A Widely Linear LMS Algorithm for MAI Suppression for ... - CiteSeerX
A Widely Linear LMS Algorithm for MAI Suppression for DS–CDMA R. Schober1 , W.H. Gerstacker2 , and L. H.–J. Lampe3
1 Department of Electrical and Computer Engineering, University of British Columbia 2 Chair for Mobile Communications, University Erlangen-N¨u rnberg 3 Chair for Information Transmission, University Erlangen-N¨urnberg Abstract — In this paper, a novel data–aided stochastic gradient algorithm for adjustment of the widely linear (WL) minimum mean–squared error (MMSE) filter for multiple access interference (MAI) suppression for direct–sequence code–division multiple access (DS–CDMA) is introduced and analyzed. We give analytical expressions for the steady–state signal–to–interference– plus–noise ratio (SINR) of the proposed WL least–mean–square (LMS) algorithm, and we also investigate its speed of convergence. Both analytical considerations and simulations show in good agreement the superiority of the novel WL adaptive algorithm. Nevertheless, the computational complexity of the proposed algorithm is lower than that of the linear LMS algorithm.
I.
I NTRODUCTION
It is well known that in non–orthogonal direct–sequence code– division multiple access (DS–CDMA) systems a substantial performance improvement can be realized by multiuser detection [1]. Because of their low complexity and good performance linear multiuser receivers are particularly interesting, e.g. [1, 2]. In addition, the filter coefficients of linear minimum mean–squared error (MMSE) receivers can be conveniently adjusted to the channel and interference situation by using adaptive algorithms, cf. e.g. [3]. Recently, it has been shown that under certain conditions the performance of linear multiuser receivers can be significantly improved if not only the received signal but also its complex conjugate is processed [4]. The resulting schemes are commonly referred to as conjugate linear [5] or widely linear (WL) receivers [6]. WL processing is particularly beneficial if all users employ signal constellations that can be interpreted as one–dimensional (real) at the receiver [4]. This embraces systems using e.g. amplitude–shift keying (ASK), binary phase– shift keying (BPSK), Gaussian minimum–shift keying (GMSK), and offset quadrature–amplitude modulation (QAM). Because of the similarity of linear and WL receivers, efficient adaptive algorithms for WL MMSE filters can be obtained. A recursive least–squares (RLS) algorithm has been proposed in [7]. However, a low–complexity and robust stochastic gradient algorithm for adaptation of WL MMSE filter has not been reported so far.
In this paper, we introduce and analyze the WL least–mean– square (LMS) algorithm. We will show that, in contrast to the WL–RLS algorithm in [7], the computational complexity of WL–LMS algorithm is not higher but lower than that of its conventional (linear) counterpart. Our analysis shows that the WL–LMS algorithm always achieves a higher steady– state signal–to–interference–plus–noise ratio (SINR) than the linear LMS algorithm. We will show that in interference limited systems the speed of convergence of the WL–LMS algorithm is always higher than that of the linear LMS algorithm. Our simulations confirm the high performance of the derived WL stochastic gradient algorithm even in overloaded CDMA systems, where linear adaptive algorithms fail to achieve a high SINR. II.
S YSTEM M ODEL
In this section, we describe the DS–CDMA system model and report some results for WL MMSE multiple access interference (MAI) suppression. A. DS–CDMA For simplicity, we assume a synchronous DS–CDMA system with K users and L chips per symbol interval T . After chip matched filtering, (perfect) time synchronization, sampling with rate L=T , and appropriate normalization, the vector r [k ℄ containing the L samples received in the interval [kT ; (k + 1)T ), k 2 ZZ , can be expressed as
r[k ℄
=
K r X E j e s b[k℄ + n[k℄
=1
E1
K X ej p b[k℄ + n[k℄
=
p1 b1 [k ℄ +
=
P b[k ℄ + n[k ℄;
=2
(1)
where E , , s , and b [℄ denote the energy per symbol, the channel phase1 , the spreading sequence (length: L), and
1 Note that although we refer to as ”channel phase”, it may be influenced by the system designer. E.g. for the downlink of a synchronous CDMA K , can system with perfect phase synchronization for user 1, , be completely specified at the transmitter.
