www.ietdl.org Published in IET Communications Received on 23rd July 2009 Revised on 12th March 2010 doi: 10.1049/iet-com.2009.0470
ISSN 1751-8628
Parallel optimisation of time-varying adaptive algorithms for interference cancellation in code division multiple access systems K. Shahtalebi1 G. Bakhshi2 H. Saligheh Rad3 1
IT Department, University of Isfahan, Isfahan, P.O. Box 81746-73441, Iran ECE Department, Yazd University, Yazad, P.O. Box 89195-741, Iran 3 Department of Radiology, University of Pennsylvania, Pennsylvania 19104-6021, USA E-mail:
[email protected] 2
Abstract: In this study, we propose a least mean square-partial parallel interference cancellation (LMS-PPIC) method named parallel LMS-PPIC (PLMS-PPIC) in which the normalised least mean square (NLMS) adaptive algorithm with optimised chip time-varying step-size is engaged to obtain the cancellation weights. The former LMS-PPIC method is based on fixed not optimised step-size, which causes propagation of error from one stage to the next one and increases the bit error rate (BER). The unit magnitude of the cancellation weights is the principal property in our step-size optimisation. To avoid computational complexity a small set of NLMS algorithms with different step-sizes are executed. In each iteration the parameter estimate of that NLMS algorithm which the elements magnitudes of its cancellation weight estimate have the best match with unit is chosen. Magnificent decrease in BER is achieved by executing the proposed method. Moreover PLMS-PPIC like former LMS-PPIC method comes to practice only when the channel phases are known. When they are unknown, having only their quarters in (0, 2p), we propose modified versions of LMS-PPIC and PLMS-PPIC to find the channel phases and the cancellation weights simultaneously. Simulation scenarios are given to compare the performance of our methods with that of LMS-PPIC in two cases: balanced channel and unbalanced channel. The results show that in both cases the proposed method outperforms LMS-PPIC, especially for high processing gains.
1
Introduction
The multiuser detectors for code division multiple access (CDMA) receivers are attractive techniques in eliminating the multiple access interference (MAI) and the near-far problem simultaneously. In CDMA systems all users receive the whole transmitted signals concurrently and they are recognised by their specific pseudo noise (PN) sequences. In such a system, there exists a limit for the number of users that can simultaneously communicate. This limitation comes from MAI (see e.g. [1, 2]). High quality detectors improve the capacity of the systems and therefore have received much significant attention [1 – 7]. Although optimum multiuser detectors can significantly eliminate the MAI and provide a substantial increase in IET Commun., 2010, Vol. 4, Iss. 16, pp. 1963 – 1973 doi: 10.1049/iet-com.2009.0470
system capacity, they suffer from this fact that their computational complexities grow exponentially with increasing the number of users and the length of the sequence [8]. Suboptimal solutions with reduced computational complexity are proposed that can mostly categorised as linear multiuser detectors and subtractive interference cancellers. Furthermore, some other suboptimum proposed methods are more effective in special applications (see e.g. [9 –12]). In linear multiuser detectors, a linear transformation is applied to the soft output of the conventional matched filters in order to produce a new set of decision variables (see e.g. [13]). In multiple stage subtractive interference cancellation the estimated interference from other users are removed from the specific user’s received signal before making data 1963
