Low Complexity Iterative Receiver for Downlink MC-CDMA System in Doppler Channels L. B. Thiagarajan §,
Y.-C. Liang ‡,
and
S. Attallah †
§ ‡ Institute for Infocomm Research, 21 Heng Mui Keng Terrace, Singapore 119613. Email: lb
[email protected],
[email protected] † School of Science and Technology, SIM University (UniSIM), Singapore 599491. Email:
[email protected]
Abstract— Block Iterative - Generalized Decision Feedback Equalizer (BI-GDFE) is an iterative receiver using previously made hard decisions to cancel out the multiple access interference (MAI) and inter-symbol interference (ISI). In this paper, we formulate the BI-GDFE for downlink multi-carrier code division multiple access (MC-CDMA) system under time varying channel conditions. The time varying coefficients of each channel tap over a data block is a finite length sequence whose maximum bandwidth depends on the Doppler bandwidth. This sequence can be represented using basis expansion models proposed for time varying channels. In this paper, Slepian basis expansion is used because the number of basis vectors required to represent the time varying channel is less compared with other basis expansion models and hence the number of parameters to be estimated is less. Thus less number of pilot symbols are required for channel estimation. BI-GDFE for fully loaded downlink MCCDMA system can be formulated as a one-tap equalizer which is simple for implementation. Furthermore, the computation of the signal to interference noise ratio (SINR) value for each iteration of BI-GDFE can be simplified using matrix manipulations for the downlink MC-CDMA system. Computer simulations show that the BER performance of BI-GDFE with channel estimation is close to the single user matched filter bound (MFB). Keywords: Channel estimation, Iterative processing, Interference mitigation, Equalization, MC-CDMA, Downlink, Multi-user detection
I. I NTRODUCTION Multi-carrier code division multiple access (MC-CDMA) is a promising candidate for multiuser wireless communications. It combines the advantages of CDMA and orthogonal frequency division multiplexing (OFDM). Due to the use of OFDM like signal structure, efficient frequency domain equalization is possible and hence it is simple to implement. For coherently detecting the transmitted data, the receiver should have the channel knowledge. This necessitates the need for channel estimation. Furthermore, when the users are mobile, the channel will be time varying and the block fading channel model assumption for this scenario will degrade the system performance [1] as the user velocity and/or the data block length increases. Thus frequent channel estimation will be required which leads to an increase in the computational complexity. The time varying coefficients of each channel tap over a data block1 is a finite length sequence at rate 1/Td 1 Data block consists of several MC-CDMA symbols with interleaved pilot symbols
978-1-4244-1645-5/08/$25.00 ©2008 IEEE
whose maximum bandwidth is given by the normalized one Td , where Td is MCsided Doppler bandwidth vD = fc vcmax 0 CDMA symbol duration, c0 is the velocity of light and vmax is the maximum velocity of the user. The finite bandwidth and finite length coefficient sequence over a data block can be represented using Slepian basis expansion. The channel can be estimated once the basis coefficients are known. The basis coefficients for linearly combining the Slepian basis vectors are estimated using the pilot symbols. Once the channel estimate is obtained the receiver can perform the data detection. The performance of linear equalizers, like zero-forcing (ZF) and minimum mean square error (MMSE), and nonlinear equalizers such as VBLAST receiver and generalized decision feedback equalizer (GDFE), is usually far away from the maximum likelihood (ML) performance bound. Use of exhaustive search methods to obtain ML performance are computationally intensive and hence not practical. Near ML performance can be achieved using QR decomposition with M-algorithm (QRD-M) [4], probability data association (PDA) method [5] and closest lattice point search (CPS), such as sphere decoding [2] [3]. However the complexity of the above receivers is still high and increases with increase in block size and order of data modulation used. MC-CDMA system is multiple