2
, 1 K , p 1 and n[k℄ is a zero–mean additive respectively. white Gaussian noise (AWGN) vector with Efn[k ℄nH [k ℄g = 4 N =E , where N is the single–sided power n2 I L , n2 = 0 1 0 the transmitted symbol sequence of user
4 j =
spectral density of the underlying continuous–time noise process. Here, [℄H , Efg, and I L denote Hermitian transposition, the expected value of the random variable in brackets, and the L L identity matrix, respectively. Without loss of generality we assume that user 1 is the desired user. We also assume perfect phase synchronization for the desired user, q i.e., 1 = 0 is
=4 [ej1 p1 : : : ejK pK ℄, and b[k℄ =4 [b1 [k℄ b2 [k℄ : : : bK [k℄℄T ([℄T : transposition). In this paper, we assume that b [k ℄ is a 4 real–valued white sequence with unit variance b2 = Ef(b [k℄)2 g = 1. b[k℄ may be e.g. an ASK or BPSK sevalid. In Eq. (1), we use the definitions p
=4
E s , P E1
quence [8]. The transmitted symbol sequences of different users are assumed to be mutually statistically independent. The spreading sequences are normalized to sH s = 1, 1 K. Although in this paper we only consider intersymbol interference (ISI) free transmission, for symbol–by–symbol detection the adopted model (Eq. (1)) can be easily generalized to asynchronous transmission over frequency–selective channels, see e.g. [2, Section II] and [9, Section II] for details. Furthermore, since some types of continuous–phase modulation (e.g. MSK and GMSK) and offset QAM can be (approximately) modeled as ASK transmitted over an ISI channel [10], our results on WL adaptive algorithms can be straightforwardly extended to these modulation formats as well, see [11] for details. B. Widely Linear MMSE MAI Suppression
given by [7]
W L
= 2(Rrr R^ rr Rrr1 R^ rr ) 1 (p1 R^ rr Rrr1 p1 )
(2)
4
([℄ : complex conjugation), where Rrr = Efr [k ℄rH [k ℄g = ^ rr =4 Efr[k℄rT [k℄g = P P T denote P P H + n2 I L and R the autocorrelation and the pseudo–autocorrelation matrix of vector r[k ℄ [12], respectively. For convenience, similar to [7] we introduce the bijective transformation T
x
T! x~ :
~= x
p1 [xT 2
xH ℄T ;
(3)
where x is a complex vector. T has the following two prop~ H y~ = y~ H x~ , ii) xH x = x~ H x~ , where erties: i)
1 Department of Electrical and Computer Engineering, University of British Columbia 2 Chair for Mobile Communications, University Erlangen-N¨u rnberg 3 Chair for Information Transmission, University Erlangen-N¨urnberg Abstract — In this paper, a novel data–aided stochastic gradient algorithm for adjustment of the widely linear (WL) minimum mean–squared error (MMSE) filter for multiple access interference (MAI) suppression for direct–sequence code–division multiple access (DS–CDMA) is introduced and analyzed. We give analytical expressions for the steady–state signal–to–interference– plus–noise ratio (SINR) of the proposed WL least–mean–square (LMS) algorithm, and we also investigate its speed of convergence. Both analytical considerations and simulations show in good agreement the superiority of the novel WL adaptive algorithm. Nevertheless, the computational complexity of the proposed algorithm is lower than that of the linear LMS algorithm.
I.
I NTRODUCTION
It is well known that in non–orthogonal direct–sequence code– division multiple access (DS–CDMA) systems a substantial performance improvement can be realized by multiuser detection [1]. Because of their low complexity and good performance linear multiuser receivers are particularly interesting, e.g. [1, 2]. In addition, the filter coefficients of linear minimum mean–squared error (MMSE) receivers can be conveniently adjusted to the channel and interference situation by using adaptive algorithms, cf. e.g. [3]. Recently, it has been shown that under certain conditions the performance of linear multiuser receivers can be significantly improved if not only the received signal but also its complex conjugate is processed [4]. The resulting schemes are commonly referred to as conjugate linear [5] or widely linear (WL) receivers [6]. WL processing is particularly beneficial if all users employ signal constellations that can be interpreted as one–dimensional (real) at the receiver [4]. This embraces systems using e.g. amplitude–shift keying (ASK), binary phase– shift keying (BPSK), Gaussian minimum–shift keying (GMSK), and offset quadrature–amplitude modulation (QAM). Because of the similarity of linear and WL receivers, efficient adaptive algorithms for WL MMSE filters can be obtained. A recursive least–squares (RLS) algorithm has been proposed in [7]. However, a low–complexity and robust stochastic gradient algorithm for adaptation of WL MMSE filter has not been reported so far.
