A Low Complexity two Stage MMSE-Based Receiver for Single-Carrier Frequency-Domain Equalization Transmissions over Frequency-Selective Channels Homa Eghbali, Sami Muhaidat and Naofal Al-Dhahir∗∗ School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6 Email:
[email protected],
[email protected] ∗∗ Dept. of Electrical Engr., The University of Texas at Dallas, Richardson, TX, USA, 75083-0688 Email:
[email protected] ∗
Abstract—In this paper, we propose a novel low complexity two-stage minimum mean square error (MMSE)-based receiver for single carrier frequency-domain equalization (SC-FDE) for space-time block coded (STBC) transmissions over frequency selective channels. We demonstrate that the proposed receiver enjoys a remarkably simple decoding scheme. We further show that, by incorporating linear processing techniques, our MMSEbased receiver is able to collect full antenna and multipath diversity gains, while maintaining low complexity, thus, eliminating the need for maximum-likelihood sequence detection (MLSD), which has certainly prohibitive complexity, specially, when the constellation size of the transmitted signals and/or the block length increases. Simulation results demonstrate that our proposed receiver significantly outperforms the conventional SC-MMSEFDE receiver, while maintaining nearly similar complexity.
I. I NTRODUCTION The increasing demand for wireless multimedia and interactive Internet services has lead to intensive research efforts on high speed data transmission. A key challenge for high-speed broad-band applications is the dispersive nature of frequencyselective fading channels, which causes the so-called intersymbol interference (ISI), leading to an inevitable performance degradation. An efficient approach to mitigate ISI is the use of single-carrier (SC) modulation combined with frequency domain equalization (FDE). It has been shown that FDE is an attractive equalization scheme for broadband wireless channels which are characterized by their long impulse response memory [1], [2]. Unlike time-domain equalization whose complexity grows exponentially with channel memory and spectral efficiency, SC-FDE enjoys lower complexity, due to its use of the computationally-efficient fast Fourier transform (FFT). Furthermore, SC-FDE has several advan-tages over orthogonal frequency-division multiplexing (OFDM) systems [2], such as large peak-to-average ratio and the sensitivity to carrier-frequency offsets. It has been shown in [3] that spatial diversity offers significant improvement in link reliability and spectral efficiency through the use of multiple antennas at the transmitter and/or receiver side. These gains are typically realized at the physical layer and require co-located antenna elements (i.e. physical
antenna arrays) at the base station and/or the mobile terminal. Multiple-antenna techniques are very attractive for deployment in cellular applications at base stations and have already been included in the 3rd generation wireless standards. An effective and practical way to realize the benefits of MIMO systems is to employ Space-time block codes (STBCs) [4], [5]. Since STBCs have been proposed for frequency-flat fading channels, it is challenging to apply them over frequency-selective channels. The dispersive nature of such channels causes performance degradation due to intersymbol interference (ISI). The integration of FDEs into various STBC systems has been investigated by several authors [6]-[10]. A comprehensive review of single-carrier frequency domain equalization (SCFDE)can be found in [11]. An elegant Alamouti-like scheme for combining space-time block-coding with single-carrier frequency-domain equalization has been proposed in [12]. Although significant diversity gains were demonstrated, the proposed scheme is not be able to fully exploit the embedded multipath diversity due to its MMSE implementation. In order collect full multipath and diversity gains, MLSD equalization is necessay[13]. However, it is known that the complexity of MLSD equalizer increases with the channel memory, signal constellation size, and the number of transmit/receive antennas. This, in turn, places significant additional computational and power consumption loads on the receiver side. Therefore, low-complexity equalization schemes are particularly desirable especially for cases where the receiver implementation supposed to be simple and required to operate on a limited battery power. In this paper, we propose a novel low complexity two stage MMSE-based receiver design for SC-FDE transmissions. Building upon the MMSE-FDE receiver proposed in [12], we show that linear processing can collect both multipath and spatial diversity gains, leading to significant improvement in performance at nearly no additional complexity. The rest of this paper is organized as follows. We start in Section II by describing our model and assumptions and reviewing the SC-MMSE-FDE. In Section III, the lowcomplexity two-stage MMSE-based receiver is proposed. Numerical results are presented Section IV and the paper is
