Modulation Recognition with Frequency Offset and Phase Offset over Multipath Channels

Mingqian Liu ,Zhaoxi Wen ,Yunfei Chen ,Ming Li

1 State Key Laboratory of Integrated Service Networks,Xidian University,Shaanxi,Xi’an 710071,China

2 Department of Engineering,University of Durham,South Road,DH1 3LE,Durham,UK

3 Guilin Changhai Development Co.,Ltd,Guangxi,Guilin 541001,China

*The corresponding author,email: mqliu@mail.xidian.edu.cn

Abstract: Modulation recognition becomes unreliable at low signal-to-noise ratio (SNR) over fading channel.A novel method is proposed to recognize the digital modulated signals with frequency and phase offsets over multi-path fading channels in this paper.This method can overcome the effects of phase offset,Gaussian noise and multi-path fading.To achieve this,firstly,the characteristic parameters search is constructed based on the cyclostationarity of received signals,to overcome the phase offset,Gaussian white noise,and influence caused by multi-path fading.Then,the carrier frequency of the received signal is estimated,and the maximum characteristic parameter is searched around the integer multiple carriers and their vicinities.Finally,the modulation types of the received signal with frequency and phase offsets are classified using decision thresholds.Simulation results demonstrate that the performance of the proposed method is better than the traditional methods when SNR is over 5dB,and that the proposed method is robust to frequency and phase offsets over multipath channels.

Keywords: cyclic characteristics;frequency and phase offset;multi-path channels;modulation recognition

I.INTRODUCTION

The fifth generation (5G) cellular networks are being rolled out across the world[1],and high speed is one of the main functions of 5G networks [2].With the rapid development of science and technology,future wireless networks are expected to support extremely high data rates and numerous users or nodes with various applications and services [3].Non-cooperative communication is a communication method based on not interfering the normal communication between the authorized users,it is a non-authorized opportunistic communication mode.It has been used widely in civilian communication,such as spectrum management [4],communication failure detecting,as well in military communication,for example communication countermeasure.There are many technologies in non-cooperative communication under multi-path channels,such as orthogonal frequency division multiplexing with subcarrier number modulation(OFDMSNM) [5].In practical non-cooperative communication systems,the signal needs to be down converted after estimating the carrier frequency at the receiver.However,because of the error when estimating carrier frequency without any prior information and the oscillator mismatch between the transmitter and receiver,there will be frequency offset and phase offset in the signal that has been down converted at the receiver.As a consequence,researching the recognition method of modulation signal with frequency and phase offset is of great practical engineering significance[6].

Recently,some researchers have worked on digital modulation and recognition with frequency and phase offset,which can be categorized into two routes.One is the researches over Gaussian channel.Qinghua Shietal.proposed a recognition method of quadrature amplitude modulation (QAM) signal by utilizing the amplitude and phase of signal’s characteristic function and the mixed recognition method of four-order cumulant amplitude after signal difference[7],which can eliminate the influence caused by frequency offset when normalized frequency offset is 0.1.However,this method can eliminate the frequency and phase offset influence totally only in high signal-tonoise ratio (SNR).Amy C.Maladyetal.recognized a set of binary phase shift keying (BPSK)/quadrature phase shift keying (QPSK) signals by estimating the signal’s cyclic time moment.The recognition rate of this method reaches 100% when the normalized frequency offset is 0.01 and SNR is 1dB,but this method can only be applied to BPSK/QPSK signals.Luping Zhangetal.proposed a method to recognize a set of QPSK/binary frequency shift keying (2FSK),16 quadrature amplitude modulation(16QAM),binary amplitude shift keying (2ASK) with frequency and phase offset by utilizing the cyclic cumulant of signal envelope.When the SNR is over 6dB,the recognition rate can be higher than 99%,but this method is only applied to different envelope signals,so it is lack of universality.Shinji Oharaetal.recognized the 16/64 quadrature amplitude modulation(64QAM)signal with frequency offset by utilizing the assemblage characteristics of signal’s amplitude and square cosine matrix,but it is invalid to other modulation signals.Bassel F.Beidasetal.proposed a method based on high-order correlation (HOC) to recognize the 32 frequency shift keying (32FSK)/64 frequency shift keying (64FSK) signals,the recognition rate of this method can reach 99%when SNR is 11dB,but it is only applied to frequency shift keying signals[8].