In this paper, we introduce and analyze the WL least–mean– square (LMS) algorithm. We will show that, in contrast to the WL–RLS algorithm in [7], the computational complexity of WL–LMS algorithm is not higher but lower than that of its conventional (linear) counterpart. Our analysis shows that the WL–LMS algorithm always achieves a higher steady– state signal–to–interference–plus–noise ratio (SINR) than the linear LMS algorithm. We will show that in interference limited systems the speed of convergence of the WL–LMS algorithm is always higher than that of the linear LMS algorithm. Our simulations confirm the high performance of the derived WL stochastic gradient algorithm even in overloaded CDMA systems, where linear adaptive algorithms fail to achieve a high SINR. II.
S YSTEM M ODEL
In this section, we describe the DS–CDMA system model and report some results for WL MMSE multiple access interference (MAI) suppression. A. DS–CDMA For simplicity, we assume a synchronous DS–CDMA system with K users and L chips per symbol interval T . After chip matched filtering, (perfect) time synchronization, sampling with rate L=T , and appropriate normalization, the vector r [k ℄ containing the L samples received in the interval [kT ; (k + 1)T ), k 2 ZZ , can be expressed as
r[k ℄
=
K r X E j e s b[k℄ + n[k℄
=1
E1
K X ej p b[k℄ + n[k℄
=
p1 b1 [k ℄ +
=
P b[k ℄ + n[k ℄;
=2
(1)
where E , , s , and b [℄ denote the energy per symbol, the channel phase1 , the spreading sequence (length: L), and
1 Note that although we refer to as ”channel phase”, it may be influenced by the system designer. E.g. for the downlink of a synchronous CDMA K , can system with perfect phase synchronization for user 1, , be completely specified at the transmitter.
2
, 1 K , p 1 and n[k℄ is a zero–mean additive respectively. white Gaussian noise (AWGN) vector with Efn[k ℄nH [k ℄g = 4 N =E , where N is the single–sided power n2 I L , n2 = 0 1 0 the transmitted symbol sequence of user
4 j =
spectral density of the underlying continuous–time noise process. Here, [℄H , Efg, and I L denote Hermitian transposition, the expected value of the random variable in brackets, and the L L identity matrix, respectively. Without loss of generality we assume that user 1 is the desired user. We also assume perfect phase synchronization for the desired user, q i.e., 1 = 0 is
=4 [ej1 p1 : : : ejK pK ℄, and b[k℄ =4 [b1 [k℄ b2 [k℄ : : : bK [k℄℄T ([℄T : transposition). In this paper, we assume that b [k ℄ is a 4 real–valued white sequence with unit variance b2 = Ef(b [k℄)2 g = 1. b[k℄ may be e.g. an ASK or BPSK sevalid. In Eq. (1), we use the definitions p
=4
E s , P E1
quence [8]. The transmitted symbol sequences of different users are assumed to be mutually statistically independent. The spreading sequences are normalized to sH s = 1, 1 K. Although in this paper we only consider intersymbol interference (ISI) free transmission, for symbol–by–symbol detection the adopted model (Eq. (1)) can be easily generalized to asynchronous transmission over frequency–selective channels, see e.g. [2, Section II] and [9, Section II] for details. Furthermore, since some types of continuous–phase modulation (e.g. MSK and GMSK) and offset QAM can be (approximately) modeled as ASK transmitted over an ISI channel [10], our results on WL adaptive algorithms can be straightforwardly extended to these modulation formats as well, see [11] for details. B. Widely Linear MMSE MAI Suppression
given by [7]
W L
= 2(Rrr R^ rr Rrr1 R^ rr ) 1 (p1 R^ rr Rrr1 p1 )
(2)
4
([℄ : complex conjugation), where Rrr = Efr [k ℄rH [k ℄g = ^ rr =4 Efr[k℄rT [k℄g = P P T denote P P H + n2 I L and R the autocorrelation and the pseudo–autocorrelation matrix of vector r[k ℄ [12], respectively. For convenience, similar to [7] we introduce the bijective transformation T
x
T! x~ :
~= x
p1 [xT 2
xH ℄T ;
(3)
where x is a complex vector. T has the following two prop~ H y~ = y~ H x~ , ii) xH x = x~ H x~ , where erties: i)
