Mimo Preamble Design With A Subset Of Sub Carriers In

MIMO Preamble Design with a Subset of Subcarriers in OFDM-based WLAN Ting-Jung Liang and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Dresden University of Technology, D-01062 Dresden, Germany {liang,fettweis}@ifn.et.tu-dresden.de, http://www.ifn.et.tu-dresden.de/MNS Abstract— This paper deals with how to design a MIMO preamble for OFDM-based WLAN with a subset of subcarriers, which decreases the performance of channel estimation in MIMO. A practical MIMO preamble design should consider mean square error (MSE) of channel estimate, peak-to-average-power-ratio (PAPR) and autocorrelation functions (ACF) together. Our results show that the phase-shift (PS) codes modified by the ”jumping design”, ”seed sequence” and the ”known scrambler” together perform good regarding MSE, PAPR and ACF respectively. The combined design is proposed as new long training symbols at the second part of a preamble.

I. I NTRODUCTION In present packet-based wireless SISO-OFDM systems, such as IEEE802.11a/g, the synchronization and channel estimation functions are accomplished by a preamble, which is composed of two parts: short training symbols (responsible for synchronization) followed by long training symbols (mainly designed for channel estimation). If we extend a preamble technique to MIMO systems, the same short training symbols can be sent from only one antenna. At the receiver, the known SISO synchronization algorithms can accomplish the packet detection and time/frequency synchronization. But, the long training symbols must be modified, because the channel impulse response (CIR) pairs between all the transmit and receive antennas have to be identified, i.e. the training sequences of different transmit antennas have to be orthogonal to avoid inter-antenna interference. In this paper, we discuss how to design long training symbols with multiple transmit antennas. In practical systems, such as IEEE802.11a/g, some subcarriers are set to zero, i.e. only a subset of subcarriers is used. We assume that the length of the MIMO preamble is the same as that of an IEEE802.11a preamble, i.e. no extra transmission delay is introduced. Under this assumption, the required orthogonality of the training sequences in a MIMO long training symbol can be established by special codes, such as Phase-Shift (PS) codes proposed by Barhumi [1] and Ye Li [2]. But, how to adapt the ideal PS codes into a specific subset of subcarriers, i.e. subcarrier index [-26:-1 1:26] with minimum decrease of channel estimation performance, is not discussed in the literature. We analyze the reasons of impairment caused by the subset of subcarriers and propose ”jumping design” to PS codes. In addition, we discuss the Peak-to-Average-PowerRatio (PAPR) and the Autocorrelation Function (ACF) of

training sequences, which are important with respect to power amplifier issues and fine timing synchronization. Notation: Upper (lower) letters will be generally used for frequency-domain (time-domain) signals; boldface letters represent matrices and column vectors; letters with both boldface and underline represent block matrices or vectors in MIMO. II. SISO- AND MIMO-OFDM S IGNAL M ODEL Our channel estimator estimates channel impulse response (CIR) taps, which we characterize by the mean square error M SEsub of the used subset of subcarriers. The reference model of a SISO-OFDM WLAN with 64 subcarriers is shown in Fig. 1(a). The sent preamble signals, CIR, noise and received preamble signals on subcarrier n are P (n), H(n), V (n) and Y (n) respectively. The H in the frequency domain is the DFT (H = F h) of the CIR h in time domain. The matrix F is the DFT matrix. The reference model of MIMO systems is shown in Fig. 1(b). The CIR in frequency or time domain is H nt nr , hnt nr respectively. The received signal in frequency domain is Y nt nr . The diagonal preamble matrix is P nt , whose diagonal vector is [P nt (−32), · · · , P nt (31)]. Nt Tx antennas are used and Nr Rx antennas, with the respective indices nt and nr . In general,

P(-32)

P(31)

H(-32)

V(-32)

x

+

H(31)

V(31)

x

+ (a)

Y(-32)

Y(31)

P1 PNt

1Nr

H

1Nr

= Fh

H Nt Nr = FhNt Nr

Y1

YNr

(b)

Fig. 1. (a) Different subcarriers are treated as independent and parallel channels, (b) Subcarrier signals from different Tx antennas are superimposed at an Rx antenna

the total Tx power is limited to one. The number of channel taps L is assumed to be known at the receiver. The MIMO channel estimation problem can be decomposed into several MISO channel estimation in parallel. An IEEE802.11a/g-like system with four transmit and receive antennas is assumed in numerical examples.