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www.ietdl.org decisions. The cancellation can be carried out either serially (successively) (see e.g. [14, 15]) or in parallel (see e.g. [2, 3, 6]). The parallel interference cancellation (PIC), a low computational complex method, causes less decision delay than the successive detection and also is much simpler in implementation. At the first stage of interference cancellation in a multiple stage system, for each user, interfering data of other users is unknown. PIC is implemented to estimate it stage by stage. In fact, the bit decisions at the (s 2 1)th stage of cancellation are used for bits detection at the sth stage, in the way that MAI is estimated for each user. Apparently, the more accurate the estimates, the more correct bit detection of the detector is. However, in the conventional multistage PIC [3], a wrong decision in one stage can increase the interference. Based on minimising the mean square error between the received signal and its estimate from the previous stage, Xue, et al. [6] proposed the least mean square-partial parallel interference cancellation (LMS-PPIC), an adaptive multistage PIC method. In LMS-PPIC a weighted value of MAI of the other users is subtracted before making the decision of a specific user. The least mean square (LMS) optimisation and the normalised least mean square (NLMS) algorithm [16] have made the structure of the LMS-PPIC method in cancellation weight estimation of each stage. However, the performance of the NLMS algorithm is mostly dependent on its step-size. Although a large step-size results in a faster convergence rate but it causes a large misadjustment. On the other hand, with a very small step-size the algorithm almost keeps its initial values and cannot estimate the true cancellation weights. In the LMS-PPIC method both of these cases cause propagation of error from one stage to the next one. In LMS-PPIC the mth element of the weight vector in each stage, is the true transmitted binary value of the mth user divided by its hard estimate value from the previous stage. Hence the magnitude of all weight elements in all stages is equal to one. This is a valuable information that can be used to improve the performance of the LMS-PPIC method. This is what shapes the structure of part of our paper. In fact, in this paper by using a set of NLMS algorithms with different step-sizes, we propose parallel LMS-PPIC (PLMS-PPIC) method, a modified LMS-PPIC method to keep this property, that is in each iteration, the parameter estimate of that NLMS algorithm which the element magnitudes of its cancellation weight estimate have the best match with unit, is chosen. This strategy is equal to have an NLMS algorithm with optimised step-size value in each iteration (i.e. chip time-varying step-size). Moreover in both LMS-PPIC and PLMS-PPIC methods, it is assumed that the receiver knows the phase of all user channels. However in practice it must estimate them. Having focus on this case we improve the LMSPPIC and PLMS-PPIC procedures in such a way that when there is only a partial information of the channel 1964 & The Institution of Engineering and Technology 2010
phases, both LMS-PPIC and PLMS-PPIC can estimate the phases and the cancellation weights simultaneously. The partial information is the quarter of each channel phase in (0, 2p). Accordingly, the paper is organised as follows. In Section 2 we review the LMS-PPIC. In Section 3 the PLMS-PPIC method is explained. In Section 4 modified versions of LMS-PPIC and PLMS-PPIC with capability of channel phase estimation are given. In Section 5 some simulation scenarios are given to compare the results of PLMS-PPIC and modified PLMS-PPIC with those of LMS-PPIC and modified LMS-PPIC. Finally the paper concluded in Section 6.
2 Multistage PIC-LMS-PPIC method Assume M users synchronously send their symbols a1 , a2 , . . . , aM , where am [ {−1, 1}, via a base band CDMA transmission system. The mth user has its own code pm (.) of length N, where pm (n) [ {−1, 1}, for all n. At receiver we assume that perfect power control scheme is applied. Without loss of generality we also assume the power gains of all channels are equal to one, user channels do not change during each symbol transmission, and the channel phase fm of mth user is known for all m = 1, 2, . . . , M (Unknown channel phase will be explored in Section 4). Define cm (n) = e j fm pm (n)
