Xiu-Hua Li，Lin Ma，Xue-Zhi Tan，Xin Liu

(1.Communication Research Center，Harbin Institute of Technology，Harbin 150001，China;2.Science and Technology on Information Transmission and Dissemination in Communication Networks Laboratory，Shijiazhuang 050002，China)

1 Introduction

In wireless communication systems，the transmission signal is greatly affected by multi-path fading channel where is usually corrupted by noise or interference. In orderto reduce the effect， the traditional single carrier system employs the timedomain equalization technology. However， it is generally considered that the technology has much too high complexity.OFDM system can reduce the equalization’s complexity and solve this problem effectively by converting a wide-band frequency selective channel into a parallel collection of narrowband frequency flat sub-channels［1］.But OFDM system has some obvious disadvantages as well，that is，it has high peak-peak average power ratio(PARP)and it is sensitive to the offset of frequency and phase.After considerable research in the past few years，the single carrier system is found to be able to employ the frequency-domain equalization technology with low complexity，which is so-called SC-FDE technology.Moreover，SC-FDE can overcome the inherent disadvantages of OFDM.SC-FDMA，which is the multiple access of SC-FDE，has been considered as a promising technology against multi-path fading and adopted in the uplink of fourth generation(4G)mobile communication systems［2］.The performance of SC-FDE system greatly depends upon the accuracy of channel estimation about channel state information(CSI)in the receiver，especially the channelimpulse response which is utilized in the frequency-domain equalization.

Good channel estimation can improve and enhance the system’performance greatly.The existing channel estimations are mainly divided into two categories according to their operational scenarios:pilot-aided where the transmitter inserts the pilots in the block-type frame and transmits them periodically，and the receiver has a priori knowledge of the transmitted pilots［3］;nonpilot-aided where the transmitter does not transmit the pilots or the receiver does not require such knowledge.To achieve better performance， most wireless communication systems employ the pilot-aided channel estimation though it costs some transmission bandwidth.Thus，the channel estimation is pilot-aided for SC-FDE system in this paper.

For SC-FDE system，the LS channel estimation［4］is a commonly adopted method which has relatively good performance.Especially when the errors follow a normaldistribution， LS estimation is unbiased，consistent and effective among the unbiased estimations［5］. Moreover， compared with minimum mean-square error(MMSE)estimator［6］and linear minimum mean-square error(LMMSE)estimator［7］，LS channel estimation needs no knowledge of channel condition’s statistics or noise power.However，when the observed signal(pilot)is corrupted abnormally so that the errors are not normal-distributed，LS estimation has bad performance of anti-jamming and stability.Then CSI is not estimated accurately and the performance ofSC-FDE system is affected and degraded.

In this paper，to overcome the disadvantage of LS estimation， ridge regression (RR)［8］channel estimation is proposed and adopted.RR estimation has relatively stronger anti-jamming.The numerical results show that the performance of the proposed RR channel estimation is better than that of LS estimation，and it has better accuracy of estimation in low CSNR level.

2 System Description and Signal Model

A baseband-equivalent model of a pilot-aided SCFDE system with perfect synchronization is depicted in Fig.1. In the transmitter， the encoded binary information data is mapped and grouped according to the modulation for signal mapping.Then pilot symbols x(n)(n=1，2，…，N)are inserted into the parallel data to assemble frames.The cyclic prefix(CP)with L symbolsis appended to reduce the inter-symbol interference(ISI)before every frame of a data block，and the structure schematic diagram of block is shown in Fig.2.Then the block is operated for parallel to serial conversion.The transmission signal is through a multi-path time-varying frequency selective fading channel with the channel impulse response h(n)which is assumed stationary within one SC-FDE data block，and corrupted by complex AWGN samples w(n)with noise mean of zero and noise power variance of σ2.In the receiver，the receiving signal is operated for serial to parallel conversion.After the CP is removed correctly，the receiving pilot symbols y(n)can be expressed as follows:

where(·)⊕(·)denotes their periodic convolution.Moreover，for P paths fading channel，h(n)can be represented as follows:

where hl，τ and τldenote the complex impulse response of the lth path，the delay spread index and the lth path delay normalized by the sampling time respectively.

