§ 4.1 System Model

Consider a synchronous
-user CDMA system in the AWGN channel. Let the spreading se-quence of the^th user denoted by^

 with processing gain^ and amplitude^

. Then the chip-sampled received signal in a certain bit interval can be represented as



















   
  (4.1)

where^ and^ are the channel gain and data bit of the^th user, and^ is the AWGN with variance^. The first stage of the partial HPIC is the matched filter output given by



where the time-averaged cross-correlation function between the^-th and the^th user is defined as

and the noise sample after multiplying the spreading code is expressed by⁵






  . We further denote the noise term after despreading as

5






. The adaptive blind partial HPIC uses an adaptive filter to estimate the channel gains and then cancel the interference produced by other users. The adaptive algorithm used is the well-known LMS algorithm, as depicted in Figure 4.1. The LMS algorithm minimizes the MSE between the received signal and the regenerated signal in a bit interval. The optimal weight vector can be obtained by

where the error signal is represented by

#

In above equations, the superscript^ denotes the corresponding variable is operated at the^th stage. Note that only one bit period is available for weight adaptation. We first express the

spreading sequence vector as

The LMS update equation for the^th HPIC stage (with^stages of interference cancellation) is then formulated as



After the weights are trained, they are used to cancel the interference from other users such that the input to the^th user’ slicer in the^-stage is

Then the^th stage output from the partial HPIC (for User^) can be formed by

 will address a two-stage partial HPIC structure and omit the superscript for the stage number^ such that^{
 } ,^$ convergent weights are

$

Thus, the convergence weights depend on whether the bit decision results in the previous stage are correct or erroneous. The adaptive algorithm allows each user can weight to attain the desired value symbol by symbol. This is the reason why the adaptive approach performs better than non-adaptive methods.

As mentioned, the adaptation period is constrained in one symbol period. This is because the optimal weight for User^ may be^or^depending on the bit decision for each sym-bol. Although the LMS algorithm is simple, its convergence may slow and the weight may not converge to the desired value in such a short period. In addition, the resultant weight heav-ily depends on the parameters used in the LMS algorithm so is the cancellation performance.

These parameters include the step size and weight initials. In the conventional approach, these parameters are determined heuristically. The weight initials are usually set as the channel gains, i.e., ^$




. This is reasonable since the bit error probability is usually low, most of the weights will start their adaptation at the optimal values; only few weights are away from their desired values by ^. A larger step size will accelerate the convergence speed for the weights with erroneous decision, but also inevitably introduces a larger variance. There is little research regarding the convergence analysis for the adaptive blind partial HPIC receiver and this is the motivation of our research.

The LMS algorithm has been analyzed and developed for over four decades. However, most results cannot be used here. This is because the step size used in this application is large and this will violate many assumptions assumed. The other reason is that we most concern the transient behavior (due to small sample size) while most works only concern steady-state behavior. We then develop a novel method to overcome this problem. We will start the analysis with a single-user scenario. In this case, there is no MAI; however, the result can serve as a base for the two-user and general
-user scenario.

§ 4.2 Exact Analysis for Single-user Scenario

§ 4.2.1 Optimal Weight Analysis

Consider the CDMA system with only one active user, i.e.,

^. Since only one user is present, we will omit the subscript^ for notational simplicity. Thus,^
 , ^
,




,^




 ,^5
5

 ,^
, and^6






 . Note that by definition,

5
  and








5  (4.11)

The matched filter output signal in a certain bit interval in (4.2) can be rewritten as










 






  








 








5

 (4.12)

where the noise samples^{5 
 }are i.i.d. random variables with zero mean and variance ^







. Note that in the following derivation, we refer to the first stage decision, the first stage correct decision, and the first stage erroneous decision as the decision, the correct decision, and erroneous decision, respectively. From (4.12), it is simple to see that the decision^, which equals sgn^, depends on the noise term ^. It is simple to derive the condition for correct or erroneous decision. Denote the set for which decision is correct as

,

and that for which decision is erroneous as
^. Then,

 



 



 7 for ^


  for ^


(4.13)

and



 







  for ^


 7 for ^


(4.14)

We will first derive the optimal weight conditioned on^ and then take the expectation on the conditional optimal weight to obtain the final result. Since the input to the LMS filter depends on ^, the optimal weight will be different for ^
 and ^
. To facilitate the derivation, we first define some notations. Let a random variable  conditioned on ^ be denoted as , i.e., ^. Also let the conditional random variable with^{ }

be denoted as
, i.e.,





 

