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Bit Error Rate Approximation of MIMO-OFDM Systems with Carrier Frequency Offset and Channel Estimation Errors
EURASIP Journal on Wireless Communications and Networking volume 2010, Article number: 176083 (2010)
Abstract
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.
1. Introduction
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 [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 MIMO-OFDM systems is proposed in [15], which considers an identical timing offset and frequency offset with respect to each transmit-receive antenna pair. In [10], 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 [10], 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 
.
2. MIMO-OFDM Signal Model
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 
.
The discrete channel response between the 
th receive antenna and 
th transmit antenna is 
, where 
is the maximum delay between the 
th transmit and the 
th receive antennas, and 
. Uncorrelated channel taps are assumed for each antenna pair 
; that is, 
when 
. The corresponding frequency-domain channel response matrix is given by 
with 
representing the channel attenuation at the 
th subcarrier. In the sequel, the channel power profiles are normalized as 
for all 
. The covariance of channel frequency response is given by
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.)
By considering the channel gains and frequency offsets, the received signal vector can be represented as
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 MIMO-OFDM Systems
This paper treats spatial multiplexing MIMO, where independent data streams are mapped to distinct OFDM symbols and are transmitted simultaneously from transmit antennas. The received vector 
at the 
th receive antenna is thus a superposition of the transmit signals from all the 
transmit antennas. When demodulating 
, the signals from the transmit antennas other than the 
th transmit antenna constitute interantenna interference (IAI). The structure of MIMO-OFDM systems is illustrated in Figure 1, where 
represents the subcarrier spacing.
Structure of MIMO-OFDM transceiver.
Here, we first assume that 
and 
for each 
have been estimated imperfectly; that is, 
and 
, where 
and 
are the estimation errors of 
and 
(
represents the estimation error of 
), respectively. We also assume that each 
is demodulated with a negligible error. After estimating 
, that is, 
, 
can be compensated for and 
can be demodulated as
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.
Let us first define the parameters 
, 
, and 
, 
. Based on (3), the 
th subcarrier 
of the 
th transmit antenna can be demodulated as
where 
is decomposed as 
, which is the ICI contributed by subcarriers other than the 
th subcarrier of transmit antenna 
. (The decomposition of ICI into the format of 
is referred to [11].) We can easily prove that 
and 
are zero-mean RVs subject to the following assumptions.
-
(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 
.
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 
.Given these assumptions, let us first define 
as the interference contributed by the 
th subcarrier of the interfering transmit antennas, that is, the co-subcarrier inter-antenna-interference (CSIAI), and define 
as the ICI contributed by the subcarriers other than the 
th subcarrier of the interfering transmit antennas, that is, the intercarrier-interantenna interference (ICIAI). Then we derive 
and 
as
where 
is given by (1). The demodulation of 
is degraded by either 
or IAI (CSIAI plus ICIAI). In this paper, we assume that the integer part of the frequency offset has been estimated and corrected, and only the fractional part frequency offset is considered. Considering small frequency offsets, the following requirements are assumed to be satisfied:
-
(1)
for all 
, -
(2)
for all 
, -
(3)
for all 
.
