Low-Complexity Turbo Equalization for Single-Carrier Systems without Cyclic Prefix 2
Dongming Wang1 , Chao Wei1 , Zhiwen Pan1 , Xiaohu You1 , Chung Hyun Kyu2 , Jeong Byung Jang2 1 National Comm. Research Lab., Southeast Univ., Nanjing, China;
[email protected] Electronics and Telecom. Research Institute (ETRI), 161 Gajeong-dong, Yuseong-gu, Daejeon, 305-350, Korea
Abstract— In this paper, we present a low complexity frequency-domain turbo equalization algorithm for single-carrier system without cyclic prefix. Comparing with the traditional time-domain turbo equalization, the proposed algorithm can be implemented in the frequency domain with fast Fourier transform (FFT), and it does not involve matrix inversion. Simulation results show that the proposed algorithm also exhibits good performance.
I. I NTRODUCTION When dealing with frequency selective channels, block transmissions such as orthogonal frequency division multiplexing (OFDM) and cyclic prefix based single-carrier block transmission (CP-SCBT, it is also called single-carrier frequency domain equalization systems [1]) have by now been well documented as the attractive means [2]. Due to the use of FFT operation, both of the techniques keep the receiver complexity significantly below the complexity of conventional single-carrier systems with time-domain equalizers. OFDM has been chosen for several broadband wireless local area network (WLAN) standards like IEE802.11a/g and European HIPERLAN/2, and terrestrial digital audio broadcasting (DAB) and digital video broadcasting (DVB-T). CP-SCBT has also been proposed in IEEE802.16. In such block transmission systems, the guard interval is used to avoid the inter-block interference. However, the introduction of the guard interval reduces spectral efficiency, especially for channels with long delay spread. Meanwhile, in the case of legacy systems cyclic prefix is not always available. Recently, turbo processing has received much attention. The basic idea of turbo detector is to exchange the soft information between detector and decoder. Then soft detection schemes which return soft, i.e., probabilistic, reliability information about the transmitted bits are in demand in this area. One of the promising low-complexity approaches is minimum meansquared-error (MMSE) based turbo detector. It is proposed by Wang and Poor [3] for turbo multiuser detection and lately is extended to turbo equalization by Reynolds [4] and [5]. In [6], a low complexity frequency-domain turbo receiver is proposed for block transmission systems with cyclic prefix. However, for traditional single-carrier systems without cyclic prefix, most work has focused on time-domain turbo receivers, This work was supported by the China High-Tech 863 Programme under Grant 2007AA01Z268, the National Natural Science Foundation of China under Grant 60702028, and Electronics and Telecom. Research Institute (ETRI), Korea.
1-4244-2424-5/08/$20.00 ©2008 IEEE
scL
Fig. 1.
sK
sccL
Block Transmission without cyclic prefix.
where the SISO detectors are trellis-based detectors [7], [8] or MMSE-based equalizers in the time-domain [9]–[11]. In this paper, we present a low complexity frequencydomain turbo equalization algorithm for single-carrier system without cyclic prefix. Comparing with the traditional timedomain turbo equalization, the proposed algorithm can be implemented in the frequency domain and does not include matrix inversion. The proposed algorithm can also be extended to single-carrier MIMO systems over frequency selective channels. II. S YSTEM M ODEL Fig. 1 depicts the transmitter structures of single-carrier systems without cyclic prefix. A stream of information bits is encoded, bit-interleaved, and mapped to symbols, and then the symbol blocks of size K are formed. We consider the following transmit-receive model. y = Hs + H s + H s + n
(1)
where y is the K + L receive vector, n is the complex additive Gaussian noise vector with covariance matrix white of E nnH = σ 2 IK+L , s , s , and s denote the current symbol block, the previous symbol block and the next symbol block, respectively, all of them are K × 1 vectors, and the channel matrices H , H and H are given by ⎡ ⎤ h0 ⎢ .. ⎥ ⎢ . ⎥ h0 ⎢ ⎥ ⎢ ⎥ .. .. ⎢ hL ⎥ . . H =⎢ ⎥ ⎢ hL h0 ⎥ ⎢ ⎥ ⎢ .. ⎥ .. ⎣ . . ⎦ hL (K+L)×K
0L×(K−L) HU H = 0K×(K−L) 0K×L ⎡ ⎤ hL hL−1 · · · h1 ⎢ hL · · · h2 ⎥ ⎢ ⎥ HU = ⎢ .. ⎥ .. ⎣ . . ⎦
1091
0
hL
L×L
ICCS 2008
0K×L HL
H = ⎡ ⎢ ⎢ HL = ⎢ ⎣
0K×(K−L) 0L×(K−L)
h0 h1 .. .
h0 .. .
hL−1
hL−2
III. SISO D ETECTOR UNDER COLORED NOISE 0
..