(1)
According to the above assumptions the received signal is r(n) =
M
am cm (n) + v(n),
n = 1, 2, . . . , N
(2)
m=1
where v(n) is the additive white Gaussian noise with zero mean and variance s2 . Multistage PIC methods use a(s−1) , a(s−1) , . . . , a(s−1) M , the bit estimates outputs of the 1 2 previous stage, s 2 1, to estimate the related MAI of each user, subtract it from the received signal r(n) and make a new decision on each user variable individually to make a (s) (s) new variable set a(s) 1 , a2 , . . . , aM for the current stage s. In the following, we explain the structure of the LMS-PIC method. [ {−1, 1} is a given estimate of am from Assume a(s−1) m stage s 2 1. Define w(s) m =
am a(s−1) m
(3)
From (2) and (3) we have r(n) =
M
(s−1) w(s) cm (n) + v(n) m am
(4)
m=1
IET Commun., 2010, Vol. 4, Iss. 16, pp. 1963 – 1973 doi: 10.1049/iet-com.2009.0470
www.ietdl.org Define (s) (s) T W (s) = [w(s) 1 , w2 , . . . , wM ]
(5)
(where T stands for transposition) and T c1 (n), a(s−1) c2 (n), . . . , a(s−1) X (s) (n) = [a(s−1) M cM (n)] 1 2
(6)
From (4) –(6) we have T
r(n) = W (s) X (s) (n) + v(n)
(7)
Given the observations {r(n), X (s) (n)} for n ¼ 1 to n ¼ N, in LMS-PPIC the NLMS algorithm is used to iteratively compute T (s) (s) W (s) (N ) = [w(s) 1 (N ), w2 (N ), . . . , wM (N )]
(8)
an estimate of W (s) after N iteration. Then a(s) m , the estimate of am at stage s is given from
a(s) m = sign real
N
qm(s) (n)cm∗ (n)
(9)
n=1
in which
∗
Figure 2 Procedure of the LMS-PPIC method
stands for complex conjugation and
qm(s) (n) = r(n) −
M m′ =1,m′ =m
(s−1) w(s) m′ (N )am′ cm′ (n)
M {a(0) m }m=1 is given by the conventional bit detection
(10)
Fig. 1 is the flowchart and Fig. 2 shows the full structure of the LMS-PPIC method. The inputs of the pre stage
a(0) m
= sign real
N
r(n)cm∗ (n)
,
m = 1, 2, . . . , M
n=1
(11) To improve the robustness of the algorithm (having no bias to 1 or 21), in all stages, the weights are initialised to zero. As the number of stage is increased the system delay in symbol estimation is increased. According to our simulations the LMS-PPIC obtains its final performance in Stage 2 and there is no need for Stage 3 or more. Using a fixed step-size causes the LMS-PPIC has no preference over the conventional method in some cases (see our simulations). To improve the performance of the LMS-PPIC method in the next section we propose a parallel version of it.
3 Multistage PIC-PLMS-PPIC method Figure 1 Flowchart of the LMS-PPIC method IET Commun., 2010, Vol. 4, Iss. 16, pp. 1963 – 1973 doi: 10.1049/iet-com.2009.0470
The NLMS (with fixed step-size), converges only in mean. In the literature, m [ (0, 2) guarantees the mean convergence of the NLMS algorithm (see [16, 17]). Based 1965
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www.ietdl.org on Crame´r – Rao bound, a sharper range was given in [18]
M −1 m [ C = 0, 1 − M
(12)
where here M is the number of users. As we explain in the proposed PLMS-PPIC method C has a critical role. From (3) we have |w(s) m| = 1
m = 1, 2, . . . , M
(13)
which is equivalent to M
||w(s) m | − 1| = 0
(14)
m=1
To improve the performance of the NLMS algorithm at time iteration n we can determine the chip time-varying step-size M (s) m(n) from C in such a way that m=1 ||wm (n)| − 1| is minimised, that is
m(n) = arg min
m[C
M
||w(s) m (n)|
Figure 3 Flowchart of the PLMS-PPIC method
− 1|
(15)
m=1
(s) where, w(s) m (n), the mth element of W (n) is given by (s) w(s) m (n) = wm (n − 1) + m(n)
a(s−1) cm∗ (n) e(n) X (s) (n)2
(16)
The complexity of determining m(n) from (15) with the restriction (16) is high. Instead of, we divide C into L subintervals and consider L individual step-sizes
Q = {m1 , m2 , . . . , mL }, where m1 = (1 − (M − 1)/M )/L, m2 = 2m1 , . . . , and mL = Lm1 . In each stage L individual NLMS algorithms are executed (ml is the step-size of the lth algorithm.). In stage s, and at iteration n, if (s) T Wk(s) (n) = [w(s) 1,k , . . . , wM,k ] the parameter estimate of the kth algorithm minimised our criteria, that is Wk(s) (n)
