Wireless Pers Commun DOI 10.1007/s11277-011-0451-z
Transceiver Design for MIMO Systems with Improper Modulations M. Raja · P. Muthuchidambaranathan · Ha H. Nguyen
© Springer Science+Business Media, LLC. 2011
Abstract This paper considers joint transceiver designs for single-user multiple-input, multiple-output systems employing improper constellations such as binary phase shift-keying and M-ary amplitude shift-keying (M-ASK). Proposed are novel joint linear transceivers that minimize the total mean squared error of the symbol estimation at the output of the decoder. The joint linear transceiver designs are carried out for both cases of perfect channel state information (CSI) and imperfect CSI at the transmitter and receiver. For the case of imperfect CSI, the channel model takes into account both transmit and receive correlations as well as the channel estimation error. The superiority of the proposed transceivers over the previously-proposed designs is verified by simulation results. Keywords Channel state information (CSI) · Mean-square error (MSE) · Multiple-input multiple-output (MIMO) · Linear precoding · Spatial multiplexing · Improper modulations 1 Introduction Multiple-input multiple-output (MIMO) architecture is a multiple-antenna technology for wireless communication systems. It has been extensively studied over the last decade [1]. Analysis has shown that the capacity of MIMO systems is significantly higher than that of single-input single-output (SISO) systems [2,3]. Depending on the channel condition, MIMO techniques can be used to achieve a higher data rate, by means of spatial multiplexing [4], or a higher diversity, by means of space-time coding [5]. These techniques do not require knowledge of the channel state information (CSI) at the transmitter (CSIT). In [6] and [7], it
M. Raja (B) · P. Muthuchidambaranathan Department of Electronics and Communication Engineering, National Institute of Technology (NIT), Tiruchirappalli, India e-mail:
[email protected] H. H. Nguyen Department of Electrical and Computer Engineering, University of Saskatchewan, 57 Campus Dr., Saskatoon, SK S7N 5A9, Canada
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is shown that additional performance gain is possible with multiple-antenna systems in the presence of CSIT. However, to design a precoder or joint transceiver, it requires both channel state information at the transmitter and channel state information at the receiver (CSIR), i.e., at both ends of the MIMO systems. In practice, to enable precoding at the transmitter, the MIMO channel has to be estimated at the receiver, and the CSIR is then feedback to the transmitter side [8]. This means that the CSIT present at the transmitter is not perfect because there is always channel estimation error and/or delay in feedback. Various performance measurements have been considered to obtain an optimum precoder or linear transceiver structure, such as minimum total meansquare error (TMSE) from all the data streams [9,10], weighted minimum mean square error (MMSE) [11] and minimum bit error rate (BER) [12,13]. Among these criteria, the minimum TMSE criterion leads to a lower BER in a single-user MIMO (SU-MIMO) system. Precoders or linear transceivers have been designed under various assumptions of CSI. The optimum joint linear transceiver using the weighted MMSE criterion subject to a transmit power constraint with perfect CSI was treated in [11]. In [9,10] optimum precoder and decoder have been designed under the minimum TMSE criterion with perfect CSI. Joint linear precoder and decoder under the minimum MMSE were designed with outdated CSIT and perfect CSIR in [14]. The channel mean information and/or channel correlation information are assumed as available CSIT in [15,16]. In [17], the minimum TMSE design is formulated as a nonconvex optimization problem under a total transmit power constraint, and the optimum linear precoder and decoder are determined by solving a nonconvex optimization problem. The optimum transceiver is obtained by considering both perfect CSI and imperfect CSI, and the impact of channel estimation error as well as channel correlations at both the transmitter and receiver. The feedback is assumed