0) K Pj ( 0)
Tb1  
b j R j ,1  1
2
2
j 2
(6)
where,
cj = Spreading code vector of length N for user j
c1 = Spreading code vector of length N for user 1
 N T
1  N  0, 0 
4 

(a) When code sequence are Orthogonal
c j c 1   j1
T
Therefore the matched filter output is,
z1 
P1 (0)
Tb1  1
2
(7)
The probability of error is given as,
 2 Eb ,1
Pb,1= Q
 N0





Thus performance is same as single user channel
(b) When users use random codes
The error probabilities are calculated in two steps, noting that cj, j=1,2….K are
random binary vectors.
Sandeepa Mukherjee
Page 5 of 18
Wireless Communication Technologies
Lecture 17 & 18
Step 1: Fix the codes and bits of other users. Then calculate the conditional
probability of error for synchronous users with fixed data bits and conditioned on
Rj, 1 and bj(0).
This conditional Probability of error is represented as,
, b j (0))
Pb,1( R j ,1 
(8)
Step 2 : Find the average Probability of error Pb,1 , by averaging over {Rj,1} and
{bj(0)}.
For fixed {Rj,1} and {bj(0)},
, b j (0)) = Pr{ z1 < 0 | b1(0) = 1}
Pb,1( R j ,1 
K
 P

Pj ( 0)
 1T 

b
R

j
j ,1
 2

2
j 2
= Q

N 0T




4


K
 2 Eb,1
2 Eb, j ( 0) R j ,1 

= Q

bj
 N0

N
T
j 2
0


The average probability of error is,
K
 2 Eb,1
2 Eb, j ( 0) R j ,1 


bj
Pb,1 = Eb0 R Q
j
j ,1
 N0

N
T
j 2
0


This average probability of error is difficult to calculate.
(9)
1.1.3 Using Gaussian approximation for the Interference
Assume Pj=P for all j and the no. of user K is very large.
The interference term for user 1 is given as,
K
I1  
j 2
P (0)
b j R j ,1
2
We approximate I1 to be Gaussian. We proceed to find the mean and variance of
I1 as follows,
 K P (0)
 K P
E I 1   E 
b j R j ,1   
E b (j 0 ) E R j ,1
 j 2 2
 j 2 2
  
(10)
bj and Rj,1 is independent. Also,
Sandeepa Mukherjee
Page 6 of 18
Wireless Communication Technologies
Lecture 17 & 18
 
E b (j 0)  0 
Therefore E [I1] = 0
VarI1  

  
2
P K
E b (j 0) Var R j ,1

2 j 2
(11)
    1
Now, E b (j 0)
R j ,1  c j c 1
T
 
2
T
N
2


2
2
T2
T 
T 
T
Var R j ,1    E c j c 1    N 
N
N
N
2
PT
K  1
Therefore VarI 1  
2 N
The output of the matched filter z1 can be written as,
P ( 0)
Tb1  I 1  1
2


P


T


2
Therefore P b ,1  Q

2
 N 0T  PT K  1 


4
2N




P


T


2
 Q

2
 PT K  1 


 2N

For large K

N 

 Q

 K 1 
z1 
(12)
1.1.4 Case of Asynchronous Users
For asynchronous user case, we have non- zero j and j. Each bit of the desired
user is affected by two bits from the interferer. Therefore at the receiver each
interferer has two terms. Recall that the matched filter output is given as:
P1 ( 0) K Pj
Tb1  
TI j ,1 b j , j ,  j   1
2
2
j 2
where Ij,1 is the interference of user j to user 1, and given by
Z1 
Sandeepa Mukherjee
Page 7 of 18
Wireless Communication Technologies

I j ,1 b j ,  j ,  j  

Lecture 17 & 18

1 ( 1)
b j R j ,1  j   b (j 0) Rˆ j ,1  j  cos j 
T

b j  b (j 1) , b (j 0)
Assume that Pj = P for all users j, that is, users are all power controlled. The
matched filter output can then be written as,
P ( 0)
(13)
Z1 
T b1  I1 b, ,   1
2





where the resultant interference I1 b, ,    I j ,1 b j , j ,  j 
b  b 2 , b 3 ,.....b K  ,
K
j 2
   2 ,  3 ,...... K 
   2 , 3 ,..... K  ,
Phase and Time as assumed to be acquired for user 1, i.e. the Rx is
synchronized in time and phase w.r.t user 1.
For fixed b, , 
-1 is transmitted is,