III. C HANNEL E STIMATION IN SISO-OFDM

MISO systems in parallel) is: 

A. Least Square(LS) Channel Estimator In SISO-OFDM with 64 subcarriers, a received OFDM long training symbol, regardless of the total number of used subcarriers N (N =64 or N =52) is: Y = P ( F

[N,1]

h )+V = X h+V

[N,N ] [N,L][L,1]

[N,L]

[N,N ]

[L,L]

and P H P = I. The mean square error of

channel estimation in frequency domain is:  L 1
 H = σV2 M SEall = tr E (F
T D ) (F
T D ) N N

(2)

B. Impact of a Subset of Subcarriers Using the Fourier matrix F corresponding to the CIR length L, the impact of a subset of subcarriers can be characterized by the mean square error on subcarrier n, as derived by Morelli [3]  L L    j2πn(k1 −k2 ) 64 [C
T D
T D ]k1 ,k2 e M SEsub (n)= k1 =0 k2 =0

(3)

the covariance matrix of the estimation error is:    −1 H H σV2 = X X C
T D
T D = E
T D (
T D ) [L,L]

[L,L]

[L,1]

[N,1]

A. Least Square(LS) Channel Estimator In MIMO-OFDM, we define X nt = P nt F and a received [N,L]

OFDM long training symbol at receive antenna nr (several

· · · X Nt

 X

nr

X

h

[N,Nt L][Nt L,1]

+V

nr

(5)

nr

B. State of Art - MIMO Long Training Symbol Design In order not to introduce extra transmission delay, the length of the MIMO long training symbol is kept the same as that of an IEEE802.11a long training symbol. The orthogonality of training sequences between different transmit antennas can be established by special codes. 1) All Subcarriers Available: The phase shift (PS) code proposed by Barhumi [1] and Ye Li [2] can be applied in long training symbols. The original papers assume that all subcarriers are available and training sequences are equipowered and equi-spaced. The PS code is derived from I for ∀ L. An example of PS code with X H X = En [Nt L,Nt L]

four transmit antennas is shown in Fig. 2. The sequences in the (nt )th row of Fig. 2 are the diagonal vector of preamble matrix P nt . The training sequence has period of four subcarriers Tx\Sub

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IV. MIMO-OFDM C HANNEL E STIMATION AND L ONG T RAINING S YMBOL D ESIGN

X1

 = (X H X)−1 X H Y nr and The MISO LS estimator is h nr  nr − hnr . the estimation error in time domain is
T D =h

4

The matrix product X H X with dimension [L,L] characterizes the mean square error on different subcarriers. For P H P = I, the properties of X H X = F H F should be analyzed. For all subcarriers used, F H and F are inverse operations. This property of the the Fourier transform is destroyed by using only a subset of all subcarriers. More clearly, if X H X = En I (En : energy of training sequence in one OFDM symbol), M SEsub (n) is constant for ∀ n. Conversely, if frequency selective channel (L > 1) exists and only a subset of subcarriers are available, X H X is no more identity matrix and the mean square error varies with subcarrier index.



=

Tx\Sub

(4)



h nr

(1)

The dimensions of vectors or matrices are written below the mathematical symbols. The Fourier matrix F has entries [F ]n,k = e−j2πnk/N , k=[0:(L-1)], and n=[-32:31] (N =64) or n=[-26:-1 1:26] (N =52). The LS channel estimator is ˆ = (X H X)−1 X H Y [3] and the estimation error in time h ˆ − h. domain is
T D = h If all subcarriers are available and the power of the training sequences on all subcarriers is the same, then F H F = N I , FFH = L I

Y nr =

 h1nr   .. n   +V r .  hNt nr

 