to be error-free for the purpose of convenient analysis in [18,19]. In [20], linear and non-linear precoding and decoding structures for frequency non-selective MIMO channels have been studied under the assumption that the channel state information is available at the transmitter. Linear structures are mostly used in order to avoid the high computation complexity of non-linear designs. Novel linear precoding schemes to improve the quality of downlink transmission in MIMO systems have been proposed with improper modulation techniques in [21]. Precoding is designed with a modified cost function that is more appropriate for an improper signal constellation. Improved zero-forcing (ZF) and MMSE precoders are shown to achieve a superior performance than the conventional linear and non-linear precoders [21]. Both cases of perfect and imperfect CSI are considered, where the imperfect CSI case takes into account the channel mean and transmit correlation information [21]. The same precoding design is also applied to the multi-user MIMO (MU-MIMO) in the case of imperfect channel estimation with a known error covariance. The conventional transceiver design under the minimum TMSE criterion yields good BER performance for proper modulation schemes, e.g., M-QAM, and M-PSK [17]. However, when applying the same design to the improper modulation schemes, e.g., BPSK and M-ASK, the BER performance degrades significantly. The improved minimum TMSE design for improper signal constellations was recently proposed in [21] and shown to give superior BER performance than the conventional design in [17]. However, to the best of our knowledge, no attention has been paid to the optimum joint linear transceiver design for the SU-MIMO systems which employ improper modulation techniques, either under the perfect CSI or imperfect CSI assumption. To fill the gap, this paper shall examine the problem of joint linear precoding/decoding design for SU-MIMO systems employing improper constellations. In both cases of perfect CSI and imperfect CSI, an improved minimum TMSE
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y = Hx + n
Input bits Improper modulation
s
Linear
x = Fs
precoder F
MIMO channel R1/R 2 H w R1/T 2
+ n
Output bits
Improper demodulation
ˆs = Re ( s )
Linear decoder G
s = Gy
Re (⋅ )
Fig. 1 Block diagram of a SU-MIMO system with improper modulation
criterion is established and used to develop an iterative design procedure for the optimum precoding and decoding matrices. The rest of the paper is organized as follows. The proposed joint linear transceiver design under perfect CSI is presented in Sect. 2. The design is then extended to the case of imperfect CSI in Sect. 3. The superiority of the proposed joint linear transceiver designs over the conventional designs is verified with simulation results in Sect. 4. Finally conclusions are given in Sect. 5. Notations: Throughout this paper, upper (lower) case boldface letters are for matrices (vectors), (·)T denotes matrix transpose, (·) H stands for matrix conjugate transpose, (·)∗ means matrix conjugate, E(·) is expectation, · is Euclidian norm, Tr(·) is the trace operation and I N is an N × N identity matrix.
2 Improved Minimum TMSE Transceiver Design With Perfect CSI This section considers transceiver design for the single-user MIMO (SU-MIMO) system, which is illustrated in Fig. 1. In this system, it is assumed that both the transmitter and receiver have perfect CSI. The design will be extended in Sect. 3 to the case with imperfect CSI that also takes into account transmit and receive correlations. Let NT be the number of antennas used at the transmitter and NR be the number of antennas used at the receiver. The information symbols to be sent are denoted by a B × 1 vector s = [S1 , . . . , S B ]T , where B is the number of data streams. The symbol vector s is precoded using a NT × B precoding matrix F to produce a NT × 1 precoded vector x = Fs, which is then transmitted simultaneously over NT antennas. The data symbols are assumed to be uncorrelated and have zero mean and unit energy, i.e., E[ss H ] = I B . The signal after the precoder satisfies the following total transmit power constraint: E[x2 ] = E[Fs2 ] = Tr(FF H ) = PT .
