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A continuous phase modulation single-carrier wireless system with frequency domain equalization

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This paper presents a continuous phase modulation (CPM) based single-carrier frequency-domain equalization (CPM-SC-FDE) structure for broadband wireless communication systems. The proposed structure combines the advantages of the frequency diversity
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  1 A Continuous Phase Modulation Single-CarrierWireless System With Frequency DomainEqualization Emad S. Hassan 1 , Xu Zhu 2 ,  Member, IEEE  , Said E. El-Khamy 3 ,  Fellow, IEEE  , Moawad I. Dessouky 1 , Sami A.El-Dolil 1 , Fathi E.Abd El-Samie 1  Abstract —This paper presents a continuous phase modula-tion (CPM) based single-carrier frequency-domain equalization(CPM-SC-FDE) structure for broadband wireless communicationsystems. The proposed structure combines the advantages of the frequency diversity and low complexity of SC-FDE andthe energy efficiency of CPM. Simulation results show that, theproposed CPM-SC-FDE structure provides a better performancethan conventional SC-FDE and the CPM based orthogonalfrequency division multiplexing (OFDM) systems, by exploitingthe channel frequency diversity, efficiently. A properly chosenmodulation index can achieve an efficient utilization of themultipath diversity, while maintaining high bandwidth efficiency.The performance analysis of the proposed structure is alsopresented in the paper.  Index Terms —Single carrier (SC), Continuous phase modula-tion (CPM), Frequency domain equalization (FDE), OFDM. I. I NTRODUCTION Future wireless communications are required to supporthigh-speed and high-quality multimedia transmission. How-ever, there exist challenges due to the data transmission overfrequency selective channels. The SC-FDE system has provedto be one of the most promising approaches for data transmis-sion over a severe frequency-selective environment due to itseffectiveness and low complexity [1]–[5]. Compared to orthog- onal frequency-division multiplexing (OFDM) system [6], [7],the SC-FDE system has a lower peak-to-average power ratio(PAPR), less sensitivity to frequency synchronization errors,and a higher frequency diversity gain when a relatively highchannel coding rate is used [3].CPM [8]–[11] is widely used in wireless communication systems, because of its constant envelope, which is neededfor power efficient transmitters, and its ability to increase thediversity of the multipath channel, which is needed to improvethe BER performance. In [8] and [10], CPM was applied in OFDM systems to solve the PAPR problem. In [12], a newlow-complexity linear FDE approach for CPM signals wasdeveloped. Novel equalization algorithms in frequency-domainfor CPM are illustrated in [13]. In [14], a new interleaving scheme for CPM systems using chaotic maps was presented. 1 Dept. of Electronics and Electrical Comm., Faculty of Electronic Eng.,Menoufia University, 32952, EGYPT. 2 Dept. of Electrical Engineering and Electronics, The University of Liver-pool, Liverpool L69 3GJ, U.K. 3 Dept. of Electrical Eng., Faculty of Engineering, Alexandria University,Alexandria 21544, EGYPT. In this paper, we propose a CPM-SC-FDE structure forbroadband wireless communication systems. The proposedstructure combines the key characteristics of CPM and SC-FDE systems to produce a constant envelope SC-FDE wave-form. Thus, this new structure has the advantages of exploitingthe channel frequency diversity and the low complexity of SC-FDE and the energy efficiency of CPM. Simulation resultsshow that, the proposed CPM-SC-FDE structure provides abetter performance than the conventional SC-FDE system andthe CPM-OFDM system, by the efficient utilization of thechannel frequency diversity. A properly chosen modulationindex demonstrates a significant utilization of the multipathdiversity when compared to the case with a single path.The rest of the paper is organized as follows. SectionII, presents the proposed CPM-SC-FDE system model. Anexplanation of the FDE process is presented in section III.Section IV, describes