Algorithm for Detecting the Number of Transmit Antennas in MIMO-OFDM Systems Eckhard Ohlmer, Ting-Jung Liang and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, TU-Dresden, Germany email:
[email protected] Abstract—In wireless MIMO-OFDM systems, knowledge of the channel between all transmit-receive antenna pairs is essential to enable the decoupling of spatial streams and coherent data detection at a receiver. Regarding a typical preamble design for MIMO channel estimation in an ad-hoc network, the preamble structure depends on the number of transmit antennas and this number must be known before the actual MIMO channel estimation is accomplished. In this work, we propose an algorithm to detect the number of transmit antennas, using only the received pilot sequences intended for MIMO channel estimation. Thus, no explicit signaling is needed, i.e. overhead and transmission latency can be reduced.
I. I NTRODUCTION MIMO-OFDM systems have been shown to provide enormous capacity gains compared to conventional SISO-OFDM systems [1] and thus have been subject to intense research. Therefore, MIMO techniques are of particular interest in current and upcoming OFDM based wireless communications standards such as 3GPP LTE, WIMAX and IEEE802.11n [2]. A key issue in practice is to estimate the MIMO channel at a receiver. This knowledge is needed for decoupling spatial streams and coherent data detection. This work focuses on an ad-hoc MIMO-OFDM system operating on a burst transmission basis. In such a system the initial acquistion of incoming packets is of major importance. Typically, a receiver would first perform packet detection, followed by timing synchronization and subsequent frequency synchronization before estimating the MIMO channel and detecting the received data [3]. Synchronization as well as channel estimation in MIMO systems is usually achieved by employing a preamble which is composed of a short training field (STF), primarily intended for synchronization, followed by a long training field (LTF), primarily intended for channel estimation. Synchronization can be performed without knowledge of the actual number of transmit antennas if the STF is carefully designed by e.g. employing a repetitive structure as in [2]. The LTF structure as well as the pilot sequences transmitted within the LTF, typically depend on the number of transmit antennas, which means that the number of transmit antennas has to be known before the MIMO channel estimation. A simple solution, adopted in [2], is the transmission of additional signaling information before the MIMO-LTF whereas this approach introduces overhead and transmission latency. In this work we propose a novel algorithm based only on
the MIMO-LTF and hypothesis testing to detect the number of transmit antennas before MIMO channel estimation. The robust performance of the our algorithm will be shown for a MIMO-OFDM system with IEEE802.11n parameters. Throughout this work, perfect receiver syncronization is assumed. The remainder of the paper is organized as follows: In section II the system model covering the signal model and the preamble design will be introduced briefly. In section III the preamble based least squares channel estimation technique will be reviewed. The algorithm for detecting the number of transmit antennas will be derived in section IV and its performance discussed in section V. Finally, conclusions are drawn in section VI. II. S IGNAL M ODEL AND P REAMBLE D ESIGN OFDM is a multicarrier modulation technique, where (in general complex valued) transmit symbols are modulated onto multiple subcarriers in frequency domain. Transformation to time domain can be achieved by means of a discrete Fourier Transform (DFT). Each time domain OFDM symbol is protected against inter symbol interference (ISI) by prepending a cyclic prefic (CP) which is a repetition of its last samples. At the receiver, after deleting the CP, the signal is converted to frequency domain by means of an inverse DFT. A. Signal Model The received time domain signal 1 can be written as # " Nt NX CIR −1 X t rt r yk = xk−b hb + vrk , t=1
(1)
b=0
where y r , xt , v r , k denote the signal received at antenna r, the signal transmitted at antenna t, complex valued, zero mean additive white gaussian noise (AWGN) and the time domain sample index. The channel impulse response (CIR) hrt between receive-transmit antenna pair rt has a true length of NCIR taps with power σh2 k at tap k and sum power σh2 . The DFT (inverse DFT) of some arbitrary vector b (B) of dimension [N × 1] in time (frequency) domain is defined by 1 B = Fb, b = FH B and FFH = N IN (2) N 1 Lowercase, uppercase, normal and boldface letters denote the time domain, frequency domain, sample and matrix or vector signal representation, respectively.
