- Research Article
- Open Access
Bit Error Rate Approximation of MIMO-OFDM Systems with Carrier Frequency Offset and Channel Estimation Errors
© Zhongshan Zhang et al. 2010
- Received: 23 February 2010
- Accepted: 16 September 2010
- Published: 23 September 2010
The bit error rate (BER) of multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems with carrier frequency offset and channel estimation errors is analyzed in this paper. Intercarrier interference (ICI) and interantenna interference (IAI) due to the residual frequency offsets are analyzed, and the average signal-to-interference-and-noise ratio (SINR) is derived. The BER of equal gain combining (EGC) and maximal ratio combining (MRC) with MIMO-OFDM is also derived. The simulation results demonstrate the accuracy of the theoretical analysis.
- Carrier Frequency Offset
- Maximal Ratio Combine
- Channel Estimation Error
- Equal Gain Combine
- Intercarrier Interference
Spatial multiplexing multiple-input multiple-output (MIMO) technology significantly increases the wireless system capacity [1–4]. These systems are primarily designed for flat-fading MIMO channels. A broader band can be used to support a higher data rate, but a frequency-selective fading MIMO channel is met, and this channel experiences intersymbol interference (ISI). A popular solution is MIMO-orthogonal frequency-division multiplexing (OFDM), which achieves a high data rate at a low cost of equalization and demodulation. However, just as single-input single-output- (SISO-) OFDM systems are highly sensitive to frequency offset, so are MIMO-OFDM systems. Although one can use frequency offset correction algorithms [5–10], residual frequency offsets can still increase the bit error rate (BER).
The BER of SISO-OFDM systems impaired by frequency offset is analyzed in , in which the frequency offset is assumed to be perfectly known at the receiver, and, based on the intercarrier interference (ICI) analysis, the BER is evaluated for multipath fading channels. Many frequency offset estimators have been proposed [8, 12–14]. A synchronization algorithm for MIMO-OFDM systems is proposed in , which considers an identical timing offset and frequency offset with respect to each transmit-receive antenna pair. In , where frequency offsets for different transmit-receive antennas are assumed to be different, the Cramer-Rao lower bound (CRLB) for either the frequency offsets or channel estimation variance errors for MIMO-OFDM is derived. More documents on MIMO-OFDM channel estimation by considering the frequency offset are available at [16, 17].
However, in real systems, neither the frequency offset nor the channel can be perfectly estimated. Therefore, the residual frequency offset and channel estimation errors impact the BER performance. The BER performance of MIMO systems, without considering the effect of both the frequency offset and channel estimation errors, is studied in [18, 19].
This paper provides a generalized BER analysis of MIMO-OFDM, taking into consideration both the frequency offset and channel estimation errors. The analysis exploits the fact that for unbiased estimators, both channel and frequency offset estimation errors are zero-mean random variables (RVs). Note that the exact channel estimation algorithm design is not the focus of this paper, and the main parameter of interest is the channel estimation error. Many channel estimation algorithms developed for either SISO or MIMO-OFDM systems, for example, [20–22], can be used to perform channel estimation. The statistics of these RVs are used to derive the degradation in the receive SINR and the BER. Following , the frequency offset of each transmit-receive antenna pair is assumed to be an independent and identically distributed (i.i.d.) RV.
This paper is organized as follows. The MIMO-OFDM system model is described in Section 2, and the SINR degradation due to the frequency offset and channel estimation errors is analyzed in Section 3. The BER, taking into consideration both the frequency offset and channel estimation errors, is derived in Section 4. The numerical results are given in Section 5, and the conclusions are presented in Section 6.
Notation. and are transpose and complex conjugate transpose. The imaginary unit is . and are the real and imaginary parts of , respectively. represents the angle of , that is, . A circularly symmetric complex Gaussian RV with mean and variance is denoted by . is the identity matrix, and is the all-zero matrix. is the all-zero vector. is the th entry of vector , and is the th entry of matrix . and are the mean and variance of .
Input data bits are mapped to a set of complex symbols drawn from a typical signal constellation such as phase-shift keying (PSK) or quadrature amplitude modulation (QAM). The inverse discrete fourier transform (IDFT) of these symbols generates an OFDM symbol. Each OFDM symbol has a useful part of duration seconds and a cyclic prefix of length seconds to mitigate ISI, where is longer than the channel-response length. For a MIMO-OFDM system with transmit antennas and receive antennas, an vector represents the block of frequency-domain symbols sent by the th transmit antenna, where . The time-domain vector for the th transmit antenna is given by , where is the total transmit power and is the IDFT matrix with entries for . Each entry of is assumed to be i.i.d. RV with mean zero and unit variance; that is, for and .
