Channel Compensation for SFBC-OFDM Systems in Broadband Wireless Channels ∗
† ¨ Berna Ozbek and Didier Le Ruyet ∗ † Izmir Institute of Technology, Izmir, Turkey
[email protected] Conservatoire National des Arts et M´etiers, Paris, France
[email protected]
Abstract— In this paper, we consider orthogonal space-frequency transmit diversity schemes for OFDM systems in broadband wireless channels. We propose a compensation method to prevent error floor caused by unequal subchannels in the space-frequency block coded OFDM systems for 2 and 3 transmit antennas while providing diversity gain. We will show that for 2 antennas case, the proposed compensation method gives the zero forcing solution. We compare the proposed method with the space-time block coded OFDM systems with respect to broadband wireless channel specifications. We also show that channel compensation method can be performed after channel estimation using a Kalman filter. Index Terms— OFDM, space-frequency block code, channel compensation, error floor.
I. I NTRODUCTION For future wireless broadband systems, orthogonal frequency division multiplexing (OFDM) is the most promising technique to provide high data rates over time varying and frequency selective channels. OFDM with cyclic prefix transforms a frequency selective channel to Nc frequency flat subchannels. Furthermore, multiple-input multiple-output (MIMO) systems can increase the data rate using multiplexing diversity schemes and also mitigate the fading effects through spatial diversity techniques such as orthogonal designs. In MIMO-OFDM, the orthogonal designs can be applied as space-time block coded (STBC)-OFDM [1] or space-frequency block coded (SFBC)-OFDM [2]. In STBC-OFDM, the symbols are transmitted over the same subcarriers by using two or more adjacent OFDM symbols. On the other hand, in SFBC-OFDM, the symbols are combined using an orthogonal matrix through the neighboring subcarriers in the same OFDM symbol. In both cases, it is generally assumed that the channel coefficients are constant over neighboring
subcarriers or OFDM symbols in the orthogonal code structure. In [3], the authors have compared these two techniques. They have shown that for high delay spread, the STBC-OFDM is superior to the SFBCOFDM. Indeed, for SFBC-OFDM over severe frequency selective channels which have high delay spreads compared to symbol duration, the channel coefficients might not be constant over neighboring subcarriers. Therefore, the orthogonality of the code structure is no longer preserved. This causes an error floor in the bit-error-rate (BER) performance due to the interference which comes from the neighboring symbols. However, SFBC-OFDM has an important advantage compared to STBC-OFDM which is the absence of decoding delay. In STBC-OFDM, we should wait Tf OFDM symbols before decoding while there is almost no decoding delay in SFBCOFDM. In this paper, we propose a channel compensation method which prevents the error floor with a computationally efficient receiver structure for the SFBCOFDM systems with 2 and 3 transmit antennas. We verify the proposed method by obtaining BER results. We also show that the channel compensation can be performed after channel estimation using Kalman filtering for time varying broadband channels. In section II, we introduce the SFBC-OFDM over broadband wireless channels. Then, in section III, we propose the channel compensation method for SFBC-OFDM system with 2 and 3 transmit antennas. In section IV, we present the channel estimation based on Kalman filtering. Finally, we give the simulation results for schemes with different receiver structures considering broadband wireless channel specifications.
II. T HE SFBC-OFDM
SN Rk+1,n =
IN BROADBAND WIRELESS
CHANNELS
We consider a space-frequency block coded OFDM system with Nc subcarriers, Nt transmit antennas and Nr receive antennas. The space frequency block code matrix consists of Q symbols over Tf subcarriers providing the transmission rate RSF BC = Q/Tf . A. Nt = 2 transmit antennas case We use the Alamouti STBC [4] through adjacent subcarriers, k and k + 1 for each OFDM symbol. At the receiver, after applying a fast Fourier transform (FFT), the received signals belonging to nth OFDM symbol can be written in the matrix/vector form Rk,n = Hk,n Sk,n + Nk,n
k = 1, 3, ..., Nc − 1 (1)
where Sk,n = [ sk,n sk+1,n ]T and Rk,n = ∗ [ Rk,n Rk+1,n ]T are transmitted and received vec∗ ]T is the tors respectively, Nk,n = [ Nk,n Nk+1,n noise vector whose elements are FFT of additive white Gaussian noise with zero mean and σn2 variance per real dimension and
Hk,n =
H1,k,n H2,k,n ∗ ∗ H2,k+1,n −H1,k+1,n
ˆ k,n = Hmf Rk,n S k,n
(3)
where Hmf k (n) is the transpose-conjugate of the Hk,n matrix in (2). Then, we have: sˆk,n sˆk+1,n
=
Dk,n εk,n ε∗k,n Dk+1,n
sk,n sk+1,n
¯ k,n +N
(4) where Dk,n = |H1,k,n |2 + |H2,k+1,n |2 and Dk+1,n = |H2,k,n |2 + |H1,k+1,n |2 are the diversity gains and ∗ ∗ εk,n = H2,k,n H1,k,n − H2,k+1,n H1,k+1,n is the interference term. At the receiver side, SNRs can be calculated as, SN Rk,n =
|Dk,n |2 |2
|εk,n +
2σ σ
2 n 2 s
Dk,n
Es N0
(5)
=
σs2 2 2σn
|εk,n |2 +
(6)
2 2σn σs2 Dk+1,n
is the SNR at the transmitter side.
