1
Differential Coding for MAC Based Two-User MIMO Communication Systems Manav R. Bhatnagar, Member, IEEE and Are Hjørungnes, Senior Member, IEEE Abstract— In this paper, we explain how to implement differential modulation in multiple access channel (MAC) based uplink two-user multiple-input multiple-output (MIMO) communication system. It is assumed that the users cannot exchange their information and transmit data over uplink MAC channels simultaneously in the same frequency band and without any orthogonal signatures (no CDMA). We derive two types of differential decoders for two-user MIMO MAC system namely 1) Partial differential decoder, 2) Heuristic based differential decoders. The partial differential decoder is able to avoid the knowledge of the channel gains of one user at the receiver for joint decoding of the data of both users. Whereas, the heuristic differential decoders are obtained heuristically and work without any channel knowledge of both users. For the heuristic differential decoder to work properly, the data of one user must be rotated before transmission. An upper bound of the pairwise error probability (PEP) is derived for the proposed scheme. The upper bound of the PEP is minimized for obtaining an optimized value of the rotation angle.
I. I NTRODUCTION In multiple-input multiple-output (MIMO) systems, differential modulation is studied extensively in the last ten years [1]–[6]. Differential modulation is useful since there is no need of channel estimation at the receiver. Hence, differential modulation is useful especially for mobile users at the cell boundary who experience bad channels and for estimating the channel gains significant amount of training data is required. It is shown in the literature [7] that by introducing orthogonality among the transmissions of multiple users through TDMA, FDMA, or CDMA, the differential modulation can be implemented in a multi-user scenario. However, these coordinated transmission schemes reduce the effective data rate or number of users which can be accommodated in the multi-user system. In order to improve the data rate of a multi-user system, we need to consider transmissions over a general multiple access channel (MAC) [8], where the users transmit their information in a non-orthogonal manner, i.e., multiple users utilize the same time interval, same frequency band, and same signature waveform for their transmissions. Since differential modulation avoids the need of training data transmission, a rate efficient multi-user system utilizes differential modulation. Corresponding author: Manav R. Bhatnagar is with Department of Electrical Engineering, Indian Institute of Technology Delhi, Hauz Khas, IN110016 New Delhi, India, email:
[email protected]. Are Hjørungnes was with UNIK – University Graduate Center, University of Oslo, Gunnar Randers vei 19, P. O. Box 70, NO-2027 Kjeller, Norway, email:
[email protected]. This work was supported by the Research Council of Norway projects 176773/S10 called OptiMO and 183311/S10 called M2M, which belong to the VERDIKT program and by the Department of Science and Technology, Government of India under SERC scheme for the Project “Interference Cancellation in MAC Based Multiuser MIMO Communication Systems” (Project Ref. No. SR/S3/EECE/0089/2009).
In [9], a two-user based differential MAC system is considered where the transmitter and receiver contain a single antenna. It is shown in [9] that for the two-user differential MAC system to work properly, an optimized rotation angle is required. The key differences between [9] and this paper are as follows: 1) In this paper, we consider a generalized MAC based set-up, where the transmitters and receiver contain multiple antennas, whereas, in [9], a single antenna based MAC differential system is considered. Differential decoders for MAC based MIMO system are obtained in this paper by using exhaustive matrix algebra. 2) Since [9] is a special case of the multiple antenna based MAC differential system considered in this paper, the proposed decoders are applicable for the single antenna setup in [9]. However, converse is not true. 3) In [9], no analytical performance analysis is provided and the optimized rotation angles between the constellation of both users are calculated heuristically. Here, we derive pairwise error probability (PEP) of the differential MAC based MIMO system and utilize this PEP bound for calculation of the optimized rotation angles. In this paper, we have the following main contributions: 1) We derive a partial differential decoder for a MAC based two-user MIMO communication system. 2) Two heuristic differential decoders are derived based on three consecutively received data blocks. 3) We derive an upper bound of PEP for the heuristic decoder. 4) Based on the PEP, we obtain optimized rotation angle for MAC based two-user MIMO system. The rest of this paper is organized as follows: In Section II, the system model is introduced. Section III derives the partial and the heuristic differential decoders for the two users based MAC MIMO system. The simulation results are discussed in Section IV. Some conclusions are drawn in Section V. This article contains four appendices. II. S YSTEM M ODEL AND D IFFERENTIAL M ODULATION Consider a wireless communication system consisting of two users, where each user is equipped with nt transmit antennas and one base-station (BS) having nr receive antennas. Let sk [n] = sk1 , sk2 , ..., skv ∈ Ψ v , where v ≤ nt and Ψ is an M -PSK constellation, be the data of User k, k = 1, 2, to be transmitted in the n-th, n = 0, 1, 2, ..., block. Both users encode their data into nt × nt square orthogonal spacetime block code (OSTBC) S k [n]. Then, the OSTBC S k [n] is differentially encoded by User k into X k [n] of size nt × nt as follows: X k [n] = X k [n − 1]S k [n].
