Simplified Local Search Algorithm for Multiuser Detection in Multipath Rayleigh Channels Leonardo D. de Oliveira and Taufik Abr˜ao DEEL-UEL State University of Londrina, Londrina, PR, 86051-990, Brazil Email:
[email protected],
[email protected]
Paul Jean E. Jeszensky Department of Telecommunications and Control Engineering Escola Polit´ecnica of University of S˜ao Paulo (LCS-PTC-EPUSP) S˜ao Paulo, 05508-900, Brazil Email:
[email protected]
Abstract—This work intends to show that the simple 1optimum local search (LS) technique is enough to solve the multiuser detection problem with blue an excellent performancecomplexity trade-off. Based on the deterministic behavior of the LS algorithm and some simplifications in the cost function calculation, a more efficient algorithm can be obtained and also a new perspective about the multiuser detector implementation. The performance and complexity analysis are evaluated considering a DS/CDMA system under multipath fading channels.
I. I NTRODUCTION The capacity of a DS/CDMA system in a multipath channel is mainly limited by the multiple access interference (MAI), self-interference (SI), near-far effect, fading and Additive White Gaussian Noise (AWGN). Rake receivers explore the path diversity in order to reduce the AWGN effect [1], but it is not able to mitigate the MAI [2]. An alternative to solve this limitation is to apply the multiuser detection (M U D). In this case the best performance is obtained using the logarithmic likelihood function (LLF) known as optimum multiuser detection (OM U D) [3], [4]. However this best performance is only achieved with high computational complexity, which increases exponentially with the number of users. Other multiuser detectors with lower complexity and suboptimum performance were proposed, such as linear detectors [2], [4], subtractive interference canceling [2] and also heuristic methods based on LLF [5], [6], [7], [8], [9], [10]. This work applies the 1-optimum local search technique [11] for the M U D problem (1-opt-LS-M U D/LS-M U D) and presents some simplifications in order to reduce the number of operations and to reach an attractive performance-complexity trade-off. II. S YSTEM M ODEL Considering a DS/CDMA system, shared by K users, the base band transmitted signal of the k-th user, with I BPSK modulated bits, is given by [1]: I sk (t) = Ek bk (i)gk (t − iT )
(1)
i=1
where Ek is the bit energy, bk (i) ∈ {±1} is the i-th transmitted information bit and gk (t − iT ) is the spreading sequence for
the k-th user, given by: gk (t) =
N −1
ak (n)p(t − nTc ),
0≤t≤T
(2)
n=0
where T is the bit period, ak (n) is a pseudo-noise sequence with N chips assuming the values {±1}, p(t) is the pulse shaping (assumed rectangular with unitary amplitude) with duration Tc , where Tc is the chip interval. The processing gain is N = T /Tc . Considering a normalized large scale fading, the received signal containing I transmitted bits by each user in multipath fading channel is given by: rk (t) =
I L K
Ek bk (i)gk (t − iT − τk, )ζk, (i) + η(t)
i=1 k=1 =1
(3) where L is the number of paths fading, admitted equal for all K asynchronous users, τk, (i) is the total delay for the signal of the k-th user, -th path and i-th bit, considering the asynchronism among the users and random delays for different paths; η(t) is the AWGN with bilateral power spectral density equal to N0 /2 and ζk, is the complex channel coefficient for the k-th user and -th path, defined as: ζk, (i) = γk, (i)ejφk, (i)
(4)
where the module γk, (i) is a random variable (RV) characterized by a Rayleigh distribution and the phase φk, (i) is a RV modeled through U(0, 2π). A slow (channel coefficients is constant along T ) and frequency selective ( T1c >> (∆B)c , the coherence bandwidth of the channel) channel was assumed. The Rake receiver consists in a KD matched filter bank, with D ≤ L, where D is the number of paths supported by the receiver. The signal for the k-th user, -th path and bit i, after despreading, is given by: yk, (i) = Ek ζk, (i)bk (i) + SIk, (i) + Ik, (i) + ηk, (i) (5) where the first term corresponds to the desirable signal, the second to the self-interference, the third to the multiple access interference and the last to the filtered AWGN.