Note that if and are satisfied simultaneously, we assume that there is no correlation between and . Otherwise the correlation between and is nonzero.
In this paper, and are used to represent the initial phase and normalized frequency offset (normalized to the OFDM subcarrier spacing) between the oscillators of the -th transmit and the th receive antennas. The frequency offsets for all are modeled as zero-mean i.i.d. RVs. (Multiple rather than one frequency offset are assumed in this paper, with each transmit-antenna pair being impaired by an independent frequency offset. This case happens when the distance between different transmit or receive antenna elements is large enough, and this big distance results in a different angle-of-arrive (AOA) of the signal received by each receive antenna element. In this scenario, once the moving speed of the mobile node is high, the Doppler Shift related to different transmit-receive antenna pair will be different.)
where , and is a vector of additive white Gaussian noise (AWGN) with . Note that the channel state information is available at the receiver, but not at the transmitter. Consequently, the transmit power is equally allocated among all the transmit antennas.
where is derived from by replacing with and and are the residual IAI and AWGN components of , respectively (When is large enough and the frequency offset is not too big (e.g., ), from the Central-Limit Theorem (CLT) [23, Page 59], the IAI can be approximated as Gaussian noise.).
3.1. SINR Analysis without Combining at Receive Antennas
The SINR is derived for the th transmit signal at the th receive antenna. The signals transmitted by antennas other than the th antenna are interference, which should be eliminated before demodulating the desired signal of the th transmit antenna. Existing interference cancelation algorithms [24–27] can be applied here.
is an i.i.d. RV with mean zero and variance for all .
is an i.i.d. RV with mean zero and variance for each .
for each .
is an i.i.d. RV with mean zero and variance for each .
, , , and are independent of each other for each .
for all ,
for all ,
for all .
where and , independent of .
For signal demodulation in MIMO-OFDM, signal received in multiple receive antennas can be exploited to improve the receive SINR. In the following, equal gain combining (EGC) and maximal ratio combining (MRC) are considered.
3.2. SINR Analysis with EGC at Receive Antennas
3.3. SINR Analysis with MRC at Receive Antennas
The BER as a function of SINR in MIMO-OFDM is derived in this section. We consider -ary square QAM with Gray bit mapping. In the work of Rugini and Banelli , the BER of SISO-OFDM with frequency offset is developed. The BER analysis in  is now extended to MIMO-OFDM.
where and are specified by signal constellation, is the average SINR of the th transmit antenna, and is the error function (Please refer to  for the meaning of and .).
where , , , and . Since obtaining a close-form solution of (23) appears impossible, an infinite-series approximation of is developed. In , the average is expressed as an infinite series of generalized hypergeometric functions.
where is based on instead of . We first define and , which will be used in the following subsections. We next give a recursive definition for for the following reception methods: (1) demodulation without combining, (2) EGC, and (3) MRC.
Note that the SINR for each combining scenario (i.e., without combining, EGC, or MRC) is a function of the second-order statistics of the channel and frequency offset estimation errors (although the interference also comprises the fourth-order statistics of the frequency offset estimation errors, they are negligible as compared to the second-order statistics for small estimation errors). Any probability distribution with zero mean and the same variance will result in the same SINR. Therefore, the exact distributions need not be specified. However, when the BER is derived by using an infinite-series approximation, the actual distribution of the frequency offset estimation errors is required. In , it is shown that both the uniform distribution and Gaussian distribution are amenable to infinite-series solutions with closed-form formulas for the coefficients. In the following sections, the frequency offset estimation errors are assumed to be i.i.d. Gaussian RVs with mean zero and variance .
4.1. BER without Receiving Combining
4.2. BER with EGC
4.3. BER with MRC
4.4. Complexity of the Infinite-Series Representation of BER
Infinite-series BER expression (27), (29), or (32) must be truncated in practice. The truncation error is negligible if the number of terms is large enough: Reference  shows that when the number of terms is as large as 50, the finite-order approximation is good. In this case, a total of multiplication and summation operations are needed to calculate the BER for each combining scheme.
Parameters for BER simulation in MIMO-OFDM systems.
