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The algorithm and the results of a nonlinear detector using a machine learning technique called support vector machine (SVM) on an efficient modulation system with high data rate and low energy consumption is presented in this paper. Simulation results showed that the performance achieved by the SVM detector is comparable to that of a conventional threshold decision (TD) detector. The two detectors detect the received signals together with the special impacting filter (SIF) that can improve the energy utilization efficiency. However, unlike the TD detector, the SVM detector concentrates not only on reducing the BER of the detector, but also on providing accurate posterior probability estimates (PPEs), which can be used as soft-inputs of the LDPC decoder. The complexity of this detector is considered in this paper by using four features and simplifying the decision function. In addition, a bandwidth efficient transmission is analyzed with both SVM and TD detector. The SVM detector is more robust to sampling rate than TD detector. We find that the SVM is suitable for extended binary phase shift keying (EBPSK) signal detection and can provide accurate posterior probability for LDPC decoding.

Increasing demand for wireless communication in various areas of human life has brought about an exponential increase in the number of wireless services. There will be a continuous increase in the demand for wireless spectrum in the foreseeable future with the introduction of internet multimedia applications such as online video, multimedia networks, and distributed gaming. This exponential increase has resulted in spectrum scarcity as the electromagnetic spectrum has become too crowded to incorporate all the upcoming wireless services. There is more of an increase in demand for spectrum than for development in technology, which aims at increasing the spectrum efficiency [_{2} emissions of the information and communications technology industry. There is a need on environmental grounds to reduce the energy requirements of wireless communications [

Recently, these problems have attracted a lot of attention of researchers and many new ideas have been proposed to mitigate the problem of spectrum scarcity and energy efficiency [

Some preliminary works on detecting the signal of SIF output have been presented in [

The contribution of this paper is to cover high bit rate, low bit error rate (BER) and low energy consumption by applying the SVM technique. A numerical example is used in giving a brief demonstration of the SVM detector and the design parameters have been considered and investigated for the purpose of optimization and simplification. Other related issues such as the kernel selection, features extraction and reducing complexity of the detector have also been analyzed. In addition, we give the analysis of state-of-the-art nonlinear detector together with the channel decoder.

The remainder of this paper is organized as follows: Section 2 is devoted to introducing the efficient modulation. We present the receiver scheme in Section 3 and briefly describe the SVM classification and PPEs for LDPC decoding. In Section 4, we include illustrative experiments to compare the performance of the SVM detector. We conclude in Section 5 with some final comments.

The increasing demand for frequency resources is becoming a tough problem when allocation and reallocation of frequency bandwidth are periodically repeated. Higher level modulations are used in solving the problem. However, these solutions are all at the expense of increased energy. Furthermore, the order grows in powers of two, while the constellation is becoming dense and difficult to divide. As a result, only binary modulation or binary keying make sense to easily and fairly measure the bandwidth efficiency.

Efficiency modulation was first proposed by Walker, who holds several patents on the technique. After his cooperation with Photron Science Company, these patents were registered as ultra spectral modulation (USM), which has pretty high bandwidth efficiency. In some other literatures, from a signal bandwidth rather than power bandwidth point of view, such a system can be referred to as a carrier-synchronized ultra-wide band system (CS-UWB) [

From a unified expression perspective, all of these techniques are actually a special EBPSK system that is defined as follows:
_{0} and _{1} are the modulation waveforms corresponding to bit “0” and bit “1”, respectively; _{c}_{c}_{c}

According to the FCC's bandwidth definition, there should be a total 99% signal power hold in the band. Such a power reservation criterion is practically equivalent to the −20 dB attenuation bandwidth, indicating that spectral attenuation from the peak power to the cutoff frequency point is no less than 20 dB. The −40 dB attenuation bandwidth of EBPSK modulation is only several Hertz [

The difference between the waveforms of “0” and “1” is very small, so a traditional IIR or FIR filter with a narrow bandwidth can erase the minute difference information and leave only a sine wave, such that we cannot perform detection in the receiver [

Therefore a simple amplitude detector can be used in separating the symbols “0” and “1” because of the existence of high impulse in coded 1 s. From reference [^{2} is the noise variance, A0 and A1 is the maximum amplitude of the filter output corresponding to code “0” and “1”, respectively, as is shown in

According to reference [_{0} and _{1} can be obtained through the following equations:

Though the SIF transforms phase modulation into amplitude changes, several signal cycles that followed the change part are distorted. Obviously, if we use short bit duration, the subsequent symbol will be interfered with as is shown in

Also, the signal amplitude of symbol “1” with short bit duration is lower than that with long bit duration, but for symbol “0” both of them are almost identical. According to the relationship between _{0} and _{1} in (

In this section, we suggest a nonlinear detection algorithm from an appealing pattern classification point of view. We detect the received signals of SIF output by using the SVM technique. The main advantage of using such a technique is that it can make full use of the characteristics of the received waveforms.

