Research Article  Open Access
Zedong Xie, Xihong Chen, Xiaopeng Liu, Yu Zhao, "MMSENPRISICBased Channel Equalization for MIMOSCFDE Troposcatter Communication Systems", Mathematical Problems in Engineering, vol. 2016, Article ID 5158406, 9 pages, 2016. https://doi.org/10.1155/2016/5158406
MMSENPRISICBased Channel Equalization for MIMOSCFDE Troposcatter Communication Systems
Abstract
The impact of intersymbol interference (ISI) on singlecarrier frequencydomain equalization with multiple input multiple output (MIMOSCFDE) troposcatter communication systems is severe. Most of the channel equalization methods fail to solve it completely. In this paper, given the disadvantages of the noisepredictive (NP) MMSEbased and the residual intersymbol interference cancellation (RISIC) equalization in the single input single output (SISO) system, we focus on the combination of both equalization schemes mentioned above. After extending both of them into MIMO system for the first time, we introduce a novel MMSENPRISIC equalization method for MIMOSCFDE troposcatter communication systems. Analysis and simulation results validate the performance of the proposed method in timevarying frequencyselective troposcatter channel at an acceptable computational complexity cost.
1. Introduction
Large capacity troposcatter communication not only plays an important role in the military communications, but also has great potential in other aspects [1]. With the further increases of the required bandwidth and highspeed data transmission, the influence of multipath delay spread of troposcatter channel is becoming more and more prominent [2]. The typical delay spread is extending over tens or hundreds of bit intervals. Furthermore, the significant timevarying Doppler shift mainly due to relative motion of the scatterer causes not only rapid fluctuation in the fading channel response but also compression or dilation of signal waveforms. Thereby, the timevarying frequencyselective fading appears to be more and more severe along with the delay’s increase.
In order to combat the multipath fading when the delay spread of channel impulse response (CIR) is large, some scenarios have been devised [3]. The traditional singlecarrier timedomain equalization (SCTDE) typically requires a number of multiplications per symbol that is proportional to the maximum expected channel impulse response length, which results in high computational complexity and rather slow convergence speed [4]. Further increasing of data rates imposes a greater challenge on SCTDE system. The orthogonal frequency division multiplexing (OFDM) technique [5] has some advantages such as good antifading capability, high spectrum efficiency, and low complexity, but it also has some disadvantages such as high peaktoaverage power ratio (PAPR) of its signal and its sensitivity to carrier frequency offset and phase noise [5], so it is not the best applicable one for troposcatter communication whose transmission power is severely limited.
To achieve the goal of favorable tradeoff between performance in severe multipath fading channel and complexity of signal processing, the singlecarrier frequencydomain equalization (SCFDE) has recently attracted increased interest because of its similar performance, efficiency, and low signal processing complexity advantages as OFDM, and in addition it is less sensitive than OFDM to radio frequency (RF) impairments such as power amplifier nonlinearities. Consequently, by performing various operations in the frequency domain, through the discrete Fourier transform (DFT), the complexity of processing can be reduced [6].
In SCFDE systems, linear equalization is simple and practical, but it does not do well in noise and intersymbol interference (ISI) suppression [7]. In the literature, the decision feedback equalization (DFE) with a hybrid structure (HDFE) [8, 9], where the feedforward (FF) filter is realized in the frequency domain (FD) while the feedback (FB) filter is realized in the time domain (TD), is derived to cancel the ISI with the disadvantages of relatively high design complexity and errorpropagation phenomena. Likewise, there is an equivalent algorithm which is called noisepredictive DFE (NPDFE) [10]; it is known as suboptimal DFE because the FF filter and the noise predictor (NP) are independently designed, while the FF filter and the FB filter in the conventional HDFE structure are jointly designed. What is more, residual intersymbol interference cancellation (RISIC) algorithm based minimum mean square error (MMSE) criteria can be utilized to mitigate residual intersymbol interference (RISI), which is also a kind of decision feedback equalization, and it avoids the computation of matrix inversion [10]. As a matter of fact, NPDFE algorithm only considers the channel noise of the system, while RISIC algorithm takes into account the RISI of the system. Therefore, nonlinear feedback and iteration mechanism are necessarily utilized to improve the performance. Iterative block decision feedback equalization (IBDFE) is an effective nonlinear algorithm with a disadvantage of rather high computational complexity [11].