Fig.1 Baseband-equivalent SC-FDE system model

Fig.2 Block structure schematic diagram

Then the data block is sent to FFT module to transform the signal from time-domain to frequencydomain.It is assumed that the size of the CP is longer than that of the channel impulse response，that is，there is almost no ISI between SC-FDE symbols，the frequency-domain pilot symbols Y(k)(k=1，2，…，N)can be expressed as follows:

where X(k)，H(k)and W(k)denote the N-point FFT of x(n)，h(n)and w(n)respectively.Moreover，the N-point FFT is expressed as follows:

With accurate channel estimation，the transmission signal is recovered by frequency-domain equalization and then transformed into time-domain signal by IFFT module.After parallel to serial conversion，the transmission encoded binary information data is finally demapped and restored back in the signal demapper module.

3 Channel Estimation

In the section，the traditionalLS channel estimation in SC-FDE system is firstly introduced and then its inherent disadvantage is analyzed.In order to overcome and improve the disadvantage，a novel RR channel estimation is proposed and adopted.

3.1 LS Channel Estimation

The principle of LS estimation can be represented as follows:

From Eq.(2)，we can get the mean square error(MSE)of LS estimation as follows:

It is obvious that LS estimation is an easy algorithm to estimate the channel response.However，when the observed signal is corrupted abnormally by noise or even interference，the performance of LS estimation gets badly，that is，it has little anti-jamming and stability in that it does not consider or utilize the influence of noise.

3.2 Ridge Regression Channel Estimation

To overcome the inherent disadvantage of LS channel estimation，RR channel estimation is proposed and adopted in this paper. Firstly，the frequencydomain signal parameter is expressed as a matrix or vector，that is，the receiving pilot vector

the transmission pilot diagonal matrix X=diag(X1，X2，…，XN)，the channel response vector

and the noise vector

Then Eq.(1)can be rewritten as follows:

According to the principle of RR estimation，the channel response vector can be estimated as follows:

where λ is a non-negative parameter;(·)Hdenotes its conjugate transposed matrix and I is a N-order unit matrix.Moreover，if λ =0，by substituting it into Eq.(4)，it can be reduced and degraded as follows:

which is the matrix expression of LS channel estimation without taking advantage of the noise power.Thus，LS estimation is a special case of RR estimation.

From Eq.(4)，the MSE of RR estimation can be expressed as follows:

where A(k)= σ2n/|X(k)|2.Thus，its MMSE is dependent on the factual channel frequency-domain response and the parameter，λ.

1.2.3 評價標(biāo)準(zhǔn) 跌倒概率范圍從0～100，0表示沒有跌倒風(fēng)險，100表示極有可能跌倒，按照以下組別來劃分級別:低跌倒風(fēng)險(0～35)、中跌倒風(fēng)險(36～57)和(58～100)高跌倒風(fēng)險。跌倒為出現(xiàn)突發(fā)的、不自主和非故意的體位改變而倒在地上或更低的平面上。按照第10版國際疾病分類(ICD-10)對跌倒的分類，跌倒包括以下兩類:①從一個平面至另一個平面的跌落;②同一平面的跌倒。包括跌倒發(fā)生次數(shù)、時間、地點、活動情況、有無先兆癥狀、著地部位和受傷部位等。傷害程度分類:跌倒對患者未造成人體傷害，未做任何處理;跌倒給患者造成人體傷害，有醫(yī)護(hù)治療處理。

The performance of RR estimation depends upon the parameter， λ.What’smore， tomakethe performance of RR estimation better than that of LS estimation，an appropriate value of λ should satisfy the following condition

By substituting Eqs.(3)and(5)into Eq.(6)，it should satisfy that

Considering［A(k)］2/［1+ λA(k)］2＞ 0 and λ ＞ 0，a feasible solution of λ can satisfy the expression as follows:Moreover，we can get

where CSNRk=|X(k)H(k)|2/is the CSNR value of the kth pilot symbol and SNRk=|X(k)|2/is the SNR value of the kth pilot symbol.Then we rewrite Eq.(7)as follows:

We can get from Eq.(8)as follows:

Considering Eqs.(9)and(10)，the solution of λ can be expressed as follows:

Moreover，for any possible CSNR level，to make MSERRless than MSELS，λ should satisfy that

Thus，the proposed RR channel estimation is prior to LS channel estimation in that the former takes the influence of noise into consideration.

