Generalised prefix for space–time block-coded ... - IEEE Xplore

www.ietdl.org Published in IET Communications Received on 1st July 2013 Accepted on 3rd February 2014 doi: 10.1049/iet-com.2013.1166

ISSN 1751-8628

Generalised prefix for space–time block-coded orthogonal frequency division multiplexing wireless systems over correlated multiple-input multipleoutput channels Hakan Yıldız1, Yusuf Acar2, Todor Cooklev3, Hakan Dogan4 1

Department of Information Technologies, Istanbul University, Fatih, Istanbul 34134, Turkey Department of Electronics Engineering, Istanbul Kultur University, Bakirkoy, Istanbul 34156, Turkey 3 Wireless Technology Center, Indiana University-Purdue University Fort Wayne (IPFW), USA 4 Department of Electrical and Electronics Engineering, Istanbul University, Avcilar, Istanbul 34320, Turkey E-mail: [email protected] 2

Abstract: Space–time block coding (STBC) orthogonal frequency division multiplexing (OFDM) is known to improve the reliability of broadband communication over wireless links. However, this technique suffers from a performance loss when the multiple-input multiple-output (MIMO) channel is correlated. This study addresses this problem. A STBC-OFDM system is proposed that is based on convolution that is skew-circular, rather than circular. This skew-circular convolution leads to one additional degree of freedom. The additional parameter can be optimised leading to improved performance for MIMO channels that are correlated. The skew-circular convolution is achieved via the introduction of a generalised prefix. The simulation results are given, illustrating the performance improvement of the proposed STBC-OFDM system. Furthermore, the proposed system is shown to be robust when channel state information is known not precisely, but imperfectly.

1

Introduction

There has been an increasing interest in advanced techniques for next generation wireless communication systems to meet the requirements of high data rate, reliability, capacity and mobility in practical wireless environments. Orthogonal frequency division multiplexing (OFDM) technique has been widely accepted for high data rate transmission because of its numerous advantages. OFDM with a sufficiently long prefix can convert a frequency selective channel into multiple independent flat-fading subchannels [1, 2]. Space–time block coding (STBC) is a well-known transmit diversity scheme. STBC provides diversity by transmitting a space–time coded signal through multiple antennas. These diversity gains are achieved with minimum computational complexity, since at the receiver very simple maximum likelihood decoding can be used, based only on linear processing. On the other hand, if a space–time trellis code is used, the decoder complexity increases exponentially with the number of transmit antennas [3, 4]. OFDM and STBC can be combined to achieve a higher spectral efficiency than each technology can achieve individually. In such STBC–OFDM system, data symbols are encoded over space (using multiple transmit antennas) and over time (using OFDM symbols). STBC–OFDM schemes have been investigated intensively [5, 6]. Various IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166

4G wireless communication systems, including worldwide interoperability for microwave access (WiMAX) and long-term evolution (LTE)-advanced [7], currently use STBC–OFDM. The performance improvement of STBC is a direct result of the assumption about the multiple-input multiple-output (MIMO) channel being uncorrelated. Unfortunately, uncorrelated MIMO channels exist only in theory; practical MIMO channels have different degrees of correlation. As a result, the performance gains promised in theory will be reduced in practice. The reasons for this correlation include limited physical size of transmitters/receivers (e.g. handsets), antenna design and the lack of sufficient scatters in the transmission environments [8, 9]. Antenna correlation is generally viewed as an obstacle to realise the desired performance of a wireless system and considered as an adverse effect that degrades the system performance [10]. Other than improving the antenna design, few solutions to the MIMO channel correlation problem are known. In this paper, we suggest a digital signal processing solution. The proposed digital signal processing technique is based on skew-circular convolution that requires OFDM symbols to have a generalised prefix (GP). As a result, the physical MIMO channel appears at the baseband level as a MIMO channel with less correlation. In the literature several prefix techniques have been proposed for OFDM. One of these techniques is 1589