where F represents the [N × N ] Fourier matrix with elements {F}n,k = W nk = exp(−j2πnk/N ), (·)H is the hermitian operator and IN is the [N × N ] identity matrix. Under the assumption of perfect receiver synchronization, the N received time domain samples, belonging to the m-th OFDM symbol, r are stacked into a column vector ym . Applying a DFT to r ym yields the frequency domain representation of the received signal in vector-matrix notation |
r Fym r Ym,0 .. . r Ym,N −1 {z
=
T
X1m Hr1 . . = .. .. t XN HrNt m } | {z }| {z
r Ym
+ }
Hr
Xm
|
r Vm,1 .. . r Vm,N R {z
. }
r Vm
(3) Similar to ykr in Eq. (1) Yr , X, Hr and Vr denote for the received signal at antenna r, the signal transmitted from all antennas t = [1 · · · Nt ], the channel between all transmit antennas and receive antenna r and AWGN. The transmitted signal t t is defined by Xtm = diag{Xm,0 . . . Xm,N −1 }, where diag{·} creates a diagonal matrix from the complex valued symbols r Xm,n modulated onto subcarriers n = [0 . . . N − 1]. Finally, ¤T £ rt . a single channel vector is given by Hrt = H0rt . . . HN −1 P N −1 2 2 Assuming σX , σV2 , σH = k=0 σh2 k = σh2 = 1 as the signal, noise and channel power per subcarrier, the signal-to-noise 2 ratio (SNR) at the receiver is obtained as SNR = σX /σV2 . B. Preamble Design In packet based OFDM systems, the transmitted data is typically preceded by a bipartite preamble. The generic structure, considered in this work, is depicted in Fig. 1. Upper and lower indices denote transmit antenna t and OFDM symbol m. STF
LTF
Tx 1
LTF11
...
1 LTFNT
Fig. 1.
...
...
...
LTF1NT
Tx NT
Data1
...
NT LTFNT
DataNT
MIMO preamble structure
Particular focus lies on the second preamble part, the LTF. The LTF comprises Nt OFDM symbols, each guarded by a cyclic prefix, as adopted in [2]. Every LTF-OFDM symbol contains pilot sequences for channel estimation. In order to minimize the channel estimation error, pilot sequences transmitted from different antennas are designed to be mutually orthogonal [4]. Orthogonality can be achieved by transmitting sequences separated in time (TO), frequency (FO) or code (CO) as illustrated in Fig. 2. In a TO design antenna t transmits pilot sequences only during the LTFtm=t . In a FO design it transmits pilot sequences only on a subset of subcarriers in every LTF-OFDM symbol which are not occupied by other transmit antennas. In a CO design each antenna transmits pilot sequences occupying all subcarriers in all LTF-OFDM
Tx 1
m
m
m
2
2
2
1
1
n
m
1
n
m
2
2
1
1
n
m 2
Tx 2 1
n
n
a) TO design
b) FO design
n
c) CO design
Fig. 2. Time, frequency and code orthogonal LTF designs, Nt = 2, (m, n) denote OFDM symbol index and subcarrier index
symbols and orthogonality has to be achieved by carefully designed codes. To further minimize the channel estimation error, all pilot symbols have constant modulus and are thus 2 [4]. equipowered with σX III. P REAMBLE BASED C HANNEL E STIMATION Stacking the Nt LTF-OFDM symbols, received at antenna r, in a column vector and using Eq. (3), the received signal can be rewritten as r r1 r H Y1 X1 V1 . . . . .. = .. .. + .. . (4) r rNt r YN t XN t VN t H | {z } | {z }| {z } | {z } r YLTF
XLTF
Hr
r VLTF
Note, that in Eq. (4) the channel is assumed to be constant during the LTF duration. If XLTF is of full rank, which has to be ensured by an appropriate preamble design, the least squares channel estimate can be calculated by ˆ r = X† Y r , H LTF LTF
(5)
X†LTF
−1 H where = (XH XLTF denotes the pseudo inverse LTF XLTF ) of the pilot sequences, transmitted within the LTF. For a comprehensive overview on preamble based channel estimation in MIMO-OFDM, refer to [5].