( ; )
Without combining; EGC; MRC
Our theoretical BER approximations are accurate at low SNR with/without diversity combining. However, the simulation and theory results diverge as the SNR increases, especially when is large. For example, in Figure 9, with 16QAM, when ( , ) and , about 1 dB difference exists between the simulation and the theoretical result for either EGC or MRC at high SNR. This discrepancy is due to several reasons. As the SNR increases, the system becomes interference limited. When , , and are not large enough, the interferences may not be well approximated as Gaussian RVs with zero mean. In addition, with either EGC or MRC reception, the phase rotation or channel attenuation of the receive substreams should be estimated, and their estimation accuracy will also affect the combined SINR. The instant large phase or channel estimation error also contributes a deviation to the BER when using EGC or MRC.
The BER of MIMO-OFDM due to the frequency offset and channel estimation errors has been analyzed. The BER expressions for no combining, EGC, and MRC were derived. These expressions are in infinite-series form and can be truncated in practice. The simulation results show that the truncation error is negligible if the number of terms is large than 50.
This paper has been presented in part at the IEEE Globecom 2007 . Although the conference paper was a brief version of this journal paper and they have the same results and conclusion, this journal paper provides a more detailed proof to each result appeared in the IEEE ICC 2007 paper.
- Foschini GJ: Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas. Bell Labs Technical Journal 1996, 1(2):41-59.View ArticleGoogle Scholar
- Foschini GJ, Gans MJ: On limits of wireless communications in a fading environment when using multiple antennas. Wireless Personal Communications 1998, 6(3):311-335. 10.1023/A:1008889222784View ArticleGoogle Scholar
- Goldsmith A, Jafar SA, Jindal N, Vishwanath S: Capacity limits of MIMO channels. IEEE Journal on Selected Areas in Communications 2003, 21(5):684-702. 10.1109/JSAC.2003.810294View ArticleMATHGoogle Scholar
- Marzetta TL, Hochwald BM: Capacity of a mobile multiple-antenna communication link in rayleigh flat fading. IEEE Transactions on Information Theory 1999, 45(1):139-157. 10.1109/18.746779MathSciNetView ArticleMATHGoogle Scholar
- Moose PH: Technique for orthogonal frequency division multiplexing frequency offset correction. IEEE Transactions on Communications 1994, 42(10):2908-2914. 10.1109/26.328961View ArticleGoogle Scholar
- Cui T, Tellambura C: Maximum-likelihood carrier frequency offset estimation for OFDM systems over frequency-selective fading channels. Proceedings of the IEEE International Conference on Communications, May 2005, Seoul, Korea 4: 2506-2510.Google Scholar
- Minn H, Bhargava VK, Letaief KB: A robust timing and frequency synchronization for OFDM systems. IEEE Transactions on Wireless Communications 2003, 2(4):822-839.View ArticleGoogle Scholar
- Zhang Z, Zhao M, Zhou H, Liu Y, Gao J: Frequency offset estimation with fast acquisition in OFDM system. IEEE Communications Letters 2004, 8(3):171-173. 10.1109/LCOMM.2004.823423View ArticleGoogle Scholar
- Zhang Z, Jiang W, Zhou H, Liu Y, Gao J: High accuracy frequency offset correction with adjustable acquisition range in OFDM systems. IEEE Transactions on Wireless Communications 2005, 4(1):228-236.View ArticleGoogle Scholar
- Besson O, Stoica P: On parameter estimation of MIMO flat-fading channels with frequency offsets. IEEE Transactions on Signal Processing 2003, 51(3):602-613. 10.1109/TSP.2002.808102MathSciNetView ArticleGoogle Scholar
- Rugini L, Banelli P: BER of OFDM systems impaired by carrier frequency offset in multipath fading channels. IEEE Transactions on Wireless Communications 2005, 4(5):2279-2288.View ArticleGoogle Scholar
- Schmidl TM, Cox DC: Robust frequency and timing synchronization for OFDM. IEEE Transactions on Communications 1997, 45(12):1613-1621. 10.1109/26.650240View ArticleGoogle Scholar
