Colorado State University, Ft. Collins
Fall 2007
EE 401: Low Complexity Signal Detection
Instructor:
Phone:
J. Rockey Luo
(970) 491-7411
Office:
E-mail:
B118, Engineering Building
rockey@engr.colostate.edu
Part II: Multiuser CDMA System and Optimal Detectors
Direct sequence spread spectrum communication
Assume the transmitter wants to transmit a symbol x   1,1. Instead of transmitting
the symbol directly, it spread the symbol to a symbol sequence of length N. Let
N
s   1,1 be a binary valued column vector of dimension N. For example, if N=4, s
can be  1,1,1,1 . We term s the signature sequence, which is known to both the
transmitter and the receiver.
T
The transmitter transmits the binary sequence xs over the channel, i.e. the channel is used
N times to transmit one source symbol. The channel output is given by
(1)
z  shwx  v
Here z is a column vector containing N symbols; v is a vector Gaussian noise with
Ev   0 , E vv T   2 I , where I is the identity matrix.
 
After receiving z , the receiver computes the following output (which is termed the
matched-filter output)
(2)
y  s T z  s T shwx  s T v  hwNx  n
T
Here
n
is
still
a
Gaussian
noise
with
and
En  E s v  0 ,
2
T
T
T
T
2 T
2
E n  E s vv s  s E vv s   s s   N . If the receiver detects the source symbol
using y, the signal to noise ratio of the system is
  
SNR 

hwN
 2N
 
 
2

N hw
2
2
(3)
Comparing to the SNR expression we have in Part I, spreading helps the system to increase
its SNR using the same transmit power.
Direct Sequence Code Division Multiple Access (DS-CDMA)
In the third-generation (3G) cellular system, direct sequence spread spectrum
communication is used to enable multiple users to share the same channel.
1
Suppose we have K users. We represent the symbols of the users using a K-dimensional
T
column vector x  x1 ,, x K  where x k is the source symbol of user k. We assume the
source symbols of different users are independently generated with
Pxk  1  Pxk  1  0.5 . Let the signature sequences of user k be s k . Let
S  s1 ,, s K  be the signature matrix whose columns are the signature sequences of the
users.
When K users transmit together to the common receiver, the channel output can be
represented by
(4)
z  SHWx  v
where H and W are K  K diagonal matrices
h1 0  
 w1 0  


H   0  0  , W   0  0 
 0 hK 
 0 wK 
with hk being the channel gain from user k to the receiver and wk being the square root of
 
the transmit power of user k. v is a vector Gaussian noise with Ev   0 , E vv T   2 I .
At the receiver, the receiver computes the following matched filter outputs for the K users
(5)
y  S T z  S T SHWx  S T v  RHWx  n
T
Here R  S S is termed the correlation matrix; n is the Gaussian noise vector with
En  E S T v  0 and E nnT  E S T vv T S  S T E vv T S   2 S T S   2 R .
 
  

 
Ideally, if the signature sequences of different users are orthogonal to each other, i.e.,
R  S T S  NI , then y k  Nhk wk xk  nk is the same as the single user case (2).
Consequently, assume hk wk  0 , the optimal decision of the receiver is
xˆ k  sgn  y k 
(6)
1. In the current 3G systems, signature sequences of the users are not orthogonal to each
other, i.e., R  S T S  NI . Given channel output y , the maximum likelihood detection
rule that maximizes the probability Pxˆ  x is given by
xˆ  min
x1, 1
K
 y  RHWx T R -1  y  RHWx 
(7)
For simplicity, let HW  I . Define the SNR of the system as
N
SNR  2
(8)

We count a detection error as long as at least one user symbol is erroneously detected.
Assume we have 8 users, K  8 . Assume the signature sequence length is N  15 . Assume
user signature sequences are randomly generated with each element taking +1 or -1 with
equal probability.
2
Use MATLAB to simulate the performance of the following three detectors in terms of
error probability versus SNR.
(1)
Matched filter detector : xˆ  sgn  y 
(2)
Decorrelation detector : xˆ  sgn R -1 y

(3)
ML detector: xˆ  min
x1, 1
K

 y  RHWx T R -1  y  RHWx 
2. What are the average numbers of float point operations for the three detectors?
Key Reference
[1]
Sergio Verdu, “Multiuser Detection”, Cambridge University Press, 1998.
3