IV. D ETECTION OF THE N UMBER OF T RANSMIT A NTENNAS A. Algorithm Derivation The algorithm for detecting the number of transmit antennas will be derived, based on the following assumptions: • The LTF is designed orthogonally as depicted in section II-B and thus allows for the direct estimation of N , instead of NCIR , taps of the CIR. • In order to distinguish different numbers of transmit antennas, pilot sequences transmitted from one antenna are choosen from e.g. different sets of code for different numbers of transmit antennas. For simplicity, the derivation is restricted to TO and FO preamble designs, although the proposed algorithm can be applied to CO preamble designs as well without any changes. Starting point of the derivation is the MIMO channel estimation, which will be performed after synchronization. At that stage,
the number of transmit antennas and therefore the particular structure of the transmitted LTF, are unknown quantities at the receiver. Nevertheless, assuming a certain number of transmit antennas Nt,u ∈ {1 · · · Ntmax } with a dedicated LTF structure and unique pilot sequences, channel estimation according to Eq. (5) can be carried out. Subsequently, we will use the statistical properties of the channel estimates for correct and incorrect assumptions to detect the number of transmit antennas. Considering only one particular estimated vector of ˆ rt and dropping the transmit and receive channel coefficients H antenna indices as well as the OFDM symbol index, we obtain £ ¤T ˆ = 1 X ∗ Xt,0 H0 , . . . , X ∗ H + p,0 p,N −1 Xt,N −1 HN −1 2 σX | {z } ˜ H
£ ∗ ¤T ∗ Xp,0 V0 , . . . , Xp,N , −1 VN −1 {z } | ˜ V
(6) where ∗ denotes the complex conjugate. The truly transmitted symbols are denoted by Xt,n and the pilot symbols which are assumed to be transmitted for a certain number of transmit antennas Nt,u are denoted with Xp,n . If the assumption on Nt was correct, Xp,n equals Xt,n and Eq. (6) yields the true vector of channel coefficients, disturbed by AWGN. If the assumption on Nt was incorrect, Xt,n might be a data symbol or a pilot symbol different from Xp,n . Hence, Eq. (6) yields a (pseudo) random vector. This observation can be employed to decide whether the assumption on Nt was correct or not. Therefore, we calculate the covariance matrix of the channel estimates h i ˆH ˆH , Φˆ ˆ = E H (7) HH
where E denotes the expectation operator [6]. In case of an incorrect assumption on Nt , Eq. (7) ideally yields a diagonal ˆ can be assumed to be uncorrelated. matrix if the elements of H In case of a correct assumption on Nt , Eq. (7) yields elements different from zero on the main and off diagonals, since adjacent channel coefficients are correlated if NCIR < N holds. To further distinguish between correct and incorrect assumptions on Nt , the time domain representation of Eq. (7) can be employed h i h i ˆh ˆ H = 1 E FH H ˆH ˆ HF E h N2 h i (8) 1 ˜H ˜H +H ˜V ˜H +V ˜H ˜H +V ˜V ˜ H F, = 2 4 FH E H N σX ˆ k,k = E[|h ˆ k |2 ] on the main diagwhere the elements {E[hh]} onal are of particular interest and will be calculated in the following. The expectation of a single element of the fourth term in Eq. (8) yields io n h £ ∗ ¤ ∗ ˜V ˜H = E Xp,n E V 0 Vn0 Xp,n00 Vn00 n0 ,n00 ( 2 2 (9) σX σV , for n0 = n00 = 0, for n0 6= n00
and hence we obtain h i ˜V ˜ H F = N σ 2 σ 2 IN . E FH V X V
(10)