(1)
It is assumed that the constellation is normalized such that S k [n]S H k [n] = I nt , where I nt is the nt × nt identity matrix.
2
In order to initiate the differential transmission, X k [0] is chosen as a unitary matrix. It can be seen from (1) that X k [n]X H k [n] = I nt . The data transmitted by the users in a block n reaches the BS simultaneously, i.e, their transmissions are assumed to be perfectly synchronized in time, frequency, and phase. Moreover, both users utilize the same frequency band for their transmission and do not use any orthogonal signature. The users do not share their information. Therefore, both users independently transmit their OSTBCs to the BS. The data received in n-th block the BS is Y [n] = H 1 X 1 [n] + H 2 X 2 [n] + E[n],
(2)
where H k is an nr × nt matrices containing the independent and identically distributed (i.i.d.) complex Gaussian channel coefficients with zero mean and σh2 variance between the User k and the BS, and E[n] ∈ Cnr ×nt contains the complex additive white Gaussian noise (AWGN) terms with zero mean and σ 2 variance. It is further assumed that H 1 and H 2 remain constant over multiple (more than two) consecutive time-intervals. In this paper, the signal-to-noise ratio (SNR) is defined as the ratio of the average signal power transmitted in one time slot and average noise power. III. D ECODING OF D IFFERENTIAL DATA FROM T WO -U SERS
˜ 2 which minimizes the ML to zero, leads to the value of h metric (5): −1 ˜ 2 = S T [n] − S T [n] ⊗ I n h 1 2 r i h (6) × S T1 [n] ⊗ I nr y[n − 1] − y[n] .
By substituting (6) in (5), it can be seen that Γ [n] = 0, ∀sk [n] ∈ Ψ v , i.e., the resulting ML metric cannot be used for estimating the data of both users. Hence, the data of both users cannot be decoded differentially on the basis of two consecutively received data blocks. ˜ 2 , we can obtain a However, with perfect knowledge of h partially differential decoder by minimizing the ML metric given in (5) as follows:
ˆ2 [n]} = {ˆ s1 [n], s arg min
y[n] − S T1 [n] ⊗ I nr {s1 [n],s2 [n]}∈Ψ 2v
×y[n − 1] +
2 ˜ 2 S T1 [n] − S T2 [n] ⊗ I nr h
. (7)
The partially differential decoder of (7) avoids the knowledge of the channel of User 1 and uses only the knowledge of the channel of User 2 for decoding of the data of both users. This partially differential decoder is useful for scenarios where one user is at the cell boundary and its signals are very weak, hence, it is difficult to estimate its channel gains. Whereas the other user can be close to the BS and its channel gains can be estimated with small errors.