(1) E[ssT ]
= 0), such as Uncoded MIMO systems using improper modulations (for which BPSK and M-ASK [22,23] are considered (in fact BPSK can be viewed as 2-ASK as well). After the precoder, the precoded data vector x is transmitted across a slowly-varying flat Rayleigh fading MIMO channel. Such a channel can be described by an NR × NT channel matrix H, and as mentioned before, this matrix is assumed to be known at both the transmitter and the receiver. At the receiver, the NR × 1 received signal vector y = Hx + n is fed to the decoder G, which is a B × NR matrix. Then the resultant vector is:
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s = GHFs + Gn
(2)
where the NT × 1 vector n represents spatially and temporally additive white Gaussian noise (AWGN) of zero mean and variance σn2 . The conventional precoder and decoder are derived by minimizing the following minimum TMSE: = E[¯s − s2 ] = E[(GHFs + Gn) − s2 ]
(3)
under the transmit power constraint specified by (1). This design criterion is optimum in the MSE sense for systems with proper modulations, such as M-QAM and M-PSK for which E[ssT ] = 0. However, with improper modulation schemes considered in this paper (for which E[ssT ] = 0), the design criterion expressed by (3) is suboptimum. This fact was pointed out earlier in [21], in which a novel linear transmit precoding strategy for MIMO systems employing improper constellations is also proposed. In this paper, the same strategy in [21] is extended to design both the precoder and decoder, i.e., a joint linear transceiver scheme. With improper constellations, the error vector is defined as follows: e = sˆ − s
(4)
where sˆ = (GHFs + Gn). Observe that the value of the received signal sˆ is modified from that used in the conventional MMSE receiver. This is because the conventional optimization approach expressed by (3) yields a complex-valued filter output. However, only the real part of this output is relevant for the decision in a system employing improper constellations.1 As such, minimization of the MSE with respect to only the real part of the received signal will result in a better design. With the newly defined error vector, the TMSE can be computed as follows: E[e2 ] = E[ˆs − s2 ] = E[(GHFs + Gn) − s2 ] = E[(GHFs + G∗ H∗ F∗ s∗ )/2 + (Gn + G∗ n∗ )/2 − s2 ] = Tr E 0.5(GHFs + G∗ H∗ F∗ s∗ ) + 0.5(Gn + G∗ n∗ ) − s × 0.5(s H F H H H G H + sT FT HT GT ) + 0.5(n H G H + n T GT ) − s H (5) From the assumptions on the statistics of the channel, noise and data, one has E[ss H ] = E[ssT ] = I B , E[nn H ] = σn2 INT and E[n] = E[nn T ] = E[n∗ n H ] = 0. Using these facts and after some manipulations (5) can be simplified to E[e2 ] = Tr 0.25 GHFF H H H G H + GHFFT HT GT + G∗ H∗ F∗ F H H H G H
+ G∗ H∗ F∗ FT HT GT − 0.5(GHF + G∗ H∗ F∗ + F H H H G H + FT HT GT ) (6) + I B + 0.25σn2 (GG H + G∗ GT ) The design objective is to find a pair of matrices F and G to minimize E[e2 ] subject to the total transmit power constraint. That is, min E[e2 ] s.t. Tr(FF H ) ≤ PT . F,G
(7)
1 This is also true for offset quadrature phase-shift keying (OQPSK) when a simple demodulation method
that alternates between the inphase and quadrature streams is employed [23].
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The formulation in (7) shall be referred to as improved minimum TMSE design for MIMO systems employing improper modulations. To find the solution to the above problem, form the Lagrangian η = E[e2 ] + μ(Tr[FF H ] − PT )
(8)
where μ is the Lagrange multiplier. By substituting (6) into (8) and then taking the derivatives of η with respect to F and G [24], the associated Karush-Kuhn-Tucker (KKT) conditions can be obtained and given in the following. ∂η First, the value of ∂G can be found by using the cyclic property of the trace function. ∂η Setting ∂G = 0 gives 0.25[G∗ (HFF H H H )T + 2GHFFT HT + G∗ H∗ F∗ FT HT ] − 0.5(FT HT + FT HT ) + 0.25σn2 (G∗ + G∗ ) = 0
(9)
Taking the complex conjugates of both sides of (9) yields GHFF H H H + G∗ H∗ F∗ F H H H + σn2 G = 2F H H H Similarly, setting