Since noise and data are mutually uncorrelated, the expectation of both, the second and third term yields h i h i ˜V ˜ H F = E FH V ˜H ˜ H F = 0N E FH H (11) where 0N is the zero matrix of dimenstion [N × N ]. It should be noted that all but the first term in Eq. (8) do not depend on whether a correct or an incorrect number of transmit antennas has been assumed. By contrast, concerning the first term in Eq. (8), two cases have to be considered: 1) The correct number of transmit antennas has been assumed and the first term results in h i h ¡ ¢ i ˜H ˜ H F = σ 4 E FH H FH H H E FH H X n o (12) 4 diag σh2 0 . . . σh2 N −1 . = N 2 σX The second line has been derived using the property FH H = N h (see Eq. (2)) and assuming uncorrelated Rayleigh fading channel taps. 2a) An incorrect number of transmit antennas has been assumed and the received signal results from data symbols. A single element of the first term in Eq. (8) then yields n h io ¤ £ ∗ ∗ ∗ ˜H ˜H E H = E Xp,n 0 Xt,n0 Hn0 Xp,n00 Xt,n00 Hn00 n0 ,n00 ( 4 2 σX σH , for n0 = n00 = 0, for n0 6= n00 , (13) ∗ due to uncorrelated pilot and data symbols E[Xt,n0 Xt,n 00 ] = 0 ∀ n0 6= n00 and thus h i ˜H ˜ H F = N σ 4 σ 2 IN . E FH H (14) X h 2b) An incorrect number of transmit antennas has been assumed and the received signal results from pilot symbols. ˜ comprises pilot symbols Xp,n , Xt,n of constant Hence, H modulus but different phase and an arbitrary channel coefficient Hn and thus, possesses random magnitude and phase. ∗ Since Xt,n is deterministic, we can not employ E[Xt,n0 Xt,n 00 ] to derive an expression as in Eq. (14). Nevertheless, if we ˜ n, assume perfect periodic autocorrelation properties [7] for H we get ( N −1 X 6= 0, for ∆n = 0 ∗ ˜ nH ˜ PACF∆n = H , (n+∆n) mod N = 0, for ∆n 6= 0 n=0 (15) where mod denotes the modulo operator and ∆n = [0...N − 1]. These properties of the periodic autocorrelation function (PACF) can hardly be ensured in practice, since they depend on the pilot sequence design and the channel coefficients. However, we will use them
0.7
for theoretic derivation purposes. Dropping the expectation operator, it can be shown that à N −1 ! N −1 n o X X 00 0 H ˜ ˜H ∗ n k ˜ n0 H ˜ 00 W H W −n k F HH F = =
n0 =0
n00 =0
N −1 ¯ X
N −1 N −1 ¯ X X ¯ ˜ 0 ¯2 ˜ n0 H ˜∗ 0 W ∆nk H ¯ Hn ¯ + (n +∆n) mod N ,
n0 =0
n0 =0
∆n=1
|
{z
(16) where ∆n replaces n00 − n0 . Using Eq. (15), the last term equals zero and the expectation of Eq. (16) thus yields "N −1 # N −1 X X ∗ ∗ ∗ ˜ n0 H ˜ n∗0 = E Xp,n0 Xp,n H 0 Xt,n0 Xt,n0 E [Hn0 Hn0 ] =
0.4 correct hypothesis 1 transmit antenna 0.3
}
PACF∆n
n0 =0
0.5
n
ˆ k,u |2 |h
k,k
0.6
0.2 wrong hypothesis 4 transmit antennas 0.1 0 20
40
60
80
100
120
k
Fig. 3. Squared estimated channel impulse response, 1x1 system, exponential PDP, SNR = ∞
n0 =0 4 2 N σX σh ,
(17) which equals the result in Eq. (14) for received data symbols. Summarizing the results in Eq. (10), (11), (12), (14) and (17), for incorrect and correct assumptions on Nt , the diagonal elements in Eq. (8) can now be calculated according to µ ¶ 1 σV2 2 (18a) N σh + σ 2 n h io X ˆh ˆH E h = k,k 1 σV2 σh2 k + (18b) 2 . N σX For an incorrect assumption on Nt (Eq. (18a)) those elements where found to be constant and do not depend on the time domain sample index k. For a correct assumption on Nt (Eq. (18b)) the diagonal elements result in the true channel PDP and an additional constant term. Since a typical OFDM system is designed such that NCIR ≤ NCP holds, the channel tap power is accumulated in a small window of maximum length NCP . These properties serve as motivation to define a metric PNCP −1 n h ˆ ˆ H io E hh k=0 n h io k,k , Θ= P (19) N −1 H ˆ ˆ E hh k=0 k,k
which, using Eq. (18a) and Eq. (18b) yields NCP N σ2 Θ= σh2 + NNCP σV2 SNR + NNCP X = . 2 σ 2 + σV2 SNR + 1 h σ
(20a) (20b)
X
For an incorrect assumption on Nt , Θ yields a constant (Eq. (20a)), while for a correct assumption (Eq. (20b)) on Nt , it yields a value depending only on the SNR. For an infinitely small SNR Eq. (20b) equals Eq. (20a), for an increased SNR it will exceed Eq. (20a) and for an infinitely high SNR it tends to 1. For a plot of the metric versus the SNR refer to Fig. 4 (dashed curves).