Pb,1  Pr Z1  0 | b1(0)  1 



P
r 1 




P1

T  1  I 1 b, ,   0 
2




 2 Eb
1  I 1 b,  , 
  Q
N
0



 








a
Average probability of error is,
P b,1  Eb, , Pb,1 
Suppose b1( 0)  1, the error probability is given as,
Pb,1  Pr Z1  0 | b1(0)  1


 2 Eb
 Q
1  I 1 b,  , 
 N
0

Pb,1  Eb, , Pb,1 




(14b)
As seen from eq.(14a) & (14b) the error probabilities are not symmetric as the
interference is not symmetric.
It is not possible to get a closed form for the average probability of error. It is very
difficult to compute exact error probability however good bounds and
approximations exist. (“Error probability for DS/SS multiple access comm. Part-I
Upper and lover bound IEEE TranCom, May 1982”).
Sandeepa Mukherjee
Page 8 of 18
Wireless Communication Technologies
Lecture 17 & 18
1.1.5 Error Probability for Gaussian Approximation


Model interference I 1 b, ,  , as a Gaussian random variable.
The other assumptions are:
-bjm and bjn are independent for all i  j & m  n
-i, j are independent for all i  j & uniform over 0, T 
-i, j are independent for all i  j & uniform over 0,2 
We now proceed to find the mean and variance of I 1 b, ,  .




E I j ,1 b j ,  j ,  j   0 for all j (because bits are independent w.r.t codes)
 2j ,1  VarI j ,1 b j , j ,  j 
 


1
E cos 2  j E j R 2j ,1  j   Rˆ 2j ,1  j 
2
T
T
1 1
2
ˆ 2   d 



R


R

j
,
1
j ,1
2T 2  T 0




1
2T 3

 R    Rˆ  d
T
2
j ,1
2
j ,1
(15)
0
From eq.(13) we have,


P ( 0)
P
Tb1 
TI1 b, ,   1
2
2
Therefore for b1(0)=-1 we have,
Z1 


E Z1 | b1( 0)  1  

 

P
P
P
T
TE I1 b, ,   E1    T  0
2
2
2

P
T
2
PT 2
PT 2
 1 
Var I 1 (b, ,  )  Var1  
2
2
 E Z1 | b1( 0)  1  
VarZ
1
(0)
1
|b



K

j 2
2
j ,1

N 0T
4
Therefore the average probability of error is given as,
Sandeepa Mukherjee
Page 9 of 18
Wireless Communication Technologies



P b,1  Q





 2 Eb
 Q
 N0


P
T
2
N 0T PT 2

4
2
K

j 2
2
j ,1

 


1


K


2 Eb
2
 j ,1  
 1 

N
0 j 2

 
Lecture 17 & 18







(16)
As seen, the performance depends critically on j,12 which in turn depend on
cross correlation properties of chip waveform or code sequence. Performance
can be made better if better code sequences are designed with good cross
correlation properties j,12.
M-sequence and gold sequence are two types of code with fairly desirable cross
correlation properties (refer to “Error probability for DS/SS multiple access
communications, Geranitis and Pursley, IEEE TranCom, May 1982”)

Sandeepa Mukherjee
Page 10 of 18
Wireless Communication Technologies
Lecture 17 & 18
2.1 Probability of Error for asynchronous users
This section contains mathematical analysis, which provides expressions for
determining the average bit error rate for users in a single channel, CDMA
system. (refer to the paper "Performance evaluation for Phase Coded Spread
Spectrum Communication - Part II, Code Sequence analysis" by Pursley, Sarvate
& Stark, IEEE TCOM Aug 1977)
2.1.1Gaussian Approximations
As seen from eq. (3), the contribution from kth interfering user, to user 1 is,
Tb
Ik, 1 =

2 Pk bk (t   k )ck (t   k )c1 (t ) cos( c t   k ) cos  c tdt
(17)
0
It is useful to simplify this equation in order to examine the effects of multiple
access interference on the average bit error probability for a single user. The
relationship between bk (t   k ) , ck (t   k ) and c1 (t ) is illustrated in Fig.1
Integration Period Tb = NTc
C1, i
C1, i+1
C1, i+N-1
C1, i+2
k
 kT c 
k
Ck, j--1
Ck, j-
...
Ck, j
...
bk, 0
bk, -1
Figure 2: Timing of the local PN sequence for user 1 and user k
The quantities k and k in Fig. 1 are defined from the delay of user k relative to
user 1, k, such that,
k = kTc + k
0 <= k < Tc
It is possible to show that, the contribution from kth interfering user to user 1 is
given as,
Sandeepa Mukherjee
Page 11 of 18
Wireless Communication Technologies