∂η ∂F
(10)
= 0 gives
0.25[(H H G H GHF)∗ + 2HT GT GHF + HT GT G∗ H∗ F∗ ] − 0.5[HT GT + HT GT ] + μF∗ = 0
(11)
Again, taking the complex conjugates of both sides of (11) has H H G H GHF + H H G H G∗ H∗ F∗ + 2μF = 2H H G H Next, by post-multiplying both sides of (10) by
GH
one obtains
GHFF H H H G H + G∗ H∗ F∗ F H H H G H + σn2 GG H = 2F H H H G H Likewise, pre-multiplying both sides of (12) by
FH
(12)
(13)
produces
F H H H G H GHF + F H H H G H G∗ H∗ F∗ + 2μF H F = 2F H H H G H
(14)
It then follows from (13) and (14) that: GHFF H H H G H + G∗ H∗ F∗ F H H H G H + σn2 GG H
= F H H H G H GHF + F H H H G H G∗ H∗ F∗ + 2μF H F
(15)
Then, by taking the traces of both sides of (15) one has: μ=
σn2 Tr(GG H ) 2PT
(16)
Although (10), (12) and (16) show explicit relationships for the optimum solutions of F and G, closed-form expressions of the solutions appear intractable. Nevertheless, similar to [21], an iterative procedure is developed to find the solutions. First, define G = GRe + jGIm H
HFF H ∗ ∗ H
H
H F F H
= ARe + jAIm
H
= BRe + jBIm
2F H H H = CRe + CIm
(17) (18) (19) (20)
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Then CRe and CIm can be expressed using (10), in a vector form as ARe + BRe + σn2 I NR AIm + BIm CRe CIm = GRe GIm BIm − AIm ARe + BRe − σn2 I NR The above expression also implies that −1 ARe + BRe + σn2 I NR AIm + BIm GRe GIm = CRe CIm BIm − AIm ARe + BRe − σn2 I NR
(21)
(22)
Similarly, define F = FRe + jFIm
(23)
H G GH = PRe + jPIm
(24)
H H G H G∗ H∗ = QRe + jQIm
(25)
H
H
H
2H G
H
= RRe + RIm
(26)
Then RRe and RIm can be expressed using (12), in a vector form as QIm − PIm PRe + QRe + 2μI NT FRe RRe = RIm PIm + QIm PRe + QRe − 2μI NT FIm Equivalently,
FRe FIm
PRe + QRe + 2μI NT QIm − PIm = PIm + QIm PRe + QRe − 2μI NT
−1
RRe RIm
(27)
(28)
Based on the above expressions, the optimum precoder and decoder can be solved by an iteration procedure as outlined in Fig. 2, where Fi denotes F in the ith iteration. Furthermore, the initial precoder F0 is chosen such that the B × B upper sub-matrix of F0 is a scaled identity matrix (which satisfies the power constraint with equality), while all the other remaining entries of F0 are zero. Furthermore, it can be shown that the algorithm described in Fig. 2 converges [21]. 3 Improved Minimum TMSE Transceiver Design with Imperfect CSI In general, the wireless channels are time-varying. As such, obtaining the channel information at both the transmitter and receiver can be difficult. Usually the obtained channel information is not the same as the instantaneous channel information. This means that the transceiver design in Sect. 2 under perfect CSI is no longer optimum for the systems operating with estimated channel information. It also suggests that a better transceiver design can be obtained by taking into account channel estimation error. This is precisely the objective of this section. 1/2 1/2 We adopt the channel model in [23], i.e., H = RR Hw RT , where Hw is a spatially white matrix whose entries are independent and identically distributed (i.i.d.) Nc (0, 1). The matrices RT and RR represent the normalized transmit and receive correlations (i.e., with unit diagonal entries), respectively. Both RT and RR are assumed to be full-rank and known to both the transmitter and the receiver. Typically, channel estimation is performed based on orthogonal training method as in [8]. The signal received at the receive antenna during the training phase is expressed as 1/2
1/2
Ytr = HXtr + Ntr = RR Hw RT Xtr + Ntr
H
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(29)
Transceiver Design for MIMO Systems Fig. 2 Iterative procedure for solving the optimum precoder and decoder
Initialize F = F0
Update G using (22)
Update μ using (16)
Update F using (28)
If Tr ( FF H ) > PT
Yes Scale F such that
No
Tr ( FF H ) = PT
If No
Tr ( ( Fi − Fi −1)( Fi − Fi −1) H ) is sufficiently small
Yes Stop