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www.ietdl.org zero-padding (ZP) [11, 12]. In [13], Muck et al. proposed the insertion of a pseudo-random postfix (PRP) between OFDM symbols. This postfix is constructed by means of a known vector weighted by a pseudo-random scalar sequence to enable semi-blind channel estimation. As described in [13], at the receiver, PRP-OFDM symbols are transformed into ZP-OFDM symbols by subtracting the contribution of the postfix. Subsequent additional processing is performed to obtain a symbol corresponding to cyclic prefix (CP) OFDM. Other approaches that have been suggested are based on the idea that the CP can be replaced by a pre-defined sequence of known symbols [14, 15]. These approaches are referred to as known symbol padding OFDM and unique word OFDM [16]. In these systems, the known samples in the guard interval could be used for channel, timing and carrier offset estimation. It must be noted that all techniques mentioned above other than the CP do not diagonalise the equivalent channel matrix and do not decompose the multicarrier system into N parallel independent systems. Since they do not diagonalise the equivalent channel matrix, they are computationally much more complex to implement and are not used by practical systems. Currently, all practical STBC–OFDM systems employ a CP because of its computational simplicity and convenience for detection and synchronisation. The technique proposed in this paper maintains all these advantages, and in addition achieves lower bit error rate (BER) for correlated MIMO channels. Recently, another prefix construction technique called generalised prefix (GP) was proposed for OFDM systems [17] that also diagonalises the channel matrix, that is, the CP is not the only prefix with this property. It was also shown that this GP-OFDM system has the lowest BER among the other prefix construction techniques, considering all other parameters identical. In this paper, the GP structure is applied for STBC–OFDM systems. When the GP is used for STBC–OFDM systems, a new degree of freedom is introduced. This degree of freedom shifts the phases of the multipath components and effectively changes the wireless channel experienced by the OFDM system into a different channel as long as the prefix length is equal to or greater than the delay spread of the channel. As technique for effectively reducing the MIMO channel correlation, this degree of freedom has not been known previously. For concreteness, we consider 2 × 2 and 4 × 1 MIMO channels. We demonstrate that the proposed STBC– OFDM system outperforms conventional STBC–CP-OFDM systems for correlated and uncorrelated MIMO channels. The simulation results also confirm that the proposed technique leads to lower BER for practical spatially correlated channels based on the IEEE 802.11n models. It is also shown that the proposed method can be applied to non-contiguous STBC– OFDM, where there are unused carriers. The paper is organised as follows: In Section 2, we discuss the OFDM system model to introduce the notations. The signal model in [17] is simplified and written as a vector-matrix form for single-input single-output (SISO) systems, which, in turn, is easier to apply to the MIMO case. In Section 3, we develop STBC–GP-OFDM. In Section 4, we describe the correlative MIMO channel model. The simulation results are presented in Section 5. Finally, Section 6 contains the conclusions. Notation: Vectors (matrices) are denoted by boldface lower (upper) case letters; all vectors are column vectors; (·)−1, (·) *, (·)T and (·)† denote the matrix inversion, the conjugate, 1590 & The Institution of Engineering and Technology 2014

transpose and conjugate transpose, respectively; IP denotes the P × P identity matrix. OP × L denotes the P × L zero matrix.

2

OFDM system model

First, we consider OFDM systems with single transmit and receive antennas. The N frequency domain constellation symbols ck can be denoted as
T c = c1 , c2 , . . . , cN

(1)

The time-domain OFDM symbol is obtained by applying the N-point inverse discrete Fourier transform (DFT)
T s = F † c = s1 , s 2 , . . . , s N

(2)

Then, the CP of length P is inserted as s˜ = Gs, where G has N + P rows and N columns  ⎡ ⎤  OP×(N −P)  I P ⎦ (3) G=⎣ IN The transmitted signal becomes    s˜ = ⎣ sN −P+1 sN −P . . . sN 



















 ⎡

⎤T s1 , s 2 , . . . , sN ⎦











(4)

N

P

The received signal is a result of linear convolution with the wireless channel impulse response, which can be represented in matrix form as r′ = H s˜ + n

(5)

where n is the additive white Gaussian noise and ⎡

h(1) ⎢ h(2) ⎢ ⎢ .. ⎢ . ⎢ ⎢ ⎢ H = ⎢ h(L) ⎢ ⎢ ⎢ 0 ⎢ ⎢ . ⎣ .. 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ h(1) ⎥ ⎥ ⎥ h(2) ⎥ ⎥ ⎥ h(L − 1) ⎦

h(1) h(2) .. . h(L) .. . 0

..

.

..