The other is the researches over fading channels.Octavia A.Dobreetal.recognized the 4 quadrature amplitude modulation (4QAM)/16QAM signals with frequency offset by utilizing the assemblage characteristics of fourth-order and sixth-order,and the recognition rate of this method can get 96.2% when SNR is 10dB in fading channels,but this method is not applied to phase shift keying (PSK) signals.Fanggang Wangetal.proposed a method based on Kolmogorov-Smirnov (K-S) test to recognize the MQAM (M=4,16,64) signals with frequency offset and MPSK(M=4,16,64)signals with frequency offset respectively.Over flat fading channel,when SNR is 20dB,the recognition rate of MQAM is over 90%,and the recognition rate of MPSK is over 85%,but the performance of this method is poor at low SNR.Hsiao-Chun Wuetal.proposed a method based on estimating the high-order cumulants of normalized coefficient over fading channel to recognize two sets of signals BPSK,QPSK and 4QAM,16QAM,64QAM respectively[9].When SNR is 10dB,the recognition rate of BPSK,QPSK is less than 90%;when SNR is 30dB,the recognition rate of 4QAM,16QAM,64QAM is less than 80%.Hence,the recognition performance of this method is poor at low SNR.William C.Headlyetal.proposed an asynchronous modulation recognition method to identify amplitude-phase modulated signals over flat fading channels.This method estimates the amplitude,delay and noise power of the signals using moment estimation firstly,and then uses the maximum likelihood algorithm to identify the modulation type.For the Nakagami fading channel,the average recognition rates of BPSK,QPSK,8-ary phase shift keying (8PSK),16QAM and 64QAM signals all reach 90% when SNR is 10dB,but the recognition performance of this method is also poor at low SNR.Qinghua Shietal.utilized suboptimal maximum likelihood algorithm to recognize the MQAM signals,and the recognition rate can get 90% when SNR is 20dB over Rayleigh fading channel,but the recognition performance is poor at low SNR[10].

In this paper,a modulation recognition method is proposed to recognize the digital modulation signal with frequency and phase offset over multi-path fading channel.The main contributions of this paper are summarized as follows:

• The cyclostationarity and cyclic spectrum of the received signal is analyzed in detail,and the fact that the cyclic spectrum density functioncan usually denote the cyclic spectrum density of frequencyfin the spectrum of cyclostationary signalx(t)is proved;

• It is deduced that the recognition character based on cyclic correlationhas the ability of resisting multi-path fading;

• The carrier frequency based on cycle characteristic is estimated,and the range precision of carrier frequency estimation is analyzed in detail.

The remainder of the paper is organized as follows.The cyclostationarity and cyclic spectrum of signal are analyzed in Section II.Section III presents the proposed recognition method based on cyclic correlation.The simulation results and performance analyses are given in Section IV.Finally,the conclusions are given in Section V.

II.CYCLOSTATIONARITY AND CYCLIC SPECTRUM OF DIGITAL MODULATION SIGNAL

Supposingx(t) is a cyclostationary signal with zero mean,its correlation function can be expressed by the Fourier series and the Fourier coefficientas

whereα=m/T0represents the cyclic frequency,is the cyclic autocorrelation function.It can be seen from(1),that the cyclic autocorrelation function is actually at different cyclic frequencyα.The cyclic spectrum density function is the Fourier transform of cyclic autocorrelation function given by

As a result,the cyclic spectrum density functioncan usually denote the cyclic spectrum density value of frequencyfin the spectrum of cyclostationary signalx(t).It can be calculated by utilizing the cross correlation between two frequency componentsf-α/2 andf+α/2,which can be proved as follows.

The spectrum of signal in period[t-T/2,t+T/2]is given by

wherex(u)denotes the cyclic stationary signal.Shifting its spectrum upward and downward byα/2 respectively,the time average cross correlation of them is

let ∆t,T→∞.Then we can get

Thus,the time-averaged cross-correlation of the two signal components obtained by shifting the specific frequency up and down between the cyclic spectrum and the signal instantaneous spectrum are equivalent.The cyclic spectrum is also called the spectral correlation function[11].

In this paper,the digital modulation signal sets are BPSK,QPSK and 8PSK signals,and the cyclic spectrum analysis of these signals are as follows.

The BPSK modulation signal can be expressed by

wherednis the baseband information sequence,ndenotes thenth data bit,Tbstands for the bit rate of binary data.This signal has cyclic spectrum

whereQ(f) stands for the Fourier transform ofg(t),σddenotes the pulse amplitude.Whendnis the unit amplitude,i.e.dn=±1,=1,we substitude it to(8)and obtain

The QPSK modulation signal can be expressed as

wheredI,nanddQ,nrepresent the baseband information bit of I and Q channels,respectively,Tsstands for the bit rate of binary data.WhendI,nanddQ,nare both the unit amplitudes,i.e.,dI,n=dQ,n=±1,=2.The cyclic spectrum of the signal is

It can be seen that the cyclic spectral density value is not equal to zero only whenαis an integer multiple of the signal rate,which indicates that the QPSK signal is only related to the baud rate.

The expression of 8PSK signal can be expressed as

For 8PSK signals,the nonzero spectral correlation density functionexists only whenα=m/T0[12].