A. Partial Differential Decoder By using the property of vec (·) [10], we can write (2) as follows: y[n] = X T1 [n] ⊗ I nr h1 + X T2 [n] ⊗ I nr h2 + e[n], (3) where y[n] = vec (Y [n]), hk = vec (H k ), and e[n] = vec (E[n]) are nt nr × 1 vectors. Since e[n] ∼ CN 0nt nr ×1 , σ 2 I nt nr , the probability density function (p.d.f.) of y[n], when H k and X k [n] are known, can be written as 1 1 f (y[n]|H k , X k [n]]) = nt nr 2nt nr exp − 2 ky[n] σ π σ 2
T T − X 1 [n] ⊗ I nr h1 − X 2 [n] ⊗ I nr h2 , (4)
where k·k stands for the Euclidean vector norm. It is shown in Appendix I that by maximizing the joint p.d.f. of y[n − 1] and y[n] with respect to (w.r.t.) H 1 , the following ML metric is obtained:
Γ [n] = y[n] − S T1 [n] ⊗ I nr y[n − 1] 2 ˜ 2 + S T1 [n] − S T2 [n] ⊗ I nr h (5)
, ˜ 2 = X T [n − 1] ⊗ I n h2 . where h 2 r Lemma: In a two-users based MIMO communication system, the data of both users cannot be decoded differentially by using two consecutively received data blocks. ˜ 2 and setting the result Proof: Differentiating (5) w.r.t. h
B. Heuristic Decoder Based on Three Consecutively Received Data Samples In this subsection, we will develop a heuristic differential decoder based on three consecutively received data blocks for the uplink two-user differential MIMO system. From (1), (2), and (3) the data received by the three consecutive blocks n − 2, n − 1, n can be written as y[n − 2] = X T1 [n − 2] ⊗ I nr h1 + X T2 [n − 2] ⊗ I nr h2 + e[n − 2], (8) y[n − 1] = S T1 [n − 1]X T1 [n − 2] ⊗ I nr h1 + S T2 [n − 1]X T2 [n − 2] ⊗ I nr h2 + e[n − 1],
(9)
y[n] = S T1 [n]S T1 [n − 1]X T1 [n − 2] ⊗ I nr h1 + S T2 [n]S T2 [n − 1]X T2 [n − 2] ⊗ I nr h2 + e[n]. (10)
By assuming perfect knowledge of S 1 [n − 1] and S 1 [n] at the receiver, we may rearrange the relations given in (8), (9), and (10) as follows: y ′ [n − 1] = y[n − 1] − S T1 [n − 1] ⊗ I nr y[n − 2] ˜ 2 + e′ [n − 1], = S T2 [n − 1] − S T1 [n − 1] ⊗ I nr h y ′ [n] = y[n] − S T1 [n] ⊗ I nr y[n − 1] ˜ 2 + e′ [n], (11) = S T2 [n] − S T1 [n] S T2 [n − 1] ⊗ I nr h
3
where the additive noises e′ [n − 1] = e[n − T ′ 1] − S 1 [n − 1] ⊗ I nr e[n − 2] and e [n] = T e[n] − S 1 [n] ⊗ I nr e[n − 1] are distributed as CN 0nt nr ×1 , 2σ 2 I nt nr . It can be seen from (11), that e′ [n − 1] and e′ [n] are are correlated. The correlation matrices of e′ [n − 1] and e′ [n] can be obtained as ( ′ H ) e′ [n − 1] e [n − 1] Λ=E e′ [n] e′ [n] −S ∗1 [n] 2I nt 2 ⊗ I nr =σ −S T1 [n] 2I nt = Σ ⊗ I nr ,
(12)
where Σ=σ
2
2I nt −S T1 [n]
−S ∗1 [n] 2I nt
.