Pk



cos( k ) X k  (1  2 k )Yk  (1  k )U k  ( k )Vk 
2
Tc
Tc
Tc


Ik, 1 = Tc
Lecture 17 & 18
(18)
Where Xk, Yk, Uk, Vk are random variables having distribution conditioned on A &
B which are given as,
pXk
A 


  l  A 2  A


 2 
B



pYk (l )   l  B 2  B


 2 
pU k (l ) 
1
2
pVk (l ) 
1
2
l = -A, -A+2,…, A-2,A
l=-B, -B+2,…, B-2,B
l = -1, +1
l = -1, +1
A and B are defined as follows:
A = Number of integers in [0, N - 2] for which (c1, l+i)(c1, l+i+1) = 1 i.e. A, measures
for a given code the no. of successive no transitions from +1 to -1 in the code c1
B = Number of integers in [0,N - 2] for which (c1, l+I)(c1, l+I+1) = -1 i.e. B, measures
for a given code the no. of successive transitions from +1 to -1 in the code c1.
Note: A + B = N - 1 since sets A and B are disjoint and span the set of total
possible signature sequences of length N in which there are a total of N - 1
possible chip level transitions
The details of these derivations are in the following papers:
[1] Lehnart & Pursley :" Error Probability for binary DS-SS communication with
Random Signature Sequence" IEEE TCOM Jan 1987
[2] Morrow & Lehnart : "Bit to bit error dependence in slotted DS/SSMa packet
systems with Random Signature Sequence" IEEE TCOM Oct 1989
For random signature sequence statistics of A and B are known. We have seen
the variance of the j th interference term is given as,
j, 12 = Var [ Ij, 1 (bj, j, j) ]
Sandeepa Mukherjee
Page 12 of 18
Wireless Communication Technologies
=
Lecture 17 & 18
 


2
1
( 1)
E cos 2  j E  b j R j ,1 ( j )  b (j 0) Rˆ j ,1 ( j ) 
2


T
(19)
For random signature sequences it is possible to model,
b
( 1)
j

N
R j ,1 ( j )  b (j 0 ) Rˆ j ,1 ( j )   X l
l 1
where X l l 1 are i.i.d random variables with distribution given as,
N

 T T
Uniform  N , N  with probabilit y1 / 2



Xl = 
 Bernoulli   T , T , 1  with probabilit y1 / 2

 N N 2
Now,
EX l   0
VarX l  
1
1  T 
  
2
3  N 
2 T 
  
3 N 
 


2

 1T 

 


 2 N 
2
Therefore, E X l2 
E cos 2  j 
2
2 T 
 
3 N 
2
1
2
Therefore the variance of the jth interferer as given in eq.(19) can be written as,
2
j, 12
1 1 2 T 
1
= 2 N   
3N
T 2 3 N 
(20)
Therefore the Average Probability of Error is,
Sandeepa Mukherjee
Page 13 of 18
Wireless Communication Technologies


1
 2 Eb
Pb,1  Q
2E K
No
2

1  b   j ,1

N o j 2

Lecture 17 & 18











1
 2 Eb

 Q

2
E
K

1
No
1 b


N
3
N
o


(21)
For large K, the average probability of error for asynchronous users is,
 3N 

Pb ,1  Q

K

1


For large K, the average probability of error for synchronous users is,

N 

Pb ,1  Q

K

1


(22a)
(22b)
Thus we see Probability of Error is less for asynchronous users than for
synchronous users which is due to the better averaging effect due to offset of
bits.
The expressions in this section are valid only if the number of users K is large.
Furthermore, depending on the distribution of the power levels for the K-1
interfering users, even when K is large, if the interferer power levels are not equal
or constant the Central Limit Theorem does not hold and the multiple access
interference contribution Ij, 1 cannot be modeled as a Gaussian Random Variable.
Therefore Gaussian approximation is not appropriate in the following cases (1)
Number of users K is not large (2) Interferers have disparate power levels.
In these situations a more in-depth analysis is required which we shall study in
the next section.
2.1.2 Improved Gaussian Approximation (IGA)
This analysis defines the interference terms Ij, conditioned on the particular
operating condition of each user. When this is done, each Ij becomes Gaussian
for large K.
Define as the variance of the multiple access interference for a specific
operating condition i.e. is the conditional variance of Ij.
Sandeepa Mukherjee
Page 14 of 18
Wireless Communication Technologies
 

Lecture 17 & 18

 Var I 1 b, ,  , c1 ,..., c K , p |  , , p, c1 ,...., c K 

 Var I 1 . |  ,{ k },{ p k }, B 



Here p k = Power of kth user
 k = Offset of kth user

If the distribution of  is known the bit error rate may be found by averaging over
all possible values of .
  P T 2 
Pb, 1 = E Q 1 b 
  2 
 PT 2 
=  Q 1 b  f (.) d
 2 
0