. ···
From [3], if we know the apriori information of the transmitted symbols, such as, means and variances, then after soft interference cancellation, (4) can be written as
⎤ ⎥ ⎥ ⎥ ⎦
rk = r − Hc s¯[k] = Hc ek sk + Hc s[k] − Hc s¯[k] + z
h0
If we assume that the interference of the previous block can be ideally cancelled, the received signal can be rewritten as y = Hs + H s + n Define matrix the K × (K + L) J ⎡ 0 ··· 0 1 0 .. .. ⎢ .. . ⎢ . . . 0 .. ⎢ ⎢ .. ⎢ 0 ··· 0 . ⎢ J =⎢ ⎢ 1 0 ⎢ ⎢ . . ⎣ . 0
(2)
where ek denotes the i-th unit vector, s¯ is the mean vector of s, s¯[k] is obtained from s¯ by setting the k-th element to zero, and so does s[k] . Now we will focus on the detection of sk from (5). The interference plus noise is defined as z˜ = Hc s[k] − Hc s¯[k] + z
as ··· .. .
0
1
0
0
1 .. .
1
..
.
..
. 0
⎤
J y = J Hs + J H s + J n
rk = Hc ek sk + z˜
(3)
(4)
Where Hc is a K × K circular matrix, and its first column is
T hL 0 · · · h0 · · · hL−1 1×K Define the noise vector as z = J H s + J n The mean of z is E (z) = 0, and the covariance matrix can be written as H
cov (z, z) = J H H J H +σ 2 J J H
˜ and it can be where Cz˜,˜ z is the covariance matrix of z expressed as H H H 2 Cz˜,˜ z = Hc V H c − vk Hc ek ek Hc + σ IK + R
and V = cov (s, s) = diag {v1 , v2 , . . . , vK } is the variance vector of the transmitted symbols. For invertable matrices A and DA−1 B , we have the following matrix identity −1 −1 −1 D (A + BCD) B D (A + BCD) −1 = DA−1 B DA−1 It can be proved by using matrix inverse lemma −1 −1 (A + BCD) = A−1 −A−1 B DA−1 B + C −1 DA−1 Then, (8) can be written as
According to the definition of J , the covariance matrix has the following form
2 0 σ I(K−L)×(K−L) cov (z, z) = 0 2σ 2 IL + HL HLH Define
(7)
The BLUE (Bayesian Linear Unbiased Estimation of (7) can be given by [13] −1 H −1 H −1 eH (8) sˆk = eH k Hc Cz k Hc Cz ˜,˜ z Hc ek ˜,˜ z rk
The operation will add the first L samples to the last L samples. Then (3) can be expressed as r = Hc s + z
(6)
Suppose that z˜ is a proper complex Gaussian random vector [12], then (5) can be rewritten as
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎦ 1
Multiplying the left-hand-side of (2) with J results
(5)
sˆk =
1
−1 H H V H H + σ2 I + R eH Hc ek c K c k Hc −1 H 2 Hc V H H · eH rk k Hc c + σ IK + R
(9)
After simplifying, we have −1 1 H H 2 ek Hc Hc V H H + σ I + R (r − Hc s¯) + s¯k sˆk = K c ρk (10)
2
cov (z, z) = σ IK + R
0 0 R= 0 RL
where
RL = σ 2 IL + HL HLH
−1 H 2 Hc V H H ρk = eH Hc ek k Hc c + σ IK + R
The covariance matrix of z can be rewritten as cov (z, z) = σ 2 IK + R In the following, we will give the soft-input soft-output detector of the linear model (4). However, we should note that, in (4), the noise is colored Gaussian noise.
(11)
If we assume the output of the detector (10) represents the output of an equivalent AWGN channel having sk as its input symbol, we may write
1092
sˆk = sk + zˆk
Since the constant γ can be normalized in the detection, we redefine
The equivalent noise variance can be given by 1 − vk ρk
sk , sˆk |sk ) = σz2ˆk = cov (ˆ
(12)
Nstage
ΓPE =
IV. L OW C OMPLEXITY SISO D ETECTOR
−1 2 Hc V H H c + σ IK + R
where expansion number is Nstage . To ensure the convergence of the expansion, we should restrict that |λmax (γIK − IK + γΣR)| < 1
(13)
where V is a diagonal matrix, and Hc V H H c is not a circulant matrix. If we assume that the diagonals of V are equal, then Hc V H H c will be a circulant matrix. However, due to the existence of matrix R, the inversion is also very complex. In the following, we will simplify (10) through two steps.
After simplification, we have γ