Considering a MRC (Maximum Rate Combining) for D paths, the i-th detected bit for the k-th user is given by: D ∗ ˆbk (i) = sign ˆ Re yk, (i)ζ (i) (6) k,
=1
where, ζˆk, (i) is the channel coefficient estimate for the k-th user, i-th bit and -th path. III. O PTIMUM D ETECTION Introduced by S. Verdu [3], the OM U D is based on the LLF in order to eliminate the MAI. In this work the oneshot asynchronous channel [4] approach is adopted, where an asynchronous scenario with K users, I bits and D paths is assumed to be a synchronous scenario with KDI users. The maximum likelihood equation for the the joint estimation of the I transmitted bits of the K users is given by [3]: H T H T ˆ B = arg max 2Re{y C AB} − B CARAC B B∈{±1}K
(7) where y is a column vector, with dimension KDI, containing the matched filter outputs for the I bits; C and A are diagonal matrices, also with dimension KDI, of the channel coefficients and amplitudes, respectively; and the cross-correlation block-tridiagonal, block-Toeplitz matrix R is composed by the R[1] and R[0] sub-matrices, with the same dimension
T and the Hermitian operator is defined ∗[4] H . The candidate vectors B are defined by (·) = (·) by B = [b1 (1) b2 (2) . . . bK (1) b1 (2) . . . bK (I)]T , where bk (i) = [bk (i) bk (i) . . . bk (i)]TD , with bk (i) ∈ {+1, −1}. The OM U D difficulty is its high computational complexity which is proportional to O(2KDI ). Heuristic approaches can be used in order to deal with this problem. In the next section the Local Search algorithm used in this work is described.
3) Evaluate the cost function for each new generated X : F (X ) = 2Re{yT CH AX } − X T CARACH X
4) X best is updated by the vector with higher cost function. Return to step 2) until there is no improvement in the search or until that a predetermined number of iterations G is reached. 5) The output vector is X best . V. C OST F UNCTION S IMPLIFICATION For the M U D problem, the complexity depends mainly on the cost function calculation. Simplifications, obtained through the deterministic characteristics of the LS algorithm, are efficient and can be used in order to reduce its complexity [7]. A. Scenario Description In the cost function calculation, (9), two terms can be evaluated outside the iterations loop: F (1) = yT CH A
X = [x1 (1) x2 (2) . . . xK (1) x1 (2) . . . xK (I)]T
and
F (2) = CARACH
So the cost function calculation inside the iterations loop is reduced to: (10) F(X ) = Re F (1) X − X T F (2) X One important characteristic is that the X vector which maximizes eq. (10) must have the same bit for all paths received from a unique transmitted signal. 1) First Term: The first term is a vectors multiplication, and one possible simplification can be done as follows: F (1) X ≡ Fr(1) Xr
IV. 1-O PTIMUM L OCAL S EARCH A LGORITHM Comparisons among heuristic techniques performed for the M U D problem were described in [9] [10] and show that they are able to achieve performances close to the OM U D with smaller complexity. In this paper the 1-optimum local search algorithm is adopted in order to explore its deterministic characteristics in this minimization and to conclude that it is the most efficient approach even in multipath channels. Consider a candidate-vector with all bits of all users defined as: (8)
where xk (i) = [xk (i) xk (i) . . . xk (i)]TD . The steps of the LS algorithm are described below: 1) The Rake output is taken as the initial and best solution, named as X best . 2) KI candidate-vectors are generated with unitary Hamming distance1 to X best . 1 A Hamming distance equal to 1 means the change of the signal of only one bit of a selected xk (i).
(9)
(11)
(1)
where Fr = [ϑ1 (1) ϑ2 (1) . . . ϑK (1) ϑ1 (2) . . . ϑK (I)] is a vector, with dimension KI, with elements ϑk (i) defined by the sum of the D elements (paths) of F (1) related to the i-th transmitted signal of the k-th user; and Xr = [x1 (1) x2 (1) . . . xK (1) x1 (2) . . . xK (I)] is composed by one bit of each transmitted signal. The sum is carried out outside the iterations loop; inside, only KI multiplications are needed instead of KDI. 2) Second Term: The simplification for the second term considers the same characteristic of X , which in this term occurs twice, so: X F (2) X ≡ Xr Fr(2) Xr (2)
(12)
where Fr is a squared matrix, with dimension KI, obtained from F (2) , where each element ψκ,k (j, i) is the sum of the sub-matrix of F (2) representing the cross-correlation between the κ-th and k-th users, in their j and i bits, respectively, for all of their paths.
B. LS Algorithm
Fr(1) Xr = Fr(1) Xrbest + 2 · Re {ϑk (i) · xk (i)}
Rake All users s−LS LS SuB (BPSK) BERAvg
Considering that the creation of a new group of candidatevectors is deterministic, it is possible to use the previous cost function calculation for Xrbest for the cost function calculation of all new generated Xr . As the difference is only 1 bit (Hamming distance equal to 1), it is sufficient to update the terms that have changed this bit. The first term of the cost function calculation, considering a vector with neighborhood changed in the i-th bit of the k-th user, is simplified as:
−2
10
(13)
Multiplying the elements of the second term of (10) and applying some algebraic properties, the incremental gain for the change of the i-bit of the k-th user can be simplified as: = Xrbest Fr(2) Xrbest + 4 · Re xk (i) · XrT Ψk (i) − ψk,k (i, i) (14)
VI. R ESULTS A. Simulations Monte-Carlo Simulations (MCS) were carried out in order to verify the performance of the simplified LS algorithm (sLS) for the M U D problem in multipath fading channel. The adopted parameters in the simulation of Figure 1 are: K = 10, N = 31, Eb /N0 = 15dB, D = L = 2 (paths with an exponential profile) and I = 7. Considering all delays, amplitudes and channel coefficients known at the receiver, Figure 1 shows the convergence curve for the s-LS algorithm. For comparison purpose the performance of the conventional Rake and the Single-user Bound (SuB) are also included. From Figure 1 it can be verified that, for the simulated conditions, a number of iterations G = 10 it is enough for the LS and the s-LS algorithms reach the convergence. The s-LS performance was also analyzed considering different Eb /N0 and errors in the channel estimates. Errors were introduced jointly being modeled through uniform distributions: γ k = U (1 ± γ ) × γk ; θk = U (1 ± θ ) × θk where γ and θ are the maximum module and phase channel coefficients errors, respectively. Figure 2 shows the BER of the s-LS algorithm, after convergence, as a function of Eb /N0
Fig. 1.