.

h(L)

(N +P+L−1)×(P+N )

(6) The wideband wireless channel is modelled with respect to the baseband by a finite impulse response (FIR) model with L channel coefficients. To remove completely the inter-symbol interference among the OFDM symbols, it is required that P ≥ L − 1. To remove the prefix, out of the N + P + L − 1 received samples only N are kept. Then the DFT is applied to obtain r = FMr′ = FMHGF† c + w

(7)

where w = FMn and
M = ON ×P

IN

ON ×(L−1)



(8)

IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166

www.ietdl.org In a simplified form, it can be written that  +w r = Hc

(9)

 = FMHGF† is a diagonal matrix because it is where H assumed that the channel is static for the duration of one OFDM symbol  = diag[H(0), H(1), . . . , H(N − 1)] H

(10)

where H(k) = H(ejwk ) =

N −1 

h(n) e−jwkn

(11)

n=0

According to the GP technique, modulation is performed by sc = D−1 F † c

(12)

where D is a diagonal matrix ⎡ ⎢ ⎢ ⎢ D=⎢ ⎢ ⎣

⎤

1

c

c2

..

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

.

(13)

cN −1

Clearly, the GP is a variable, while the CP is fixed: the CP corresponds to f = c = 1. As a variable, the prefix can be optimised, minimising the BER. It was shown in [17] that the GP leads to better performance for the SISO case. The improvement is bigger for channels that have deep fades or, in the extreme case, spectral nulls.

3

STBC–OFDM

Fig. 1 shows a schematic of a STBC–GP-OFDM system with Alamouti’s scheme for operation with two transmit and two receive antennas. It is well known that the Alamouti’s scheme is the only method which retains orthogonality and full rate [3]. The purpose of this paper is to demonstrate that the GP approach proposed for SISO–OFDM systems [17] can be extended to the STBC case. 3.1

Alamouti’s scheme

Assume that c1 and c2 are the two vectors of complex data symbols. Each of these is modulated according to (12) and then prefixes are added according to (14). Note that the same GP is applied for every spatial stream. Assume also that Hij is the channel between the ith receive and jth transmit antennas. Then the received signal vectors are r1 = H 11 G c D−1 F † c1 + H 12 G c D−1 F † c2 + w1 r3 = H 21 G c D−1 F † c1 + H 22 G c D−1 F † c2 + w3

To add a GP s˜c = G c sc

(14)



OP×(N −P) IN

f.I P

 (15)

After successive multiplication with M (to remove the prefix), then D, and finally F, we have

In general, f and c are any complex numbers for which f = cN. However, to maintain the same peak-to-average power ratio (PAPR), their absolute values must be equal to one. At the receiver, the prefix is removed in an identical fashion as in the CP case. This is followed by multiplication with D and the DFT matrix r = FDMHGc D−1 F † c + w

(16)

The above equation can be written as c ·c+w r=H

(17)

 c = FDMHGc D−1 F † . The GP results are not in where H circular, but skew-circular convolution. It is shown in [17] that the channel frequency responses become    c = diag Hc (0), Hc (1), . . . , Hc (N − 1) H

(20)

r4 = −H 21 G c D−1 F † c∗2 + H 22 G c D−1 F † c∗1 + w4

where Gc =

r2 = −H 11 G c D−1 F † c∗2 + H 12 G c D−1 F † c∗1 + w2

(18)

12 ˜1 r˜ 1 = H 11 c c1 + H c c2 + w 22 ˜2 r˜ 3 = H 21 c c1 + H c c2 + w 12∗ ˜3 r˜ ∗2 = −H 11∗ c c2 + H c c1 + w

(21)

22∗ ˜4 r˜ ∗4 = −H 21∗ c c2 + H c c1 + w

⎡

⎤

⎡

⎤

⎡ ⎤ ˜1 w r˜ 1 ⎢ 21  22 ⎥  ⎢ r˜ 3 ⎥ ⎢ H c ⎢w ⎥ H c ⎥ c1 ˜ ⎥ ⎢ ∗⎥=⎢ ⎢ 3⎥ ⎣ r˜ 2 ⎦ ⎢ H 12∗ −H 11∗ ⎥ c2 + ⎣ w˜ ∗2 ⎦ ⎣ c c ⎦ 

r˜ ∗4 w˜ ∗4 22∗ 21∗ c H −H







c c













rm ˜ w H 12 c

H 11 c

(22)

Ec

The decoded signal for rm = Ec c + w˜ can be obtained as † cˆ = Ec rm

(23)

Equation (23) could be written as follows

and Hc (k) =

N −1 

c h(n)e n

−jwkn

n=0

IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166

21∗ 12 ∗ 22 ∗ cˆ 1 = H 11∗ c r˜ 1 + H c r˜ 3 + H c r˜ 2 + H c r˜ 4

(19)

22∗ 11 ∗ 21 ∗ cˆ 2 = H 12∗ c r˜ 1 + H c r˜ 3 − H c r˜ 2 − H c r˜ 4

(24)

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Fig. 1 Block diagram of STBC–OFDM system (2 × 2) with GP