access interference (MAI) limited and in general the MAI is treated as noise for partially loaded system. The ratio of the number of users to the spreading gain is called system load ω, where usually ω < 0.5. Using iterative multi-user receivers, the system load can be increased [6] thereby increasing the system’s throughput. Recently a iterative receiver called Block Iterative - Generalized Decision Feedback Equalizer (BI-GDFE) [7] has been proposed to mitigate interference in MIMO systems assuming a block fading channel model and perfect channel knowledge. It has been shown in [7] that the BI-GDFE’s performance after a few iterations is close to single user matched filter bound (MFB). In this paper, we consider a time varying channel, use Slepian basis expansion to represent the time varying channel as in [6] [8] and estimate the channel using pilot symbols. Using the estimated channel, BI-GDFE is constructed to iteratively detect the transmitted data. Furthermore, it will be shown that through matrix manipulations the signal to interference noise ratio (SINR) required for each iteration of BI-GDFE can be easily computed for downlink MC-CDMA system. Computer
724
simulations shows that the BI-GDFE even with the estimated channel has bit-error rate (BER) performance close to the single user matched filter bound (MFB). The notations followed throughout this paper are summarized in the following. All vector (matrices) are given in small (capital) boldface. The (N ×N ) identity matrix is given by IN . The column vector with N zeros is denoted by 0N . Transpose, conjugate and conjugate transpose operations are respectively denoted by (·)T , (·)∗ and (·)H . The operator diag(·), if (·) is a square matrix returns a column vector containing the diagonal elements of matrix (·). If (·) is a vector, then diag(·) returns a diagonal matrix with the elements in the column vector (·) as it diagonal elements. The operator E[.] denotes statistical expectation and Tr(.) denotes the trace of the square matrix (.). II. D OWNLINK MC-CDMA S YSTEM M ODEL We consider a downlink MC-CDMA system with T users and N subcarriers. The data symbol si (n) from ith user is spread in frequency domain by spreading code ci = [ci (0), ci (1), · · · , ci (N − 1)]T with spreading gain N , such that cH i ci = 1. A data block (frame) consists of M number of MC-CDMA symbols with J training symbols interleaved between (M − J) symbols. Let the set P contain the symbol indices n for which pilot symbols p(n) are transmitted. The transmitted signal before IDFT operator can be expressed as d(n)
= Cs(n) + p(n),
(1)
s(n) = where C = [c0 , c1 , · · · , cT −1 ], / P, s(n) = 0T [s0 (n), s1 (n), · · · , sT −1 (n)]T for n ∈ for n ∈ P and p(n) = 0N for n ∈ / P. The pilot symbols are chosen such that (diag(p(n)))H diag(p(n)) = IN . The transmitted signal after IDFT operation can be represented as x(n) = WH N d(n), where WN is the N -point DFT matrix. A cyclic prefix of length L − 1 is added to the output from the IDFT processor, where L − 1 is the maximum channel order. After cyclic prefixing, the ((N + L − 1) × 1) vector is converted from parallel to serial form and transmitted into the channel. Let Td be the data symbol duration and Ts be the sampling duration, therefore Ts = Td /(N + L − 1). The transmitted signal propagates through a multipath fading channel, whose finite length channel impulse response is given by h(n) = [h0 (n), h1 (n), · · · , hL−1 (n)]T . The channel is assumed to be time varying but remains unchanged within one MC-CDMA symbol duration. The received signal after the removal of cyclic prefix and N -point DFT operation is √ given by r(n) = Λ(n)d(n) + v(n), where Λ(n) = diag( N WN ×L h(n)) with WN ×L denoting the first L column vectors of matrix WN and v(n) is the (N × 1) additive white Gaussian noise vector. The received signal r(n) is ISI free and the only interference to tackle is the MAI. The time varying channel induces inter-carrier interference (ICI) and it is neglected by assuming the normalized Doppler bandwidth vD to be much smaller than (N + L − 1)/N [9], i.e. vD (N + L − 1)/N < 10−2 [8]. Under the above condition, the orthogonality between the subcarriers is still
preserved even when the channel is time varying. Therefore, r(n) = Λ(n)Cs(n) + v(n) for n ∈ / P, r(n) = Λ(n)p(n) + v(n) for n ∈ P.