It is possible to show that,

N
 
 =  T Pk cos  k   2 B  1  k
2
  Tc
k 2


where,
K
2
c
2
(23)
2

  K
  k     Z k
Tc   k  2


(24)
Tc2
Pk U k Vk
2
Uk = 1 + cos(2k)
Zk =
  2  
N
Vk =  2 B  1  k   k 
  Tc 
2
Tc 


Closed form expression for fu (.) and fv | B(.) can be found so the distribution of Zk
can be determined.
PDF of  can be obtained by K-1 fold convolution of Zk,
f .  f z2    f zk
This equation may be used in eq. (23) to determine the average bit error
probability. This technique has been shown (refer to paper by Morrow and
Lehnert - TCOM Oct 1989) to be accurate for a very small number of interfering
users for the case of perfect power control.
This results in an Improved Gaussian Approximation.
Sandeepa Mukherjee
Page 15 of 18
Wireless Communication Technologies
Lecture 17 & 18
2.1.3 Simple Improved Gaussian Approximation (SIGA)
The expressions presented in the previous section are complicated and require
significant computational time to evaluate. A simplified expression for IGA, is
given by Holtzman in the paper "A simple accurate method to calculate SSMA
error probabilities", IEEE, TCOM Mar 1992.
The simplified bit error probability expressions are based on the fact that a
continuous function f(x) may be expressed as,
f (x)  f    x    f   
1
x   2 f    
2
If x is a random variable and  is the mean of x i.e. E[x] = 
Therefore,
(25)
1
E  f  x   E  f    0   2 f    
2
1
 f     2 f  
2
If derivatives are expressed in terms of finite differences,
 f   h   f   h  1
2  f   h   2 f    f   h  
f  x   f     x   
  x    

2h
h2

 2


Neglecting higher order terms,
E f x   f   
 2  f   h  2 f    f   h 


2 
h2

In [Hol92] it is suggested that an appropriate choice for h is
(26)
3 which yields,
E  f  x  
2
1
1
f    f   3   f   3 
(27)
3
6
6
If we consider Q (.)  f ( x) , then using eq. (27) , the average probability of error is
given by,
  P T 2 
Pb ,1  E Q 1 b 
  2 
2  P1Tb2
= Q
3  2
Sandeepa Mukherjee
 1 
P1Tb2
 + Q
 6  2   3 




 1 
P1Tb2
 + Q
 6  2   3










(28)
Page 16 of 18
Wireless Communication Technologies
Lecture 17 & 18
 Mean of variance of Multiple Access Interference (MAI) conditioned on
parameters
2 = Variance of 
Fig. 3 and Fig.4 illustrate the average bit error rates for single-cell CDMA
systems using the different analytical methods discussed in this lecture (i.e. GA,
IGA and SIGA). For results shown the processing gain is N=31.
1.00E+00
IGA
SIGA
1.00E-01
1.00E-02
Bit Error Rate
GA
1.00E-03
1.00E-04
1.00E-05
1.00E-06
1.00E-07
0
5
10
15
20
25
30
31
Total Number of Users, K
Figure 3: Comparison of BER for a desired user assuming that power level for all K users are fixed
In this comparison it is assumed that the power levels for all K users are fixed.
K2 = [K/2] users have a power level of P1/4. The remaining K1 = K -1 -K2 users
each have power level equal to that of the desired user, P 1.
Note that SIGA provides a very close match to IGA. For small number of users
GA falls apart.
Sandeepa Mukherjee
Page 17 of 18
35
Wireless Communication Technologies
Lecture 17 & 18
IGA
1.00E-01
SIGA
1.00E-02
Bit Error Rate
GA
1.00E-03
1.00E-04
1.00E-05
0
5
10
15
20
25
30
31
Total Number of Users, K
Figure 4: Comparison GA, IGA and SIGA when interfering users have random but identically
distributed power levels
The interferer power levels obey a log-normal distribution with a standard
deviation of 5dB. The power level of the desired user is constant and equal to the
mean value of the power level of each individual interfering user.
References:
[1] G. Stuber, “Principles of Mobile Communication”
[2] Theodore S. Rappaport “Wireless Communications-Principles & Practice”
[3] Narayan Mandayam, Lecture notes for “16:332:559”, Spring 2002
[4] Kemal Karakayali, Lectures notes for “16:332:559”, Spring 2001
Sandeepa Mukherjee
Page 18 of 18
35