These combined signals are then sent to the maximum likelihood detector. 3.2

r˜ 7 = −H 1c c∗3 + H 2c c∗4 + H 3c c∗1 − H 4c c∗2 + w˜ 7 r˜ 8 = −H 1c c∗4 − H 2c c∗3 + H 3c c∗2 + H 4c c∗1 + w˜ 8

Tarokh’s scheme

STBC was expanded in [4] for an arbitrary number of antennas. Fig. 2 shows a schematic of a STBC–GP-OFDM system with Tarokh’s scheme for operation with four transmit and one receive antenna. The received signals are easily shown to be r˜ 1 = H 1c c1 + H 2c c2 + H 3c c3 + H 4c c4 + w˜ 1 r˜ 2 = −H 1c c2 + H 2c c1 − H 3c c4 + H 4c c3 + w˜ 2 r˜ 3 = −H 1c c3 + H 2c c4 + H 3c c1 − H 4c c2 + w˜ 3 r˜ 4 = −H 1c c4 − H 2c c3 + H 3c c2 + H 4c c1 + w˜ 4 r˜ 5 = H 1c c∗1 + H 2c c∗2 + H 3c c∗3 + H 4c c∗4 + w˜ 5 r˜ 6 = −H 1c c∗2 + H 2c c∗1 − H 3c c∗4 + H 4c c∗3 + w˜ 6 1592 & The Institution of Engineering and Technology 2014

(25)

In matrix form this is equivalent to ⎡ ⎡

H 1c

H 2c

H 3c

H 4c

⎤

⎡ ⎤ ⎢ 2 ⎥ w˜ 1 r˜ 1 ⎢ H c −H 1c H 4c −H 3c ⎥ ⎥ ⎢ r˜ 2 ⎥ ⎢ ⎢ w˜ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ H 3c −H 4c −H 1c ⎥ H 2c ⎥ ⎡ ⎤ ⎢ ⎢ r˜ 3 ⎥ ⎢ ⎢ w˜ 3 ⎥ c ⎥ 1 ⎢ ⎥ ⎢ 4 ⎢ ⎥ ⎢ r˜ 4 ⎥ ⎢ H c ⎢ w˜ 4 ⎥ H 3c −H 2c −H 1c ⎥ c2 ⎥ ⎥⎢ ⎢ ⎥ ⎢ ∗⎥=⎢ ⎢ ⎥⎣ ⎦+⎢ ˜∗⎥ ⎢ r˜ 5 ⎥ ⎢ H 1∗ H 2∗ ⎥ ⎥ c3 H 3∗ H 4∗ ⎢ ∗⎥ ⎢ c ⎢ w5∗ ⎥ c c c ⎥ ⎢ r˜ 6 ⎥ ⎢ 2∗ ⎢ ⎥ ˜ c w ⎥ 1∗ 4 ⎢ ∗ ⎥ ⎢H ⎢ 6∗ ⎥ H 4∗ −H 3∗ c c ⎥



⎣ r˜ ⎦ ⎢ c −H c ⎣w ⎦ ˜ ⎥ 7 7 ⎢ H 3∗ −H 4∗ −H 1∗ H 2∗ ⎥ c ˜ ∗8 r˜ ∗8 w ⎣ c c c c ⎦







H 4∗ H 3∗ −H 2∗ −H 1∗ rm w˜ c c c c



























⎤

Ec

(26) IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166

www.ietdl.org

Fig. 2 Block diagram of STBC–OFDM system (4 × 1) with GP

Similarly, the transmitted spatial streams can be recovered by (23) and written as (27) and (28). Then these combined

signals are sent to the maximum likelihood detector. (see (27))

⎡

⎡ 1∗ ⎤ Hc cˆ 1 ⎢ 2∗ ⎢ cˆ 2 ⎥ ⎢ H c ⎢ ⎥=⎢ ⎣ cˆ 3 ⎦ ⎢ H 3∗ ⎣ c cˆ 4 H 4∗ c ⎡

H 2∗ c

H 3∗ c

H 4∗ c

H 1c

H 2c

H 3c

−H 1∗ c

−H 4∗ c

H 3∗ c

H 2c

−H 1c

−H 4c

H 4∗ c

−H 1∗ c

−H 2∗ c

H 3c

H 4c

−H 1c

−H 3∗ c

H 2∗ c

−H 1∗ c

H 4c

−H 3c

H 2c

IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166

⎤ r˜ 1 ⎤⎢ r˜ 2 ⎥ ⎥ H 4c ⎢ ⎢ r˜ 3 ⎥ ⎥ ⎢ ⎥ H 3c ⎥⎢ r˜ 4 ⎥ ⎥⎢ ∗ ⎥ ⎢ ⎥ −H 2c ⎥ ⎦⎢ r˜ 5∗ ⎥ ⎢ r˜ 6 ⎥ ⎥ −H 1c ⎢ ⎣ r˜ ∗7 ⎦ r˜ ∗8