B. Nt = 3 transmit antennas case We apply the orthogonal STBC in [5] through adjacent subcarriers, k , k + 1, k + 2 and k + 3 (k = 1, 5, 9, ..., Nc − 3) using the symbols for k , k + 1, k + 2 (k = 1, 4, 7, ..., 3/4Nc − 2) at each OFDM symbol. At the receiver, after applying FFT, we write the received signals in the matrix/vector form as, Rk,n = Hk,n Sk ,n + Nk,n
(7)
where Sk ,n = [ sk ,n sk +1,n sk +2,n ]T and ∗ ∗ ∗ Rk+2,n Rk+3,n Rk,n = [ Rk,n Rk+1,n ]T are transmitted and received vectors respectively, and ∗ ∗ ∗ Nk+2,n Nk+3,n Nk,n = [ Nk,n Nk+1,n ]T is the noise vector. The channel transfer matrix is given by
Hk,n =
(2)
is the channel transfer matrix. Note that the channel coefficients belonging to adjacent subcarriers are not equal for each antenna pair. The matched filter (MF) is used to obtain the reconstructed symbols by multiplying the transposeconjugate of the channel transfer matrix with the received signals in (1) and then symbols can be detected separately.
where
|Dk+1,n |2
H1,k,n H2,k,n H3,k,n ∗ ∗ H2,k+1,n −H1,k+1,n 0 ∗ ∗ −H3,k+2,n 0 H1,k+2,n ∗ ∗ 0 −H3,k+3,n H2,k+3,n
(8) Similar to (3), the MF can be used to reconstruct the symbols: ˆ k ,n = Hmf Rk,n S k,n
(9)
where Hmf k,n is the transpose-conjugate of the Hk,n matrix in (8). Then, we have:
Dk ,n εk ,n εk +1,n ∗ ¯ k ,n ˆ k ,n = Dk +1,n εk +2,n S εk ,n Sk ,n +N ∗ ∗ εk +1,n εk +2,n Dk +2,n (10) where Dk +j,n and k +j,n (j = 0, 1, 2) are the diversity gains and the interference terms respectively similar to the 2 transmit antennas case. The SNRs for each symbol can be written as: SN Rk ,n =
|Dk ,n |2 |εk ,n |2 + |εk +1,n |2 +
SN Rk +1,n =
SN Rk +2,n =
2 2σn σs2 Dk ,n
(11)
|Dk +1,n |2 |εk ,n |2 + |εk +2,n |2 +
2 2σn σs2 Dk +1,n
|Dk +2,n |2 |εk +1,n |2 + |εk +2,n |2 +
(12)
2 2σn σs2 Dk +2,n
(13)
For both cases, the MF detection achieves maximum diversity, however interference terms coming from neighboring symbols add up to the noise and cause an error floor. It is possible to apply a maximum likelihood (ML) decoding which prevents the error floor at the expense of computational complexity especially for high data rates (16QAM, 64QAM, etc). Another solution is to perform zero forcing (ZF) which implies a 2 × 2 inversion or a 4 × 3 pseudoinversion of the channel transfer matrix respectively given in (2) and (8). III. T HE P ROPOSED C HANNEL C OMPENSATION M ETHOD We propose a channel compensation method which eliminates the interference terms in (4) and (10) and provides diversity gains as in ZF while avoiding the inverse operation. Instead of using only transpose conjugate of channel transfer matrix as in MF, we propose to compensate the necessary channel coefficients first, reconstruct a new channel transfer matrix, and then take the transpose conjugate of this matrix. Let’s define the gain differences between channel coefficients belonging to the same block of subcarriers. Hi,kx,n Ki,kx,n = i = 1, ..., Nt (14) Hi,k,n kx = k + 1, ..., k + Tf − 1 For the 2 transmit antennas case, we present the system model as: ˆ k,n = Hmzf Rk,n S k,n