III.MODULATION RECOGNITION BASED ON CYCLIC CORRELATION

3.1 Recognition Character Based on Cyclic Correlation

Supposing a base-band received signaly(t),with phase offsetθ0,frequency offsetfc,additive Gaussian white noisen(t)and multi-path fading channel influence,which can be expressed as

whereαlejϕlandtlare the channel response and receiving delay of thelth path,respectively,x(t) is the modulation signals.One has

wherey1(t)is uncorrelated to noisen(t).In addition,the received signal powercan be calculated from the cyclic spectrum character of non-correlated additive signal,which can be expressed as

From (17),it can be seen that the fixed phase offset has been eliminated since correlation operation is taken on the received signals with phase offset.denotes the correlation of noise.The power spectrum of white noise is uncorrelated inα/2 and-α/2(α≠0),so the white Gaussian noiseis 0.As a result,the spectrum character can eliminate the influence caused by white Gaussian noise[13].

For high SNR,n0/2≈0 and we haveso

whereH(f)stands for the transport function of multipath channel [14].The spectrum coherence equation can be gotten by utilizing the spectrum correlation equation,which can be expressed as

and we have

Using(18),(19),(20),(21),and(22),we can obtain

Whenα=0,(23) equals to 1.Despite its antiinterference features,it still cannot recognize different modulated signals,so we assume thatα≠0 in this paper.In conclusion,is free from multi-path interference,which is to say,it has the character of anti multi-path interference.As a consequence,is a valid anti multi-path modulation recognition characteristic parameter in high SNR theoretically[15].

In the low SNR,we have

so we can get

where

Using(25),(26),and(27),we can obtain

Two special cases are discussed as follows.

Whenα=0,(28)can be expressed as

whereγandβare two parameters given by

f(β) is varying withβ.When SNR is larger than 0 dB,≥n0/2,and the change off(β) is trivial.Hence,these characters can hardly be influenced at low SNR.In order to show this more visually,we showof BPSK,QPSK and 8PSK[17].

In Figure 1,we can see that the eigenvalueis free from the influence caused by multi-path,and only influenced by noise.When noise is over 10dB,and the range ofis narrow,the theoretic signal characteristic value tend to steady gradually,which isf(β)→0.So this eigenvalue has the ability of resisting multi-path,which can reflect the differences of modulation signals.

Figure 1.The change of| (f)|over multi-path channel.

According to the analyses on cyclic spectrum correlation function of received signal,the cyclic correlation function has value only in part of cyclic frequencies and power spectrum frequencies,which are all in the nearby of signal carrier frequencies.As a result,carrier frequencies should be estimated in order to get the values on carrier frequencies[18].

3.2 Carrier Frequency Estimation Based on Cycle Characteristic

A subclass of high-order cyclic moment is defined in this paper,and a MPSK signal carrier estimation method is proposed by utilizing the cyclic cumulant educed from this high-order cyclic moment.The MPSK signal cyclic moment is

Whent1=t2=t3=0,we can obtain

It can be derived from characteristics of cyclic cumulant,that the cyclic cumulant equals zero when the order of stationary (or non-stationary) Gaussian white noise is more than two.We haveUsing (34),the fourth-order cyclic cumulant of the MPSK signal can be expressed as

Because BPSK/QPSK signals have two phase or four phase,respectively [19],|exp[j(2θ)]=1| (θis the phase of PSK signal).We can obtain

In conclusion,the fourth-order cyclic cumulant of MPSK signal can be expressed as

The fourth-order cyclic cumulant of MPSK signal is not zero only at the carrier and double carrier positions,that is,the cyclic frequency is carrier frequency or double carrier frequency[20].The steps of the estimation method of carrier frequency using the fourorder cyclic cumulant are as follows:

Step 1: Calculate the fast Fourier transform (FFT)of the received signalr(t),detect the signal spectrum position and determine the search range of cyclic frequencyαas the signal spectrum range;

Step 2: Calculate the four-order cyclic cumulant of received signal.On the cyclic frequency axis,BPSK/QPSK has a peak,while 8PSK has no peak.The condition needs to be considered whether the peak exists inβ-curve;

Step 3: When a peak exists,the carrier frequencies of BPSK/QPSK signals can be estimated by detecting the peak position inβ-curve;

Step 4: When no peak exists,square these signals firstly,and then calculate the fourth order cyclic cumulant of the output signals through high pass filtering.After that,detect the peak position in thecurve.Finally,multiply the cycle frequency corresponding to the peak position by 0.5 to achieve carrier frequency estimation of 8PSK signals[21].

The mean-square error(MSE)curves of carrier frequency estimation for six signals over multipath channels are shown in Figure 2.

Figure 2.Frequency estimation performance with different SNRs.