After recalling the general methods to antagonize serious multipath fading, a novel MMSENPRISIC scheme with advantage of alleviating the effect of both noise and RISI is proposed in this paper. The proposed MMSENPRISIC scheme is the conjunction of the noisepredictive MMSEbased equalization and RISIC algorithms as mentioned previously. Applying the FF filters based on MMSE criterion, which compensates for the frequencyselective fading channel’s variations of amplitude and phase, we obtain the elementary timedomain sequence. The following procedure is subdivided into two segments. The noise term is predicted by exploitation of the deterministic characteristic of unique word (UW) and the correlation of the noise at the output for frequencydomain equalization (FDE). And the RISI term can be estimated by the fast Fourier transform (FFT) and the inverse FFT (IFFT) algorithms. Simulation results show that the proposed method can achieve better performance at a modest cost.
The rest of the paper is organized as follows. In Section 2, we give summarization about the multipath effect on the highcapacity troposcatter communication systems and the solutions to combat the resulting ISI. Then, the multiple input multiple output (MIMO) system model and equalization schemes are presented in Section 3. And in Section 4, a novel MMSENPRISIC scheme is proposed. Finally, the extensive simulations are presented in Section 5 and the conclusions are drawn in Section 6.
Notation. Matrices and column vectors are denoted by bold uppercase and lowercase letters, respectively. denotes the expectation operator. Superscripts and stand for the transpose and conjugate transpose operators. denotes the identity matrix of dimension M. is the Kronecker product.
2. Multipath Effects on Troposcatter Channel
We know that the specific type of fading for the corresponding receiver depends on both the transmission scheme and channel characteristics. The transmission scheme is specified with signal parameters such as signal bandwidth and symbol period. Meanwhile, troposcatter channels can be characterized by two different channel parameters, multipath delay spread and Doppler spread, each of which causes time dispersion and frequency dispersion, respectively [12].
Note that any received signal in the propagation environment for a troposcatter channel can be considered as the sum of the received signals from an infinite number of scatters. By the center limit theorem, the received signal can be represented by a Gaussian random variable. The amplitude, phase, and multipathdelayed components from different multipaths are timevariant, which are known as multipath effects.
In terms of time dispersion, a transmit signal may undergo fading over a frequency domain either in a selective or nonselective manner, which is referred to as frequencyselective fading or frequencynonselective fading, respectively. Due to time dispersion according to multipaths, channel response varies with frequency. As mentioned earlier, transmit signal undergoes frequencynonselective fading when the troposcatter channel has a constant amplitude and linear phase response only within channel bandwidth narrower than the signal bandwidth [2]. In this case, the channel impulse response has a larger delay spread than a symbol period of the transmit signal. Because of the short symbol duration as compared to the multipath delay spread, multipledelayed copies of the transmit signal are significantly overlapped with the subsequent symbol, incurring ISI.
For the troposcatter channel, multipath delay spread usually can be described by bilateral multipath delay spread, denoted as , which constantly varies from 10 to 500 nanoseconds with the system parameters such as communication distance and antenna aperture. Particularly, in troposcatter communication systems, the empiric formula of applicable to engineering is given as [13]where is the communication distance, denotes the transmitter frequency, and and are antenna aperture of the transmitter and the corresponding earth equivalent radius, respectively.
The ratio of bilateral multipath delay spread to symbol period of the transmit signal, denoted as , is the most effective parameter to measure ISI of troposcatter communication systems, with denoting the symbol period. In the following and without loss of generality, we can assume that 150 km, GHz, m, and km. Here, by substituting them into (1), bilateral multipath delay spread is equal to ns. Given a troposcatter communication system with the data rate of , we can figure out that . This in turn means that how to mitigate ISI is a question for troposcatter communication systems with highspeed data transmission.
As long as is small enough, the current symbol does not affect the subsequent symbol as much over the next symbol period, implying that ISI is not significant and thereby we can take into account some straightforward measures such as adaptive predistortion technique to combat it. However, when is large, some more complex equalization technology must be utilized to mitigate the ISI of the system aroused by multipath effects existing in troposcatter communication. As a matter of fact, under the condition that channel fading is not severe, SCTDE can do well in eliminating ISI. With the increase of transmission rate, the computational complexity of SCTDE becomes unacceptable. As mentioned above, SCFDE may be applicable to practical systems over troposcatter multipath channel with favorable tradeoff in performance and complexity.
3. System Model and Equalization Schemes
3.1. The System Model
We consider a spatial multiple MIMO system. The baseband equivalent system model of the MIMOSCFDE transceiver is shown in Figure 1. Let and be the number of transmit and receive antennas, respectively.