for all 
,
for all 
,
for all 
.Condition 1 requires that each frequency offset should be much smaller than 1, and conditions 2 and 3 require that the sum of any two frequency offsets (and the frequency offset estimation results) should not exceed 1. The last two conditions are satisfied only if the estimation error does not exceed 0.5. If all these three conditions are satisfied simultaneously, we can represent 
, 
, 
, and 
as
Therefore, the interference due to the 
th subcarrier of transmit antennas (other than the 
th transmit antenna, i.e., the interfering antennas) is
with 
representing the higher-order item of 
and 
. It is easy to show that 
and 
are zero-mean RVs and that their variances are given by
respectively. After averaging out frequency offset 
, frequency offset estimation error 
, and channel estimation error 
for all 
, the average SINR of 
(parameterized by only 
) is
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
In order to demodulate the signal transmitted by the 
th transmit antenna, the 
received signals are cophased and combined to improve the receiving diversity. Therefore, the EGC output is given by
where 
. After averaging out 
, 
, and 
for each 
, the average SINR of 
is given by
When 
is large enough, (17) can be further simplified as
3.3. SINR Analysis with MRC at Receive Antennas
In a MIMO-OFDM system with 
receive antennas, based on the channel estimation 
for each 
, the received signal at all the 
receive antennas can be combined by using MRC, and therefore the combined output is given by
where 
. After averaging out 
, 
, and 
for each 
, the average SINR of 
is
where we have defined 
, and the noise part can be represented as 
. When 
is large enough, (20) can be further simplified as
4. BER Performance
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 [11], the BER of SISO-OFDM with frequency offset is developed. The BER analysis in [11] is now extended to MIMO-OFDM.
As discussed in [11, 28, 29], the BER for the 
th transmit antenna with the input constellation being 
-ary square QAM (Gray bit mapping) can be represented as
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 
.).
Note that in MIMO-OFDM systems, the SINR at each subcarrier is an RV parameterized by the frequency offset and channel attenuation. In order to derive the average SINR of MIMO-OFDM systems, (22) should be averaged over the distribution of 
as
where 
, 
, 
, and 
. Since obtaining a close-form solution of (23) appears impossible, an infinite-series approximation of 
is developed. In [11], the average is expressed as an infinite series of generalized hypergeometric functions.
From [30, page 939], 
can be represented as an infinite series:
Therefore, (23) can be rewritten as
where 
depends on the type of combining. Note that 
has been derived in Section 3 and that for the 
th subcarrier 
, 
, 
and 
for each 
have been averaged out. Therefore, 
in (25) can be replaced by 
; that is, the average BER can be expected over subcarrier 
(
), and finally 
can be simplified as
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 [31], 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 
[10].
4.1. BER without Receiving Combining
The BER measured at the 
th receive antenna for the 
th transmit antenna can be approximated by (25) with 
instead of 
being used here; that is,
When 
, we have 
, as derived in Appendix A. The initial condition is given by
4.2. BER with EGC
For a MIMO-OFDM system with EGC reception, the average BER can be approximated by (25) with 
instead of 
being used here; that is,
Defining 
, 
, 
, and 
, when 
, we have
as derived in Appendix B. The initial condition is given by
4.3. BER with MRC
For a MIMO-OFDM system with channel knowledge at the receiver, the receiving diversity can be optimized by using MRC, and the average BER can be approximated by (25) with 
instead of 
being used here; that is,
By defining 
, 
with 
is given by
as derived in Appendix C. The initial condition is given by
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 [31] 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.
5. Numerical Results
Quasistatic MIMO wireless channels are assumed; that is, the channel impulse response is fixed over one OFDM symbol period but changes across the symbols. The simulation parameters are defined in Table 1.
The SINR degradation due to the residual frequency offsets is shown in Figure 2 for 
and 
10 dB. The SINR degradation increases with 
. Because of IAI due to the multiple transmit antennas, the SINR performance of MIMO-OFDM with 
is worse than that of SISO-OFDM, even though EGC or MRC is applied to exploit the receiving diversity. IAI in MIMO-OFDM can be suppressed by increasing the number of receive antennas. In this simulation, when 
, the average SINR with either EGC or MRC will be higher than that of SISO-OFDM system. For each MIMO scenario, MRC outperforms EGC.
SINR reduction by frequency offset in MIMO-OFDM systems.
The BER degradation due to the residual frequency offsets is shown in Figure 3 for 
and 
10 dB (
is the bit energy per noise per Hz). The BER for 4-phase PSK (QPSK) or 16QAM subcarrier modulation is considered. Just as with the case of SINR, the BER degrades with large 
. For example, when 
and 
for QPSK (16QAM), a BER of 
(
) or 
(
) is achieved with EGC or MRC at the receiver, respectively. When 
is increased to 
, a BER of 
(
) or 
(
) can be achieved with EGC or MRC, respectively.