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Fig. 2. Design example of phase shift (PS) codes with period 4 subcarriers (Larsson’s proposal [4] sets the non-shaded subcarriers to zero)

and repeats through all subcarriers. The mean square error of channel estimation is minimized as: M SEall =

LNt 2 σ N V

(6)

The mean square error is Nt times that of the mean square error for the SISO case. 2) Impact of a Subset of Subcarriers: Based on the model of several MISO channel in parallel, Morellis’work [3] can be extended to the mean square error on different subcarriers for

antenna pair (nt ,nr ).  L L    nt nr C
TntDnr
TntDnr M SEsub (n)= k



1 ,k2

k1 =0 k2 =0

e

j2πn(k1 −k2 ) 64

(7)

The covariance matrix of the estimation error is:    C
nT rD
nT rD = E [Nt L,Nt L]



nr H r
n T D (
T D ) [Nt L,1]

C
1nr
1nr TD TD  .. = .  C
Nt nr
1nr TD

TD

= X

H

−1

X

[Nt L,Nt L]

σV2

 C
1nr
Nt nr TD TD  ..  (8) .  C
Nt nr
Nt nr

··· .. . ···

TD

TD

Using the known training sequence, the block matrices on the diagonal above can be used to calculate the channel estimation error on different subcarriers in the frequency domain. According to similar arguments as for the SISO case, if X H X = En I (En : total energy of training sequence from nt nr (n) is all transmit antennas in one OFDM symbol), M SEsub constant for ∀ n. If not, the loss of effective power decreases the channel estimation performance. When all subcarriers are available, the condition of minimized mean square error of channel estimation proposed nt nr . by Barhumi [1] coincides with that of constant M SEsub If only a subset of subcarriers is available, since matrix product X H X characterizes the mean square error of channel estimation on different subcarriers, we look for an adaptation of the design of proposed training sequences, which minimizes the energy in the off-diagonal of the matrix product X H X. C. Analysis of the Matrix Product X H X We assume Nt = 4 in the following discussion. 1) Structure of the Matrix Product X H X: The structure of matrix product X H X is shown in Fig. 3. The matrix D-OBM

O-CBM

O-OBM

D-CBM

X X= H

Nt L

NtL

Fig. 3. The matrix product is composed of nonshaded Central Block Matrix(CBM)and shaded Off-diagonal Block Matrix(OBM)

product is composed of two parts: nonshaded Central Block Matrix(CBM) and shaded Off-diagonal Block Matrix(OBM). The element in diagonal and off-diagonal of CBM and that of OBM are defined as D-CBM, O-CBM, D-OBM and O-OBM. The elements of four CBMs in MIMO (equivalent four MISO channel in parallel) is the same as that of SISO. The element in diagonal of CBM (D-CBM) and the element in th off-diagonal of CBM (O-CBM) listed in equation (9) have parameters: θn = 0, n = [−32 : 31] and parameters:

θn = −2π(n/64), n = [−32 : 31],  = [1 : L − 1] respectively.   −1     −27 D-CBM   jθn jθn jθ0 or e e + + e = n=−32 n=−26 O-CBM  26   31    jθn jθn + e e + (9) n=1

n=27

The amplitude of these elements is constant due to equipowered training sequence. The element in diagonal of OBM (D-OBM) and the element in th off-diagonal of OBM (O-OBM) listed in equation ∗ (10) have parameters: ant1 nt2 (n) = P nt1 (n) (P nt2 (n)) , θn = 0, n = [−32 : 31] and parameters: ant1 nt2 (n) = ∗ P nt1 (n) (P nt2 (n)) , θn = −2π(n/64), n = [−32 : 31],  = [1 : L − 1] respectively.   −27 D-OBM  nt1 nt2 jθn or a (n)e = n=−32 O-OBM  −1     nt1 nt2 jθn nt1 nt2 jθ0 + a (n)e (0)e + a  +