where Xtr is a NT × NT training signal matrix with the training power of tr(Xtr XtrH ) = Ptr and Ntr is the noise matrix. Since RT and RR are assumed to be known, the task is to estimate the −1/2 random channel matrix Hw . A typical training design is Xtr = RT X, where X is NT × NT unitary matrix scaled by Ptr /tr(RT−1 ). 2 = Tr(R−1 )σ 2 /P . Then it is simple to show that the ¯ w = R−1/2 Ytr X−1 and σce Define H tr n T R MMSE estimation of Hw is given by [8]: 2 −1 −1 ¯ ¯ w. ˆ w = [I NR + σce RR ] Hw = Re,R H H
(30)
2 R−1 ]−1 represents the effect of the receive correlation on the chanwhere Re,R = [I NR + σce R nel estimation error. Furthermore, the true channel can be expressed as a sum of the estimated channel and an estimation error matrix. That is 2 −1 −1/2 ˆ w + R−1/2 [I NR + σce Hw = H RR ] Ew R
ˆ w + R−1/2 R1/2 Ew =H R e,R
(31)
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M. Raja et al. 2 ). The true overall channel matrix H = where the entries of Ew are i.i.d. Nc (0, σce 1/2 1/2 RR Hw RT can then be expressed as: 1/2
1/2
H = RR Hw RT
1/2 ˆ 1/2 1/2 −1/2 1/2 1/2 RR Re,R Ew RT = RR H w RT + RR ˆ +E =H
(32)
ˆ = R1/2 H ˆ w R1/2 is the estimated overall channel matrix (i.e., the channel mean), where H R T 1/2 1/2 and E = Re,R Ew RT is the channel estimation error matrix. It should be pointed out that 2 reflects the quality of the channel estimate, which is inversely proportional the parameter σce to the signal-to-noise ration during training phase, i.e., SNRtr = Ptr /σn2 . In the extreme case, 2 = 0 implies perfect channel estimation. setting σce By modeling the true channel as in (32) under the MMSE channel estimation, the TMSE function for joint transceiver design can be evaluated for improper modulation as follows: E[e2 ] = E[ˆs − s2 ] ˆ + E)Fs + Gn) − s2 ] = E[(G(H ˆ + E)Fs + G∗ (H ˆ + E)∗ F∗ s∗ )/2 + (Gn + G∗ n∗ )/2 − s2 ] = E[(G(H ˆ + E)∗ F∗ s∗ ) + 0.5(Gn + G∗ n∗ ) − s ˆ + E)Fs + G∗ (H = Tr E 0.5(G(H ˆ + E) H G H + sT FT (H ˆ + E)T GT ) × 0.5(s H F H (H (33) + 0.5(n H G H + n T GT ) − s H 1/2
1/2
Substituting E = Re,R Ew RT in (33) and after taking expectation with respect to s, Ew , and n, (33) becomes:2 H ˆH H ˆ ˆ TH ˆ ˆ T GT − 0.5GHF E[e2 ] = Tr 0.25GHFF H G + 0.25GHFF 2 + 0.25GRe,R G H Tr(RT FF H )σce + 0.25GG H σn2 ˆ ∗ F∗ F H H ˆ H G H + 0.25G∗ H ˆ ∗ F∗ F T H ˆ T GT + 0.25G∗ H ∗ 2 ˆ ∗ F∗ + 0.25G∗ Re,R − 0.5G∗ H GT {Tr(RT FF H )}∗ σce
ˆ H G H − 0.5FT H ˆ T GT + I B + 0.25G∗ GT σn2 − 0.5F H H
(34)
As before, to obtain equations that can be solved iteratively for the optimum precoder F and decoder G, form the following Lagrangian that takes into account the total power constraint: η = E[e2 ] + μ[Tr(FF H ) − PT ]
(35)
By substituting (34) in (35) and taking the derivatives of η with respect to G and F, it can be shown that the associated Karush-Kuhn-Tucker (KKT) conditions are as follows: H ˆH 2 ˆ ∗ F∗ F H H ˆ H + σn2 G = 2F H H ˆH ˆ Tr(RT FF H )) + G∗ H H + Re,R σce G(HFF 2 ˆ H G H GH ˆ + RT σce ˆ H G H G∗ H ˆ H GH ˆ ∗ F∗ + 2μF = 2H (H Tr(Re,R G H G))F + H
(36) (37)
2 In performing the expectation, the following results are used: E[E ] = E[E H ] = 0, E[E AE H ] = w w w w 2 tr(A)I and E[E AET ] = 0. σce w N w
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The expression of the Lagrange multiplier is still the same as in (16), i.e., μ=
σn2 Tr(GG H ) 2PT
(38)
The above set of equations can be solved iteratively as outlined in Sect. 2. Specifically, define G = GRe + jGIm H ˆH 2 ˆ Tr(RT FF H ) = JRe + jJIm HFF H + Re,R σce ˆ ∗ F∗ F H H ˆ H = KRe + jKIm H
(40)
ˆ H = LRe + LIm 2F H H
(42)
(39) (41)
Then GRe and GIm can be expressed as −1 JRe + KRe + σn2 I Nr JIm + KIm GRe GIm = LRe LIm KIm − JIm JRe + KRe − σn2 I Nr
(43)
Likewise, define F = FRe + jFIm H H 2 ˆ ˆ Tr(Re,R GH G) = URe + jUIm H G GH + RT σce H H ∗ ˆ G G H ˆ ∗ = VRe + jVIm H ˆ G 2H H
Then
FRe FIm
H
(44) (45) (46)
= WRe + WIm
VIm − UIm URe + VRe + 2μI NT = UIm + VIm URe + VRe − 2μI NT
(47) −1
WRe WIm
(48)
Finally, the flowchart of Fig. 2 can be applied to obtain the optimum precoder and decoder pair {F, G} by replacing Eqs. (16), (22), (28) by Eqs. (38), (43), (48), respectively.