The received signal can only result from a transmission from Nt,u ∈ {1 . . . Ntmax } antennas, each coupled with a dedicated LTF structure and unique pilot sequences. Therefore, Eq. (19) has to be calculated for all hypothesis Nt,u and the hypothesis is assumed to be correct that maximizes Θu . ˆT = arg max {Θu } . N u
(21)
B. Practical Realization ˆh ˆ H ]}k,k is not available Eq. In a real system where {E[h (19) can be approximated by PNt PNr PNCP −1 ¯¯ˆ rt ¯¯2 ¯hk,u ¯ ˆ u = t=1 r=1 k=0 ¯ Θ (22) PNt PNr PN −1 ¯ˆ rt ¯¯2 h ¯ ¯ t=1 r=1 k=0 k,u incorporating the estimated CIR between all transmit-receive antenna pairs which belong to a certain hypothesis Nt,u . A ˆ rt |2 for a system which employs one transmitrealization of |h k,u receive antenna pair (denoted with (1 × 1)), is plotted in Fig. 3 for the TO LTF design, σV2 = 0 and the system parameters described in section V-A. If the receiver tests the incorrect hypothesis Nt,u = 4, the estimated channel tap power is evenly spread all over the complete length N , whereas if it tests the correct hypothesis Nt,u = 1, the estimated channel tap power is accumulated in a small window. This property allows to distinguish between correct and incorrect hypothesis using eq. (22). V. P ERFORMANCE E VALUATION A. Simulation Setup A OFDM system with IEEE802.11n physical layer parameters [2] is considered. The DFT size and cyclic prefix length were N = 128 and NCP = 32. A multipath Rayleigh fading channel with a PDP decaying with exp(−k × 0.36) and k = [0 . . . 15] has been chosen. The system comprises Nt ∈ {1, 2, 4} transmit and Nr ∈ {1, 2, 4} receive antennas. The LTF consists of Nt OFDM symbols (see section II-B) and we consider TO, FO and CO designs. For the TO or FO design, all antennas transmit the same random, BPSK modulated base
0
1
10 1 Tx antennas assumed 2 Tx antennas assumed 4 Tx antennas assumed
0.9
TO LTF ♦
theoretical value correct hypothesis
0.8
FO LTF CO LTF
−1
10
1 − PA
correct hypothesis 1 transmit antenna
0.6
C
ˆu Θ, Θ
0.7 −2
10
1x1
0.5
C
−3
0.4
10
2x2
wrong hypotheses
0.2 −20
C
4x4
0.3
C
theoretical value wrong hypothesis −15
−10
−5
0
5
10
−4
15
20
10 −15
SNR [dB]
Fig. 4. Metric for detecting the number of transmit antennas, 1x1 transmission, TO LTF design
sequences shown as circles in Fig. 2(a) and Fig. 2(b). Different base sequences have been used, depending only on the number of transmit antennas. For the CO design, shown in Fig. 2(c), the LTF is obtained by the following operation " # 1 1 0 XLTF = XLTF ◦ (DNt ⊗ IN ) , e.g. D2 = , (23) 1 −1 where XLTF in Eq. (4) is composed of the matrices Xtm (see Eq. (3)), which contain the same, random BPSK modulated base sequence on the diagonal of the matrix Xtm ∀ m, t. The operator ◦ denotes the elementwise matrix product. The matrix DNt is a Hadamard matrix of dimension [Nt × Nt ] to ensure the full rank of XLTF . The operator ⊗ denotes the Kronecker product. Finally, the spectrum mask according to [2] has been adopted and hence the outermost subcarriers and the DC subcarrier have been set to be zero. B. Results Fig. 4 depicts the theoretical metric (Θ in Eq. (20a), (20b)) ˆ u according to Eq. compared to its simulated counterpart Θ ˆ (22), which is the average of Θu over 104 realizations. The true number of transmit antennas was Nt = 1, one receive antenna has