. Similarly,^{
 
}. Also let







,^









,










, and ^













 where the subscript ⁸ denotes the corresponding variable is a mean value,^denotes the expectation operated on^ and^ , and^denotes the expectation operated on ^. Let^denote the expectation operated on all random variables, and we have ^{


}
, and ^{

} . Using the similar rule, we define the optimal weight conditioned on^{ }

as^$
, and that on^{ 
} as



$





. Also, let the optimal weight for correct decision be^$



and that for erroneous decision be^$^ . We then have^$






$



and^$^





$





. The conditional optimal weight is given by



$















(4.15)

where<sup>

</sup>

The conditional mean for^5
 can be obtained by taking the conditional expectation on both sides of (4.11).



Thus, (4.16) can be rewritten as



Assuming^
, we can obtain the optimal weight for correct decision as (the result is identical for^
)

Note that^ is a Gaussian random variable with zero mean and variance^



9 denote a probability density function. Thus, the second term in the righthand side of (4.21) can be expressed as



Similarly, from (4.16), we can obtain the conditional optimal weight for the erroneous decision as

The optimal weight is then

$









$















 (4.24)

where^





can be evaluated as that in (4.22).

§ 4.2.2 Weight Error Mean Analysis

The LMS update equation for the single-user scenario can be formulated as




6 $

%
  

$
$ % 6 

(4.25)

Define two weight errors as

 :


$

 $

 (4.26)

 :




$



 $





 (4.27)

Our objective is to find close-form expressions for the mean values of ^:
 and ^: . Using the notations defined above, we have ^:




 :

  and ^:




 :



 . We first consider the scenario of correct decision and rewrite (4.26) as

 :


$

 $



$

 $



$



$



 (4.28)

We then define



;


$

 $



(4.29)



Æ

$



$



 (4.30)

From (4.28), we can have

 :


;

 



Æ

 (4.31)

It is simple to see that ^Æ

. Thus, ^:

Iterating (4.32), we can obtain

 we have



Using the result from (4.19), we have



where^$

$ and^{:

$ $}
. The same result can be obtained for the weight error mean of erroneous decision.

:

§ 4.2.3 Weight Error Variance Analysis

In this section, we will find close-form expressions for^{ :
 }and that of^{ : }. Let

. From the conditional random variable property, we have ^:

 first consider the scenario of correct decision. From (4.31) and (4.33), we have



Note that the second term in the righthand side of (4.40) is just^;
 . This can be seem from (4.33) and (4.35). We now evaluate this term.



From (4.41), we can see that we have to find the autocorrelation function of ^5
 , which is

 To solve the problem, we first consider a simple two-chip case in which^



where the unconstrained variables ^5
 and ^5
 are two i.i.d. random variables with zero mean and variance^



. We can evaluate the conditional joint probability function of^{5
 5
 } as

where^! is a normalization constant. From (4.44), we can obtain⁵

. Multiplying ^5
 and taking expectation on the both sides of (4.43), we can obtain

Instituting the result of⁵



Direct extension of the above derivation to^{ 7} is difficult since we have to evaluate multi-dimensional integrations. We now use a simple method to overcome this problem. First, we let

 be even and rewrite the^-constrained equation as



where the unconstrained variables ^=
 and ^=
 are i.i.d. random variables with the same distribution. We can then apply the result in (4.45) and obtain



Thus, from (4.49) we can obtain the crosscorrelation^/ for^{ 7} as

/

Multiplying^5
 on both sides of (4.47) and taking expectation, we have

Finally, we obtain

.

Simulation results show that the result (derived for an even^) is also very accurate for an odd

. We can then have an explicit expression of the first summation term on the righthand side of (4.41) as

Thus, combining (4.41), (4.50), (4.52) and (4.53), we have



By definition and (4.40), we have

:

Note that the result in (4.54) is independent of^; it is a function of noise variance and the step size only. Thus,^;



;

 
;

 . Thus, the second term in the righthand side of (4.55) can be evaluate using (4.54). Denote the first term in (4.55) as^Æ
. Then,

:

The term^Æ
can be further evaluated as

Æ

where the second moment of^$
is given by

Thus, we can obtain the weight error power shown in (4.38) using (4.36), (4.54), (4.55) and (4.57) as

Similarly, the weight error power for the erroneous decision described in (4.39) can be obtained as

The second term in the righthand side of (4.60) can be expanded as

Æ

 . The third term in the righthand side of (4.60) can be evaluated using (4.54).