with
Hmzf k,n =
∗ H1,k,n ∗ H2,k,n
(15)
H2,k+1,n Ak,n − H1,k+1,n A∗k,n
(16)
∗ K2,k+1,n . where Ak,n = K1,k+1,n In this model, the diversity gains become
¯ k,n = |H1,k,n |2 + |H2,k+1,n |2 D Ak,n |2 ¯ k+1,n = |H2,k,n |2 + |H1,k+1,n D ∗
(17)
Ak,n
Note that the proposed solution satisfies the equal ¯ k,n 0 D mzf ity Hk,n Hk,n = ¯ k+1,n . In order to 0 D obtain equal and real valued diversity gains for each symbol, by suppressing the noise power, we propose:
Hzf k,n
=
1 ¯ k,n D
0
0 1
¯ k+1,n D
Hmzf k,n
(18)
The SNRs at the receiver side are: ¯ k,n |2 σs2 |D SN Rk,n = 2 2,k+1,n | (|H1,k,n |2 + |H|A )2σn2 2 k,n | SN Rk+1,n =
¯ k+1,n |2 σs2 |D (|H2,k,n |2 +
|H1,k+1,n |2 2 |Ak,n |2 )2σn
(19)
(20)
We extend the method for 3 transmit antennas by defining the matrix Hmzf k,n as:
∗ H1,k,n
H2,k+1,n Ak,n − H1,k+1,n A∗k,n
∗ H2,k,n ∗ H3,k,n
− H3,k+2,n Bk,n
0
− H3,k+3,n Ck,n
0
H1,k+2,n ∗ Bk,n
0
H2,k+3,n ∗ Ck,n
(21) =
∗ K2,k+1,n , Bk,n where Ak,n = K1,k+1,n ∗ ∗ K3,k+3,n . K1,k+2,n K3,k+2,n and Ck,n = K2,k+3,n
The achievable diversity gains become:
¯ k,n = |H1,k,n |2 + |H2,k+1,n |2 + |H3,k+2,n |2 D Ak,n Bk,n 2 |2 ¯ k+1,n = |H2,k,n |2 + |H1,k+1,n + |H3,k+3,n | D ∗ ¯ k+2,n = |H3,k,n |2 + D
Ak,n |H1,k+2,n |2 ∗ Bk,n
Ck,n |H2,k+3,n |2 ∗ Ck,n
+
(22) Similar to (18), we can obtain system matrix as:
Hzf k,n =
1 ¯ k,n D
0 0
0 1 ¯ k+1,n D
0
0 0
mzf H k,n
1
(23)
¯ k+2,n D
The SNRs for each symbol are written as, SN Rk ,n =
¯ k,n |2 σ 2 |D s
(|H1,k,n |2 +
SN Rk +1,n = SN Rk +2,n =
|H2,k+1,n |2 |Ak,n |2
+
|H3,k+2,n |2 2 |Bk,n |2 )2σn
¯ k+1,n |2 σs2 |D
(|H2,k,n |2 +
|H1,k+1,n |2 |Ak,n |2
+
|H3,k+3,n |2 2 |Ck,n |2 )2σn
¯ k+2,n |2 σs2 |D
(|H3,k,n |2 +
|H1,k+2,n |2 |Bk,n |2
+
(24)
(25)
|H2,k+3,n |2 2 |Ck,n |2 )2σn
(26)
The proposed channel compensation method in (18) and (23) eliminates the interference terms coming from neighboring subcarriers while providing diversity gains for the SFBC-OFDM systems with 2 and 3 transmit antennas. The proposed method gives the ZF solution since it satisfies the equality Hzf k Hk = INt . Note that, this method avoids the channel transfer matrix inversion. Hence, we reduce the classical ZF receiver complexity especially for 3 transmit antennas case for unequal subchannels. However, since we don’t perform exactly pseudoinverse which is rather complex operation, there would be difference between the results of classical and proposed ZF receiver.