(27)

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www.ietdl.org (see (28)) 3.3

Calculation of optimum c

The GP transforms the wideband MIMO channel with FIR transfer function N −1 

H(z) =

h(n)z−n

(29)

n=0

into an equivalent channel with transfer function H c (z) =

N −1 

cn h(n)z−n

(30)

n=0

The value of c can be determined using an optimisation procedure with the BER as the objective function. Since |c| = 1, we can write c = ejα, that is, α is like a frequency shift. Therefore, the minimisation problem could be written as follows

a∗ = arg min Pe (c)|c=e ja

(31)

a[[0,2p]

where Pe(c) is the bit error probability (BER) of the proposed system and α* is the optimum frequency shift. The search space is limited to [0, 2π] because the objective function is periodic in α, with the period being 2π. Since the same prefix is used on every spatial stream, this is a one-dimensional optimisation. The average BER can be derived analytically. To accomplish this, we can write, for the 2 × 2 case  G ON (32) ON G 2           21 2  12 2  22 2 where Γ equals to H 11 c  + H c  + H c  + H c  . † Ec Ec =



       2    21 2  12 2  22 2    21 2 H c,k  + H c,k  + H c,k  and Gk = H 11 c,k  +H c,k  +      12 2  22 2 H c,k  +H c,k  are for 2 × 2 and 4 × 1, respectively. The BER is a continuous function of α and many optimisation algorithms can be used. One efficient algorithm is the golden section search method [17]. This optimisation requires channel state information (CSI) and can be done at the receiver or the transmitter. Providing full CSI to the transmitter represents significant overhead, but is still sometimes done in practice (such as in IEEE 802.11n). The optimisation may be done at the receiver, with only the optimum value sent to the transmitter. The overhead for sending the optimum value to the transmitter is negligible resulting few overheads for the feedback of the phase c data, especially compared to the overhead of providing full CSI at the transmitter. The optimum value can be used only while the channel for which it was calculated is static. Note that the proposed prefix may not be globally optimal considering all linear and non-linear prefix constructions. This question may be appropriate for a future study.

4

MIMO channels with correlation

MIMO systems employ multiple antennas. However, the signals received by the receiver’s antennas in practice are not independent, but correlated. The correlation depends on parameters such as the mean angle of arrival (AoA), the mean angle of departure (AoD), the angular spread (AS) at the receiver and the angular spread at the transmitter [18]. Antennas for MIMO systems are geometrically placed to minimise the spatial correlation. One solution is to make antennas equally spaced by d = λ/2, where λ is the wavelength. Also, the propagation environment for MIMO channels in practice does not exhibit independent scattering, leading to spatial correlation and degradation in the system performance. 4.1 Generation of correlated MIMO channel coefficients

Similarly, for the 4 × 1 case ⎡

G ⎢ ON ⎢ Ec Ec = ⎣ ON ON †

ON G ON ON

ON ON G ON

The uncorrelated MIMO channel gain matrix can be written as

⎤

ON ON ⎥ ⎥ ON ⎦ G

(33)

where Γ equals to |H 1c |2 + |H 2c |2 + |H 3c |2 + |H 4c |2 . Based on (32), (33), and the well-known closed-form expressions for the BER of, for example, binary phase shift keying (BPSK), the average BER of the system is  N 1 Gk Es Pe = Q s2 N k=1 where

Q(·)

denotes

the

Q-function.


hl = h(1)(l)

h(2)(l)

h(MN )(l)

T

(35)

where h (l ) is a complex Gaussian random variable with zero mean for the lth path and its uncorrelated channel coefficients. By multiplying the uncorrelated MIMO fading channel vector by an MN × MN matrix C, the correlated MIMO channel coefficients can be generated [18]. The matrix C can be called correlation-shaping matrix or symmetric mapping matrix. The correlated coefficients can be obtained according to

(34) 2    Gk = H 11 c,k  +

...