(13)
C. Approximate Differential Decoder Let us ignore the noise correlation of (12), i.e., S 1 [n] is ignored in (12) and obtain an approximate decoder. In Appendix II, an approximate differential decoder which completely avoids the knowledge of the channels of both users is derived as ˆ2 [n − 1], s ˆ1 [n], s ˆ2 [n]} {ˆ s1 [n − 1], s =
{s1 [n−1],s2 [n−1],s1 [n],s2 [n]}∈Ψ 4v 2
kAl y[n] + B l y[n − 1] + C l y[n − 2]k ,
(14)
l=1
¯ T [n − 1]S ∗ [n − 1]S ¯ ∗ [n] ⊗ I n , S 2 r ∗ 1 ¯T ¯ = ⊗ I nr , B 1 A2 = d S [n]S [n] − I nt T ∗ T ∗ ∗ T 1 ¯ ¯ ¯ [n − 1]S [n − 1]S ¯ [n]S [n] S 2 1 d S [n − 1]S [n− 1]− ¯ ∗ [n − 1] − S ¯ T [n] ¯ T [n]S T [n − 1]S −I nt )⊗I nr , B 2 = d1 S 2 ¯ T [n − 1] ¯ ∗ [n]S T [n] + S T [n] ⊗ I n , C 1 = − 1 S ×S 1 1 r d ¯ ∗ [n − 1]S T [n − 1] + S T [n − 1] ⊗ I n , C 2 = ×S 1 1 r ∗ T T 1 ¯T ¯ [n − 1]S [n − 1] ⊗ I n , d = − d S [n]S 2 [n − 1]S 1 r n o ∗ T ∗ T T ∗ 1 ¯ ¯ ¯ ¯ Tr [n − 1] , [n − 1] S [n] S [n]S S [n − 1] S [n − 1]+S 2 2 nt ¯ and S[i] , S 2 [i] − S 1 [i], i = n − 1, n. It can be seen from (14), that if S 2 [i] = S 1 [i], then the differential decoder fails. This situation can be avoided by choosing the data of User 1 from a rotated constellation ejθ Ψ , where θ ∈< −π, π]. In [11], [12], it is shown that for a coherent decoder of two-user data over single-input singleoutput (SISO) Gaussian MAC channels to work properly, the data of both users must be chosen from relatively rotated constellations. In [12, Table I], optimized values of the rotation angles for different M -PSK constellations and different SNR values are obtained by maximizing the capacity of two-user Gaussian multiple access channel. However, it is difficult to find the capacity of the differential MAC MIMO system considered in this paper. Therefore, for finding the optimized rotation angles we will utilize an upper bound of the PEP. where
A1
=
1 d
In Appendix III, it is shown that by considering the noise correlation given in (12), an exact differential decoder can be derived as ˆ2 [n − 1], s ˆ1 [n], s ˆ2 [n]} {ˆ s1 [n − 1], s =
arg min {s1 [n−1],s2 [n−1],s1 [n],s2 [n]}∈Ψ 4v
2
−1/2 ⊗ I nr z ′ [n] ,
Π⊥ Σ −1/2 Z T [n] Σ
(15)
iT h T T where z ′ [n] = (y ′ [n − 1]) , (y ′ [n]) , Z[n] = [S 2 [n − 1]−S 1 [n − 1], S 2 [n − 1](S 2 [n]−S 1 [n])], = I 2nt − Σ −1/2 Z T [n] and Π⊥ Σ −1/2 Z T [n] −1 × Z ∗ [n]Σ −1 Z T [n] Z ∗ [n]Σ −1/2 . The exact differential decoder does not work well if both users utilize the same signal constellation, hence, a rotation will be used here as well. The heuristic decoders of (14) and (15) are derived under the assumption of perfect knowledge of the data of User 1, i.e., S1 [n] and S1 [n − 1]. However, it can be seen from (14) and (15) that decoding of the data of both users is performed, i.e., it is assumed that S1 [n] and S1 [n − 1] are not known while decoding. E. PEP Performance Analysis of the Heuristic Decoder
arg min
2 X
D. Exact Differential Decoder
In this subsection, we will derive an upper bound of the PEP (UBPEP) of the proposed exact differential decoder (15). The UBPEP will be used to demonstrate the diversity of the proposed decoder and calculating the optimized values of the rotation angle θ. We can rewrite (11) as ! # " ′ S T2 [n − 1] − ST1 [n − 1] y [n − 1] ⊗I nr = y ′ [n] S T2 [n] − S T1 [n] S T2 [n − 1] ′ e [n − 1] ˜ × h2 + . (16) e′ [n] We can treat differential detection as the problem of non-coherent detection of the following n-th data block: Z[n] = [S 2 [n − 1] − S 1 [n − 1], S 2 [n − 1] (S 2 [n] − S 1 [n])]. A UBPEP for the proposed heuristic decoder is derived in Appendix IV as T ∗ UBPEP = Z 0 [n] Σ −1/2 Π ⊥ (Σ −1/2 ) Z H [n] ∗ H −nr γ 2 −nt nr −1/2 0 , (17) × Σ Z [n] 4 where Z 0 [n] is the wrongly decoded data matrix and γ 2 = σh2 /σ 2 . It can be seen from (17) that the proposed differential scheme achieves full diversity of nt nr . The rotation angle optimization problem can be expressed as min
UBPEP.
{−π