= arg
min
Wl(s) (n)[IW (s)
M
||w(s) m,l (n)| −
1|
(17)
m=1
where Wl(s) (n) = W (s) (n − 1) + ml (X (s) (n)/X (s) (n)2 )e(n), l = 1, 2, . . . , k, . . . , L − 1, L and IW (s) = {W1(s) (n), . . . , WL(s) (n)}, then it is considered as the parameter estimate at time iteration n, that is W (s) (n) = Wk(s) (n) and all other algorithms replace their weight estimate by Wk(s) (n). Fig. 3 is the flowchart and Fig. 4 shows the details of the PLMSPPIC method. As Fig. 4 shows, in stage s and at time iteration N where W (s) (N ) is computed, like LMS-PPIC, the PLMS-PPIC method computes a(s) m from (9). In each iteration of each stage of PLMS-PPIC, we must Compute ml Z(n) = ml (X (s) (n)/X (s) (n)2 ), for l = 1, 2, . . . , L and M (s) m=1 ||wm,l (n)| − 1|. It causes the number of multiplications and summations in each iteration of the NLMS algorithm in 1966 & The Institution of Engineering and Technology 2010
Figure 4 Procedure of the PLMS-PPIC method IET Commun., 2010, Vol. 4, Iss. 16, pp. 1963 – 1973 doi: 10.1049/iet-com.2009.0470
www.ietdl.org PLMS-PPIC to be L times of them in LMS-PPIC. The number of summations and multiplications in NLMS algorithm in LMS-PPIC are almost equal to M and therefore in our proposed method they are almost equal to LM. Although it means that our proposed method is L times more complex than LMS-PPIC method, the proposed method like LMS-PPIC has polynomial order of complexity which is much lower than exponential complexity of optimum solutions. On the other hand, simultaneously running L individual NLMS algorithms in PLMS-PPIC method causes both LMS-PPIC and PLMS-PPIC methods have the same time response. It is also to be notified that in practice a small L not greater than 10 is sufficient and therefore implementation of the proposed method is practical. Because the step-sizes of all individual NLMS algorithms of the proposed method are given from a stable operation range, fast or slowly, all of them converge. Hence the PLMS-PPIC is a stable method (see [17]). As we expected from interval of (12) and based on our simulations, choosing the step-size as a decreasing function of system loads M, improves the performance of both NLMS algorithm in LMS-PPIC and parallel NLMS algorithms in PLMS-PPIC methods in such a way that there is no need for the third stage (i.e. both the LMS-PPIC and PLMS-PPIC methods obtain the optimum weights in the second stage.). Although increasing L, the number of parallel NLMS algorithms in PLMS-PPIC method, increases the complexity, it improves the performance, as well. From our simulations we found out that the LMS-PPIC method is more sensitive to the channel loss or unbalanced channel gain than the PLMS-PPIC method.
T X (s) (n) = [a(s−1) p1 (n), a(s−1) p2 (n), . . . , a(s−1) M pM (n)] 1 2
(s) (s) T With W (s) = [w(s) 1 , w2 , . . . , wM ] and from new definitions (s) (s) of wm and X (n), while (4) is changed to
r(n) =
In exploiting the conventional, the LMS-PPIC and the PLMS-PPIC methods it is assumed that the channel phase fm is known for all m = 1, 2, . . . , M. In practice, these procedures must use its estimated value. When the receiver has only succeeded to find the quarter of fm in (0, 2p), the challenge remains for LMS-PPIC and PLMSPPIC methods to estimate the exact value of it. Based on this partial information, modified versions of LMS-PPIC and PLMS-PPIC are given in this section. The modified procedures are capable of estimating the channel phases and the cancellation weights simultaneously. Consider R ¼ (0, 2p) and divide it into four parts R1 = (0, p/2), R2 = (p/2, p), R3 = (p, 3p/2), and R4 = (3p/2, 2p), the partial information is that each fm (m = 1, 2, . . . , M) belongs to which of the four quarters Ri , i ¼ 1, 2, 3, 4. In this case, we change our definitions of W (s) and X (s) (n) as follows w(s) m =
am a(s−1) m
ejfm
IET Commun., 2010, Vol. 4, Iss. 16, pp. 1963 – 1973 doi: 10.1049/iet-com.2009.0470
(18)
M
(s−1) w(s) pm (n) + v(n) m am
(20)
m=1