- Morelli M, Mengali U: Improved frequency offset estimator for OFDM applications. IEEE Communications Letters 1999, 3(3):75-77. 10.1109/4234.752907View ArticleGoogle Scholar
- Ma X, Tepedelenlioǧlu C, Giannakis GB, Barbarossa S: Non-data-aided carrier offset estimators for OFDM with null subcarriers: identifiability, algorithms, and performance. IEEE Journal on Selected Areas in Communications 2001, 19(12):2504-2515. 10.1109/49.974615View ArticleGoogle Scholar
- Mody A, Stuber G: Synchronization for MIMO OFDM systems. Proceedings of IEEE Global Telecommunications Conference (GLOBECOM '01), November 2001, San Antonio, Tex, USA 1: 509-513.View ArticleGoogle Scholar
- Minn H, Al-Dhahir N, Li Y: Optimal training signals for MIMO OFDM channel estimation in the presence of frequency offset and phase noise. IEEE Transactions on Communications 2006, 54(10):1754-1759.View ArticleGoogle Scholar
- Ghogho M, Swami A: Training design for multipath channel and frequency-offset estimation in MIMO systems. IEEE Transactions on Signal Processing 2006, 54(10):3957-3965.MathSciNetView ArticleGoogle Scholar
- Loyka S, Gagnon F: Performance analysis of the V-BLASt algorithm: an analytical approach. IEEE Transactions on Wireless Communications 2004, 3(4):1326-1337. 10.1109/TWC.2004.830853View ArticleGoogle Scholar
- Loyka S, Gagnon F: V-BLAST without optimal ordering: analytical performance evaluation for rayleigh fading channels. IEEE Transactions on Communications 2006, 54(6):1109-1120. 10.1109/TCOMM.2006.876875View ArticleGoogle Scholar
- Li Y, Cimini LJ Jr., Sollenberger NR: Robust channel estimation for OFDM systems with rapid dispersive fading channels. IEEE Transactions on Communications 1998, 46(7):902-915. 10.1109/26.701317View ArticleGoogle Scholar
- Cui T, Tellambura C: Robust joint frequency offset and channel estimation for OFDM systems. Proceedings of the 60th IEEE Vehicular Technology Conference (VTC '04), September 2004, Los Angeles, Calif, USA 1: 603-607.Google Scholar
- Minn H, Al-Dhahir N: Optimal training signals for MIMO OFDM channel estimation. IEEE Transactions on Wireless Communications 2006, 5(5):1158-1168.View ArticleGoogle Scholar
- Proakis JG: Digital Communications. 4th edition. McGraw-Hill, New York, NY, USA; 2001.MATHGoogle Scholar
- Dào DN, Tellambura C: Intercarrier interference self-cancellation space-frequency codes for MIMO-OFDM. IEEE Transactions on Vehicular Technology 2005, 54(5):1729-1738. 10.1109/TVT.2005.853477View ArticleGoogle Scholar
- Tang T, Heath RW Jr.: Space-time interference cancellation in MIMO-OFDM systems. IEEE Transactions on Vehicular Technology 2005, 54(5):1802-1816. 10.1109/TVT.2005.851299View ArticleGoogle Scholar
- Giangaspero L, Agarossi L, Paltenghi G, Okamura S, Okada M, Komaki S: Co-channel interference cancellation based on MIMO OFDM systems. IEEE Wireless Communications 2002, 9(6):8-17. 10.1109/MWC.2002.1160076View ArticleGoogle Scholar
- Stamoulis A, Diggavi SN, Al-Dhahir N: Intercarrier interference in MIMO OFDM. IEEE Transactions on Signal Processing 2002, 50(10):2451-2464. 10.1109/TSP.2002.803347View ArticleGoogle Scholar
- Cho K, Yoon D: On the general BER expression of one- and two-dimensional amplitude modulations. IEEE Transactions on Communications 2002, 50(7):1074-1080. 10.1109/TCOMM.2002.800818View ArticleGoogle Scholar
- Yang L-L, Hanzo L: Recursive algorithm for the error probability evaluation of M-QAM. IEEE Communications Letters 2000, 4(10):304-306. 10.1109/4234.880816View ArticleGoogle Scholar
- Gradshteyn IS, Ryzhik IM: Table of Integrals, Series, and Products. 5th edition. Academic Press, New York, NY, USA; 1994.MATHGoogle Scholar
- Zhang Z, Tellambura C: The effect of imperfect carrier frequency offset estimation on an OFDMA uplink. IEEE Transactions on Communications 2009, 57(4):1025-1030.View ArticleGoogle Scholar
- Zhang Z, Zhang W, Tellambura C: BER of MIMO-OFDM systems with carrier frequency offset and channel estimation errors. Proceedings of IEEE International Conference on Communications (ICC '07), June 2007, Glasgow, Scotland 5473-5477.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.