n=−26 26  nt1 nt2

a

n=1

 jθn

(n)e

 +

31 

 nt1 nt2

a

jθn

(n)e

n=27

(10) The phasors of D-OBM and O-OBM vary as in CBM, but all phasors are modulated by the product of training sequence and conjugate of its corresponding training sequences of different transmit antenna on the same subcarrier. 2) Impairment caused by A Subset of Subcarriers: If all subcarriers are available and an ideal PS code is used as the training sequence, the matrix product X H X is precisely an identity matrix. If only a subset of subcarriers available, Larsson [4] has proposed a ”truncate design” shown in Fig. 2, which generates ideal PS codes through all subcarriers and sets directly the unused subcarriers to zero. In the case that subcarrier indices [-26:-1 1:26] are used, the orthogonality in equation (9) and (10) is destroyed by ”truncate design”. According to equation (9) and (10), the ”truncate design” can be further improved. Consecutive subsequences, which enforce the diagonal of OBM (D-OBM) to be zero, are proposed to reduce crosscorrelation values between different PS codes. Fig. 4 shows one example of our proposal ”jumping design” to PS codes with four transmit antennas. The principle of our proposal is to keep the continuity of training sequence in used subset of subcarriers. More clearly, the periodic block starts from sub-carrier indices [-26,-25,-24,-23], repeats to [-6,-5,-4,3], jumps over DC, i.e. in subcarrier indices [-2,-1,1,2], and repeats again up to [23,24,25,26]. The matrix product X H X for the ”jumping design” and the ”truncate design” formed by PS codes is characterized by the normalized energy in the elements of the matrix product relative to the main diagonal. Fig. 5 shows that the average of normalized energy outside of the main diagonal by ”truncate

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Example of ”jumping design” based on PS code (Nt = 4) Fig. 6.

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Average mean square error vs. Subcarrier Index [n]

design” is generally larger than that by ”jumping design” in different off-diagonals. It is also verified by integral that the 0

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Fig. 7. Fig. 5. The average of normalized energy in different off-diagonals of matrix product XH X by the ”truncate design” and the ”jumping design”

total energy outside of the main diagonal in the matrix product X H X of ”jumping design” is smaller than that of ”truncate design” by 5.8 percent. D. Channel Estimation Performance in MIMO OFDM In all our numerical examples, we assume that only one OFDM symbol of the long training symbol is available. In Fig. 6 and 7, the channel model is assumed to have length L, which is known at the receiver, and to appear always on the system sampling points. The channel power in all taps is statistically limited to ”1”, but the distribution of channel power in taps is not specified in our model. The mean square error on different subcarriers (M SEsub ) is the expectation value of random realizations averaging through all antenna pairs. In Fig. 6, the mean square error at the subcarrier edge is worse due to a subset of subcarriers. The ”jumping design” outperforms the ”truncate design” on subcarriers near DC and the two edge in this case. M SEavg is defined as the average of the mean square error M SEsub on 52 used subcarriers, i.e. subcarrier index [-26:-1 1:26]. Fig. 7 shows, the ”jumping design” outperforms the ”truncate design” from L=1 (flat fading) up to L=13 by M SEavg . The channel taps larger than

Average mean square error vs. L [tap]

If the estimated length of CIR (Lest ) is not the same as the true L, the channel estimation performance is investigated by simulation using a channel model with exponential power delay profile (τ = 2, the length of CIR=L and the channel power in all taps normalized to ”1”). Two cases are considered: low SNR case (SNR=10dB; noise dominates all) and high SNR case (SNR=30dB; noise dominates, only when L overestimated) shown in Fig. 8 and 9 respectively. In Fig. 8, the performance depends only on the estimated length at the receiver, which includes Lest taps of noises. The channel estimator with fewer knowledge of received taps (L > Lest ) performs better than that with perfect knowledge, because the end taps in exponential power delay profile occupy only a very few portion of the total power, the loss of which has relatively smaller effects compared to the decrease of noise power. Fig. 9 shows that, if L < Lest (L overestimate), the estimated length at the receiver dominates the performance, too. But, if L > Lest , the channel estimation performance is worse than the MSE with the true L < Lest , because of the loss of useful channel information in taps. The scale of performance loss depends on the channel model. But, in all investigated cases, the ”jumping design” always outperforms the ”truncate design”.

significantly. But, the ratio of central lobe energy to total energy called F in [5] in SISO (F=1.73) is still better than that in MIMO (F=0.61) between [-63:63]. In practice, because of

0

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*

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Energy of ACF

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Fig. 8.