equation (7) remains unchanged. The LMS-PPIC and the PLMS-PPIC methods are restructured based on this new model. T (s) (s) is Assume W (s) (N ) = [w(s) 1 (N ), w2 (N ), . . . , wM (N )] the weight estimate of the algorithm (either NLMS in LMS-PPIC or PNLMS in PLMS-PPIC) at time instant N in stage s. From (18) we will have
a(s−1) fm = / m w(s) am m
(21)
(s) We estimate fm by fˆ m , where
(s) fˆ m
a(s−1) = / m w(s) (N ) am m
(22)
/am = 1 or 21 we will have Because a(s−1) m
(s) fˆ m =
4 Modified LMS-PPIC and PLMSPPIC methods
(19)
⎧ ⎪ (s) ⎪ ⎪ ⎨ /wm (N ), ⎪ ⎪ ⎪ ⎩ +p + /w(s) m (N ),
a(s−1) m =1 am a(s−1) if m = −1 am if
(23)
(s) (s) (s) Hence fˆ m [ P (s) = {/w(s) m (N ), /wm (N ) + p, /wm (N )− (s) p}. If wm (N ) has sufficiently converged to its true value ˆ (s) w(s) m , the same region for fm and fm is expected. In this case one of the three members of P (s) has the same region (s) as fm . For example if fm [ (0, p/2), then fˆ m [ (0, p/2) (s) (s) and therefore /w(s) m (N ) or /wm (N ) + p or /wm (N ) − p (s) belongs to (0, p/2). If, for example /wm (N ) + p is such a member, between all three members of P (s) , it is the best candidate for phase estimation, that is (s) fm ≃ fˆ m = /w(s) m (N ) + p
We admit that when there is a member of P (s) in the quarter of fm , then w(s) m (N ) has converged. What would happen when non of the members of P (s) has the same quarter as fm ? This situation will happen when the absolute difference between /w(s) m (N ) and fm is greater than p. It means that w(s) (N ) has not converged yet. In this case, the m expected value is the optimum choice for channel phase estimation, that is if fm [ (0, p/2) then p/4, if fm [ (p/2, p) then 3p/4 and so on is the estimation of 1967
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Figure 5 Flowchart of the modified LMS-PPIC method
Figure 6 Flowchart of the modified PLMS-PPIC method
the channel phase fm . The results of the above discussion is summarised in (24) (see (24))
Figs. 5 and 6 are the flowcharts and Figs. 7 and 8 show the structures of the modified LMS-PPIC and modified PLMS-PPIC, respectively. It is to be notified that modified LMS-PPIC and PLMS-PPIC have the same computational complexity as LMS-PPIC and PLMSPPIC respectively, because in modified forms of them, channel phase estimation is done only at the end of each stage. Like in LMS-PPIC and PLMS-PPIC, in modified versions of them, as the processing gain, N, is increased the BER is decreased. However, it increases the system delay.
The rest of the modified LMS-PPIC and modified PLMSˆ (s) PPIC is given by (9) in which cm (n) is replaced by ejf m pm (n), that is N (s) (s) (s) −jfˆ m am = sign real qm (n)e pm (n) (25) n=1
and (10) in which cm (n) is replaced by pm (n), that is qm(s) (n) = r(n) −
M m′ =1,m′ =m
(s−1) w(s) m′ (N )am′ pm′ (n)
(26)
In the following section, some scenarios are given to illustrate the usefulness of our proposed methods.
M The inputs of the first stage {a(0) m }m=1 is given by
a(0) m
= sign real
N
ˆ (0) −jf m
r(n)e
pm (n)
(27)
n=1
where assuming fm [ Ri , then (i − 1)p + ip (0) fˆ m = 4
(s) fˆ m =
(28)
5
In this section, we have considered some scenarios. In Scenario 1, we compare NLMS and PNLMS procedures in a simple autoregressive model. Scenarios 2 and 3 compare the LMS-PPIC and PLMS-PPIC methods in two cases: balanced channels and unbalanced channels. In these cases the channel phases are known. Both of them
⎧ /w(s) ⎪ m (N ), ⎪ ⎪ ⎪ (s) ⎪ ⎨ /wm (N ) + p,
if /w(s) m (N ), fm [ Ri ,
/w(s) ⎪ m (N ) ⎪
if
− p, ⎪ ⎪ (i − 1) p + ip ⎪ ⎩ , 4
if
/w(s) m (N ) (s) /wm (N )
Simulations
i = 1, 2, 3, 4
+ p, fm [ Ri ,
i = 1, 2, 3, 4
− p, fm [ Ri ,
i = 1, 2, 3, 4
(s) if fm [ Ri , /w(s) m (N ), /wm (N ) + p Ri ,
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(24) i = 1, 2, 3, 4
IET Commun., 2010, Vol. 4, Iss. 16, pp. 1963 – 1973 doi: 10.1049/iet-com.2009.0470