At the transmitter, the data stream is subdivided into independent branches by serialparallel converter. And then they are grouped by the length of . At the beginning of transmit signal block and in the ending of each grouped data stream, a fixed UW sequence known to the receiver is inserted periodically with the length of . For avoiding interblock interference (IBI), we assume that is not less than , the length of CIR.
The data stream at each branch can be denoted as , for . At the receiver, the timedomain MIMO system can be expressed as [14]where the input vector is and the additive noise vector is , where for denotes white Gaussian noise vector with zero mean and covariance . The output matrix withand the timedomain matrix is an matrix whose th subchannel impulse response can be expressed as an circulant matrix with the main diagonal value equal to and the first column equal to , where for is the th tap coefficient of Llength channel timedomain impulse response between the tth transmit antenna and the rth receive antenna.
Since is a circulant matrix, it has the eigendecomposition [15]:where is the normalized DFT matrix of size ; that is, its th element is given byIt is easy to prove that is the corresponding matrix performing IDFT operation. is an diagonal matrix with the main diagonal entries given by the channel frequency responses, which can be expressed asDefine . With the property , we can obtain the frequencydomain MIMO system model aswhere the channel frequencydomain response matrix is defined byHowever, the overall channel matrix is not diagonal, which does not lead to computational savings in the present form. Here, rearrange the input and output vectors by each frequency binwhere as the rearranged input and as the output one. Similarly, the channel matrix for the th frequency bin iswhere The channel matrix is converted into a block diagonal matrix, that is, . The channel model for each frequency bin is then defined as
In the next equalization process, is equalized by some frequencydomain equalization (FDE) schemes, which can be performed separately for each . The equalized signal is converted from the FD back to the TD by the IFFT and the resulting signal is finally detected by the receiver. Thereby, the design of the equalizer is of great importance to detection performance. Next section, some equalization schemes are discussed in detail at the th frequency tone in order to simplify the derivation and computation.
3.2. Some Equalization Schemes
3.2.1. NPDFE Scheme
We first consider the NPDFE scheme in MIMO systems without channel coding or interleaving. The coefficients of the feedforward NPDFE and the feedback NPDFE are obtained by minimizing the MSE. We prove that in uncoded systems this scheme has exactly the same performance as that of the conventional FDDFE scheme. The advantages of NPDFE are also discussed.
We concentrate on the receiver structure in Figure 2, which consists of a feedforward FDE processed in the frequency domain and a group of NPs processed in the time domain [16]. For simplicity, we assume that all NPs have the same order B. Thus, the coefficients of NPs for different data streams can be derived together in a matrix and vector form.
By multiplying an matrix , the output is given bywhere is the coefficient of the feedforward FDE. By converting to the time frequency, we will haveThen, the data vector before detection can be represented bywhere and is an matrix representing the coefficients of NPs at the th tap. Assuming that the feedback symbols are always correct, that is, , then the error vector is given bywherewith its average autocorrelation matrix beingThe coefficients of the feedforward FDE and the NP can be obtained by minimizing the MSE, which is the trace of (18). Through access to relevant references and noting that , we getwhereand where
This shows that the proposed NPDFE is also an optimal design in the sense of MMSE. Furthermore, by taking advantage of the MIMO architecture, different data streams can be reliably detected. The structure of the NP can be dynamically changed without affecting the feedforward FDE.
3.2.2. MMSERISIC Scheme
Here, when MMSE criterion is adopted for FDE, the equalized signal can be given by
In fact, the MMSEbased equalization algorithm is tradeoff between the channel noise and the RISI. Given the existence of RISI, FDDFE and NPDFE are proposed to alleviate or eliminate residual intersymbol interference. But their performance greatly depends on the order of feedback filter operated in the time domain. Consequently, computational complexity becomes rather high with the increase of the order.
MMSERISIC equalization is on the base of MMSE equalization, which is a new kind of decision feedback equalizer. However, this equalization can only be used in SISO system. Here, we propose novel MMSERISIC equalization for MIMO system for the first time.
Figure 3 shows the structure of MMSERISIC equalization, in which the RISI of the system is estimated and eliminated in the frequency domain. In order to estimate the RISI of the MIMO system, some manipulations are carried out after the feedforward MMSEFDE. Specifically, we give the simple mathematical derivations.