Since we consider higher Doppler frequencies, we propose to apply Kalman filtering [6] to estimate and track the channel coefficients. For each subcarrier block, k = 1, Tf + 1, ..., Nc − Tf + 1, the changes of the channel coefficients in the time domain can be represented by first order AR model [7]: Hk,n+1 = AHk,n + Wk,n
(27)
where Hk (n) = [H1,k,n H2,k,n H1,k+1,n H2,k+1,n ]T is the channel vector for the system with Nt = 2. Similarly it can be written for the system with Nt = 3. The matrix A = αIQNT is the state transition and Wk,n is the state noise matrix. These channel variations are mainly caused by a Doppler effect and mismatch between the transmitter and receiver frequency. Assuming perfect synchronization, the changes in the channel coefficients between the OFDM symbols are defined as, α = J0 (2πfd Ts )
2 σw = 1 − |α|2
(28)
2 is the state noise variance per real dimenwhere σw sion in time domain, fd is the Doppler frequency, Ts is the duration of one OFDM symbol and J0 (.) is the zeroth order Bessel function. The pilot symbol matrices used in the algorithm for the system with 2 transmit antennas is defined as,
Spk,n
=
sk,n sk+1,n 0 0 ∗ ∗ 0 0 −sk+1,n sk,n
(29)
and similarly it can written for the system with 3 transmit antennas. We perform the Kalman estimation and tracking algorithm using pilot and estimated symbol matrices for SFBC-OFDM with 2 and 3 transmit antennas case in time varying broadband wireless channels. V. S IMULATION R ESULTS In this section, we give the simulation results for SFBC-OFDM with the proposed solution compared to STBC-OFDM MF and SFBC-OFDM MF and ZF for Nt = 2 and Nt = 3 using broadband wireless channels. The symbol duration of OFDM symbol is set to TOF DM = Nc Tsc where Nc = 1024 is the number of subcarriers and Tsc = 7.4ns is the symbol duration as given in [3]. The total OFDM symbol duration including guard interval of length TOF DM /4 is equal to Ts = TOF DM +Tg . For QPSK
and 16QAM, 1536 and 3072 bits per OFDM symbol are employed respectively. Wireless channels from each transmit to each receive antenna experience Rayleigh fading with independent propagation paths. The Doppler spectrum is obtained by using the Jakes’ model. The power loss and delay profiles are taken from the Channel Model B given in [3] which includes 24 paths. The time and power difference between the paths is set as ∆Tt = 8Tsc and ∆At = 0.5dB , respectively. (The power of the first path is 0dB ). The rms delay spread of Channel Model B is calculated as στ = 3.5µs. It should be noted that, since the criterion Bc = 5σ1τ > TTfs given in [3] is not satisfied for Nc = 1024, the subchannel coefficients are not constant during 2 and 4 subcarriers. Firstly, we obtain the received SNR values for SFBC-OFDM system with 3 transmit antennas at Es /N0 = 15dB considering only 192 subcarriers per OFDM symbol block. According to results given in Figure 1, the proposed solution increases the received SNR compared to the MF solution at each subcarrier. Moreover, the classical ZF receiver achieves about 0.6dB more SNR than the proposed channel compensation method. (3,1) SFBC−OFDM system, Channel Model B, Es/N0=15dB 20
18
Classical ZF Receiver Proposed ZF Receiver Matched Filter
16 Received SNR (dB)
IV. C HANNEL E STIMATION BASED ON K ALMAN F ILTERING
14
12
10
8
0
20
40
60
80
100
120
140
160
180
200
subcarrier index, k
Fig. 1.
Received SNR values of SFBC-OFDM with Nt = 3
In Figure 2, we compare the BER performance of the proposed SFBC-OFDM receiver with the SFBC-OFDM MF and ZF receivers and the STBCOFDM MF receiver for 2 transmit antennas. The receiver speed is chosen as v = 228km/h which provides us fd Ts = 0.01. 16QAM is employed by using 3072 bits per OFDM symbol. According to the simulation results, the error floor observed using the MF solution is prevented by using the proposed channel compensation method and gives exactly the same performance with the classical SFBCOFDM ZF solution. Moreover, the performance of
the SFBC-OFDM system with the proposed receiver approaches the performance of the STBC-OFDM system.