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
concluded in Section V. −
T
H
Notation: (.), (.) , and(.) denote conjugate, transpose, and Hermitian transpose operations, respectively. [.]k,l denotes the (k, l)th entry of a matrix, [.]k denotes the k th entry of a vector. IN denotes the identity matrix of size N, and 0M ×M denotes all-zero matrix of size M × M . For a vector a = [ a0 · · · aN −1 ]T , [PqN a]s = a ((N − s + q) mod N ) where PN is a N × N permutation ma-trix. Q represents the N × N √ FFT matrix whose (l,k) element is given by Q(l, k) = 1/ N exp(−j2πlk/N ) where 0 ≤ l, k ≤ N − 1. Bold upper-case letters denote matrices and bold lower-case letters denote vectors. II. T RANSMISSION M ODEL We consider space-time block coding transmission assuming a single receive antenna and two transmit antennas over a frequency-selective fading wireless channel, although extension to multiple receive and transmit antennas is straightforward. Specifically, we consider the single-carrier frequency domain equalization scheme proposed in [12]. The channel impulse response (CIR) over block j between the mth transmit antenna and the receive antenna is modeled as a finite impulse response (FIR) filter with coefficients hjm = j T hm [0], ..., hjm [L] , where m = 1, 2 and L denotes the corresponding channel memory length. The random vectors hjm are assumed to be independent zero-mean complex Gaussian with two choices of power delay profile θ(τk ): Uniform power delay profile with variance 1/ (L + 1), and an exponentially decaying power-delay profile θ(τk ) = Ce−τk /τrms with delays τk that are uniformly and independently distributed [13]. The CIRs are assumed to be constant over two consecutive blocks, i.e., j and j + 1, and vary independently every two blocks. Information symbols are first parsed to two streams of M × 1 blocks xji , i = 1, 2, and then multiplied by T T a zero-padding (ZP) matrix Ψ = IT of size M , 0M ×L N × M , where N is the frame length. As explained in [13], the use of zero-padding as the precoding method in a single-carrier transmission scenario ensures that the available multipath diversity is fully exploited. To further remove interblock interference and make the channel matrix circulant, a cyclic prefix (CP) with length L is added between adjacent information blocks. Due to the adopted precoding form, i.e. zero padding, we simply insert additional zeros at the start of the frame as CP. The transmitted blocks are generated ¯ j2 , dj+1 = Jd ¯ j1 , where = −Jd by the encoding rule dj+1 1 2 j j di = Ψxi , m = 1, 2 is the zero-padded information vector and J = PM N is a N × N a partial permutation matrix [14]. With two transmit and one receive antenna and assuming that the channel coefficients remain constant over two consecutive blocks, i.e. Hjm = Hj+1 m = Hm , received blocks j and j + 1 and are given by rj1 = H1 dj1 + H2 dj2 + nj1 j
(1)
j
¯ 2 + H2 Jd ¯ 1 + n2 j+1 r2 j+1 = −H1 Jd
(2)
are complex Gaussian with zero mean and where nj1 and nj+1 2 variance of N0 /2 per dimension. In (1)-(2), H1 , H2 are N ×N circulant matrices with entries [Hm ]k,l = hm ((k − l) mod N ) m = 1, 2, respectively. III. T WO - STAGE MMSE- BASED RECEIVER FOR SC-FDE-STBC The proposed receiver show in Fig.1 includes an MMSE equalizer, followed by linear processing operations. Specifically, the first stage consists of the SC-MMSE-FDE scheme which was originally proposed in [12]. First, the received signals (cf. (1)-(2)) are transformed to the frequency domain by applying the DFT (Discrete Fourier Transform) 1 , i.e. multiplying by the Q matrix, as follows Qrj1 = QH1 dj1 + QH2 dj2 + Qnj1
(3)
j H j = −QHH nj+1 QJ¯ rj+1 1 d2 + QH2 d1 + QJ¯ 2 2 .
(4)
Exploiting the circulant structure of the channel matrices, we have Hm = QH Λm Q
(5)
where Λm , m = 1, 2, is a diagonal matrix whose (n, n) element is equal to the nth DFT coeffi-cient of hm . Using (5) and dropping the superscript j for brevity, we can write (3) and (4) in matrix form as Λ2 Λ Qd1 Qn1 Qr1 ˜ ˜ = ¯1 + = ΛU+ N. ¯1 QJ¯ r2 Qd2 QJn2 Λ2 −Λ (6) ˜ is an orthogonal matrix of size 2N × 2N , we can Since Λ ˜ H , resulting in multiply (6) by Λ 2 2 (7) ri,out = |Λ1 | + |Λ2 | Qdi + ni,out where njout,m is a noise vector with Gaussian each entry still 2 2 with zero-mean and variance of |Λ1 | + |Λ2 | N0 /2 per dimension. The resulting data streams can be now detected by applying the MMSE (minimum mean square error) equalizer [12], i.e., multiplying (7) by −1 1 2 2 IN , (8) |Λ1 | + |Λ2 | + SNR where SNR represents the signal-to-noise ratio. Next, the signals in (7), after multiplying by (8), are fed into a minimum Euclidean distance decoder, yielding x ˆ1 and x ˆ2 , i.e., a decoded version of the transmitted symbols. The decoded signals are then fed into the second stage which is responsible for exploiting the underlying multipath diversity. Specifically, consider the generation of the matrix Πm,l , l = 0, 1, 2, ..., L, as follows: 1 In this paper, we assume that the DFT size N is a power of 2, thus, the terms FFT and DFT are interchangeable.