4 Numerical Results This section presents numerical results to illustrate performance improvement, in terms of the bit error rate (BER), by our proposed linear transceiver designs. In particular, the following comparisons are made: 1. Performance of the proposed joint precoding/decoding design is compared with the linear transmit precoding strategy in [17] for SU-MIMO systems employing the same improper signal constellations. This comparison is to illustrate the benefit of performing decoder optimization in SU-MIMO systems that is not considered in [17]. 2. The proposed joint linear transceiver is also compared with the previously-designed joint linear transceiver strategy in [21], but without taking into account specific property of improper modulations. In simulating the MIMO channel, the transmit and receive correlation matrices are generated by the exponential model [25,26]. Specifically, the transmit and receive correlation |i− j| |i− j| matrices are defined as RR (i, j) = ρR for i, j = 1, 2, . . . , NR and RT (i, j) = ρT for i, j = 1, 2, . . . , NT . For all simulation results reported in this section, the numbers of
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10
−1
BER
10
−2
10
−3
10
4−ASK, precoding only BPSK, precoding only 4−ASK, precoding and decoding BPSK, precoding and decoding
−4
10
0
2
4
6
SNR= P T /
8 2 n
10
12
(dB)
Fig. 3 Performance comparison of transmit precoding and joint precoding/decoding for BPSK and 4-ASK 2 = 0, ρ = ρ = 0.0 modulations. NT = NR = 4, B = 4, σce T R
transmit and receive antennas are set to be NT = NR = 4. Except for Fig. 6 where the numbers of data streams are B = 3 or B = 4, all other figures consider B = 4. In all figures, the signal-to-noise ratio is defined as SNR = σPT2 . To have a fair comparison with the results n
reported in [17], the SNR in training phase is set to SNRtr = σPtr2 = 26.016 dB. n It should also be pointed out that, for the proposed joint transceiver design, the number of iterations needed to solve the optimum precoder and decoder as outlined in Fig. 2 depends on the channel condition and the SNR. As mentioned before, the proposed algorithm converges since the value of the objective function is reduced at each iteration and is bounded from below. For all the simulation results reported in this paper, the average numbers of iterations required to converge to the vicinity of the minimum point is 5 for NT = NR = 4. In particular, it was observed that 4 to 6 iterations were adequate in all simulations. These numbers are well within the range of the numbers of iterations needed for conventional MIMO transceiver design algorithm to converge (which vary between 2 to 20) [27–30]. First, Fig. 3 compares the performance of the linear transmit precoding design in [21] with that of the proposed joint precoding/decoding for BPSK and 4-ASK modulations and when perfect CSI is available at both the transmitter and receiver. Both the designs under comparison take into account the one-dimensional property of improper modulations. It is clear from the figure that a significant performance improvement is achieved by performing joint precoding and decoding. For example, with BPSK the SNR saving is about 3 dB at the BER level of 10−2 . As expected with either precoding only or joint precoding/decoding, BPSK performs better than 4-ASK. Note also that, since perfect CSI is available at both the transmitter and receiver ends, the performance curves improve exponentially with SNR (similar to that over an AWGN channel) and there is no error floor in all performance curves. Figure 4 shows performance comparisons of the conventional joint transceiver design in [17] and the proposed joint transceiver design under perfect CSI for both BPSK and 4-ASK. As mentioned before the conventional design does not take into account the one-dimensional property of improper modulations. As can be seen from the figure, the proposed joint linear