the phase demodulator. The performanceanalysis of the CPM-SC-FDE system in frequency-selectivechannels is considered in Section V. Section VI provides thesimulation results and the discussion. Finally, Section VIIprovides some concluding remarks.II. CPM-SC-FDE SYSTEM MODELIn this section, the model of the proposed CPM-SC-FDEsystem is presented. The block diagram of this system isillustrated in Fig. 1. A block length of   N   symbols is assumedwith  x ( n )  ( n  = 0 , 1 ,...,N   −  1 ), representing the data se-quence after symbol mapping. During each  T  − second symbolinterval,  x ( n )  passes through a phase modulator (PM) to getthe constant envelope sequence,  s ( n ) . After PM, a cyclic prefix(CP) is added at the beginning of each data block to mitigatethe inter-block interference (IBI), which is assumed to havea longer length than the channel impulse response. Then, theCPM-SC-FDE signal,  s ( t )  is generated at the output of thedigital-to-analog (D/A) converter. According to [15], [16],  s ( t ) can be written as: s ( t ) =  Ae jφ ( t ) =  Ae j [2 πhm ( t )+ θ ] (1)where  A  is the signal amplitude,  h  is the modulation index, θ  is an arbitrary phase used to achieve continuous phasemodulation [8], and  m ( t )  is the real-valued message signalgiven by: m ( t ) =  C  K  − 1  k =0 I  k q  k ( t )  (2) 978-1-4244-584 4-8  /09/$26.00 ©2009 IEEE  599 Authorized licensed use limited to: Akademia Gorniczo-Hutnicza. Downloaded on July 31,2010 at 23:02:24 UTC from IEEE Xplore. Restrictions apply.  Fig. 1. Block diagram of the CPM-SC-FDE system. where  I  k  are the  M  − ary real-valued data symbols, whichhave an average energy of   E  I  ,  M   is the number of constel-lation points,  q  k ( t )  are the orthogonal subcarriers. The real-valued data symbols,  I  k  can be written as [10]: I  k  =  ℜ{ x ( n ) } , n ≤ N/ 2 −ℑ{ x ( n − N/ 2) } , n > N/ 2 (3)where ℜ{ x ( n ) } and ℑ{ x ( n − N/ 2) } are the real and imaginaryparts of   x ( n ) , respectively.In (2),  C   is a normalizing constant used to normalize thevariance of the message signal,  m ( t ) , and consequently thevariance of the phase signal,  σ 2 φ  = (2 πh ) 2 . This requirementis achieved by setting  C   as: C   =    2 N E  I  (4)The transmitted signal  s ( t )  is then passed through themultipath channel. The channel impulse response is modeledas a wide-sense stationary uncorrelated scattering (WSSUS)process consisting of   L  discrete paths: h ( t ) = L − 1  l =0 h [ l ] δ  ( t − τ  l )  (5)where  h [ l ]  and  τ  l  are the channel gain and delay of the  lth path, respectively. The continuous-time received signal  r ( t ) ,shown in Fig. 1, is expressed as: r ( t ) = L − 1  l =0 h [ l ] s ( t − τ  l ) + n ( t )  (6)where  n ( t )  is a complex additive white Gaussian noise(AWGN) with single-sided power spectral density  N  0 .Let  J   denote the oversampling factor. There are  N  DFT   = JN   samples per block. The output of the analog-to-digital(A/D) converter is sampled at  t  =  iT/ ( JN  ) , and the CPis discarded. The  ith  ( i  = 0 , 1 ,...,JN   − 1)  sample of thereceived signal  r ( t )  is given by r ( i ) = LJ  − 1  n =0 h ( n ) s ( i − n ) + n ( i )  (7)Defining  N  DFT   =  JN  , the received signals  r ( i )  aretransferred into the frequency domain by using the  N  DFT  -point discrete Fourier transform (DFT). The received signalon the  mth  ( m  = 0 , 1 ,...,N  DFT   − 1)  subcarrier is given by R ( m ) =  H  ( m ) S  ( m ) + N  ( m )  (8)where  R ( m ) ,  H  ( m ) ,  S  ( m ) , and  N  ( m )  are the  N  DFT  -pointDFTs of   r ( i ) ,  h ( i ) ,  s ( i ) , and  n ( i ) , respectively.III. EQUALIZER DESIGNIn this section, the design of SC-FDE is described. Asshown in Fig. 1, the received signals are equalized in thefrequency-domain after the DFT block. The equalized signalsare then transferred back into the time domain by using inverseDFT (IDFT). Letting  W  ( m ) ( m  = 0 , 1 ,...,N  DFT  − 1)  denotethe equalizer coefficient for the  mth  subcarrier, the time-domain equalized signal   s [ i ] , which is the soft estimate of  s ( n ) , can be expressed as:  s ( n ) = 1 N  DFT N  DFT  − 1  m =0 W  ( m ) R ( m ) e j 2 