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
Fig. 1.
Proposed receive block diagram.
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
−1
10
[Πm,l ]p,q =
0 [Hm ]p,q
p − q mod N = l p − q mod N = l,
SC−MMSE−FDE
(9)
Proposed receiver −2
10
where 1 ≤ p, q ≤ N, then, (1) and (2) can be re-written as y1l = r1 − (Π1,l Ψˆ x1 + Π2,l Ψˆ x2 )
(10)
y2l = r2 − −Π1,l JΨˆ x2 + Π2,l JΨˆ x1
(11)
−3
BER
10
−4
10
Under the high SNR assumption, we can safely assume that x1 ≈ x ˆ1 and x2 ≈ x ˆ2 . Therefore, we can write (10) and (11) as follows: y1l = H1,l Ψx1 + H2,l Ψx2 + n1 ,
−5
10
(12) 0
y2l = −H1,l JΨˆ x2 + H2,l JΨˆ x1 + n2 ,
(13)
p − q mod N = l p − q mod N = l.
8 SNR(dB)
12
14
16
−1
SC−MMSE−FDE Proposed receiver
10
(14)
(15)
10
10
−2
¯ m,l J = HH , we Conjugating Jyl2 and using the fact JH m,l obtain H Jyl2 = −HH n2 , 1,l Ψx2 + H2,l Ψx1 + J¯
6
−3
10 BER
[Hm,l ]p,q =
[Hm ]p,q 0
4
Fig. 2. BER performance of SC-MMSE-FDE and proposed receiver (L =1).
where ni (i=1,2) is complex Gaussian with zero mean and variance of N0 /2 per dimension and
2
−4
In matrix form, we can write (12) and (15) as l H1,l H2,l y1 Ψx1 n1 = + . (16) HH −HH Ψx2 J¯ n2 Jyl2 2,l 1,l Multiplying (16) by HH eq,l , we observe that the output streams are decoupled (due to orthogonality of Heq,l ) allowing us to write 2 2 ˜i zli = |H1,l | + |H2,l | Ψxi + n =
2
2
|hm,l | IN Ψxi + n ˜ i , i = 1, 2
where the noise term is still complex Gaussian. Interestingly, the frequency selective channels are now converted in to parallel independent frequency flat channels. The signals zli , l = 0, 1, ..., L, can now be combined, resulting in 2 L
2
−5
10
0
2
|hm,l | IN Ψxi + (L + 1) n ˜i .
(18)
m=1 l=0
The signals in (18) are then fed into a maximum likelihood decoder to recover the transmitted symbols. It is observed from (18) that the maximum achievable diversity order is given by 2 (L + 1). This illustrates that our proposed receiver is able to fully exploit the underlying spatial and multipath diversity gains relying on simple linear processing operations only.
4
6
8 SNR(dB)
10
12
14
16
Fig. 3. BER performance of SC-MMSE-FDE and proposed receiver (L =2).
IV. N UMERICAL R ESULTS
(17)
m=1
˜ zi =
10
In this section, we present Monte-Carlo simulation results for the proposed receiver assuming a quasi-static Rayleigh fading channel . In Figures 1, 2, and 3, the channel impulse responses CIRs are modeled as frequency-selective channels with memory lengths L = 1, 2, 3 and a uniform delay power profile assuming 4-PSK modulation. Fig. 2 illustrates the bit error rate (BER) performance of the proposed receiver for L=1. As a benchmark, we also include the performance of the SC-MMSE-FDE receiver which was proposed in [12]. It is clear from the slope of the curves that our new receiver is able to collect full diversity gains, i.e., 2(L+1)=4, confirming our earlier conclusion. Our simulation results indicate that the proposed receiver outperforms the MMSE-FDE equalizer by ≈ 3.3dB at BER = 10−4 . In Figs. 3 and 4, we provide further results on the performance of the proposed receiver for L=2, 3, respectively.
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
−1
10
−1
10
SC−MMSE−FDE Proposed receiver
SC−MMSE−FDE Proposed receiver
−2
10
−2
10
−3
10
BER
−3
BER
10
−4
10 −4
10
−5
10
−5
10
−6
10
0
5
10
15
SNR(dB) 0
5
10
15
SNR(dB)
Fig. 4. BER performance of SC-MMSE-FDE and proposed receiver (L =3).