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BER
10
10
10
10
0
−1
−2
−3
4−ASK, conventional BPSK, conventional 4−ASK, propsed BPSK, proposed
−4
0
5
10
SNR= P T /
15 2 n
20
(dB)
Fig. 4 Performance comparison of the conventional transceiver and proposed transceiver with perfect CSI, 2 = 0, ρ = ρ = 0.0 for BPSK and 4-ASK. NT = NR = 4, B = 4, σce T R 0
10
−1
BER
10
−2
10
−3
10
4−ASK, conventional BPSK, conventional 4−ASK, propsed BPSK, proposed
−4
10
0
5
10
15
20
SNR= P T /σ n2 (dB)
Fig. 5 Performance comparison of the conventional transceiver and the proposed transceiver with imperfect 2 = 0.015, ρ = ρ = 0.5 CSI for BPSK and 4-ASK. NT = NR = 4, B = 4, σce T R
transceiver leads to a very large performance improvement, especially for BPSK modulation (an SNR improvement of about 10 dB is observed for BER of 10−3 ). Similar comparisons are illustrated in Fig. 5, but for the case of imperfect CSI and ρT = ρR = 0.5. Note that, 2 = 0.015. Again, the performance with SNRtr = σPtr2 = 26.016 dB and ρT = 0.5, one has σce n improvement by our proposed design over the conventional design is clearly observed from Fig. 5. The purpose of Fig. 6 is to show that, similar to the conventional transceiver design in [17], the proposed transceiver design also enjoys further gain if the number of data streams is reduced from B = 4 to B = 3. Note that the performance curves shown in
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10
10
10
10
0
−1
−2
−3
4−ASK, B=3 BPSK, B=3 4−ASK, B=4 BPSK, B=4
−4
0
5
10
SNR= P T /
15 2 n
20
(dB)
Fig. 6 Performance comparison of the proposed transceiver with B = 3 and B = 4 for BPSK and 4-ASK. 2 = 0, ρ = ρ = 0.0 NT = NR = 4, B = 3 or B = 4, σce T R 10
−1
BER
10
0
10
−2
ρ =0.9, ρ =0.9 T
R
ρT=0.9, ρR=0.5 ρ =0.5, ρ =0.9 T
10
R
ρT=0.5, ρR=0.5
−3
0
5
10
SNR= P T /
15 2 n
20
(dB)
Fig. 7 Effect of transmit and receive correlations on the performance of the proposed transceiver design with 2 are 0.015, 0.0739 for ρ = 0.5 and ρ = 0.9, respectively 4-ASK. NT = NR = 4, B = 4. The values of σce T T
Fig. 6 are for the case of perfect CSI, but the same observation holds for imperfect CSI as well. Figure 7 examines the effect of channel correlations on the system BER performance under imperfect CSI. For this figure, 4-ASK modulation is employed with the number of data streams B = 4. Various sets of transmit/receive correlations considered are {ρT = 0.9, ρR = 0.9}; {ρT = 0.9, ρR = 0.5}; {ρT = 0.5, ρT = 0.9}; and {ρT = 0.5, ρT = 0.5}. 2 is 0.015 and 0.0739 for ρ = 0.5 and With SNRtr = σPtr2 = 26.016 dB the parameter σce T n ρT = 0.9, respectively. In general, Fig. 7 shows that higher values of the transmit and receive correlations cause bigger performance losses.
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−1
4×4 BPSK, σ2 =0.0 ce 2
4×3 BPSK, σce=0.0 2
3×4 BPSK, σce=0.0 2 ce
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Fig. 8 Performance of the proposed transceiver design with BPSK under symmetrical and asymmetrical 2 = 0, ρ = ρ = 0.0 MIMO configurations. NT = NR = 3 or 4, B = 3, σce T R
Fig. 9 Performance of the proposed transceiver design with BPSK under symmetrical and asymmetrical 2 = 0.015, ρ = ρ = 0.5 MIMO configurations. NT = NR = 3 or 4, B = 3, σce T R
Finally, Figs. 8 and 9 demonstrates that the proposed transceiver design performs well under both symmetrical (3 × 3 or 4 × 4) and asymmetrical (3 × 4 or 4 × 3) MIMO configurations. In particular, Fig. 8 shows the results with perfect CSI, while Fig. 9 plots the results with imperfect CSI. For both figures, BPSK modulation is employed with the number of data streams B = 3. It is clear from these two figures that a larger performance gain is achieved when number of antennas is higher, at either the transmitter or receiver. Furthermore, it is intuitively satisfying to see that choosing NT = 4 and NR = 3 yields the same average performance as choosing NT = 3 and NR = 4.