(2) (3)
Equation (2) can be expressed as = H(n)s(n) + v(n),
r(n)
(4)
where H(n) = Λ(n)C is a (N × T ) matrix. The objective is to estimate Λ(n) using (3) and detect the transmitted data using the estimated channel. The following assumptions are made in [7] for the formulation of BI-GDFE and are used in this paper to formulate BI-GDFE for downlink MC-CDMA system in Doppler channels. AS1 The channel has unit gain, i.e., E h(n)2 = 1. AS2 The transmitted data symbols si (n) are iid, zero mean and E[|si (n)|2 ] = σs2 . AS3 The AWGN vector v(n) satisfies E[v(n)v(n)H ] = σn2 IN . AS4 The detected data symbols sˆi (n) are also zero mean, iid and with power E[|ˆ si (n)|2 ] = σs2 . III. S LEPIAN BASIS AND C HANNEL E STIMATION A. Slepian Basis Expansion Equation (3) can be re-written as r(n) = diag(p(n))f(n) + v(n), (5) √ where f(n) = N WN ×L h(n). The finite length sequence tTk = [hk (0), hk (1), · · · , hk (M −1)] for the kth channel tap is bandlimited by the maximum Doppler bandwidth vDmax for k = 0 to L−1. Every realization of tk spans a low dimensional subspace [10]. The Slepian basis vectors also span the same L−1 can be expressed low dimensional subspace. Thus {tk }k=0 using Slepian basis expansion as [6], [8] tk
≈
D−1
uj γk,j ,
(6)
j=0
where γk,j are the basis coefficients, uj is a (M × 1) Slepian basis vector and D is the dimension of the subspace spanned by tk which is given by D
= 2vD M + 1,
(7)
where x returns a minimum integer greater than x. Let the Slepian basis vector uj be represented as uj = [uj (0), uj (1), · · · , uj (M − 1)]T .
(8)
From (6) and (8), the channel h(n) can be approximated as h(n)
≈ A(n)q,
where
725
A(n) = IL ⊗ b(n), b(n) = [u0 (n), u1 (n), · · · , uD−1 (n)], q = [gT0 , gT1 , · · · , gTL−1 ]T ,
(9)
gk
= [γk,0 , γk,1 , · · · , γk,D−1 ]T .
IV. BI-GDFE
Substituting (9) into (5), we obtain √ r(n) ≈ diag(p(n)) N WN ×L A(n)q + v(n).
Using (14) the matrix H(n) is estimated as
B. Channel Estimation It is noted that the matrix A(n) for n = 0 to M − 1 can be pre-computed as the maximum velocity of the user can be obtained a priori. Therefore the only unknown parameters in (10) is the (LD × 1) dimensional vector q and the AWGN vector v(n). To estimate the channel, we need to estimate q. At least D pilot symbols are required to estimate the (LD × 1) dimensional q vector, i.e. J ≥ D. Let {ni }J−1 i=0 denote the symbol indices for which pilot symbols are transmitted, i.e P = {n0 , n1 , · · · , nJ−1 }. The received signal corresponding to the pilot indices are stacked to form a (JN ×1) dimensional vector rp = [r(n0 )T , r(n1 )T , · · · , r(nJ−1 )T ]T , which can be expressed as rp
≈
Rq + vp ,
(11)
where vp = [v (n0 ), v (n1 ), · · · , v (nJ−1 )] and √ diag(p(n0 )) N WN ×L A(n0 ) .. R = . √ diag(p(nJ−1 )) N WN ×L A(nJ−1 ) T
T
T
ˆ H(n)
(10)
T
. (12)
It should be noted that the pilot symbol indices are chosen such that matrix R is of full rank. It can be shown that matrix R is of full rank LD, if the T
is (JL × LD) matrix AT (n0 ), AT (n1 ), · · · , AT (nJ−1 ) of full rank LD. This is due to the fact that matrix (WN ×L )H diag(p(ni ))H diag(p(ni ))WN ×L = IL for i = 0 to J − 1. Using least-squares (LS) approach, the vector q is estimated as −1 H ˆ = RH R R rp . (13) q
ˆ = Λ(n)C.
