As mentioned, the adaptive blind partial HPIC essentially performs system identification. As a consequence, if the training period is long enough (all weights converge), the mean value for theth weight will be
In the previous chapter, we have analyzed the adaptive two-stage partial HPIC receiver. The result reveals that the performance of the adapted weights are determined by several factors listed as follows.
Number of weights
Step size
Number of training data
Noise variance
Weight initials
Note that these factors may interact one another. Here, we will manipulate the first two factors, the weight numbers and the step size to obtain improved performance. We propose an algorithm that can reduce the number of adapted weight as well as its variance. At the same time, the step size can be increased to accelerate convergence. First, we will show that the MSE of the adaptive blind partial HPIC is proportional to the number of weights adapted in the LMS algorithm. Assume that the first user is the desired user. From (4.150), we have the output MSE as
Since " is usually no more than , the total MSE is dominated by the part with correct decision. Note that$
is usually close to the desired weight value even when is large.
Thus the MSE can be simplified to
can be neglected since optimal weights and initials are close to. The term; is a function of step size and noise variance only.
;
Thus the major term in the output MSE isÆ . We first represent this variance function given
as
Taking the expectation on (5.28), we have
For the correct decision scenario, the conditional optimal weights are close to the ideal values, i.e.,$"
. In that case, we can observe that the second moment in (5.29) is increased with
$ for . In summary, we know that the MSE of the adapted weights increases with $, and thus with. One way to improve the system performance is to reduce the weight number trained in the LMS algorithm. This is possible if we know the channel gains. We then propose a procedure to do that. If a user’s matched output magnitude exceeds a threshold
in the
th stage, the corresponding decided bit is deemed reliable and the weight corresponds to this
( )i
yk ( )i ( )
φs ⋅
( )i
k s
aξ
( )i k s
aξ
−
ak
( )i
k f
aξ
(a) (b)
( )i
µ
( )i ( ) wk N
( )i( ) φf ⋅
Figure 5.2: Functions used in the proposed algorithm. (a) Weight selection function. (b) Weight post filtering function.
bit is deactivated. In other words, this weight will not be included in the training process. This algorithm can be easily expressed using a two step-size scenario. Let the step size for User be
. Then,
if
7
if
(5.30)
The step-size decision function, denoted as @
, is shown in Figure 5.2(a). Note that there must be some users whose weights are erroneously decided. If this happens, it will increase the noise variance (in the computation of; ). The variance increased can be calculated as
$
(5.31)
Weight selection
Interference cancellation/Despreading Weight post filtering
LMS algorithm
Bit decision
( 1) ( 1)
1i , , Ki y− " y −
( ) ( )
1i( ), , Ki( )
w N "w N
( ) ( )
1i, , Ki y " y
1 i= +i
( )i( ) φs ⋅
( )i( ) φf ⋅
( ) ( )
1i( ), , Ki( ) w n "w n
( )
ˆ , ,1i
b " ˆ( )i bK
Figure 5.3: Flow chart for the proposed algorithm.
In (5.31),
is obtained by
(5.32)
We call this procedure as the weight selection procedure.
It is well known that the convergent weights in the LMS algorithm are random. Thus, if we know the weight distribution, we can perform weight post filtering (estimation). This will enhance the PIC performance furthermore. Figure 5.4 shows a typical probability function for the LMS convergent weight. It is clear that some of weights are greater than the channel gains and some weights are less than the channel gains. Note that given a binary random variable embedded in AWGN, the MMSE estimate corresponds to a transformation with a hyperbolic
tangent function. We can then apply the estimation theory here. To ease the derivation, we make a simplified alternative where a piece-wise linear decision function is used for weight post filtering; we denote this function as@
. It is shown in Figure 5.2(b) in which a threshold
is required. If a trained weight is greater than some threshold, it is decided to be. Note that no decision is made below. This is because the probability that the weights appear in the region is low and it has little impact in overall performance. We call this the weight post filtering procedure.
As mentioned, the weight distribution has different mean values for correct/erroneous de-cision (in the previous stage). The weight means for erroneous dede-cision bits will approach the corresponding optimal weights if the processing gain is large. However, in a practical sys-tem, is usually not large enough. Thus, we prefer to use a large step sizeto speed up the weight adaptation for users with erroneous decisions. However, a larger step size will enlarge the weight variance which adversely affect the final performance. The two procedures propose above can reduce the number of active weights and further filter the convergent weights. As a result, it is possible to use a larger step size without significantly increasing the weight variance.
By careful examination, we can find a good compromise among the parameters
(
) such that the weights are determined in an optimal way. The flow chart for the proposed algorithm is depicted in Figure 5.3.
§ 5.2.1 Gradient Guided Search Algorithm
It is well known that the HPIC was proposed based on the ML principle. The HPIC decides the desire user bit polarity with larger likelihood while estimate other user data bits from the previous stage. The procedure of likelihood maximization is performed simultaneously for all users. When the MAI is strong, the full HPIC output will not converge but oscillate in subsequent stages. The partial HPIC relieves the limit cycle phenomenon and finds a local maximum with likelihood higher than that of the full HPIC. There are many methods that can increase the likelihood. One method applied in the full HPIC is to flip parts of the user bits in
one stage and output the pattern giving the highest likelihood. [69],[70]. We call this method the gradient guided search (GGS) algorithm [71], whose procedure is outlined as follows.
(a) Let . Obtain the initial input bits. This is usually performed by the matched filter output as
sgn
(5.33) (b) Flip the user bit one by one and compute the log-likelihood functions #
using (2.3). The input bit sequence is then
#
(5.34)
(c) Choose one pattern whose log-likelihood function is the largest amonglikelihood func-tions. Note that this likelihood must be greater than that for the initial bit pattern.
#
(5.35) If no one log-likelihood function exceeds that of the original bit pattern, all user bits are keep unchanged and the algorithm terminates.
(b) Update the initial bit pattern with the new one and proceed to the next stage from (b) with stage number.
In this chapter, the GGS algorithm is utilized as a post processing algorithm to further improve the performance of the adaptive partial HPIC.