1
A Continuous Phase Modulation SingleCarrierWireless System With Frequency DomainEqualization
Emad S. Hassan
1
, Xu Zhu
2
,
Member, IEEE
, Said E. ElKhamy
3
,
Fellow, IEEE
, Moawad I. Dessouky
1
, Sami A.ElDolil
1
, Fathi E.Abd ElSamie
1
Abstract
—This paper presents a continuous phase modulation (CPM) based singlecarrier frequencydomain equalization(CPMSCFDE) structure for broadband wireless communicationsystems. The proposed structure combines the advantages of the frequency diversity and low complexity of SCFDE andthe energy efﬁciency of CPM. Simulation results show that, theproposed CPMSCFDE structure provides a better performancethan conventional SCFDE 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 modulation (CPM), Frequency domain equalization (FDE), OFDM.
I. I
NTRODUCTION
Future wireless communications are required to supporthighspeed and highquality multimedia transmission. However, there exist challenges due to the data transmission overfrequency selective channels. The SCFDE system has provedto be one of the most promising approaches for data transmission over a severe frequencyselective environment due to itseffectiveness and low complexity [1]–[5]. Compared to orthog
onal frequencydivision multiplexing (OFDM) system [6], [7],the SCFDE system has a lower peaktoaverage 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 newlowcomplexity linear FDE approach for CPM signals wasdeveloped. Novel equalization algorithms in frequencydomainfor 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 Liverpool, Liverpool L69 3GJ, U.K.
3
Dept. of Electrical Eng., Faculty of Engineering, Alexandria University,Alexandria 21544, EGYPT.
In this paper, we propose a CPMSCFDE structure forbroadband wireless communication systems. The proposedstructure combines the key characteristics of CPM and SCFDE systems to produce a constant envelope SCFDE waveform. Thus, this new structure has the advantages of exploitingthe channel frequency diversity and the low complexity of SCFDE and the energy efﬁciency of CPM. Simulation resultsshow that, the proposed CPMSCFDE structure provides abetter performance than the conventional SCFDE system andthe CPMOFDM 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 CPMSCFDE system model. Anexplanation of the FDE process is presented in section III.Section IV, describes the phase demodulator. The performanceanalysis of the CPMSCFDE system in frequencyselectivechannels is considered in Section V. Section VI provides thesimulation results and the discussion. Finally, Section VIIprovides some concluding remarks.II. CPMSCFDE SYSTEM MODELIn this section, the model of the proposed CPMSCFDEsystem 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 sequence 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 interblock interference (IBI), which is assumed to havea longer length than the channel impulse response. Then, theCPMSCFDE signal,
s
(
t
)
is generated at the output of thedigitaltoanalog (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 realvalued message signalgiven by:
m
(
t
) =
C
K
−
1
k
=0
I
k
q
k
(
t
)
(2)
97814244584
48
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Fig. 1. Block diagram of the CPMSCFDE system.
where
I
k
are the
M
−
ary realvalued data symbols, whichhave an average energy of
E
I
,
M
is the number of constellation points,
q
k
(
t
)
are the orthogonal subcarriers. The realvalued 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 widesense 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 continuoustime 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 singlesided power spectral density
N
0
.Let
J
denote the oversampling factor. There are
N
DFT
=
JN
samples per block. The output of the analogtodigital(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 SCFDE is described. Asshown in Fig. 1, the received signals are equalized in thefrequencydomain 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 timedomain 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 SCFDE. 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 demodulator to obtain the estimate of the data symbol
x
(
n
)
.IV. PHASE DEMODULATORIn this section, we present the design of the phase modulator, which follows the FDE. It is illustrated in Fig. 2.
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Fig. 2. Phase demodulator at the receiver of CPMSCFDE system.
First, a ﬁnite impulse response (FIR) ﬁlter is applied toremove the outofband 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 discretetime 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, theCPMSCFDE demodulator operates in the discretetime domain. It is convenient however, to consider the continuoustimemodel for the analysis [10].
A. Multipath Diversity
In this subsection, we study the multipath diversity of thephase modulation based SCFDE system. This property is seenby viewing the Taylor series expansion of the CPMSCFDEsignal 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 CPMSCFDE 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 CPMSCFDE signal does not have the frequency spreading given by the higherorder terms. In this case, the CPMSCFDE signal is essentially equivalent to a conventional SCFDEsignal, 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 CPMSCFDE structure. For simplicity, we assumethat the FIR ﬁlter impulse response is
g
(
t
)
= 1, and nophase unwrapper is needed. Assuming high signaltonoiseratio (SNR), the continuoustime 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)
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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 discretetime 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 zeroforcing(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 performance of the proposed CPMSCFDE 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 CPMSCFDE system, at a ﬁxedSNR = 20 dB, for both single path and multipath cases.As discussed in Subsection VB, the BER of CPMSCFDEsystem decreases with the increase of the modulation index
h
.In multipath channels, the CPMSCFDE system outperformsthe single path case for
2
πh >
0
.
4
, which veriﬁes the analysisin Subsection VA 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 following results, we use
2
πh
= 1 to make a good tradeoff 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 CPMSCFDE system for both single path and multipath cases using
4
−
PAM at SNR= 20 dB.
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Fig. 4 shows the performance comparison between theproposed CPMSCFDE system, the CEOFDM system described in [10], the conventional QPSKOFDM system, andthe conventional SCFDE system. It is clear that, the proposedCPMSCFDE system signiﬁcantly outperforms all the othersystems by the best utilization of the channel frequencydiversity. For example, at a BER
= 10
−
3
, the CPMSCFDE system provides SNR gains of 5.2 dB, 8 dB, and 12dB over CEOFDM system [10], the conventional SCFDEsystem [1], and the conventional OFDM system, respectively.Fig. 4 also demonstrates a good match between the numericalresults of the CPMSCFDE system and the analytical resultsgiven by (28), especially at high SNRs. As discussed inSubsection VB, 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 CPMSCFDE, CEOFDM, SCFDE, andOFDM systems.
Fig. 5 shows the performance of the proposed CPMSCFDE system, the CEOFDM system [10], the conventionalOFDM system, and the conventional SCFDE 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 CEOFDM system and the CPMSCFDEsystem converge with a small performance loss compared toconventional OFDM and SCFDE systems, due to the effect of phase demodulator threshold. In frequencyselective channels
(
τ
rms
>
0)
, however, the CPMSCFDE system achieves asigniﬁcant performance gain over the other systems.VII. CONCLUSIONIn this paper, we proposed a CPMSCFDE structure forbroadband wireless communication systems, which not onlyhas a low complexity at the receiver, but also achieves anefﬁcient utilization of the frequency diversity in frequencyselective 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 CPMSCFDE, CEOFDM, SCFDE, and OFDM systems at SNR = 20 dB.
performance improvement of the proposed CPMSCFDE system over the conventional OFDM, the conventional SCFDE,and the CEOFDM 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
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