For the binary classification problem, during the training stage, the goal of SVM is to seek a separation plane which maximizes the margin between the two classes of 1 and 0. Each input training sequence _{i}^{n}_{i}_{i}_{i}_{i}^{T} Ψ(_{i} into the high-dimensional feature space, and b is a bias term of the decision hyperplane.

Define a coefficient vector _{i}

The output is a reduced set of those training data, because most training data _{i}_{i}_{i}

Usually, four kernel functions are used in different cases. The RBF kernel non-linearly maps samples into a higher dimensional space, so it can handle the case when the relation between class labels and attributes is nonlinear. Compared to the RBF kernel, the polynomial kernel has more hyper-parameters, which influences the complexity of model selection. In addition, the sigmoid kernel behaves like RBF for certain parameters. Furthermore, there are some situations where the RBF kernel is not suitable. Thus, one may just use the linear kernel which is the simplest one. In this paper, the SVM detector uses two types of kernel functions to compare the performance with each other. The first is the simplest linear kernel, shown as:

The optimal selection of discriminant features is an issue of the greatest importance in EBPSK system.

If we define the area below y(n) as the energy, then we may note this energy item is quite concentrated at the left range of the characteristic waveform “1” when the received signals pass through the SIF. This energy is relatively dispersed while the symbol “0” passed through the SIF with only channel noise. As a result, we define the first feature as:
_{s}_{s}

It is noticeable that in

From

The total received energy can be also utilized to differentiate the two symbols, Therefore, we add it to our feature set as the fourth feature:

By taking full advantage of the developed characteristic waveforms, we have constituted a feature set which is dedicated to separating the two symbols. It is noteworthy that we do not need to estimate the channel noise power ^{2}, and choose only four features of SVM for training and testing, which can reduce the complexity significantly.

Based on above elaborations, we have taken full advantage of the EBPSK waveform and established a quantitative feature set Fl = [_{1}_{2}_{3}_{4}_{1}_{2}_{3}_{4}

An appropriate selection of discriminant features is carried out in order to determine the best performing features for the signal detection as Fl and Fs. By using the method, the original higher-dimensional inputs (the number of sampling points is M) will be transformed into lower-dimensional features. In our scheme, four remarkable features are chosen to separate the symbols “0” and “1”.

Scaling before applying SVM is very important. The main advantage of scaling is to avoid attributes in greater numeric ranges dominating those in smaller numeric ranges. Another advantage is to avoid numerical difficulties during the calculation. Because kernel values usually depend on the inner products of feature vectors, e.g., the linear kernel and the polynomial kernel, large attribute values might cause numerical problems. We use linearly scaling each attribute to the range [0,1].

The initial training stage only needs to be performed once unless the channel condition has varied significantly. Some training examples are given to the machine to create certain decision functions in order to differentiate the different types of objects, or so-called classes.

During the testing stage, the SVM detector is ready for estimating the source bit based on classifying an unforeseen object, which is a new noisy data stream, and then classified by those decision rules. The detection task then becomes a pattern classification problem. The transmitted message bit is estimated by making a hard-decision from the decision function formed earlier in (

We have made a hard-decision by using SVM classification, in some cases, such as channel decoder needs a posterior probability to achieve capacity. Platt has proposed that the SVM output can be transformed into posterior probabilities [_{i}_{i}_{i}_{A,B}_{i}_{i}_{i}

Unfortunately, log and exp could easily cause an overflow. If _{i}_{i}_{i}_{i}

From (

We employ low-density parity-check (LDPC) codes [

Initialization:
_{n} = x | y_{n})is the PPEs of detector outputs.

Horizontal Step: the MAP output from _{m}_{n}

Vertical Step: updating the message from _{n}_{m}

Tentative output:

Although, for classic digital modulation technologies the BP decoding is analyzed in [

In this part, we evaluate the performance of the proposed SVM detector and its soft output for LDPC decoding. For all simulations, unless specified otherwise, the system had 3,000 random symbols for training and the reported BER are computed using 10^{6} symbols and we average the results over 1,000 independent trials with random training and test data. We use

In this subsection, the performance of the SVM detector, using the kernel functions (10) and (11), introduced in Section 3, is compared. The 10-fold cross-validation sweep from the training samples was used to find the optimum parameters of C and

When

We have analyzed the BER performance of the SVM detector with linear kernel which is superior to the RBF one. Nevertheless, the solution for such a problem is computationally complex by consuming a mass of energy. The complexity of training an SVM for binary classification is O(n2), using the sequential minimal optimization [

The BER performance of the EBPSK system will be diverse with different bit durations and sampling rates. To prove the effectiveness of the proposed method, various simulations were conducted. In

The BER performance comparison of the SVM with TD by different sampling rates is plotted in _{s}_{c}_{s}_{c}_{s}_{c}^{−3}, respectively. This means the performance of the SVM detector improved significantly while the sampling rate is low, and it is more robust to sampling rate than TD.