been employed and the TO LTF design was used. The simulation results show a good match with the theoretical derivation, thus justifying the applicability of Eq. (22). Since the mean metric for a correct hypothesis exceeds the mean metric for an incorrect hypothesis significantly even at low SNR, the performance will mainly depend on the variance ˆ u , which depends on the SNR, the channel of the metric Θ statistics, the LTF structure and the pilot sequence design. The figure of merit, to benchmark the proposed metric, is the probability of detecting the incorrect number of transmit antennas, denoted by 1 − PA . Results are plotted in Fig. 5. It can be seen that the proposed algorithm works well with all three LTF designs, i.e. TO, FO and CO, and the differences in performance between the three LTF designs are negligable. The worst case is a (1 × 1) transmission, in which allready 1 − PA < 10−4 at SNR = 0 dB is achieved. Increasing the
−10
−5
0
SNR [dB]
Fig. 5.
Probability of detecting the incorrect number of transmit antennas
number of transmit and receive antennas, performance gains of roughly 5 dB in a 2x2 MIMO transmission and 8 dB in a 4x4 MIMO transmission respectively, can be achieved at 1−PA = 10−4 , compared to a SISO transmission. These gains result mainly from combining the channel estimates (Eq. (22)) on different antenna pairs, where diversity can be exploited to reduce the variance of the metric. VI. C ONCLUSION We derived an algorithm for detecting the number of transmit antennas in burst-wise transmitting MIMO-OFDM systems, where each burst is preceded by a preamble. The proposed algorithm employs only the LTF part of the preamble. It can successfully detect the number of transmit antennas by estimating channels with different hypothetical numbers of transmit antennas. The results show that our algorithm achieves error probabilities of 1 − PA < 10−4 at SNR = 0 dB for the worst case of a SISO transmission. This performance supports us to further adopt this detection algorithm in an ad-hoc MIMO-OFDM system in order to reduce signaling overhead and transmission latency, i.e. no explicit signaling of the number of transmit antennas is required. R EFERENCES [1] H. Boelcskei, D. Gesbert, and A. J. Paulraj, “On the Capacity of OFDMBased Spatial Multiplexing Systems,” IEEE Transactions on Communications, vol. 50, pp. 225–234, 2002. [2] IEEE, “802.11n/D0.04 - Part 11: Wireles LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications: Enhancements for Higher Throughput,” Tech. Rep., 2006. [3] H. Minn, V. Bhargava, and K. Letaief, “A Robust Timing and Frequency Synchronization for OFDM systems,” Wireless Communications, IEEE Transactions on, vol. 2, no. 4, pp. 822–839, Jul 2003. [4] T.-L.Tung and K. Yao, “Channel Estimation and Optimal Power Allocation for a Multiple-Antenna OFDM System,” EURASIP Journal on Applied Signal Processing, 2002. [5] I. Barhumi, G. Leus, and M. Moonen, “Optimal Training Design for MIMO OFDM Systems in Mobile Wireless Channels,” IEEE Transactions on Signal Processing, vol. 51, 2003. [6] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes. McGraw-Hill, 2002. [7] B. M. Popovic, “Generalized Chirp-Like Polyphase Sequences with Optimum Correlation Properties,” IEEE Transactions on Information Theory, vol. 38, pp. 1406–1409, 1992.