l h˜ =

 l Pl Ch

(36)

where Pl is the average power of the lth path. Now, the

2∗ 3∗ 4∗ 1 ∗ 2 ∗ 3 ∗ 4 ∗ cˆ 1 = H 1∗ c r˜ 1 + H c r˜ 2 + H c r˜ 3 + H c r˜ 4 + H c r˜ 5 + H c r˜ 6 + H c r˜ 7 + H c r˜ 8 1∗ 4∗ 3∗ 2 ∗ 1 ∗ 4 ∗ 3 ∗ cˆ 2 = H 2∗ c r˜ 1 − H c r˜ 2 − H c r˜ 3 + H c r˜ 4 + H c r˜ 5 − H c r˜ 6 − H c r˜ 7 + H c r˜ 8 4∗ 1∗ 2∗ 3 ∗ 4 ∗ 1 ∗ 2 ∗ cˆ 3 = H 3∗ c r˜ 1 + H c r˜ 2 − H c r˜ 3 − H c r˜ 4 + H c r˜ 5 + H c r˜ 6 − H c r˜ 7 − H c r˜ 8

(28)

3∗ 2∗ 1∗ 4 ∗ 3 ∗ 2 ∗ 1 ∗ cˆ 4 = H 4∗ c r˜ 1 − H c r˜ 2 + H c r˜ 3 − H c r˜ 4 + H c r˜ 5 − H c r˜ 6 + H c r˜ 7 − H c r˜ 8

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IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166

www.ietdl.org correlated MN × 1 MIMO channel vector coefficients can be written as

Table 1 IEEE 802.11n MIMO channel model parameters in non-line-of-sight mode

 T l (l) (l) (l) (l) (l) (l) (l) , h , . . . , h , h , h , . . . , h , h , . . . , h h˜ = h(l) MN 11 21 M1 12 22 M2 13

path delays, in seconds

(37) Actually, the correlation-shaping matrix C in (36) defines the spatial correlation coefficients, and furthermore V = CC

T

(38)

where Ω is a root-power correlation matrix that is non-singular  √ R, for field type V= (39) R, for complex type where R is a spatial correlation matrix and given as  RRx ⊗ RTx : downlink R= RTx ⊗ RRx : uplink

(40)

where ⊗ denotes the Kronecker product. RTx and RRx are symmetric spatial correlation matrices for the transmitter and receiver, respectively. 4.2

IEEE 802.11n channel model

The correlated MIMO channels can be generated by the correlation matrices RRx and RTx for different environments. For example, correlation matrices for the IEEE 802.11n indoor wireless channel models can be calculated using the following assumptions: † The power azimuth spectrum (PAS) and the power delay spectrum (PDS) are separable: each tap is modelled independently. † The PAS and the Doppler spectrum for each tap are separable: the spatial correlation (correlation matrices) and temporal correlation (Doppler spectrum) for each tap are modelled independently. † Each tap is modelled using the Kronecker model for Rician channels, hence it is assumed that the transmit and receive correlation matrices are separable for each tap. The IEEE 802.11n channel models are devised for indoor wireless local area networks operating in the 2.4 and 5 GHz bands and occupying a total bandwidth of up to 100 MHz. The specified six spatially correlated channel models, labelled A to F, are defined in [19]. Each channel model also has a path loss model, MIMO multipath fading model, the Doppler spectrum and a certain number of taps, where each tap is characterised by a relative delay with respect to the first path delay. Moreover, each model further comprises a number of clusters, which correspond to overlapping subsets of the tap delays. In this section, we focus on model B (residential with maximum delay of 80 ns and delay spread of 15 ns) that has two clusters: cluster 1 corresponds to tap delays 0–40 ns (in steps of 10 ns), whereas cluster 2 corresponds to tap delays 20–80 ns (also in steps of 10 ns). Hence, clusters 1 and 2 comprise 5 and 7 tap delays, respectively, and overlap in 3 tap delays (20, 30 and 40 ns). Each cluster is assigned a set of spatial properties such as AoA, AoD and IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166

[0 10 20 30 40 50 60 70 80] × 10−9 [0 −5.4 −10.8−16.2 −21.7 −inf −inf −inf −inf]; 0.25

average path gains of cluster 1, in dB spacing at the transmit antennas (normalised by the wavelength) spacing at the receive antennas (normalised by the wavelength) maximum Doppler shift for all paths (identical) angular spreads – cluster 1

0.25 3

angular spreads – cluster 2 mean angles of departure – cluster 1 mean angles of departure – cluster 2 angular spreads on receiver side – cluster 1 angular spreads on receiver side – cluster 2 mean angles of arrival on receiver side – cluster 1 mean angles of arrival on receiver side – cluster 2