As mentioned previously, (22) can also be rewritten as follows:whereThe observation of (23) indicates that we cannot separate out from just in the method used in SISO system. Here, we consider that by multiplying the relationship between and RISI can become increasingly clear. That is,whereThen, the data vector before detection can be represented bywhere
The RISI estimate value can be greatly used to combat residual intersymbol interference of the MIMO system. In addition, the complexity is low because it mainly needs once FFT and IFFT operations.
4. The Proposed MMSENPRISIC Scheme
The algorithm of the NPDFE scheme only takes into account channel noise. Likewise, the MMSERISIC equalization mainly focuses on elimination of the residual intersymbol interference of the MIMO system. Based on what is mentioned above, a novel MMSENPRISIC equalization scheme is proposed.
Figure 4 shows the structure of the proposed MMSENPRISIC equalization. The NP part and MMSERISIC part are both based on MMSE criterion and operated in the frequency domain. Therefore, the NP part is different from the NPDFE scheme mentioned above, where the feedback filter is operated in the time domain. We analyze and derive the frequencydomain coefficients of both the feedforward FDE and feedback FDE in the MMSE sense. As is shown in Figure 4, the output vector of the NP part, , can be obtained as Suppose ; then the detection error of the NP part in the frequency domain will be given by
The autocorrelation matrix of the detection error will beThe MSE is the trace of (31). By substituting (30) into (31), differentiating the trace with respect to , and setting the result to zero, we getwhere is the frequencydomain coefficients of the feedforward FDE.
By introducing the constraint thatand taking the similar manipulation, we can obtainwhere is the frequencydomain coefficients of the feedback FDE.
Here, we note that the NP part is different from the traditional NPDFE scheme. In the MMSENPRISIC scheme, FF FDE and FB FDE are both operated in the frequency domain.
Then the MMSERISIC part is utilized to further eliminate the residual ISI existing in the output vector of the NP part. Here, we refer to the MMSERISIC scheme proposed above and the coefficient matrix can be obtained in a similar approach.
Actually, the input vector of the MMSERISIC part, , can be rewritten aswhereThen, , the frequencydomain coefficients of the RISI estimation, can be given bywhen calculating , and should be obtained before according to (32) and (34), respectively.
The novel MMSENPRISIC equalization scheme can eliminate the influence of both the channel noise and the residual ISI at the same time, which is the advantage that the existing other methods do not have. It is of great importance especially for the highcapacity MIMOSCFDE troposcatter communication systems.
5. Simulation Analysis
In this section, we evaluate the performance of the proposed method in terms of the normalized mean squared error (NMSE) and bit error rate (BER) and compare the computational load with other equalization methods.
In the simulated MIMOSCFDE system, let the number of transmit and receive antennas be . A standard convolutional code with code rate 1/2, constraint length 5, and octal generator polynomials (23, 35) is applied. The coded bits are mapped to QPSK for data (BPSK for UW). The block size has been set to , while the UW extension size has been set to and the data size is .
By reference to a large amount of measured data, we can choose a typical 300kilometerlong troposcatter communications link in North China, which is subject to frequencyselective fast fading. Table 1 presents its channel parameters [17, 18], in which nine different multiple signal paths are characterized by their relative delay and average power. The length of channel impulse response is . We assumed that each channel has a fixed implied response for each block period and that the receiver has perfect synchronization and channel state information. Nevertheless, in the process of the equalization and channel decode, iterative algorithm can only provide a relatively small improvement in the accuracy, but the complexity of it will inevitably increase in multiples. Iterations are not used to obtain the results in simulations.

Here, for convenience of discussion, the implementation method based on the traditional MMSE linear equalization is called MMSEFDE. And the NPDFE scheme is noisepredictive decisionfeedback detection proposed in [10]. The MMSERISIC denotes residual ISI cancellation based minimum mean square error criterion proposed in [11]. To illustrate the degradation due to error propagation phenomenon, the performance of MMSENPRISIC and NPDFE with corrected symbols fed back is provided. Furthermore, the performance of the matched filter bound (MFB) is also provided as a useful metric to compare equalizer structures. Here, Figures 5 and 6 show the average BER and NMSE performance of the equalization methods in the troposcatter channel mentioned above, respectively.
