 Research Article
 Open Access
Bit Error Rate Approximation of MIMOOFDM Systems with Carrier Frequency Offset and Channel Estimation Errors
 Zhongshan Zhang^{1}Email author,
 Lu Zhang^{2},
 Mingli You^{2} and
 Ming Lei^{1}
https://doi.org/10.1155/2010/176083
© Zhongshan Zhang et al. 2010
 Received: 23 February 2010
 Accepted: 16 September 2010
 Published: 23 September 2010
Abstract
The bit error rate (BER) of multipleinput multipleoutput (MIMO) orthogonal frequencydivision 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 signaltointerferenceandnoise ratio (SINR) is derived. The BER of equal gain combining (EGC) and maximal ratio combining (MRC) with MIMOOFDM is also derived. The simulation results demonstrate the accuracy of the theoretical analysis.
Keywords
 Carrier Frequency Offset
 Maximal Ratio Combine
 Channel Estimation Error
 Equal Gain Combine
 Intercarrier Interference
1. Introduction
Spatial multiplexing multipleinput multipleoutput (MIMO) technology significantly increases the wireless system capacity [1–4]. These systems are primarily designed for flatfading MIMO channels. A broader band can be used to support a higher data rate, but a frequencyselective fading MIMO channel is met, and this channel experiences intersymbol interference (ISI). A popular solution is MIMOorthogonal frequencydivision multiplexing (OFDM), which achieves a high data rate at a low cost of equalization and demodulation. However, just as singleinput singleoutput (SISO) OFDM systems are highly sensitive to frequency offset, so are MIMOOFDM 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 SISOOFDM systems impaired by frequency offset is analyzed in [11], 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 MIMOOFDM systems is proposed in [15], which considers an identical timing offset and frequency offset with respect to each transmitreceive antenna pair. In [10], where frequency offsets for different transmitreceive antennas are assumed to be different, the CramerRao lower bound (CRLB) for either the frequency offsets or channel estimation variance errors for MIMOOFDM is derived. More documents on MIMOOFDM 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 MIMOOFDM, 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 zeromean 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 MIMOOFDM 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 [10], the frequency offset of each transmitreceive antenna pair is assumed to be an independent and identically distributed (i.i.d.) RV.
This paper is organized as follows. The MIMOOFDM 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 allzero matrix. is the allzero vector. is the th entry of vector , and is the th entry of matrix . and are the mean and variance of .
2. MIMOOFDM Signal Model
Input data bits are mapped to a set of complex symbols drawn from a typical signal constellation such as phaseshift 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 channelresponse length. For a MIMOOFDM system with transmit antennas and receive antennas, an vector represents the block of frequencydomain symbols sent by the th transmit antenna, where . The timedomain 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 zeromean i.i.d. RVs. (Multiple rather than one frequency offset are assumed in this paper, with each transmitantenna 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 angleofarrive (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 transmitreceive 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.
3. SINR Analysis in MIMOOFDM Systems
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 CentralLimit 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.
 (1)
is an i.i.d. RV with mean zero and variance for all .
 (2)
is an i.i.d. RV with mean zero and variance for each .
 (3)
for each .
 (4)
is an i.i.d. RV with mean zero and variance for each .
 (5)
, , , and are independent of each other for each .
 (1)
for all ,
 (2)
for all ,
 (3)
for all .
where and , independent of .
For signal demodulation in MIMOOFDM, 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
4. BER Performance
The BER as a function of SINR in MIMOOFDM is derived in this section. We consider ary square QAM with Gray bit mapping. In the work of Rugini and Banelli [11], the BER of SISOOFDM with frequency offset is developed. The BER analysis in [11] is now extended to MIMOOFDM.
where and are specified by signal constellation, is the average SINR of the th transmit antenna, and is the error function (Please refer to [28] for the meaning of and .).
where , , , and . Since obtaining a closeform solution of (23) appears impossible, an infiniteseries approximation of is developed. In [11], 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 secondorder statistics of the channel and frequency offset estimation errors (although the interference also comprises the fourthorder statistics of the frequency offset estimation errors, they are negligible as compared to the secondorder 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 infiniteseries approximation, the actual distribution of the frequency offset estimation errors is required. In [31], it is shown that both the uniform distribution and Gaussian distribution are amenable to infiniteseries solutions with closedform 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 [10].
4.1. BER without Receiving Combining
4.2. BER with EGC
4.3. BER with MRC
4.4. Complexity of the InfiniteSeries Representation of BER
Infiniteseries 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 [31] shows that when the number of terms is as large as 50, the finiteorder approximation is good. In this case, a total of multiplication and summation operations are needed to calculate the BER for each combining scheme.
5. Numerical Results
Parameters for BER simulation in MIMOOFDM systems.
Subcarrier modulation  QPSK; 16QAM 

DFT length  128 
 ; 


MIMO parameters  ( ; ) 
Receiving combining  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.
6. Conclusions
The BER of MIMOOFDM 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 infiniteseries 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.
Appendices
A. BER without Combining
B. BER of EGC
C. BER of MRC
Declarations
Acknowledgments
This paper has been presented in part at the IEEE Globecom 2007 [32]. 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.
Authors’ Affiliations
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