BER degradation due to the residual frequency offset in MIMO-OFDM systems.
Figures 4 to 9 compare BERs of QPSK and 16QAM with different combining methods. Figures 4 and 5 consider SISO-OFDM. The BER is degraded due to the frequency offset and channel estimation errors. For a fixed channel estimation variance error 
, a larger variance of frequency offset estimation error, that is, 
, implies a higher BER. For example, if 
, 
20 dB and 
, the BER with QPSK (16QAM) is about 
(
); when 
increases to 
, the BER with QPSK (16QAM) increases to 
(
).
BER with QPSK when (
).
BER with 16QAM when (
).
IAI appears with multiple transmit antennas, and the BER will degrade as IAI increases. Note that since IAI cannot be totally eliminated in the presence of the frequency offset and channel estimation errors, a BER floor occurs at the high SNR. IAI can be reduced considerably by exploiting the receiving diversity by using either EGC or MRC, as shown in Figures 6, 7, 8, and 9. Without receiver combining, the BER is much worse than that in SISO-OFDM, simply because of the SINR degradation due to IAI. For example, when 
and 
, the BER with QPSK is about 
when 
, which is three times of that of SISO-OFDM (which is about 
), as shown in Figure 6. For a given number of receive antennas, MRC can achieve a lower BER than that achieved with EGC, but the receiver requires accurate channel estimation. For example, in Figure 7, when 
with 
and 16QAM, the performance improvement of EGC (MRC) over that without combining is about 5.5 dB (6 dB), and that performance improvement increases to 7.5 dB (8.5 dB) if 
is increased to 
. By increasing the number of receive antennas to 4, this performance improvement is about 8.2 dB (9 dB) for EGC (MRC), with 
, or 11 dB (13.9 dB) for EGC (MRC), with 
, as shown in Figure 9.
BER with QPSK when (
).
BER with 16QAM when (
).
BER with QPSK when (
).
BER with 16QAM when (
).
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 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.
Appendices
A. BER without Combining
Without loss of generality, the signal transmitted by the 
th transmit antenna is assumed in this subsection to be demodulated at the 
th receive antenna. For each 
, 
has a probability density function (PDF) 
. When the number of receive antennas 
is larger than 2, 
can be represented as
where 
is defined in (15), 
, 
, and 
. Equation (A.1) can be further derived as
From the last step of (A.1), 
can be represented as a function of 
and 
:
By resolving (A.3), 
can be represented as
By replacing 
in (A.2) with (A.4), 
can be finally simplified as
B. BER of EGC
Without loss of generality, consider the demodulation of the signal transmitted by the 
th transmit antenna. Define
and 
. As in Appendix A, when 
, 
can be represented as
where 
, 
, 
, and 
. Equation (B.2) can be further simplified as
From the last step of (B.2), 
can be represented as a function of 
and 
:
By resolving (B.4), 
can be represented as
By replacing 
in (B.3) with (B.5), 
can be finally simplified as
C. BER of MRC
Without loss of generality, consider the demodulation of the signal transmitted by the 
th transmit antenna. Define 
. When 
, 
can be represented as
where 
and 
. (C.1) can be further simplified as
From the last step of (C.1), 
can be represented as a function of 
and 
:
By resolving (C.3), 
can be represented as
By replacing 
in (C.2) with (C.4), 
can be finally simplified as
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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.
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Zhang, Z., Zhang, L., You, M. et al. Bit Error Rate Approximation of MIMO-OFDM Systems with Carrier Frequency Offset and Channel Estimation Errors. J Wireless Com Network 2010, 176083 (2010). https://doi.org/10.1155/2010/176083
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DOI: https://doi.org/10.1155/2010/176083