INTRODUCTION
Significant amount of research has been conducted (Molisch
et al., 2002; Guey et al., 1999;
Alamouti, 1998; Tarokh et al.,
1998; Telatar, 1999) for MultipleInput MultipleOutput
(MIMO) based communication architectures to improve the information capacity,
signal detection reliability and spectral efficiency without increasing the
transmit power or bandwidth as compared to single antenna systems. In MIMO systems,
multiple independent transmission channels are available which increase the
channel capacity as the number of antenna elements increases. It is discussed
by Molisch and Win (2004) and Sanayei
and Nosratinia (2004) that the robustness in terms of bit error rate and
capacity of the MIMO system can be improved by exploiting antenna diversity
and spatial multiplexing, respectively.
The conventional purpose of using multiple antenna combinations at both transmitter
and at the receiver either to achieve transmitreceive diversity or to achieve
enhanced information throughput as discussed by Zheng and
Tse (2003) and Telatar (1999). Regardless of the
aim, appropriate selection of detection schemes is crucial to obtain maximum
achievable capacity and diversity of MIMO systems. Therefore, significant amount
of research has been carried out to find detection schemes that are computationally
efficient and can obtain maximum achievable diversity and capacity. With the
given resources, the most commonly used detection schemes found in the existing
literature are Zero Forcing (ZF), Minimum Mean Square Error (MMSE) and Maximum
Likelihood (ML).
For the ease of system design to achieve a predefined capacity or quality of
service requirement, analytical frameworks are extremely useful that provide
a performance bench mark. Multiple antenna based future communication system
is expected to be adaptive with available capacity or quality of service to
offer as trade off with each other. A simplified analytical framework is expected
to lead towards designing such resource adaption algorithm easier. However,
to achieve the effectiveness of such framework, a tighter bound is required.
Most of the available lower bounds in the existing literature are lacking tightness
with the actual performance. Several performance analysis frame works for ZF
and MMSE detection has been presented by Tse and Viswanath
(2005), Jiang et al. (2011), Matthaiou
et al. (2011) and Han et al. (2010).
The contribution in this study is proposition of simplified analytical frameworks
of MIMO receiver performance, which provides tighter lower bound in comparison
to existing bounds for ZF, MMSE and ML detection schemes within MIMO wireless
communication systems. Channel state information is assumed to be known at receiver.
SYSTEM MODEL
The system model considered in this study is shown in Fig. 1.
A wireless multiple input multiple output communication system with N_{t}
number of transmit antennas is assumed with N_{r} number of receiving
antennas at fusion center receiver.
The signal vector x transmitted from N_{t} number of transmit antennas
is defined as:
x = [x_{1}, x_{2}, ...,
x_{Nt}]^{T} 
(1) 
where, x is with dimensions (N_{t}x1). The received signal vector is
expressed as:
where, y is the received signal vector with dimensions (N_{t}x1), H
is the rayleigh fading channel matrix of size (N_{r}xN_{t})
and n is the noise vector with dimensions (N_{r}x1). The noise is considered
to be additive white Gaussian noise with zero mean and unity variance σ^{2}.
The Rayleigh fading channel matrix is defined as:
where, h_{i,j} denotes the channel coefficients from ith transmitter
antenna with i∈{1, 2, ..., Nt} to jth receiving antenna at the fusion center
with j ∈ {1, 2, ..., Nr}. A symmetric transmitter receiver MIMO communication
system model is considered.
DETECTION METHODS UNDERCONSIDERATION
A brief description of the detection schemes as mentioned in introductory section
of this study is provided in this section. In ZF detection, the received signals
are send through ZF filter denoted as G_{ZF} and can be defined
as:
G_{ZF} = (H^{H}H)^{1}H^{H} 
(3) 
Subsequently, the recovered spatially multiplexed data streams recovered from
the detected received signal are denoted as _{ZF} and can be written
as:
During the detection process, ZF detector is aimed to null out interfering
components, which can cause noise amplification. Subsequently, it is well established
that ZF is not the best possible detection scheme, although, it is simple and
easy to implement. MMSE is another widely used detection scheme which provides
a tradeoff between minimizing intersymbol interference and noise amplification.
MMSE filter matrix is denoted as G_{MMSE} and defined as:
G_{MMSE} = (H^{H}H+σ^{2}I)^{1}
H^{H} 
(5) 
Hence, the estimation of the transmitted signal vector can be written as:
ML detector is known to be the optimal detector in terms of minimizing the
probability of bit error rate. The criterion required to satisfy maximum likelihood
detection can be defined as:
where, x_{k} is the k^{th} candidate symbol vector out of
number of possible symbol vectors. However, computational complexity of these
detection schemes grows exponentially with the number of antenna elements when
used within MIMO systems. While designing MIMO system, selection of detection
scheme along with the resources required to be provided to achieve a given quality
of service is challenging. Moreover, to design adaptive receiver with power
constrain, lower number of iterations are desirable to converge true performance
pattern from initial approximation. One possibility of approximating the performance
of these systems is to design a framework which provides tighter error performance
bound.
ANALYTICAL FRAMEWORK
Bit error rate is a critical measure of system performance which defines the
quality of service of any telecommunication system. An intended achievable quality
of service threshold is required to be determined, to allocate resources during
any given telecommunication system. To find such threshold, analytical frameworks
have been studied in the literature that provide with a benchmark of required
resources. The following subsection provides a brief overview of such frameworks
within the existing literature, followed by the proposed analytical frameworks.
Existing framework: Most commonly used linear detection schemes e.g.,
ZF, MMSE and ML have been the prime topics of interest for such analytical performance
measure. Recent work in Jiang et al. (2011)
and Tse and Viswanath (2005) present a frame work for the
analysis of error performance for ZF and MMSE detection schemes which is defined
as:
Where:
where, U_{n} is the upper triangular matrix and Λ is the eigen
values matrix of H_{n} and H_{n} is the submatrix obtained
by taking h_{n} out of H. h_{n} is the nth column of H. Equation
7 can be simplified into Eq. 11 for a symmetric MIMO
system i.e.:
ML detection is widely known to be optimum in terms of bit error rate performance
with the cost of intensive computational complexity. Different upper bounds
on Symbol Error Rate (SER) and Bit Error Rate (BER) probability of ML detection
within MIMO communication systems have been presented by Zhu
and Murch (2002), Kuchi and Ayyar (2011) and Gritsch
et al. (2004). The upper bounds for the probability of bit error
rate defined in the existing literature are function of input signal to noise
ratio and number of receive antennas. A generalized model found in the existing
literature is given as follows:
As mentioned earlier, the frame work presented in Eq. 12
provides the error performance upper bound for ML detection. According to author’s
knowledge, there is no framework which provides error performance lower bound
for ML detection without error correction code in the existing literature.
Proposed framework: For symmetric MIMO system, the existing approximated
performance bounds for ZF and MMSE are quite loose with respect to actual simulation
results. In this context, simple analytical frameworks that provide tighter
lower bounds for ZF, MMSE and ML detection schemes are proposed in this subsection.
The proposed frameworks are simple and accurate in the context of performance
tightness that depends on the MIMO dimension as well as input signal to noise
ratio. Denoting to be the symmetric MIMO dimension, the proposed analytical
framework of bit error rate performance lower bound with ZF detection at the
receiver can be written as:
The error performance bound for receiver with MMSE detection is proposed to
be as in Eq. 14 which depends on input signal to noise ratio
and MIMO dimension:
As mentioned earlier, there is no error performance lower bound framework for
receiver with ML detection in the existing literature; a framework is proposed
in Eq. 15 which defines error performance lower bound:
SIMULATION AND RESULTS
To evaluate the performance of the proposed framework a MIMO communication
system with Rayleigh fading channel is considered. It is assumed that the channel
is changing after every transmitting symbol vector x_{Nt} with dimension
(N_{t}x1) and Binary Phase Shift Keying (BPSK) modulation schemes is
considered for simplicity. Figure 24 present
the comparative results for performance lower bound for MIMO systems with dimension
ranges d = {2, 4, 6, 8} for ZF, MMSE and ML detection schemes, respectively.
Tightness of the analytical frameworks with respect to actual simulation results
were the main focus of this work. The proposed performance lower bound provides
tighter lower bound with respect to simulation results as compared to existing
framework for receiver with ZF detection.
At 5 dB of signal to noise ratio with MIMO dimension ranges d = {2, 4, 6, 8,},
the proposed performance bound is 4 dB tighter than the existing lower bound
as compared to the actual simulated results.