(2,1) SFBC−OFDM, QPSK, fdTOFDM=0.001
0
10
Matched Filter, Perfect CSI Proposed ZF Solution, Perfect CSI Proposed ZF Solution, Channel Estimation
−1
10
(2,1) SFBC−OFDM, 16QAM, fdTs=0.01
0
10
BER
OC SFBC−OFDM, Matched Filter OC SFBC−OFDM, Classical ZF receiver OC SFBC−OFDM, Proposed ZF receiver OC STBC−OFDM, Matched Filter
−2
10
−1
10
Uncoded bit error rate
−3
10
−2
10
−4
10
0
2
4
6
8
10 Eb/N0 (dB)
12
14
16
18
20
−3
10
Fig. 4. Channel estimation performance for the SFBC-OFDM with 2 transmit antennas −4
10
0
2
4
6
8
10 Eb/N0 (dB)
12
14
16
18
20
Fig. 2. The BER performance of SFBC-OFDM and STBCOFDM with 2 transmit antennas
The BER performance for orthogonal SFBCOFDM with 3 transmit antennas is shown in Figure 3 by employing 16QAM . Again, the performance of the SFBC-OFDM system with the proposed ZF solution approaches the performance of the SFBCOFDM system with classical ZF receiver while decreasing receiver structure complexity and the performance of STBC-OFDM system while decreasing the delay time in decoding. Finally, we consider the system performance using the channel estimation based on Kalman filtering. The receiver speed is v = 23km/h corresponds to fd Ts = 0.001 and QPSK is employed. We use Np = 10 pilot OFDM symbols before Nd = 90 data OFDM symbols. As shown in Figure 4, the BER performance of the proposed ZF receiver with (3,1) SFBC−OFDM, 16QAM, fdTs=0.01
0
10
−1
Uncoded bit error rate
10
−2
10
−3
10
OC SFBC−OFDM, MF OC SFBC−OFDM, Proposed ZF receiver OC SFBC−OFDM, ZF OC STBC−OFDM, MF
−4
10
0
2
4
6
8
10 Eb/N0 (dB)
12
14
16
18
20
Fig. 3. The BER performance of orthogonal SFBC-OFDM and STBC-OFDM with 3 transmit antennas
Kalman filtering approaches to perfect CSI results with approximately 1.5dB difference at 10−3 . VI. C ONCLUSIONS While it is generally assumed that SFBC-OFDM using orthogonal designs is a technique limited to channels where the delay spread is small compared to the symbol duration, we have shown that using a ZF receiver, it is possible to extend the range of application of SFBC-OFDM. In this paper, we have proposed a new ZF receiver for orthogonal scheme with 2 and 3 antennas avoiding the matrix inversion. Based on BER performances, we have shown that the proposed method can fill the gap between STBC-OFDM and SFBC-OFDM for high frequency selective wireless channels. R EFERENCES [1] K. F. Lee, D. B. Williams, A space-time coded transmitter diversity technique for frequency selective fading channel, , Proc. IEEE Sensor Array and Multichannel Signal Proc. Workshop, vol.1 , pp.149-152, 2000. [2] H. Bolcskei, A. J. Paulraj, Space-frequency coded broadband OFDM systems, Proc. IEEE WCNC’2000, vol.1, pp.16, 2000. [3] G. Bauch, Space-time block codes versus space-frequency block codes, VTC 2003-Spring, vol.1, pp.567-571, 2003. [4] S. M. Alamouti, A simple transmitter diversity scheme for wireless communications, IEEE J. Select. Areas Communication, vol.16, pp.1451-58, 1998. [5] B. M. Hochwald, T. L. Marzetta, C. B. Papadias, A transmitter diversity scheme for wideband CDMA systems based on space-time spreading, IEEE J. Select. Areas Communication, vol.19, pp.48-60, 2001. [6] S. Haykin, Adaptive Filter Theory, 4th Edition, Prentice Hall, 2002. [7] Z. Liu, X. Ma, G. B. Giannakis, Space-time coding and Kalman filtering for time selective fading channels, IEEE Trans. on Communication, vol.50, pp.183-186, 2002.