§ 4.3 Exact Analysis for Two-user Scenario

Extending the procedure developed in the previous section, we now proceed to analyze the two-user case. Only the optimal weights and convergent weight error means are considered since the closed form expression for the weight error variance is difficult to obtain. In most cases, we only represent the result for the correct decision (denoted with superscript ‘
’). The derivation for erroneous decision is summarized in Appendix D.

§ 4.3.1 Optimal Weight Analysis

Define^
  

  

 and the matrix formed by^ as^{   }. Then, the time-averaged correlation matrix is obtained as












  (4.63)

The time-averaged correlation between these two users’ codes is given by










  



  (4.64)

Note that^is an integer. It is simple to show that





 +

,

 

 

-.

 (4.65)

The matched filter output vector, denoted by^












, is then










 






 











 











 









 









 (4.66)

where

^







 is the data bit vector,^
diag^





is the channel amplitude matrix, and^
5

 5

  

 is the noise vector after code multiplication. Let the second term in the righthand side of (4.66) be denoted as

^







. Then






  (4.67)

and







 (4.68)

As that in the single user case, the decision in the first stage depends on the value of
. However, the problem here become more involved since the distribution of
depends on ^. It can be shown that the joint probability density function for the random vector
is Gaussian and

9

where the covariance matrix is given as



Note that now the number of bits for decision is two and the number of the decision patterns becomes four. Let^
and^can be 1 or 2. Define the set for which User^’s decision is correct as

Similarly the noise subset for making erroneous decision is represented as

We then extend our notations defined in the previous section. Let a random variable  con-ditioned on
and then on ^ be denoted as , i.e., ^
. Also let 

. Using the similar rule, we define the optimal weight conditioned on
and then on^as, the optimal weight conditioned on^ as^", and the optimal weight as^. We then have^{"
­}and^
#".

The optimal weight conditioned on
and then on^can be represented as

where the correlation matrix of input signals is expressed by



The crosscorrelation vector is given by

 Thus, the conditional optimal weight vector is



As we can see from (4.76), the optimal weights depends on the decision patterns in ^. There are four decision patterns, i.e.,^

. We can have similar notation for optimal weights. Let the optimal weight conditioned on
^{ } and then on^ as ^ and ^"^

where^denotes the^th decision pattern, and
^ denotes the^th bit pattern.

If we further assume that^

, we have

The optimal weights for ^becomes

"

The complete set of
for all decision and bit patterns is shown in Table 4.1. The complete set of conditional optimal weights is given in Table 4.2.

Our objective is to determine
 and ^ by taking expectation on ^"
 and^". As seen in Table 4.1, the region of^for correct decision is different from that of^. Thus we have to determine the components of ^"
 user by user. The union of noise subsets for the first user to have correct decision is then



The occurrence probability for^ is obtained as

"

Table 4.1: Sets of
for all decision and bit patterns

The first user optimal weight for correct decision and a given^is

"

The second user optimal weight for correct decision and a given^is

"

Table 4.2: Complete list of conditional optimal weights

The optimal weight is obtained through averaging ^"



over all^values by

$ where^
 and the distribution for the correlation coefficient is given by

"

The optimal weights for erroneous decision can be obtained in a similar way and summarized in Appendix D.

§ 4.3.2 Weight Error Mean Analysis

The LMS update equation for a two-user scenario is rewritten as



. Define two weight errors as



From the optimal weight results of the two-user case, we know^"
and^"are obtained from

"



’s. Thus we also give the conditional weight errors as



where the conditional weights are defined as

As in the single-user case, our goal is to determine close-form expressions of ^:


  and

We then define



From (4.99) and (4.100) we have



It is obvious that



Thus we obtain that



where^{
  }and the parameter^{ } in the second term on the righthand side of the above equation is defined as



It can be easily shown by deduction that the recursive weight error given
and^is



>  (4.106) By combining (4.105) and (4.106) with the institution that

0 we obtain the conditional weight error as

 Note that by some algebraic computations we obtain

 and then we can express the weight error as



From the above definition, we know^" expecta-tion for the first term on the righthand side of (4.110) can be obtained as



 is a deterministic term. The conditional expectation for the second term can be obtained as

The conditional expectation for the third term can be obtained as



where the independence between^0> and^> is assumed. Combining the expectation terms through (4.111)-(4.113) the weight error mean vector conditioned on
and^is given by

"

It has to be noted that in (4.114) the evaluation of^ varies for different^ as expressed by



