Channel Equalization Techniques
for Space Time Block Codes
in Non Quasi-Static Channels
Wookbong Lee1, Chang-Kyung Sung2, Inkyu Lee2, Min-Seok Oh1, and Jin-Sung Choi1
Page 1 (1)
Abstract— Conventional space time block codes (STBC) assume quasi-static channel environments. If this assumption is not satisfied, then these schemes show serious performance degradation. In this paper, we propose a linear equalizer method to mitigate the degradation due to channel offsets. To employ this linear equalizer method, we introduce a lattice representation for the STBC schemes. Simulations confirm that the proposed scheme is effective in non quasi-static channels.
Index Terms—STBC, non quasi-static, time selective, equalization
INTRODUCTION
S
ince Alamouti introduced a transmit diversity scheme with two transmit antennas [1] , several schemes have been developed to improve the link performance by using multiple antennas as a source of spatial diversity [2], [3]. When the space-time code scheme is based on a block code structure, such codes are called space time block codes (STBC). One constraint in the STBC is the assumption that the channel response does not change during the space-time block code time period (i.e. quasi-static channels). Especially the orthogonal STBC schemes exploit the orthogonality of the code matrix to detect symbols with linear processing which provides maximum likelihood detection [3].
However, the quasi-static condition can be violated when the system is operated with high carrier frequency and high mobility, which are often assumed for next generation wireless communication systems. In this case, a linear processing technique can be adopted to mitigate non quasi-static channel effect [4]. In order to employ signal processing techniques, it is required to have a channel model in a linear superposition form. Unfortunately, STBC cannot in general be expressed as a simple complex linear superposition form. Thus, we introduce a general linear superposition form (a lattice representation), which can be applied to any STBC. Then we apply a channel equalization technique to mitigate the channel effect.
MATRIX REPRESENTATIONS FOR SPACE TIME BLOCK CODES
Consider a wireless communication system with N transmit antennas and M receive antennas. The channel is assumed to be flat fading with the propagation gain , where n, m, and l represent indices for transmit antenna, receive antenna, and time, respectively. The channel coefficient is a Rayleigh distributed random variable, independent complex Gaussian with variance 0.5 for each dimension.
We consider a STBC transmission system where the code is defined by the code matrix C and the space-time block code time period is L. Assuming T is the symbol duration, the received signal at the receive antenna m at time lT is given as
, (1)
where denotes the (l,n) component of the L by N code matrix C, and is an additive white complex Gaussian noise term with zero-mean and variance . In the quasi-static channel case, the time index l in can be omitted.
Denote , , as the transpose, the complex conjugate, and the Hermitian transpose, respectively. By collecting the received signal in the vector form, equation (1) for quasi-static channel can be written as
, (2)
where R, H, N denote an L by M matrix with the (l,m) component , an M by N channel matrix with the (m,n) element , and an L by M noise matrix, respectively.
It can be noted that it is difficult to adopt signal processing techniques to this matrix representation (2), since the code matrix C is normally not expressed as a linear combination of the STBC input symbols. Furthermore, for non quasi-static channels, equation (1) cannot be transformed into equation (2). Thus, it is important to have a general expression form regardless of the code structure or the channel condition. From a signal processing point of view, in order to apply signal processing techniques it is necessary to have a linear superposition form, such as
, (3)
where denotes the LM by 1 received signal vector, z is defined as the K by 1 transmitted signal vector , represents the LM by K channel matrix, and indicates the LM by 1 noise vector.
There exist some STBC schemes which can be transformed into a linear superposition form. The code matrix C for the Alamouti orthogonal scheme [1] is given as
.
This code can be converted into a linear superposition form as follows. By applying the conjugate operation to the second half of the received signal, the linear superposition form for the Alamouti scheme can be expressed as
where , , and are M by 1 column vectors at time lT and are defined as
, , and , respectively.
The optimal detection for the Alamouti scheme for quasi-static channels is achieved by just multiplying the vector by the matched filter (MF) and applying the symbol by symbol detector. We refer to the full Maximum Likelihood (ML) search as an exhaustive search for solving an equation, , by checking all possible combinations of transmit symbols.
For general STBC schemes, however, it is not possible to form a complex linear superposition expression (3). For example, for the four transmit antenna cases, we consider a rate 3/4 code matrix C proposed in [3] as
. (4)
This code matrix cannot be converted to a complex linear superposition form as done in the Alamouti code. However, if we expand matrix elements to real and imaginary components independently, the received vector can be expressed in a real linear superposition form. In general, every block codes can be expressed in this way by dividing into real and imaginary parts. This is called the lattice representation, and will be illustrated in the following section.
CONSTRUCTION OF LATTICE REPRESENTATION
In this section, we propose a general construction of the lattice representation for the STBC schemes. In order to construct the real lattice channel matrix, we divide the code matrix C into two parts, which can be expressed as where and denote the real and the imaginary parts of a vector or a matrix, respectively. Let us denote . Suppose that the (l,n) component of C is , where and are scalar coefficients. Then the lth row vector of and are denoted by , where and represent the real and imaginary part of , respectively. Here and are located at the nth position in and , respectively. For example, the second row vectors of matrix and with respect to the code matrix (4) can be obtained as , .
Similar to the lattice representation rule of converting a complex channel matrix equation into a real matrix equation in [5], the structure of the real lattice channel matrix of the modified code matrix becomes
where the elements in and are determined by the corresponding elements of and, respectively. That is, the kth column vector in the lth row block of the matrix (or ), (or ), is determined by the (l,n) component of (or ), (or ). For example, the second row components of and corresponding to the row vectors and are obtained as , . Here, the dimension of the real lattice channel matrix becomes 2LM by 2K.