10
15 Iterations
20
25
30
Convergence curve for the s-LS algorithm.
for K = 10 users, with (10% and 25%) and without errors in the channel coefficients estimates.
−1
10
−2
10 Avg
(2)
where Ψk (i) is the row obtained from Fr , related to the k-th user and i-th bit, defined by Ψk (i) = [ψ1,k (1, i) ψ2,k (1, i) . . . ψK,k (1, i) ψ1,k (2, i) . . . ψK,k (I, i)]. Therefore, the calculation of the cost function for the candidate vector X with i-th bit of k-th user is simplified to:
F (Xr ) = F Xrbest + 2 · xk (i) · Re {ϑk (i) − 2 · Ψk (i)Xr0 } (15) where Xr0 is the Xr vector with element xk (i) null. Equation (15) is equivalent to equation (9) but with lower complexity enhancing the LS algorithm advantage against other heuristic techniques.
5
BER
Xr Fr(2) Xr
0
Rake All Users −3
10
s−LS − εγ = 0; εφ = 0; s−LS − ε = 0.1; ε = 0.1; γ
φ
s−LS − ε = 0.25; ε = 0.25; γ
−4
10
0
φ
SuB (BPSK) 5
10 E /N [dB] b
15
20
0
Fig. 2. Performance of s-LS with errors in the channel coefficients estimates.
Without errors the performance is close to the SuB case. With errors the degradation is due to the LLF calculation, and even with joint errors in the module and phase up to 25%, there is a performance improvement compared to Rake with perfect estimation. B. Complexity The s-LS detector has a very reduced complexity when compared to the OM U D and also to other heuristic M U D. Table I exhibits the complexity for the OM U D, LS and s-LS detectors, where g is a particular number of iterations G until the convergence. The relevant operations are: multiplications, comparisons and transpositions. Figure 3 enhances the difference in the number of operations between the OM U D and the LS-M U D algorithm, and also the advantage of the s-LS algorithm over the LS, considering
TABLE I C OMPUTATIONAL COMPLEXITY FOR THE M U D. MUD
Number of operations 2KI (KID)2 + 3KID
OM U D
(g + 1)KI[(KID)2 + 3KID]+ +gKI + 2(KID)2 + 6KID + 2gKI g(7KI + 1) + 5(KI)2 + 11KI
LS s-LS
the number of operations for convergence. The number of iterations for the convergence was gs-ls = gls = [6 10 14 23 28] for the following channel and system conditions: K = [5 10 15 20 25] users; I = 7 bits L = D = 2 paths and fingers.
OMuD LS s−LS Cost Function
50
10
40
Operations
10
30
20
10
10
10
40
60
80
100 120 Users⋅ Bits [K⋅ I]
140
160
100 120 Users⋅ Bits [K⋅ I]
140
160
9
10
LS s−LS Cost Function
8
10
7
10 Operations
MUD OM U D LS s-LS F (X )
KI = 70 8.26 × 1023 1.55 × 107 3.02 × 104 1.54 × 104
KI = 140 1.95 × 1045 2.55 × 108 1.21 × 105 6.02 × 104
VII. C ONCLUSIONS The 1-optimum local search algorithm shows to be efficient for the M U D problem. The s-LS multiuser detector, obtained through simplifications of the cost function calculation, was adopted in this paper and has reached a performance very close to the SuB with a low complexity. On the other hand, errors in the channel coefficient estimates cause similar performance degradation in all of the heuristic detectors and this is inherent to the cost function calculation. Finally, the gain with the complexity reduction increases when the system’s load increases. R EFERENCES
10
6
10
5
10
4
10
3
10
TABLE II N UMBER OF OPERATIONS FOR THE M U D CONVERGENCE .
40
60
80
Fig. 3. Comparison of the number of operations for the convergence of the algorithms.
Table II shows, more precisely for two conditions of K · I, the number of operations needed for the LS and s-LS detectors in order to reach the convergence, considering a system with Eb /N0 = 12dB, N = 31, I = 7 e D = L = 2. It is also displayed the number of operations of the OM U D and the number of operations for the evaluation of one cost function F(X ).
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