πmn/N  DFT  (9)In this paper, we consider the minimum mean square error(MMSE) based SC-FDE. The equalizer coefficients  W  ( m )  aredetermined to minimize the mean square error between theequalized signal   s ( n )  and the srcinal signal  s ( n ) ,  i.e., ∆ = E  |  s ( n ) − s ( n ) | 2 . For the MMSE equalizer,  W  ( m )  is given by: W  ( m ) =  H  ∗ ( m ) | H  ( m ) | 2 + N  0 /A 2  (10)where  ( . ) ∗ denotes complex conjugate. Substituting (8)and (10) into (9) yields  s ( n ) = 1 N  DFT N  DFT  − 1  m =0 | H  ( m ) | 2 S  ( m ) | H  ( m ) | 2 + N  0 /A 2 e j 2 πmn/N  DFT              signal + 1 N  DFT N  DFT  − 1  m =0 H  ∗ ( m ) N  ( m ) | H  ( m ) | 2 + N  0 /A 2 e j 2 πmn/N  DFT              noise(11)The equalized signals are then passed to the phase demod-ulator to obtain the estimate of the data symbol  x ( n ) .IV. PHASE DEMODULATORIn this section, we present the design of the phase modula-tor, which follows the FDE. It is illustrated in Fig. 2. 600 Authorized licensed use limited to: Akademia Gorniczo-Hutnicza. Downloaded on July 31,2010 at 23:02:24 UTC from IEEE Xplore. Restrictions apply.  Fig. 2. Phase demodulator at the receiver of CPM-SC-FDE system. First, a finite impulse response (FIR) filter is applied toremove the out-of-band noise. The filter is designed by usingthe window technique [17], using a window length of   L f   anda normalized cutoff frequency of   f  nor  (0  < f  nor  ≤  1) . TheFIR filter impulse response  g [ n ]  is given by: g [ n ] =sin  2 πf  nor  n −  L f  − 12  π  n −  L f  − 12   ,  0 ≤ n ≤ L f   − 1  (12)In (12), if   n  = ( L f   − 1) / 2 , g [ n ] = 2 πf  nor /π , the outputof the FIR filter can be expressed as: f  [ i ] = L f  − 1  n =0 g [ n ]  s [ i − n ]  (13)The phase of the filtered signal  f  [ i ]  is obtained as: ϕ [ i ] =  arg ( f  [ i ]) =  φ [ i ] + δ  [ i ]  (14)where  φ [ i ]  denotes the phase of the desired signal, and  δ  [ i ] denotes the phase noise. Then the phase unwrapper is usedto minimize the effect of phase ambiguities and make thereceiver insensitive to phase offsets caused by the channel andthe memory term.Finally, a bank of   N   matched filters is used to obtain thesoft estimates of the data symbols  x ( n ) ( n  = 0 ,...,N   − 1) .Each filter is of length  N  DFT  , and the  kth  filter is matched to q  k [ n ] ( n  = 0 ,...,N  DFT  − 1) , the discrete-time representationof   q  k ( t ) . Let  ϕ [ i ]  represent the output of the phase unwrapper,the soft estimate of   x ( n )  is expressed as:  x ( n ) = N  DFT  − 1  i =0 ϕ [ i ] q  k [ i ]  (15)  x ( n )  is then passed through a decision device to obtain thehard estimate of   x ( n ) .V. PERFORMANCE ANALYSISAs indicated by the block diagram shown in Fig. 1, theCPM-SC-FDE demodulator operates in the discrete-time do-main. It is convenient however, to consider the continuous-timemodel for the analysis [10].  A. Multipath Diversity In this subsection, we study the multipath diversity of thephase modulation based SC-FDE system. This property is seenby viewing the Taylor series expansion of the CPM-SC-FDEsignal described in (1) when  θ  = 0 , s ( t ) =  Ae j 2 πhm ( t ) =  A ∞  n =0  (  j 2 πh ) n n !  m n ( t )  (16)The phase modulator mixes and spreads the data symbols inthe frequency domain, which gives the CPM-SC-FDE systemthe potential to exploit the frequency diversity of the channel.This is not necessarily the case, however. For small values of the modulation index, where only the first two terms in (16)contribute to the expansion, that is: s ( t ) =  Ae jφ ( t ) =  A [1 +  j 2 πhm ( t )]  (17)The CPM-SC-FDE signal does not have the frequency spread-ing given by the higher-order terms. In this case, the CPM-SC-FDE signal is essentially equivalent to a conventional SC-FDEsignal, and therefore does not have the ability to exploit thefrequency diversity of the channel.  