Fig. 2(ab): 
Error performance bound comparison for ZeroForcing (ZF) detection
with MultipleInput MultipleOutput (MIMO) dimension (a) d = 2 and d = 6
(b) d = 4 and d = 8 
At higher signal to noise ratio, the proposed lower bound becomes tighter
with respect to actual simulation results.
A comparative study of the existing and proposed analytical frameworks along
with actual simulated bit error rate performance for a MIMO receiver with MMSE
detector is presented in Fig. 3. As shown in the figure, at
10 dB of signal to noise ratio the proposed performance bound is 4 dB tighter
than the existing error performance bound in comparison with actual simulated
results for d = {2, 4, 6, 8}.
Figure 4 presents the performance bound of receiver with
ML detection for d = {2, 4, 6, 8}.

Fig. 3(ab): 
Error performance bound comparison for Minimum Mean Square
Error (MMSE) detection with MultipleInput MultipleOutput (MIMO) dimension
(a) d = 2 and d = 6 (b) d = 4 and d = 8 
At 10 dB of signal to noise ratio, the proposed performance bound for MLD is
19 and 30 dB tighter than the existing analytical frameworks for d = 2 and d
= 4, respectively in comparison to actual simulated results.
CONCLUSION
New analytical frameworks, which provide error performance lower bounds for
MIMO system with ZF, MMSE and ML detection schemes have been presented. Tighter
approximation have been obtained for the receiver with all three intended detection
schemes, in comparison to approximation methods within the existing literature;
considering simulated results with respective detection schemes as reference.

Fig. 4(ab): 
Error performance bound comparison for Maximum Likelihood
(ML) detection with MultipleInput MultipleOutput (MIMO) dimension, (a)
d = 2 and d = 6 (b) d = 4 and d = 8 
The proposed frameworks are expected to be helpful for engineers to approximate
the system performance accurately for a symmetric transmitter receiver MIMO
communication model. For the future extension of this work, authors are intended
to extend this work for asymmetric transmitter receiver MIMO systems.