. The weight error mean for the first user conditioned on only the correct decision and^is represented by

"

where^ is given in (4.86). Similarly the weight error mean for the second user is represented as

where^ is given in (4.91). Then the averaged weight error mean for correct decision over^ can be obtain by

:

§ 4.4 Approximate Analysis for

^

-user Scenario

In prior two sections, we have derived the exact analytical results for the optimal weight, the weight error mean, and the weight error variance for the single-user case, and the optimal

weights, the weight error means for the two-user case. In this section, we will extend the results to accommodate the general
-user case. Due to the difficulty of the problem, we will seek approximate rather exact solutions. In most cases, we will only give the result for correct decision (denoted with superscript ‘
’) and omit the derivation of erroneous decision.

First note that the received despread signal of each user composes of three parts, i.e., the desired signal, the MAI, and noise. The key to reduce the analysis complexity is to consider each user individually and treat all other
^interfering users as an equivalent user. By doing so, we can transfer the general
-user case to a two-user case. In other words, we let

















$



$

 (4.119)

where ^$

 and^$











!. Here, ^$ represent the equivalent amplitude and ^ the equivalent correlation. Using this model, we can have equivalent interference second order statistics. Also note that^$ is virtual and we do not need its actual values in derivation. In the following analysis, we assume that the desired user is the first user. Thus, the matched filter output is then

















$



$





 (4.120)

Thus, we can keep the computational complexity comparable to the two-user case.

§ 4.4.1 Optimal Weight Analysis

We use two methods to approximate optimal weights. The first method directly uses the two-user model in (4.120). All we have to do is to let the amplitude of the second two-user be equal to ^$











!; optimal weights can be obtained readily. In what follows the similar derivation for optimal weights applies, which is termed as



for correct decision and^



for erroneous decision. The associated equations are listed below with the same noise integration ranges^ in Table 4.1. The conditional optimal weights given^and^ is represented as

"

































­





 (4.121)

where^
diag^



$

. The optimal weights for correct decision of the first user given^ is similar to (4.88) as expressed by

"

with^given in (4.86). The approximate optimal weight of correct decision analogous to (4.93) is obtained by

$ with^"# defined in (4.94). This method is referred to as the optimal weight approximation one (OWA1). Similar procedures for^$can be easily repeated.

The second method simplifies the result one step further. In the preceding optimal weight approximation, it is necessary to derive the optimal weights ^" according to different noise subspaces^. It can be seen that the optimal weights of the two-user case are coupled with each other. For this reason the optimal solution for the first user requires bit decision information pertaining to the second user, thus the long list of Table 4.1 results. If we can ignore some coupling relationship, the optimal weight can be calculated more easily. Here we ignore the decision coupling between two users. In other words, the first user decision is independent of the second user decision. In this case, the decision patterns are degenerated into two,^







and ^
. We denote these patterns as the fifth and the sixth pattern. For each decision pattern, we have two bit patterns, i.e.,^



$ and^


$. The noise space can be partitioned into two subsets accordingly. Thus, for^
, we have two sets as (^=1)

Hence the conditional optimal weight on^{ } for correct decision is obtained to be

"

Note that (4.125) only involves one-dimensional integration instead of two-dimensional integra-tion. Then the optimal weight for correct decision of the first user conditioned on^in (4.122) can be approximated as

" where the noise integration region and the corresponding occurrence probabilities are defined as

The optimal weight is then

$ where^"# is defined in (4.94). This optimal weight approximation method is referred to as the optimal weight approximation two (OWA2).

§ 4.4.2 Weight Error Mean Analysis

We also develop two methods for weight error mean approximation. The first method follows the same derivation of the mean weight error vector for the two-user case. The weight error mean for the
-user case can be obtained through the direct substitution of^$, and are referred to as^:


 and^:

 for correct and erroneous decision, respectively. The weight error mean vector given^and^ are represented from (4.114) as

"

 if the OWA2 is used. The conditional weight error mean given^ for correct decision is given by

"

where the union of noise subsets^ is^ in (4.86) for the OWA1 or^ in (4.127) for the OWA2.

Also^" is^"



for the OWA1 or^" for the OWA2. Finally, the averaged weight error mean is obtained as

:

We call this approximation as the weight error mean approximation 1 (WEMA1).

The second method further explores simpler approximation. Note that^":



 differs ac-cording to different^values. From (4.94), we can find that most of the correlation values fall in

 differs ac-cording to different^values. From (4.94), we can find that most of the correlation values fall in

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