The ZFFDE cancels the interference completely without regard to noise amplification. Thereby, the MMSEFDE improves on this strategy by finding the optimal balance between interference cancellation and noise reduction that minimizes the total MSE. The NPDFE consists of a linear detector and a linear prediction mechanism that reduces noise variance. The noisepredictive implementation makes it easy to upgrade an existing linear detector by appending relatively simple additional processing. The MMSERISIC takes into consideration the fact that there still exists the residual ISI due to imperfect channel equalization, which will severely degrade the system performance. With the initial estimates of MMSE equalizer, the RISIC algorithm is adopted to alleviate the residual ISI. The proposed MMSENPRISIC scheme takes into account the noise term and RISI term of the signal equalized by MMSEbased equalizer, which can further improve the equalizer performance.
For the ideal NPDFE and ideal MMSENPRISIC, perfect decisions, that is, errorfree decisions, are fed back to the FBF. However, in the presence of noise and residual ISI, decision errors are inevitable. The first noise and residual ISI induced error is known as a primary error. As the decision error is fed back through the FBF, instead of cancelling the postcursor ISI components, it also adds additional interference.
As shown in Figure 5, the proposed MMSENPRISIC method outperforms the other schemes mentioned above. For a BER of 10^{−4}, the performance degradation of the proposed structure is about 1 dB when compared to the ideal MMSENPRISIC. Compared with the NPDFE and MMSERISIC, the new proposed structure yields SNR gain of about 2 dB and 3.5 dB at BER = 10^{−3}, respectively. It can be explained as the consequence of two factors: the improved noise predictive part and the RISIC process. In fact, in the proposed structure, the noise predictive part can gradually increase the reliability of the detected data, thus reducing the effects of the error propagation that limits the traditional DFE performance, especially at low SNRs. Moreover, the RISIC process is able to remove the residual ISI existing in the equalized signal. What is more, under the condition SNR = 10 dB, the MMSENPRISIC can decrease the average BER by almost one order of magnitude compared with the NPDFE. In the simulations, we also find that the gaps in BER performance will grow over the increase of the average SNR, which illustrates that the advantage of the proposed scheme is more obvious at relative high SNRs.
However, the proposed MMSENPRISIC still has a gap from the MFB. The performance degradation of the DFE with respect to the MFB can be decomposed into two parts. The first is the error propagation gap. The second component is known as the gap from the MFB. Note that the gap from the MFB increases with increasing SNR. This is due to the fact that at high SNR scenarios ISI is the dominant factor. We can also come to the conclusion that the MMSERISIC and NPDFE can only slightly improve the BER performance of the system, while the MMSENPRISIC will bring a significant increase.
In addition, Figure 6 illustrates NMSE performance of the proposed scheme. When average SNR < 6 dB, MMSENPRISIC exhibits similar performance with MMSERISIC and NPDFE, although their related implementation costs differ significantly, whereas, under the condition that SNR > 8 dB, the proposed method has a smaller NMSE than the others, especially at relative high SNRs. What is more, error propagation results in an increased probability of error during the subsequent symbol decision. Hence, the NMSE performance degradation of the proposed structure is severe when compared to the ideal MMSENPRISIC.
The computational complexity of the various schemes is mainly evaluated in terms of the number of complex multiplications (CMULs), for both the signal processing and filter design. Table 2 presents the CMUL per output sample needed for signal processing, and Table 3 shows the CMUL for the equalizer design [8]. As shown in the tables, the overall computational load of the proposed MMSENPRISIC is higher than the other methods mentioned above. The improvements of the performance are at an increased complexity cost. From the theoretical analysis and the time complexity collected during the run, the increase of the proposed scheme is acceptable. The frequencydomain equalization can be performed for each frequency bin to further reduce the computational load.


6. Conclusion
In this paper, we have mainly presented a novel channel equalization scheme for MIMOSCFDE troposcatter communication systems. Here, we firstly analyze the wellknown methods to combat the ISI existing in large capacity troposcatter communication. Then, we focus on the MIMOSCFDE system model and some equalization schemes. Based on the MMSE criterion, the NPDFE equalization and MMSERISIC equalization schemes are proposed to make it applicable to MIMO system for the first time, respectively. Furthermore, we present a new MMSENPRISIC channel equalization scheme to combat the ISI of the troposcatter channel. The frequencydomain channel estimation and equalization can be performed separately for each frequency bin to further reduce the computational complexity of the proposed method. Numerical results show that compared with the existing methods mentioned above the proposed equalization scheme achieves better performance at an acceptable computational load cost in MIMOSCFDE troposcatter communication systems.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by National Natural Science Fund of China under Grant no. 60971100.
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Copyright
Copyright © 2016 Zedong Xie et al. 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.