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Figure 7 Procedure of the modified LMS-PPIC method are repeated in Scenarios 4 and 5, where the receiver has only the quarter of each channel phase. Scenario 1: A simple autoregressive model. In this scenario, we compare the performance of the NLMS and parallel NLMS algorithms in parameter estimation of an autoregressive model, when the magnitude of the parameter weight elements are equal to one. Consider a simple model described by r(n) = W T X (n) + v(n), where the elements of the 10 × 1 input vector X(n) are complex√ numbers, taking their real and imaginary parts from + 0.5. The zero mean noise v(n) is a complex white additive noise with variance s2v . We assumed wm = e2jmp/10 , where wm is the mth element of W. In our simulation, the traditional NLMS algorithm with m ¼ 0.03 and PNLMS with Q ¼ {0.0125, 0.025, 0.0375, 0.05}, were executed. Fig. 9 compares the average of the error squared of parameter estimate W − Wn 2 , over 10 runs of both procedures for s2v = 0.01 and s2v = 1. As it shows, PNLMS outperforms the NLMS algorithm in both cases. Scenario 2 is given to compare LMS-PPIC and PLMSPPIC in the case of balanced channels. Scenario 2: Balanced channels, known phases. Consider the system model (4) in which M users, each having their own IET Commun., 2010, Vol. 4, Iss. 16, pp. 1963 – 1973 doi: 10.1049/iet-com.2009.0470
Figure 8 Procedure of the modified PLMS-PPIC method codes of length N, send their own bits synchronously to the receiver through their channels. The signal-to-noise ratio (SNR) is 0 dB. By assumption in this case there is no power-unbalanced or channel loss. The step-size of the NLMS algorithm in LMS-PPIC method was
m = 0.1(1 − (M − 1)/M ) and the set of step-sizes of the parallel NLMS algorithms in PLMS-PPIC method
was Q = {0.01, 0.05, 0.1, 0.2,√ . . . , 1}(1 − (M − 1)/M ), that m1 = 0.01(1 − (M −
1)/M ), . . . , m 4 = 0.2(1 − is
(M − 1)/M), . . . , m12 = (1 − (M − 1)/M ). Fig. 10 shows the average of bit error rates (BER) over all users against M, using only two stages, when N ¼ 64 (Fig. 10a) and N ¼ 256 (Fig. 10b). As it is shown, while there is no remarkable performance difference between all three methods for N ¼ 64, the PLMS-PPIC outperforms the conventional and the LMS-PPIC methods for N ¼ 256. This example was repeated, when LMS-PPIC and PLMSPPIC completed their bit estimations in three stages. The results shown there is no remarkable difference in this case with that of two stages. 1969
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Figure 9 Comparison of the weight square error of the NLMS and the proposed parallel NLMS algorithm a s2v ¼ 0.01 b sv2 ¼ 1
Figure 10 BER of the conventional, the LMS-PPIC, and the PLMS-PPIC methods against the system load in balanced channel, using two stages a N ¼ 64 b N ¼ 256
Although LMS-PPIC and PLMS-PPIC are structured based on the assumption of no near-far problem, as Scenario 3 shows, these methods and especially the second one have remarkable performance in the case of unbalanced channels. Scenario 3: Unbalanced channels, known phases. Consider Scenario 2 where now there is power unbalanced and/or channel loss in transmission system, that is the true model at stage s is r(n) =
M
(s−1) bm w(s) cm (n) + v(n) m am
(29)
m=1
where 0 , bm ≤ 1 for all 1 ≤ m ≤ M. Both the LMS-PPIC and the PLMS-PPIC methods assume the model (4), and 1970 & The Institution of Engineering and Technology 2010
their estimations are based on observations {r(n), X (s) (n)}, instead of {r(n), GX (s) (n)}, where the channel gain matrix is G = diag(b1 , b2 , . . . , bM ). In this case we repeated Scenario 2. Each element of G is a uniform random variable from the interval [0, 0.3]. The results are given in Fig. 11. As it is shown, in all cases the PLMS-PPIC method outperforms both the conventional and the LMS-PPIC methods. In the above cases, we assumed that the channel phase fm is known for all m = 1, . . . , M. We repeated them, when the channel phases are unknown and we only know their quarters in (0, 2p). Scenario 4: Balanced channels, unknown phases. Consider Scenario 2 where now the channel phases (except their IET Commun., 2010, Vol. 4, Iss. 16, pp. 1963 – 1973 doi: 10.1049/iet-com.2009.0470