(15)
The BI-GDFE detects the transmitted data in each MCCDMA symbol in an block-iterative manner. It consists of a hard decision detector, feed-forward equalizer (FFE) Kl and feedback equalizer (FBE) Dl which are given by [7] −1 ˆ ˆ ˆ H (n) + 1 IN H(n), (16) H Kl (n) = (1 − ρ2l−1 )H(n) µ Dl (n)
ˆ = ρl−1 (Bl (n) − KH l (n)H(n)),
(17)
where l denotes the iteration index, ˆ Bl (n) = diag(KH l (n)H(n)), 2 2 µ = σs /σn , s∗l−1,i (n)]. ρl−1 = (1/σs2 )E[si (n)ˆ
(18)
The parameter ρl−1 is the input decision correlation (IDC) at the (l − 1)th iteration and sˆl−1,i (n) is the hard decision for ith user’s nth transmitted symbol during the (l−1)th iteration. Let ˆsl−1 (n) = [ˆ sl−1,0 (n), sˆl−1,1 (n), · · · , sˆl−1,T −1 (n)]T denote the detected symbols at the (l − 1)th iteration. The vector ˆsl (n) is obtained as hard decisions made from the vector zl (n), which is obtained by combining the outputs from FFE and FBE as zl (n)
= KH sl−1 (n). l (n)r(n) + Dl (n)ˆ
(19)
The convergence of BI-GDFE is very sensitive to the selection of the IDC coefficient. The IDC coefficient ρl is estimated using the algorithm proposed in [7] which guarantees convergence. For fully loaded system (ω = 1), i.e. T = N ,hence footnoteAssuming perfectly orthogonal codes, say WH codes. CCH = IN . Therefore, ˆ ˆ H (n) H(n) H
2 ˆ ˆ H (n) = |Λ(n)| ˆ = Λ(n) Λ .
(20)
The Slepian basis expansion is used in time domain instead of frequency domain in order to reduce the size of the square matrix RH R from N D to LD thereby reducing the computational complexity [6]. The frequency domain estimate ˆ of the channel for every symbol duration is obtained using q as √ ˆ q). (14) Λ(n) = diag( N WN ×L A(n)ˆ
Substituting (20) into (16), we obtain −1 1 2 ˆ ˆ Kl (n) = H. (1 − ρ2l−1 )|Λ(n)| + IN µ
For example, if M = 100 and vD = 0.005, then D = 2. If fc = 2GHz and 1/Td = 48.6 × 103 s−1 . The maximum Doppler frequency is vD /Td = 243Hz and it corresponds to the maximum user velocity vmax = 131.22 Km/hr. That is, the MC-CDMA system with the above parameters can support users with velocity from 0 to 131.22 Km/hr and requires a minimum of 2 pilot symbols to estimate the time varying channel over the entire data block of 100 symbols.
Therefore, matrix Kl (n) = Fl (n)C. Substituting (20) into (17), we obtain ˆ Dl (n) = ρl−1 diag(CH FH l (n)Λ(n)C)
ˆ −CH FH (23) l (n)Λ(n)C .
Let us define a (N × N ) diagonal matrix Fl as −1 1 2 2 ˆ ˆ Fl (n) = (1 − ρl−1 )|Λ(n)| + IN Λ(n). µ
(21)
(22)
ˆ As matrices Λ(n) and Fl (n) are diagonal and the elements in matrix C are all of same amplitude, matrix Dl (n) can be
726
written as [7] Dl (n)
= ρl−1 CH
ˆ αl (n)IN FH (n) Λ(n) C, l
(24)
where 1 ˆ Tr(FH l (n)Λ(n)). N Let Ql (n) be a (N × N ) diagonal matrix defined as
ˆ Ql (n) = ρl−1 αl (n)IN − FH l (n)Λ(n) . αl (n)
where the (N × N ) diagonal matrix M(n) is given by 1 2 2 ˆ M(n) = (1 − ρl−1 )|Λ(n)| + IN . µ
(30)
ˆ are (N × N ) diagonal As matrices Fl (n), M(n) and Λ(n) matrices, it can be shown that
=
FH l (n)Fl (n)M(n) (25)
Fig.1 shows the BI-GDFE constructed using (22) and (25). For
ˆ H (n)[M(n)]−2 Λ(n)M(n) ˆ = Λ ˆ ˆ H (n)[M(n)]−1 Λ(n). = Λ
ˆ H (n)[M(n)]−1 , we have As Fl (n) = Λ FH l (n)Fl (n)M(n) and hence deno
ˆ = FH l (n)Λ(n)
=
2 2 ˆ Tr FH l (n)Λ(n) − (1 − ρl−1 )N αl (n)
=
N αl (n) − (1 − ρ2l−1 )N αl2 (n).