[14.4 14.4 14.4 14.4 14.4 −inf −inf −inf −inf] [−inf −inf 25.4 25.4 25.4 25.4 25.4 25.4 25.4] [225.1 225.1 225.1 225.1 225.1 −inf −inf −inf −inf] [−inf −inf 106.5 106.5 106.5 106.5 106.5 106.5 106.5] [14.4 14.4 14.4 14.4 14.4 −inf −inf −inf −inf]; [−inf −inf 25.2 25.2 25.2 25.2 25.2 25.2 25.2]; [4.3 4.3 4.3 4.3 4.3 −inf −inf −inf −inf] [−inf −inf 118.4 118.4 118.4 118.4 118.4 118.4 118.4]

AS. These parameters assume the same values for all tap delays pertaining to a given cluster and determine the transmit and correlation matrices associated with each tap delay. The model B channel parameters are summarised in Table 1.

5

Simulation results

In this section, we compare the proposed STBC–GP-OFDM with STBC–CP-OFDM. Alamouti 2 × 2 and Tarokh 4 × 1 schemes are considered. In these simulations, perfect knowledge of the channel transfer function at the receiver is assumed. The numbers of subcarriers are set to 64 and the length of the prefix is assumed to be equal to 16. Furthermore, we assume non-contiguous OFDM, where not all carriers are used (carriers 1–5, 35–45 and 59–64 are unused). Zeros are assigned to the unused carriers. We investigate here different scenarios to assess the performance because WiMAX/LTE-like systems have different antenna configurations to optimise the performance of the system. 5.1

Scenario 1

To evaluate the performance of the proposed system, we consider a 2 × 2 STBC–OFDM system where the total transmitted power is normalised for the transmit antennas. The data symbols are uncorrelated BPSK symbols. The MIMO wireless channels are modelled by correlated complex Gaussian channel model of length L. The spatial correlation matrices for the transmit and receive antennas are 

R2×2 Rx

1 = a∗

 a , 1



RT2×2 x

1 = b∗

b 1



where α and β are the correlation values between the antenna elements at the receiver and the transmitter, respectively. We 1595

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www.ietdl.org evaluate the performance for three correlation values: α = β = 0, representing no correlation, α = β = 0.3, representing low correlation, α = β = 0.6, representing medium correlation, α = β = 0.9, representing high correlation. In practice, high correlation values are typical. In Fig. 3, the BERs of the STBC–CP-OFDM and the proposed STBC–GP-OFDM systems are shown as functions of signal-to-noise ratio for channels with no or low correlation. It is seen that STBC–GP-OFDM achieves superior performance; in particular, it has a gain of about 3.25 and 3.5 dB at BER = 10−5 for α = β = 0 and α = β = 0.3, respectively. Fig. 4 shows additional simulation results when the spatial correlation is higher. In this case the overall performance is decreased for both systems, but the proposed system maintains its advantage. Fig. 4 shows that gains of about 3.6 and 5 dB are obtained at BER = 10−5 for α = β = 0.6 and

α = β = 0.9. Therefore, the higher the correlation, the bigger the performance improvement. Fig. 4 also shows that the new system for α = β = 0.9 performs almost as STBC– CP-OFDM for α = β = 0.6, effectively reducing the correlation. Note that the channel matrices (Ec) in (22) and (26) are employed to optimise the performance of STBC–OFDM system. If c = 1 then the system is equal to CP-OFDM. In SISO systems, it was demonstrated that c = 1 is not adequate when the channel frequency response has deep fades [17]. By using the optimised value, a channel with spectral nulls or deep fades was converted into a channel with no spectral nulls or fades that are less deep in SISO systems. 5.2

Scenario 2

Full code rate and full diversity is not possible for more than two transmit antennas. Tradeoffs have to be made to either lower the code rate or have a reduced diversity order for STBC designs. In this subsection, we evaluate the BER performance of the proposed scheme in 4 × 1 systems. As shown in Fig. 5, the proposed scheme has a performance gain of 3 dB over the conventional scheme at a BER of 10−5 for channels with no and low correlation. Similarly, it is shown in Fig. 6 that the gain is about 3.5 and 4.5 dB at the same BER = 10−5 for α = β = 0.6 and α = β = 0.9, respectively. We conclude from these figures that the proposed scheme outperforms the conventional scheme for different antenna configurations and spatial correlations. 5.3

Fig. 3 BER comparison of GP STBC–OFDM and CP STBC– OFDM when Alamouti (2 × 2) code is employed (α = β = 0 and α = β = 0.3)

Fig. 4 BER comparison of GP STBC–OFDM and CP STBC– OFDM when Alamouti (2 × 2) code is employed (α = β = 0.6 and α = β = 0.9) 1596 & The Institution of Engineering and Technology 2014

Scenario 3

In this scenario, we consider other practical channel models such as the IEEE 802.11n channel model with parameters such as AoA, AoD and AS given in Table 1. The results are shown in Fig. 7. It is seen again that the proposed system outperforms the state-of-the-art and is a strong candidate for next-generation wireless communications systems. In particular, Fig. 7 shows that gains of about 3.5 and 5.4 dB are obtained at BER = 10−5for λ = 0.5 and λ = 0.1.

