C1
1 ,
C2
1 ,
C3
2 ,
C3 3 ,
C1
3 ,
C2
Respreading Matched
Filter
Matched Filter
Matched Filter
Respreading
Respreading
Matched Filter
Matched Filter
Matched Filter
ˆ1
b
ˆ2
b
ˆ3
b ( )
r n
Figure 3.1: General partial SPIC receiver structure.
In this case, all regenerated signals are first weighted and then summed. Thus, each regenerated interference signal in (3.9) has an individual PCF and the signal to be estimated is a function of all PCFs. We call this structure the coupled structure. The other structure is one in which
!
!
. In this case, all regenerated signals are summed first and then weighted. Thus, there is one PCF for the signal to be estimated. We thus call this structure the decoupled structure. A thorough discussion of both structures is not available in the literature. Optimal PCFs have only been derived for the coupled structure under power balanced scenarios [61]. In what follows, we focus on a two-stage partial SPIC receiver with a decoupled structure. Primary simulation results (in Section 5) show that both PIC structures with optimal PCFs perform similarly.
§ 3.3 Optimal PCFs for AWGN Channels
In this section, we derive the optimal PCFs for a two-stage partial SPIC under an AWGN chan-nel. For ease of description, we only give the results associated with synchronous transmission.
Periodic and aperiodic spreading codes are both considered.
§ 3.3.1 Periodic Code Scenario
Assuming perfect chip synchronization, we first sample the received continuous-time signal in (3.1) with period . Let
be the received signal sample vector,
be the th user’s spreading sequence vector, and
be the noise sample vector. From (3.1), we have
(3.10)
Thus, we can obtain the matched filter output as
(3.11)
Note that
is a discrete version of the correlation termshown in (3.4). Similarly,
is a discrete version of the noise-related termin (3.5). For notational simplicity, we still use
to represent
andto represent
. Thus, (3.11) can be re-written as
(3.12)
For the second stage of a partial SPIC (with the decoupled structure), the regenerated signal for Useris
!
(3.13)
where
. The second stage output is then
The bit error probability for User, denoted as"
, can be written as
In (3.15), we assume that the occurrence probabilities for
and
are equal, and the error probabilities for
and
are also equal. As we can see, there are three terms in (3.14). The first term corresponds to the desired user bit. If we let
, it is a deterministic value. The second term corresponds to noise interference which is Gaussian distributed. The third term corresponds to the interference from other users and each interference is Binomial distributed. Note that correlation coefficients in (3.14) are small and CDMA systems are usually operated in low signal-to-noise ratio (SNR) environments. The variance of the third term is then much smaller than that of the second term. Thus, we can assume that conditioned on
is Gaussian distributed. The error probability is then
" whereis the Q-function and
Note that the expectations in (3.17) and (3.18) are operated on interfering user bits and noise.
. Evaluating (3.17), we obtain
Similarly, we obtain the variance as
where the coefficients ofare represented by
The optimal PCF for Usercan be found as
!
Substituting (3.19) and (3.21) into (3.25) and simplifying the result, we have the following equation.
We have two possible solutions now. The first solution for the first parenthesis is trivial since it makes the squared mean valuein (3.19) zero. The optimum PCF is then
!
(3.27)
We also derived optimal PCFs for an asynchronous CDMA system. The results are summarized in Appendix A. In what follows we discuss some special cases to give a better understanding of the optimal PCF characteristics. Let the correlations between any two user spreading codes be equal (
for ) and the power control be perfect (
and
). The optimal PCF can then be expressed as
!
(3.28)
As we can see from (3.28), the optimal PCF is smaller whenoris larger, because when the correlations between user codes are higher and the number of users is larger, the MAI is larger and the regenerated signal is unreliable. As a result, the PCF should be smaller. Also, when the user power is larger or the noise is smaller (is larger), the optimal PCF is larger. If we assume that the noise is much smaller than the signal power ( ), the optimal PCF can be further simplified to
!
(3.29)
Now the optimal PCF is independent of the transmission signal power. The bit error perfor-mance would also be saturated in this interference-limited region. From (3.28), we can also see that when the noise is large (), the optimal PCF tends to be small (!
). Note that the effect of the processing gain is reflected in the receiving SNR. If is larger, the receiving SNR will become smaller.
§ 3.3.2 Aperiodic Code Scenario
In commercial CDMA systems, the users’ spreading codes are often modulated with another code having a very long period. As far as the received signal is concerned, the spreading code
is not periodic. In other words, there will be many possible spreading codes for each user. If we use the result derived above, we then have to calculate the optimum PCFs for each possible code and the computational complexity will become very high. Since the period of the modulating code is usually very long, we can treat the code chips as independent random variables and approximate the correlation coefficient, , as a Gaussian random variable. As a result, the expectations in (3.17) and (3.18) can be further operated on . This greatly simplifies the optimal PCF evaluation. We now rewrite (3.16) as
"
denotes the expectation operator over the spreading code setand
and
are the expected squared mean and variance of , respectively, given the&th possible code in
. Letting'
, consideringas a Gaussian random variable, and evaluating (3.17) and (3.18), we have
In the above expressions, the notation( denotes the(value given the&-th possible spreading code in. Equation (3.25) can be re-expressed as
! Substituting (3.31)-(3.36) into (3.37) and simplifying the result, we finally obtain
!
As we can see, (3.38) only involves (3.32) and (3.34)-(3.36) and these expressions are easy to work with. We further consider the case in which noise is small ('
). Equation (3.38) can be simplified to
!
This result is remarkably simple. We only require and to calculate optimal PCFs; this will be useful in real-world applications.