4 Simulation and Analysis

4.1 Pilot’s Composition

In this paper，Zadoff-Chu(ZC)sequence is adopted as the pilot symbols.ZC sequence is complex and of constant modulus. Moreover， it has perfect performance of auto correlation and cross correlation［9］.

Then ZC sequence(the pilot symbols)can be expressed as［10］

The indexhasaquadraticinEq. (11)，therefore，ZC sequence is similar to linear frequency modulated(LFM)signal and then it is a kind of Chirp-Like signals［11］.

4.2 Simulation Parameters of SC-FDE System

Thekey simulation parametersofbasebandequivalent SC-FDE system are shown in Table 1.In addition，the parameters of the multi-path frequency selective channel and AWGN channel are shown in Table 2.

Table 1 Simulation parameters of baseband-equivalent SC-FDE system

Table 2 Channel parameters

4.3 Comparision of LS Estimation and RR Estimation

Figs.3 and 4 compare the MMSE performance obtained by employing the current LS channel estimation and the proposed RR channel estimation in the baseband-equivalent SC-FDE system where the time-varying frequency selective channel’s maximum Doppler shift fdis 50 Hz or 100 Hz respectively.It is obviously shown in Figs.3 and 4 that the proposed RR estimation outperforms LS estimation in terms of MMSE performance in that the former utilizes the influence of noise while the latter does not，and that the estimation accuracy decreases with the increase of the maximum Doppler shift fdas expected.Furthermore，at CSNR values less than about 5.0 dB，the MMSE performance has been greatly improved by RR estimation compared with LS estimation.In addition，the advantage of RR estimation is not much too obvious over LS estimation when 5.0 dB≤CSNR≤15.0 dB because the noise at the CSNR values has less influence on the signals than that at CSNR values less than about 5.0 dB.Thus，compared with LS estimation，the proposed RR estimation has better accuracy of estimation especially in low CSNR level where the observed signal is corrupted abnormally by the noise.

Figs.5 and 6 show the comparison of bit error ratio(BER)performance obtained adopting the perfect channel estimation，the current LS channel estimation and the proposed RR channelestimation in the baseband-equivalent SC-FDE system where the timevaryingfrequency selective channel’smaximum Doppler shift fdis 50 Hz or 100 Hz respectively.It is shown in Figs.5 and 6 that the BER performance of RR estimation is better than that of LS estimation but worse than that of the perfect estimation，and that the BER performance getsworse with the increase ofthe maximum Doppler shift fdas expected.In addition，compared to the perfect channel estimation at BER0=10－4，the proposed RR estimation has the loss of about 1 dB while LS estimation has the loss of about 2 dB，that is，RR estimation can get the gains of about 1 dB compared with LS estimation.

Fig.3 Comparison of channel estimation performance in terms of MMSE under fd=50 Hz

Fig.4 Comparison of channel estimation performance in terms of MMSE under fd=150 Hz

Fig.5 Comparison of channel estimation performance in terms of BER under fd=50 Hz

Fig.6 Comparison of channel estimation performance in terms of BER under fd=150 Hz

5 Conclusions

By considering and using the influence of noise，we propose a novel pilot-aided RR channel estimation for SC-FDE communications system on multi-path timevarying frequency selective fading channel.Its performance has been examined and compared with the perfect channel estimation and LS estimation.The numerical results have shown that the new estimation improves the accuracy of estimation especially in low CSNR level and it outperforms LS estimation，and that the performance gets worse when the channel’s maximum Dopplershiftincreases. Moreover， the proposed RR estimation can get the gains of about 1dB compared with LS estimation.

The future researches may involve the following aspects:

1)The maximum Doppler shift’s influence on the performance of channel estimation methods.

2)Designing and adopting better frame structure to improve the performance of channel estimation methods.

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