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Figure 11 BER of the conventional, the LMS-PPIC and the PLMS-PPIC methods against the system load in unbalanced channel, using two stages a N ¼ 64 b N ¼ 256
Figure 12 BER of the conventional, the modified LMS-PPIC and the modified PLMS-PPIC methods against the system load in balanced channel, using two stages a N ¼ 64 b N ¼ 256
quarters) are unknown. Because of phase ambiguity the modified versions of the LMS-PPIC and the PLMS-PPIC methods (explained in Section 4) and the conventional method (27) and (28) were executed. Fig. 12 illustrates the BER for the cases of two stages. Table 1, compares the average channel phase estimate of the first user in each stage, over 10 runs of modified LMS-PPIC and PLMSPPIC, when the the number of user is M ¼ 15. Scenario 5: Unbalanced channels, unknown phases. Consider again Scenario 3, where now we assume the quarters of the channel phases are unknown. In this case, the modified versions of LMS-PPIC and PLMS-PPIC were executed. IET Commun., 2010, Vol. 4, Iss. 16, pp. 1963 – 1973 doi: 10.1049/iet-com.2009.0470
Fig. 13 illustrates the BER against the number of users. Table 2 compares the channel phase estimate of the first Table 1 Channel phase estimate of the first user (Scenario 4) N (Iteration)
fm =
3p , 8
NLMS
PNLMS
64
3.24p ˆ (s) 3.18p (s) fˆ m = fm = 8 8
256
2.85p ˆ (s) 2.88p (s) fˆ m = fm = 8 8
M ¼ 15
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Figure 13 BER of the conventional, the modified LMS-PPIC and the modified PLMS-PPIC methods against the system load in unbalanced channel, using two stages a N ¼ 64 b N ¼ 256
Table 2 Channel phase estimate of the first user (Scenario 5) N (Iteration)
fm =
3p , 8
NLMS
PNLMS
64
2.45p (s) fˆ m = 8
2.36p (s) fˆ m = 8
256
3.09p (s) fˆ m = 8
2.86p (s) fˆ m = 8
M ¼ 15
user in each stage, over 10 runs of modified LMS-PPIC and modified PLMS-PPIC for M ¼ 15.
6
Conclusion
Finding the optimum step-size value in each iteration of the NLMS algorithm engaged in partial interference cancellation in CDMA systems is so complex that usually a fixed value is replaced by it. However, it increases the overall BER of the system in such a way that there is no remarkable performance between LMS-PPIC and conventional methods in some cases. In this paper and based on the fact that the magnitude of the cancellation weights are equal to one, we proposed a partial interference cancellation method in which a set of NLMS algorithms with different step-sizes is engaged. In the proposed method in each iteration, the parameter estimate of that algorithm, which has the best compatibility with the true parameter is chosen. It is equal to have an NLMS algorithm with optimised step-size in each iteration. While the computational complexity of the proposed method was L times of the LMS-PPIC method, our parallel approach in executing L algorithms simultaneously caused no more 1972 & The Institution of Engineering and Technology 2010
delay in parameter estimation. As our simulations shown simultaneously executing L algorithms was the cost we paid to decrease the BER for even more than 20 dB in some cases. Our simulations shown that specially for low system load, high processing gain and/or unbalanced channels the proposed method considerably outperforms the LMSPPIC method. However, in other cases also the proposed method had a better performance. In addition in natural situation the receiver does not know the channel phase. However, when the receiver has only the quarter of the channel phase of each user, we improved the LMS-PPIC and the proposed methods to estimate the cancellation weights and the channel phases simultaneously. Simulation results in this case shown that the new method has succeeded to estimate the channel phase and users’ symbols simultaneously.
7
Acknowledgment
This work has been presented in part in IST2008 and WCNC2008.
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References
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