Fig. 5 BER comparison of GP STBC–OFDM and CP STBC– OFDM when Tarokh’s scheme (4 × 1) code is employed (α = β = 0 and α = β = 0.3) IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166

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Fig. 6 BER comparison of GP STBC–OFDM and CP STBC– OFDM when Tarokh’s scheme (4 × 1) code is employed (α = β = 0.6 and α = β = 0.9) Fig. 8 Pilot pattern for dedicated time slots (block-type)

Therefore, the estimated CSI is employed for the following B − 1 OFDM symbols in the frame. The pilot-assisted channel estimation scheme that we use is based on least square (LS) techniques. Finally, Figs. 9 and 10 compare the BER performance of STBC–GP-OFDM and STBC–CP-OFDM for different antenna separations while taking into account the effects of imperfect channel estimation at the receiver. While both systems exhibit reduced performance when the channel coefficients are known only approximately, it is observed that the BER performance of the STBC–GP-OFDM is always superior to that of an STBC–CP-OFDM. It is shown that training-based channel estimation exhibited a 1-dB performance degradation in comparison to perfect CSI, due to the effect of CE errors for both STBC–CP-OFDM and STBC–GP-OFDM systems. Fig. 7 BER comparison of GP STBC–OFDM and CP STBC– OFDM when Alamouti (2 × 2) code is employed for IEEE 802.11n channel model for different antenna separations

5.4

Scenario 4

Since the performance improvement of the proposed system is due to optimisation based on the channel coefficients, it is of considerable interest to investigate the effect of imperfect channel estimation. Will the performance advantages hold when the optimisation is done using imprecise channel coefficients? To investigate this question, channel estimation using pilot tones is performed assuming the parameters as in Scenarios 1–3. The pilot pattern is shown in Fig. 8, where the OFDM frame is assumed to consist of B OFDM symbols, the first of which is dedicated to pilots. In other words, a block-type estimation, which inserts pilot tones into all of subcarriers is considered in the first OFDM symbol duration. Then the optimal c was returned to the transmitter. We also consider a quasi-static fading model, where the channel state is considered to be constant during the transmission of a block consisting of several OFDM symbols. IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166

Fig. 9 Impact of estimated channel impulse response on BER performance of the compared systems when Alamouti (2 × 2) code is employed for IEEE 802.11n channel model, distance = 0.1 Λ and = 0.25 Λ 1597

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www.ietdl.org the Research Fund of the University of Istanbul under project number 27706.

8

Fig. 10 Impact of estimated channel impulse response on BER performance of the compared systems when Alamouti (2 × 2) code is employed for IEEE 802.11n channel model, distance = 0.5 Λ and 1 Λ

6

Conclusions

In this paper, a new STBC–OFDM system is proposed. The proposed system achieves skew-circular and not circular convolution between the transmitted data vector and the MIMO channel coefficients. This skew-circular convolution is, in turn, achieved by the use of GP in the OFDM symbol construction. This GP provides an additional degree of freedom, which can be optimised. Extensive simulation results for different antenna configurations and environments are presented. Under identical assumptions the proposed system always achieves superior performance compared with currently known STBC–OFDM systems. Its effectiveness is especially high in the practical case when the spatial correlation of the MIMO channel is high. The performance advantages are maintained when the channel coefficients are known only approximately. Therefore, it is concluded that the proposed technique is a candidate for next-generation practical STBC–OFDM wireless standards.

7

Acknowledgments

The authors are grateful to the anonymous reviewers and Prof. X. Shen, the editor-in-chief of this journal, for their insightful comments and suggestions which improved the quality of the paper greatly. Corresponding author is Assoc. Prof. Hakan Doğan and this work was supported in part by

1598 & The Institution of Engineering and Technology 2014

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IET Commun., 2014, Vol. 8, Iss. 9, pp. 1589–1598 doi: 10.1049/iet-com.2013.1166