Correlation with the next repeated long training symbol

With Scrambler

SISO PS codes (4x4)

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VII. S UMMARY OF P ROPOSED MIMO P REAMBLE

SNR=30dB 7 8 L [tap]

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the typical 0-4 samples offset after fine timing [6], we suggest that descrambling is operated in frequency domain and the residual time offset can be corrected by channel estimation.

10 11 12 13

Average mean square error vs L [tap] of high SNR case

V. P EAK - TO -AVERAGE -P OWER -R ATIO (PAPR) Based on Li [2], the PAPR can be significantly improved without loss of channel estimation performance by multiplying the PS codes in different transmit antennas with the same ”seed sequence” with good PAPR characteristics. We investigate the case of the IEEE802.11a long training symbol (PAPR=4.1dB) as the ”seed sequence”. The PAPR of original PS codes in MIMO 4x4 systems are 17.2, 16.8, 15.6 and 16.8dB at transmit antenna 1, 2, 3, 4 respectively. The improved PAPRs of PS code using ”jumping design” in MIMO 4x4 systems are 4.1, 4.5, 3.9 and 4.5 dB at transmit antenna 1, 2, 3 and 4 respectively. VI. AUTOCORRELATION F UNCTION (ACF) In addition to PAPR, the PS codes severely reduce the resolution range of the autocorrelation function. A new scrambling architecture is proposed to combat this effect. After IFFT, the PS codes is, sample by sample, multiplied by a known scrambling sequence, such as the first 64 samples of IEEE802.11a scrambler. At the receiver, the received signals before descrambling are used in fine timing and the channel estimation is performed after descrambling and FFT. In Fig. 10, we correlate the known long training symbols, which are designed by PS codes using ”jumping design” and ”seed sequence”, with the received signals, i.e. short training symbols followed by long training symbols. The result shows that our scrambling architecture improves the resolution range

A preamble, composed of two parts: short training symbols followed by long training symbols, can be extended from SISO to MIMO. The same short training symbols as in IEEE802.11a can be sent from only one antenna. The phase-shift (PS) codes modified by the ”jumping design”, ”seed sequence” and the ”known scrambler” together are proposed as new MIMO long training symbols, which perform good in practice regarding MSE, PAPR and ACF respectively. Furthermore, the expense of performance improvement is a small extra-complexity of descrambling in frequency domain at the receiver. ACKNOWLEDGMENT This work was partly supported by the German ministry of research and education within the project Wireless Gigabit with advanced multimedia support (WIGWAM) under grant 01 BU 370. The authors will also thank Wolfgang Rave and Ralf Irmer for their valuable comments. R EFERENCES [1] 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, pp. 1615–1624, June 2003. [2] Y. Li and H. Wang, “Channel estimation for MIMO-OFDM Wireless Communications,” in Proc. IEEE PIMRC, vol. 1, pp. 535–539, Sept. 2003. [3] M. Morelli and U. Mengali, “A Comparison of Pilot-Aided Channel Estimation Methods for OFDM Systems,” IEEE Transactions on Signal Processing, vol. 49, pp. 3065–3073, Dec. 2001. [4] E. G. Larsson and J. Li, “Preamble Design for Multiple-Antenna OFDMbased WLANs with Null Subcarriers,” IEEE Signal Processing Letters, vol. 8, pp. 285–288, Nov. 2001. [5] M. Golay, “A Class of Finite Binary Sequences with Alternate Autocorrelation Values Equal to Zero,” IEEE Transactions on Information Theory (Correspondence), vol. 18, pp. 449–450, May 1972. [6] J. Terry and J. Heiskala, OFDM WIRELESS LANs: A Theoretical and Practical Guide. SAMS, 2001.