B. BER Analysis In this subsection, we analyze the BER performance of theproposed CPM-SC-FDE structure. For simplicity, we assumethat the FIR filter impulse response is  g ( t )  = 1, and nophase unwrapper is needed. Assuming high signal-to-noiseratio (SNR), the continuous-time form of (11) is approximatedby:  s ( t ) ≈  12 π    N/T  0 S  ( f  ) e j 2 πft df  +12 π    N /T  0 H  − 1 ( f  ) N  ( f  ) e j 2 πft df  =  s ( t ) + n r ( t ) (18)where  n r ( t ) =  12 π   N/T  0  H  − 1 ( f  ) N  ( f  ) e j 2 πft df  , representsthe noise at the FDE output.Using (14), the phase of the FDE output signal   s ( t )  can beexpressed as: ϕ ( t ) =  φ ( t ) + δ  ( t ) =  θ  + 2 πhm ( t ) + δ  ( t )  (19)Assuming high SNR,  δ  ( t )  can be approximated by whiteGaussian noise [8] with zero mean and variance σ 2 δ  ≈  βN  0 A 2  (20) 601 Authorized licensed use limited to: Akademia Gorniczo-Hutnicza. Downloaded on July 31,2010 at 23:02:24 UTC from IEEE Xplore. Restrictions apply.  where β   = 12 π    N/T  0 | H  [ f  ] | − 2 df   (21)As explained in Section IV, the output of phase demodulatoris processed by the matched filters. Assume a memorylessmodulation,  i.e.,θ  = 0 . The soft estimate of   x ( n )  in (15) canbe rewritten as:  x ( n ) =    NT  0 ϕ ( t ) q  k ( t ) dt =    NT  0 φ ( t ) q  k ( t ) dt +    NT  0 δ  ( t ) q  k ( t ) dt =  S  k  + N  k (22)Using (2) and (19), the signal component  S  k  in (22) can bewritten as: S  k  =    NT  0 φ ( t ) q  k ( t ) dt = 2 πhC     NT  0 N  − 1  n =0 [ I  n q  n ( t )] q  k ( t ) dt (23)Due to the orthogonality of the subcarriers, only the  n  =  k term in (23) contributes, and therefore the signal componentsimplifies to: S  k  =  C  1 × I  k  (24)where  C  1  =  πhC   =  πh    2 NE  I .The noise component  N  k  in (22) is given by: N  k  =    NT  0 δ  ( t ) q  k ( t ) dt  (25)For high SNR, the noise  N  k  can be approximated by AWGNwith zero mean and variance σ 2 N  k = 12 C  21 σ 2 δ  =  KE  I  4 π 2 h 2 βN  0 A 2  (26)Using (21) and (26), the BER with a given channel can beapproximated by: BER  =  Q    E  I  σ 2 N  k   =  Q  2 πh    A 2 /N  0 N  2 π   N/T  0  | H  [ f  ] | − 2 df   (27)By using the discrete-time representation of   β  , the BER in (27)can be written as: BER  =  Q  2 πh    A 2 /N  01 J   JN  − 1 m =0  | H  [ m ] | − 2   (28)This implies that the BER decreases with the increase of themodulation index  h , and the increase of the SNR  ( A 2 /N  0 ) .The BER is independent on  E  I  , the energy of symbol  I  k .Note that the above analysis assumes high SNR, where theMMSE based FDE can be approximated by the zero-forcing(ZF) based FDE. At a low SNR, the MMSE based FDE isexpected to provide a slightly better performance than theresult given by (28), due to its enhanced ability to suppressthe noise.VI. SIMULATION RESULTSIn this section we use simulations to demonstrate the perfor-mance of the proposed CPM-SC-FDE system. We use  4 − ary ( M   = 4)  pulse amplitude modulation (PAM), with a data rateof 2  Mbps . Each block contains  N   = 64  symbols, and eachsymbol is sampled  J   = 8  times. We employ a channel modelfollowing the exponential delay profile [10] with root meansquared (RMS) delay spread  τ  rms  = 2 µs  except in Fig. 5.The channel is assumed to be known perfectly at the receiver.The SNR is defined as the average ratio between the receivedsignal power and the noise power. It is given by SNR =  A 2 /N  0 .The FIR filter has a length of   L f   = 11  and a normalized cutoff frequency of   f  n  = 0 . 2  [10].Fig. 3 shows the effect of the modulation index on theperformance of the proposed CPM-SC-FDE system, at a fixedSNR = 20 dB, for both single path and multipath cases.As discussed in Subsection V-B, the BER of CPM-SC-FDEsystem decreases with the increase of the modulation index  h .In multipath channels, the CPM-SC-FDE system outperformsthe single path case for  2 πh >  0 . 4 , which verifies the analysisin Subsection V-A that, the multipath diversity can be achievedwith a relatively large modulation index. On the other hand, asshown in [18], the signal bandwidth grows with  2 πh , wherethe useful bandwidth expression for  M  − ary CPM signalis  BW   = max(2 πh, 1) /T Hz , which in turn reduces thespectral efficiency. Since the bit rate is  R  = (log 2 M  ) /T bps ,the spectral efficiency,  SE   can be expressed as: SE   =  RBW   = log 2 M  max(2 πh, 1)  bps/Hz  (29)Based on the performance shown in Fig. 3 and the bandwidthefficiency given by (29), it can be deduced that a moderatemodulation index achieves a significant frequency diversity,while maintaining a high bandwidth efficiency. In the follow-ing results, we use  2 πh  = 1 to make a good trade-off betweenperformance and spectral efficiency. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210  6 10  5 10  4 10  3 10  2 10  1 10 0 2 π h    B   i   t   E  r  r  o  r   R  a   t  e MultipathSingle path Fig. 3. Impact of the modulation index on the performance of the CPM-SC-FDE system for both single path and multipath cases using  4 − PAM at SNR= 20 dB. 602 Authorized licensed use limited to: Akademia Gorniczo-Hutnicza. Downloaded on July 31,2010 at 23:02:24 UTC from IEEE Xplore. Restrictions apply.  Fig. 4 shows the performance comparison between theproposed CPM-SC-FDE system, the CE-OFDM system de-scribed in [10], the conventional QPSK-OFDM system, andthe conventional SC-FDE system. It is clear that, the proposedCPM-SC-FDE system significantly outperforms all the othersystems by the best utilization of the channel frequencydiversity. For example, at a BER  = 10 − 3 , the CPM-SC-FDE system provides SNR gains of 5.2 dB, 8 dB, and 12dB over CE-OFDM system [10], the conventional SC-FDEsystem [1], and the conventional OFDM system, respectively.Fig. 4 also demonstrates a good match between the numericalresults of the CPM-SC-FDE system and the analytical resultsgiven by (28), especially at high SNRs. As discussed inSubsection V-B, the simulation of the FDE is based on theMMSE criterion, while the analysis is literally based on theZF criterion, which may enhance the noise power at low SNRs.Therefore, there is a slight performance gain of the numericalresults over its analytical counterpart at low SNRs. 0 5 10 15 20 25 30 35 4010  4 10  3 10  2 10  1 10 0 Eb/No [dB]    B   E   R OFDMSC FDE [1]CE OFDM [10]Proposed CPM SC FDE Analytical CPM SC FDE Fig. 4. BER performance of CPM-SC-FDE, CE-OFDM, SC-FDE, andOFDM systems. Fig. 5 shows the performance of the proposed CPM-SC-FDE system, the CE-OFDM system [10], the conventionalOFDM system, and the conventional SC-FDE system [1], interms of the BER versus the RMS delay spread (which isnormalized to the symbol period), at a fixed SNR = 20 dB.As shown from this figure, in flat fading  ( i.e.,τ  rms  = 0) , theperformance of the CE-OFDM system and the CPM-SC-FDEsystem converge with a small performance loss compared toconventional OFDM and SC-FDE systems, due to the effect of phase demodulator threshold. In frequency-selective channels ( τ  rms  >  0) , however, the CPM-SC-FDE system achieves asignificant performance gain over the other systems.VII. CONCLUSIONIn this paper, we proposed a CPM-SC-FDE structure forbroadband wireless communication systems, which not onlyhas a low complexity at the receiver, but also achieves anefficient utilization of the frequency diversity in frequency-selective channels. The obtained results show a noticeable 0 0 2 0 4 0 6 0 8 1 1 2 1 4 1 6 1 8 210  4 10  3 10  2 10  1 10 0 Normaized RMS delay    B   i   t   E  r  r  o  r   R  a   t  e OFDMSC FDE [1]CE OFDM [10]CPM SC FDE Fig. 5. Effect of RMS delay on the performance of CPM-SC-FDE, CE-OFDM, SC-FDE, and OFDM systems at SNR = 20 dB. performance improvement of the proposed CPM-SC-FDE sys-tem over the conventional OFDM, the conventional SC-FDE,and the CE-OFDM systems, especially with high RMS delayspreads. It has also been shown that, a moderate modulationindex achieves an efficient utilization of channel frequencydiversity, while maintaining high bandwidth efficiency.R EFERENCES[1] D. Falconer, S. Ariyavisitakul, A. Benyamin-Seeyar, and B. Eidson,“Frequency domain equalization for single-carrier broadband wirelesssystems,”  IEEE Commun. Mag. , vol. 40, no. 4, April 2002.[2] A. Gusmo, R. Dinis, and N. 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Vitetta, “Equalization algorithms in the frequencydomain for continuous phase modulations,”  IEEE Transactions onCommun. , vol. 54, no. 4, 2006. 603 Authorized licensed use limited to: Akademia Gorniczo-Hutnicza. Downloaded on July 31,2010 at 23:02:24 UTC from IEEE Xplore. Restrictions apply.
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