In the next experiment we face a bandwidth efficient communication model which is proposed via a NBPF at the transmitting end of the system. Though we can achieve a bandwidth efficient transmission and suppress the interference to other channels, the transmitted signals would be distorted and the amplitude of SIF output signal would be not as high as usual. We use the SVM detector to solving such issue and give the BER performance comparison between SVM detector and TD. The bandwidth of the linear phase NBPF is designed to be

PSD of the modulated signals is plotted in

The performance comparison of SVM detector and TD are presented in ^{−3} and for the TD, the gain over the NB-TD is around 4 dB with the BER = 10^{−3}.

This can be explained by the fact that when the signal was filtered by the NBPF it was distorted significantly and it was difficult to detect the signal through TD, but for the SVM detector which can make full use of the characteristics of SIF output signals, such as energy and waveforms, thus the SVM-NB is about 2 dB from the performance achieved by the SVM. Moreover, the NB-SVM is even better than TD and the former outperforms the latter by about 3 dB and outperforms the NB-TD by 7 dB with the BER = 10^{−3}. This demonstrates that the performance can be improved by using the SVM detector in a band efficient transmission system. We have shown that there are more advantages in the SVM method than TD and the BER performance was significantly improved by the former compared with the latter. Moreover, the SVM method is not as sensitive to the sampling rate as the threshold method. Thus, SVM is an effective method for EBPSK detection.

In previous subsection, we have discussed the detector based on SIF together with SVM classifier, when we compare performances at low BER. In this section, we focus on the performance after the sequence has been corrected by an LDPC decoder and the ability of SVM detector to provide accurate posterior probability estimates instead of measuring the performance of the demodulator at low BER, because the channel decoder can achieve those BER values at significantly lower signal power. Earlier this year a study has been undertaken to give the approximate LLR for LDPC decoding [

In

To understand the difference in PPEs, we have plotted the curves for the SVM and the MAPPE in

In the last experiment we will compare the ability to provide PPEs between SVM and MAPPE with different sampling rates. In _{s}_{c}_{s}_{c}_{s}_{c}

In this paper, we introduced a new approach for nonlinear detection based on a SVM classifier. A simulator of the system with high data rate and high spectra efficiency was designed. We show that the performance can be significantly improved by using a linear SVM kernel for detection, which has less computational complexity and thus saves computation time and energy. Moreover, we only use four features for training and testing, which makes full use of the characteristics of SIF output signals and reduces the complexity significantly. Furthermore, we concentrated not only on reducing the BER of detection, but also on providing accurate PPEs. The BER performance was significantly improved for the SVM-LDPC compared with the MAPPE-LDPC approach. Also, the SVM method is more robust to sampling rate than the MAPPE method, and the former is proposed for use in detection when the sampling rate is low. In addition, we analyzed the SVM detector for a bandwidth efficient communication system with the NBPF added at the transmitting end. Such a system can meet the requirements of bandwidth limitation and achieve the desired performance. As a by-product, we have shown that the short bit duration (^{2}, which reduces the complexity.

In fact, the features selection procedure is somewhat elementary as an early work, and we haven't considered the resort to the feature combination technique which can reduce the corresponding problem dimensiond. If the optimal feature combination is used, accompanying the well-established features selection procedure, the gains achieved with this technique can be further enhanced, which also remains as an attractive area for future research.

The authors would like to thank the support of the National Natural Science Foundation of China (NSFC) under the Grant (61271204), and the anonymous reviewers for helpful comments.

EBPSK modulation with

The threshold decision with σ^{2} = 0.

The envelop of symbol “1” and its followed symbol “0” with bit duration N = 4.

The signals envelope of SIF output with SNR = −2 dB, N = 20 in (

The SIF output of symbol “1” and “0”, respectively, in (

Cross-validation result of the SVM detector in RBF kernel.

The BER performance of the SVM detector in different kernels. We use ◊ for the SVM with RBF kernel and Δ with linear kernel.

The BER performance of SVM detector with difference bit duration. We represent the threshold decision for bit duration N = 5 with ▼) and N = 20 with ○, respectively; the SVM for N = 4 with *, N = 5 with ◊ and N = 20 with ✩ , respectively.

The BER performance comparison of the SVM with threshold decision by different sampling rate. We use dashed-dotted lines for the SVM BER, solid lines for the threshold decision BER. We represent the BER for _{s}_{c}_{s}_{c}_{s}_{c}

The power spectrum density of modulated signals and its filtered signals, respectively, in (

The BER performance comparison of SVM and threshold decision for general system and narrow band system with N = 5. We use dashed-dotted lines for the SVM BER, solid lines for the threshold decision BER. We represent the BER of narrow band system with ◊) and general system with ○, respectively.

We plot the BER performance at the output of LDPC decoder with soft-inputs using a SVM and MAPPE detector with N = 5, N = 20 and narrowband system in (

We plot the SVM PPE and MAPPE, respectively, in (

The BER performance at the channel decoder with SVM PEEs (dashed line) and with MAPPE (solid line). We represent the BER for _{s}_{c}_{s}_{c}_{s}_{c}

Comparison of SVM models.

RBF | Linear | |

C | 8 | 2 |

8 | - | |

SVs | 462 | 381 |