3.3 Carrier Frequency Estimation Range Analysis

In different ranges,the search range can be obtained from the sampling statistics of the estimation error.Here we havewhereis the estimated carrier frequency,fcdenotes the true value of carrier frequency,nis the range multiple,∆fstands for the product of the root of max MSE andfc,andwhich can be expressed as

After estimating the carrier frequency,the maximum value of the characteristic parameters around the carrier frequency is searched as the recognition feature to identify the received signal,which can overcome the influence of frequency offset to a certain extent and is expressed as

Thus,the identical characteristicC=based on cyclic spectrum correlation can overcome multi-path caused by frequency offset,phase offset and noise.Finally,the different modulation signals can be recognized by the decision threshold.The decision threshold is set as

whereδlimstands for the threshold value of distinguishing adjacent signalsY1andY2.max(ρY1)is the mean value of the maximum value of the characteristic parameter ofY1,and min(ρY2) is the mean value of the minimum value of the characteristic parameter ofY2[22].The procedures of the recognition method of digital modulation signals with frequency and phase offset over multi-path fading channel are summarized in Algorithm 1.

IV.SIMULATION RESULTS AND PERFORMANCE ANALYSES

In order to verify the effectiveness of this new method,computer simulations are performed via MATLAB.Suppose the simulation conditions are: the signal sets to be recognized are BPSK,QPSK,8PSK,16QAM,64QAM,128 quadrature amplitude modulation (128QAM) digital modulation signals,the noise is white Gaussian noise,the channels areandmulti-path channels.The code rate of modulated signal is 3kBaud,carrier frequency is 30kHz,sampling rate is 120kHz,the size of sampling is 512 and the number of Monte Carlo simulation runs is 1000.

Simulations are taken when there is phase offset but no frequency offset to recognize three kinds of digital modulation signals in order to get the correct recognition rate of each signal,i.e.,is the ratio of the time of correct recognition to total time.It can be seen from Figure 3,the recognition rates are all over 90.83%whenSNR≥5dB,which indicates that the proposed modulation recognition method can reach a good performance when there is phase offset[23].

Figure 3.Recognition performance of MPSK signal with phase offset.

Simulations are taken when there is frequency offset but no phase offset to recognize three kinds of digital modulation signals.It can be seen from Figure 4,the recognition rates are all over 89.67% whenSNR≥5dB,which manifests that the proposed method can also reach a good performance with frequency offset.

Figure 4.Recognition performance of MPSK signal with frequency offset.

Simulations are taken under the condition of different frequency offset and phase offset.It can be seen from Figure 5,the recognition rate are all over 86.11%whenSNR≥5dB,which shows that the proposed method is also valid with different frequency offset and phase offset.

Figure 5.Recognition performance of MPSK signal with different frequency offset and phase offset.

Simulations are taken under the condition of having frequency and phase offset over multi-path fading channels for different standard protocols to recognize three kinds of digital modulation signals.It can be seen from Figure 6,the recognition rate are all over 91.22%whenSNR≥5dB,which indicates that the proposed method is robust over different multi-path fading channels for different standard protocols[24].

Figure 6.Recognition performance of MPSK signal over different multipath channels.

Simulations are taken under the condition of same experiment environments,same signal parameter settings and having frequency and phase offset.Figure 7 shows the compare results between the proposed method and three kinds of traditional methods.The proposed method includes intra-class and inter-class recognition,Fanggang Wang’s method includes only intra-class recognition (fading channel),William C.Headley’s method includes intra-class and inter-class recognition and Hsiao-Chun Wu’s method includes only intra-class recognition(fading channel).It can be seen from Figure 7,that the recognition rate of proposed method has been improved evidently compared to three kinds of traditional methods whenSNR≥5dB,when the proposed method is better than the traditional methods[25].

Figure 7.Recognition performance comparison with different methods.

V.CONCLUSION

This paper introduces a novel method to recognize the digital modulation signal with frequency and phase offset over multi-path fading channel,which has important engineering practical significance.First,the cyclostationarity and cyclic spectrum of signal are analyzed in detail.Then,the recognition character based on cyclic correlation is derived,the carrier frequency based on cycle characteristic is estimated and the range precision of carrier frequency estimation is analyzed.Finally,simulation results show that the proposed method can reach a good performance when there are phase offset and frequency offset.And this method is robust over different multi-path fading channels of different standard protocols.In addition,this proposed method is better than the traditional methods,which shows the effectiveness of the method.

ACKNOWLEDGEMENT

This work was supported by the National Natural Science Foundation of China under Grant 62071364 and 62231027,in part by the Key Research and Development Program of Shaanxi under Grant 2023-YBGY-249,in part by the Key Research and Development Program of Guangxi under Grant 2022AB46002,and in part by the Fundamental Research Funds for the Central Universities under Grant KYFZ23001.