V. C ONCLUSION
SC−MMSE−FDE Proposed receiver
−1
10
We propose a novel low complexity two-stage MMSE-based receiver for single carrier frequency-domain equalization (SCFDE) for space-time block coded (STBC) transmissions. We show that, by incorporating linear processing techniques, our MMSE-based receiver is able to collect full antenna and multipath diversity gains, while maintaining low complexity, thus, eliminating the need for maximum-likelihood sequence detection (MLSD), which has certainly prohibitive complexity in practical scenarios. Simulation results demonstrate that our proposed receiver outperforms the conventional SC-MMSEFDE receiver, while maintaining nearly similar complexity.
−2
10
BER
Fig. 6. BER performance of SC-MMSE-FDE and proposed receiver with exponential power delay profile (L =3).
−3
10
−4
10
−5
10
0
5
10
R EFERENCES
15
SNR(dB)
Fig. 5. BER performance of SC-MMSE-FDE and proposed receiver with exponential power delay profile (L =2).
We also include the performance of SC-MMSE-FDE as a benchmark. In Fig. 3, our results indicate that the proposed receiver outperforms the SC-MMSE-FDE equalizer by 4.5dB at BER = 10−4 . In Fig. 4, the performance improvement is 5dB at BER = 10−4 . Interestingly, it observed from Figs. 2, 3 and 4 that performance improvement of the proposed receiver, in comparison to the benchmark, increases as L increases. It should be emphasized here that this improvement comes at almost no additional cost. In Fig. 5 and Fig. 6, the BER performance of the proposed receiver for L= 2, 3 are presented, assuming a quasi-static Rayleigh fading channel with exponential power delay profile and 16-PSK modulation. In Fig. 5, our results indicate that the proposed receiver outperforms the SC-MMSE-FDE equalizer by 4.6 dB at BER = 10−4 . This performance improvement is 5 dB for Fig. 6 at BER = 10−4 .
[1] M. V. Clark,“Adaptive frequency-domain equalization and diversity combining for broad-band wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1385-1395, Oct. 1998 [2] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., pp. 100-109,Feb. 1995. [3] J. G. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi element antennas,” Bell Labs Tech. J., vol.2, pp.41-59, Autumn 1996. [4] S. M. Alamouti, “Simple transmit diversity technique for wireless communications,” IEEE Journal on Select Areas in Communications, vol. 16, pp. 1451-1458, 1998. [5] V. Tarokh, H. J. Jafarkhani and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Transactions on Information Theory, vol.45, no.5, p.1456-1467, July 1999. [6] N. Al-Dhair, “Overview and Comparison of Equalization Schemes for Space Time Coded Signals with application to EDGE”, IEEE Trans. on Signal Proc, vol. 50, no. 10, Oct. 2002. [7] M. Uysal, N. Al-Dhahir and C. N. Georghiades, “Three Space-Time Block Coding Schemes for Frequency-Selective Fading Channels with Application to EDGE”, IEEE VTC’01 Fall, Atlantic City, USA, October 2001. [8] F. Pancaldi and G.M. Vitetta, “Frequency domain equalization for spacetime block coded systems,” IEEE Trans. Wireless Commun., vol. 4, no.6, pp. 2907-2916, Nov. 2005. [9] R. Kalbasi, R. Dinis, D. Falconer, and A. Banihashemi, “Hybrid timefrequency layered space-time receivers for severe time-dispesive channels,” in Proc. IEEE 5th Workshop Signal Processing Advances in Wireless Communications (SPAWC’04), May 2004, pp. 218-222.
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
[10] R. Dinis, R. Kalbasi, D. Falconer, and A.H. Banihashemi, “Iterative layered space-time receivers for single-carrier transmission over severe time-dispersive channels,” IEEE Commun. Lett., vol. 8, no. 9, pp. 579581, Sept. 2004. [11] F. Pancaldi, G. M. Vitetta, N. Al-Dhahir, R. Kalbasi, M. Uysal, and H. Mheidat, “Single Carrier Frequency Domain Equalization: A Review”, IEEE Signal Processing Magazine, vol. 25, no. 5, p. 37-56, September 2008. [12] N. Al-Dhahir, “Single Carrier Frequency Domain Equalization for Space Time Block Coded Transmissions over Frequency Selective Fading channels” IEEE Comm. Lett., vol. 5, no. 7, July 2001. [13] O. Edfors, M. Sandell, J.-J. van de Beek, S. K. Wilson, and P. O. Brjesson,“OFDM channel estimation by singular value decomposition” IEEE Comm. Lett., vol. 46, no. 7, pp. 931-939, July 1998. [14] S. Zhou and G. B. Giannakis,“Space-Time Coding with Maximum Diversity Gains over Frequency-Selective Fading Channels,” IEEE Signal Processing Letters, vol. 8, no. 10, pp. 269 272, October 2001.
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