(31)
(32)
Using (32), the SINR ηl (n) in (26) can be written as Fig. 1.
ηl (n)
BI-GDFE for Downlink MC-CDMA
the first iteration (0th iteration) we have ρ0 = 0, this implies that D0 (n) = 0 for all n and the BI-GDFE is the MMSE receiver proposed in [8]. The diagonal matrices Fl (n) and Ql (n) lead to a simple one-tap equalizer thereby significantly reducing the computational complexity. The IDC (ρl ) is computed using the algorithm in [7] which requires the value of SINR for each iteration (l) and symbol duration (n). The SINR ηl (n) for the lth iteration during the nth symbol duration is given by [7] ηl (n) =
Tr
(1−ρ2
l−1 )
ρ2l−1
N αl2 (n)
. (26) 1 H Ql (n)QH l (n) + µ Fl (n)Fl (n)
ˆ are Assuming that matrices Fl (n), Ql (n) and Λ(n) available, the number of multiplications involved in computing ηl (n) using (26) is O(3N ). Let (1−ρ2l−1 ) H 1 H deno = Tr Q (n)Q (n) + F (n)F (n) . 2 l l l µ l ρl−1 Using (25), we have Ql (n)QH l (n) ρ2l−1
ˆ = αl2 (n)IN − αl (n)FH l (n)Λ(n)
ˆ H (n)Fl (n) +αl (n)Λ 2 ˆ (27) +FH l (n)Fl (n)|Λ(n)| . H 2 ˆ As N αl (n) = Tr FH (n) Λ(n) , Tr Q (n)Q (n)/ρ l l l l−1 can be obtained from (27) as Ql (n)QH l (n) H 2 ˆ = Tr F (n)F (n)| Λ(n)| Tr l l ρ2l−1 −N αl2 (n).
(28)
Using (28), deno can be expressed as 2 2 deno = Tr FH l (n)Fl (n)M(n) − (1 − ρl−1 )N αl (n), (29)
=
αl (n) . 1 − (1 − ρ2l−1 )αl (n)
(33)
The number of multiplications involved in computing ηl (n) using (33) is only O(N ). As the IDC is to be computed for every iteration during every MC-CDMA symbol duration, equation (33) presents a computationally simple expression for the computation. V. S IMULATION R ESULTS We consider a fully loaded downlink MC-CDMA system with N = T = 256 (ω = 1). The data symbols from each user are QPSK modulated and spread in the frequency domain using Walsh-Hadamard codes. The maximum length of the channel impulse response is L = 65. The number of MCCDMA symbols in a data block is M = 50. The number of training symbols used for channel estimation is J = 4 and the corresponding pilot symbol indices are P = {4, 19, 34, 49}. We assume the carrier frequency used in the system to be fc = 2GHz. We use the channel model in [11], to simulate the time varying channel. The maximum normalized one-sided Doppler bandwidth of the system and the MC-CDMA symbols duration are respectively assumed to be vD = 0.01 and Td = 1/(48.6 × 103 ) s. This results in D = 2. The maximum velocity of the user which corresponds to the maximum onesided normalized bandwidth vD = 0.01 that the system can support is vmax = 262.44 Km/hr. The MSE for channel estimation when vD = 0.0026 or when the user velocity is 68.23 km/h, which is a typical user velocity in metro cities, is shown in Fig.2 From Fig.2, it is clear that the Slepian basis expansion of the time varying channel is valid and the MSE decreases with increase in SNR. The BER performance of BIGDFE for vD = 0.0026 is shown in Fig. 3. From Fig. 3, it can be clearly seen that BI-GDFE performs better than the MMSE receiver used in [8] and the BER performance of BI-GDFE approaches close to the single user MFB as the number of iteration increases. The performance of BI-GDFE for various Doppler bandwidth vD for a fixed SNR of 13dB is shown in
727
0
−1
10
10
−1
−2
10
10
BER
MSE of Channel Estimation
MMSE BI−GDFE (1 iteration) BI−GDFE (2 iterations) BI−GDFE (5 iterations)
−2
−3
10
10
−3
10
−4
0
2
4
Fig. 2.
6
8 SNR (dB)
10
12
14
10
16
MSE of the channel estimates
0
Fig. 4.