The lattice representation is finally written as where
, .
Following this construction rule, the real lattice channel matrix for the code in equation (4) is expressed as
Another case we consider here is the code where several 's are combined in the (l,n) element of C in an additive form, or where appears more than once in one row of C. This kind of codes can be expressed as follows by utilizing the linear superposition relationship. For the code matrix [2]
, the lattice channel matrix is represented as
and
.
For the quasi-static channel response, if the code matrix is orthogonal, then the corresponding channel matrix is also orthogonal [6]. Thus, for orthogonal code matrices, the transpose of the real lattice channel matrix corresponds to the matched filter. After matched filtering, the symbol by symbol ML detection is optimal in this case. The result of the matched filtering for the symbol is
where is a new additive noise term with zero mean and variance , and represents the channel response power.
EFFECT OF NON QUASI-STATIC CHANNEL
In this section, we analyze the performance degradation in the orthogonal STBC schemes when the channel condition is not static.
The conventional orthogonal space time codes usually assume quasi-static fading environments. However, in practice, this quasi-static assumption may not be satisfied due to Doppler spread or offsets in a timing synchronization. Taking those effect into account, for the 2 X 1 Alamouti case, the received signal can be represented by
where and are the two received signals at time T and 2T respectively, and for one receive antenna, is reduced to by dropping the receive antenna index.
We now assume that consecutive channel coefficients for each transmitted antenna are modeled as [4],
for (5)
where is a Rayleigh distributed random variable with zero mean, and is a channel offset factor (0 < < 1). We normalize , also a Rayleigh distributed random variable, to have a unit variance resulting in where is the variance of . When , the channel model in equation (5) becomes equivalent to the quasi-static channel model. When , s are independent from each other.
In practice, for fast fading channels, the value of is determined by the Doppler spread. For example, is considered as a fast fading model [4]. Especially is modeled to illustrate the offset in the channel estimation.
We now analyze the effect of and . In order to detect and for the conventional system, the matched filter is applied to . Here, neglects the changes in the channel coefficients, thus this results in the performance degradation. We refer to this conventional system as the matched filter (MF) system. Then the combining operation is
.
After linear processing, the extracted estimate of the data symbol becomes
where and are new additive noise terms. The noise variance is equal to .
Let be the average signal energy, then the corresponding Mean Square Error (MSE) for the symbol is .
For equally probable symbols, the matched filtered Signal-to-Noise Ratio (SNR) is
(6)
where SNR is defined as . Note that even if SNR goes to infinity, is bounded by when the channel is non quasi-static ().
Figures 1 and 2 show the effect of non-quasi static channel responses of the Alamouti scheme and the four transmit antenna scheme (4) respectively. The actual is evaluated based on the Monte Carlo simulations with ten thousand independent channel realizations. Note that in Fig. 1, the equation (6) matches exactly with the Monte Carlo simulation result. Thus, we substitute all the other MSE calculations (or ) by the Monte Carlo simulations. We can see that even if the input SNR becomes larger, the matched filter SNR gets saturated when the channel offset exits.
The figures show that becomes larger as the number of receive antenna increases. The reason for this is that with the matched filter, the detected symbol energy becomes , but the MSE is proportional to . As the receive antenna increase, so does the value of . As a result, the corresponding increases.
EQUALIZATION METHOD FOR NON QUASI-STATIC CHANNELS
As shown in previous section, the performance of the system is dependent on . However, even for a fast fading channel, we will show that a linear equalizer is good enough to mitigate the offset factor.
It is well known that the Minimum Mean Square Error (MMSE) solution exhibits a better performance than the Zero-Forcing (ZF) solution. Thus, we will derive the MMSE solution in this section. The MMSE solution is obtained by finding the equalizer matrix W which minimizes the MSE, . From the orthogonality principle, the MSE is minimized if and only if an error signal is orthogonal to the observed vectors, that is .
Now, we can find the equalizer matrix W which satisfies the above condition as. Then, we have where is the cross correlation matrix of and , and is the auto correlation matrix of . They can be computed as and . Finally the equalizer solution becomes
.
Using the matrix inversion, we can further show that
.
The error auto correlation matrix
Thus, if is not orthogonal, the noise will become colored.
SIMULATION RESULTS
In this section, we will show the simulation results with the rate 3/4 orthogonal STBC using four transmit antennas, in flat fading channels with various channel offsets. We compare the frame error rate (FER) for 384 bits per frame with uncoded QPSK modulation, which results in 1.5 bps/Hz.
Figure 3 shows that the performance degradation of the matched filter and the MMSE equalized system become smaller as the number of receive antenna increases. Still, the performance of the matched filter system is not acceptable.
In contrast, with the same channel offset factor , the performance degradation of the proposed MMSE equalization scheme with respect to the quasi-static case reduces from 2 dB to 0.4 dB, as one more receive antenna is added. As we mentioned in previous section, the MMSE equalizer is good enough to mitigate the offset factor even for very fast fading.
Figure 4 exhibits the FER performance comparison with different filter settings. We assume that the carrier frequency is 5.8 GHz and the symbol rate is 40 kHz. For example, at 150 km/h the Doppler frequency becomes 806 Hz. Figure 4 shows that for non quasi-static channel, the MMSE filter performance is only a few tenth of a dB away from the ML performance even for high mobile speed, while the MF performance degrades substantially. Thus, the proposed lattice representation rule allows us to apply the MMSE filter to the STBC schemes, which results in a significant gain over the conventional methods.