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5 Conclusions This paper addressed the designs of joint linear transceivers with improper constellations for SU-MIMO systems. Specifically, it made the following important extensions to the previous designs: (i) Obtaining a joint linear transceiver for SU-MIMO systems under perfect CSI assumption and with improper modulations, and (ii) Obtaining a joint linear transceiver for SU-MIMO systems under imperfect CSI assumption and with improper modulations. In both cases, the transceiver designs are accomplished with an iterative procedure, which typically converges in 4 to 6 iterations. Performance advantage of the proposed designs in terms of the system’s BER was thoroughly demonstrated with simulation results. Finally, it is pointed out that the design proposed in this work can be extended to the multi-user MIMO (MU-MIMO) systems.
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Author Biographies M. Raja received his B.Eng. Degree in Electronics and Communication Engineering from Anna University, Chennai, India, in 2005, and the M.Eng. Degree in Digital Communication and Networking from Anna University, Chennai, India, in 2007. Currently he is pursuing his Ph.D. in Wireless Communication in the Department of Electronics and Communication Engineering, National Institute of Technology (NIT), Tiruchirappalli, India. He is a recipient of Canadian Commonwealth Scholarship Award-2010 for Graduate Student Exchange Program in the Department of Electrical and Computer Engineering, University of Saskatchewan, Saskatoon, SK, Canada. His research interests include orthogonal frequency division multiplexing (OFDM), multiple-input and multiple-output (MIMO) systems, diversity and beamforming techniques, and channel estimation. He published his research papers in refereed international journals, and international and national conferences.
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M. Raja et al. P. Muthuchidambaranathan received his B.Eng. Degree in Electronics and Communication Engineering from Government College of Technology, Coimbatore, India, in 1992, the M.Eng. Degree in Microwave and Optical Engineering, from A.C. College of Engineering and Technology, Karaikudi, India, in 1994. He obtained his Ph.D. degree in optical communication from the National Institute of Technology (NIT), Tiruchirappalli, India in 2009. He is currently working as an Associate Professor in the Department of Electronics and Communication Engineering, National Institute of Technology (NIT), Tiruchirappalli, India. His research interests include wireless communications, and optical communications. He published his research papers in refereed international journals, and international and national conferences. He is an author of the textbook “Wireless Communications” (published by Prentice Hall of India).
Ha H. Nguyen (M’01–SM’05) received the B.Eng. degree from Hanoi University of Technology Hanoi, Vietnam, in 1995, the M.Eng. degree from the Asian Institute of Technology, Bangkok, Thailand, in 1997, and the Ph.D. degree from the University of Manitoba, Winnipeg, MB, Canada, in 2001, all in electrical engineering. He joined the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK, Canada, in 2001, where he is currently a full Professor. He holds adjunct appointments at the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada, and TRLabs, Saskatoon. He was a Senior Visiting Fellow in the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia, during October 2007–June 2008. His research interests include digital communications, spreadspectrum systems, and error-control coding. He is a coauthor (with E. Shwedyk) of A First Course in Digital Communications (Cambridge, U.K.: Cambridge University Press, 2009). Dr. Nguyen was an Associate Editor for the IEEE Transactions on Wireless Communications during 2007–2011. He currently serves as an Associate Editor for the IEEE Transactions on Vehicular Technology and the IEEE Wireless Communications Letters. He was a Co-chair for the Multiple Antenna Systems and Space-Time Processing Track, IEEE Vehicular Technology Conferences (Fall 2010, Ottawa, ON, Canada and Fall 2012, Quebec, QC, Canada). He is a Registered Member of the Association of Professional Engineers and Geoscientists of Saskatchewan (APEGS).
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