0.002
0.004 0.006 Normalized Doppler Frequency
0.008
0.01
BER vs vD for BI-GDFE with SN R = 13dB
0
10
simulations have shown the convergence of BI-GDFE’s BER performance to the single user MFB. The low computational complexity and the better BER performance of BI-GDFE makes it an ideal candidate for equalization in downlink transmission of MC-CDMA systems.
MMSE BI−GDFE (1 iteration) BI−GDFE (2 iterations) BI−GDFE (5 iterations) MFB
−1
10
−2
BER
10
R EFERENCES −3
10
−4
10
6
8
10
12
14
16
SNR (dB)
Fig. 3.
BER performance of BI-DFE for vD = 0.0026
Fig.4. Till vD = 0.006, which corresponds to the user velocity of 157.5 km/h, the BER almost remains unchanged. As the user velocity increases from 157.5 km/h to 262.44 km/h, the BER also increases. However, even as the BER increases, BI-GDFE performs better than the MMSE receiver. For user velocity less than 157.5 km/h, the BER performance of BIGDFE improves with increase in the number of iterations. For user velocity closer to 262.44 km/h, the BER performance improvement with increase in the number of iterations is not significant. This is due to the fact that the Slepian basis expansion is just an approximate representation of a finite time and finite bandwidth sequence. VI. C ONCLUSION In this paper, a channel estimation scheme is obtained by representing the time varying channel using Slepian basis expansion. Slepian basis expansion is used as the number of basis vectors used to represent the channel is less. Thus only a few pilot symbols are required for the channel estimation. Furthermore, the computation of SINR for each iteration is shown to be simple by using matrix manipulations. Computer
[1] B. Sklar, “Rayleigh fading channels in mobile digital communication systems part II: Mitigation,” IEEE Commun. Mag., pp. 148-155, Sept. 1997. [2] Loc Brunel, “Multiuser detection techniques using maximum likelihood sphere decoding in multicarrier CDMA systems,” IEEE Trans. Wireless Commun., vol. 3, No. 3, pp. 949-957, May 2004. [3] M. O. Damen, H. E. Gamal and G. Caire, “On maximum likelihood detection and the search for the closest lattice point,” IEEE Trans. Inf. Theory, vol. 49, No. 10, pp. 2389-2401, Oct. 2003. [4] K. J. Kim and R. A. Iltis, “ Joint detection and channel estimation algorithms for QS-CDMA signals over time-varying channels,” IEEE Trans. Commun., vol. 50, No. 5, pp. 845-855, May 2002. [5] J. Luo, K. R. Pattipati, P. K. Willett and F. Hasegawa, “Near-optimal multiuser detection in synchronous CDMA using probabilistic data association,” IEEE Commun. Lett., vol. 5, No. 9, pp. 361-363, Sep. 2001. [6] T. Zemen, C. F. Mecklenbr¨ auker, J. Wehinger and R. R. M¨ uller, “Iterative joint time-variant channel estimation and multi-user detection for MCCDMA,” IEEE Trans. Wireless Commun., vol. 5, pp. 1469-1478, June 2006. [7] Y.-C. Liang, S. Sun and C. K. Ho, “Block-iterative generalized decision feedback equalizers for large MIMO systems: Algorithm design and asymptotic performance analysis,” IEEE Trans. Signal Process., vol. 54, No. 6, pp. 2035-2048, June 2006. [8] T. Zemen and C. F. Mecklenbr¨ auker, “Time-variant channel estimation using discrete prolate spheroidal sequences,” IEEE Trans. Signal Process., vol. 53, pp. 3597-3607, Sep. 2005. [9] Y. G. Li and L. J. Cimini, “Bounds on the interchannel interference of OFDM in time-varying impairments,” IEEE Trans. Commun., vol. 49, No. 3, pp. 401-404, Mar. 2001. [10] D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainity – V: The discret case,” Bell Syste. Tech. J., vol. 57, no.5, pp. 1371-1430, May-Jun. 1978. [11] T. Zemen and C. F. Mecklenbr¨ auker, “ Doppler diversity in MC-CDMA using Slepian basis expansion model,” in Proc. 12th Eur. Signal Process. Conf., Vienna, Austria, Sep. 2004.
728
