1

Timing, Carrier, and Frame Synchronization ofBurst-Mode CPM

Ehsan Hosseini, Student Member, IEEE, and Erik Perrins, Senior Member, IEEE

Abstract—In this paper, we propose a complete synchro-nization algorithm for continuous phase modulation (CPM)signals in burst-mode transmission over additive white Gaus-sian noise (AWGN) channels. The timing and carrier recoveryare performed through a data-aided (DA) maximum likelihoodalgorithm, which jointly estimates symbol timing, carrier phase,and frequency offsets based on an optimized synchronizationpreamble. Our algorithm estimates the frequency offset via aone-dimensional grid search, after which symbol timing andcarrier phase are computed via simple closed-form expressions.The mean-square error (MSE) of the algorithm’s estimatesreveals that it performs very close to the theoretical Cramer-Rao bound (CRB) for various CPMs at signal-to-noise ratios(SNRs) as low as 0 dB. Furthermore, we present a framesynchronization algorithm that detects the arrival of bursts andestimates the start-of-signal. We simulate the performance ofthe frame synchronization algorithm along with the timing andcarrier recovery algorithm. The bit error rate results demonstratenear ideal synchronization performance for low SNRs and shortpreambles.

Index Terms—Continuous Phase Modulation, Estimation The-ory, Synchronization.

I. INTRODUCTION

CONTINUOUS phase modulation (CPM) [1] is ahighly bandwidth and power-efficient digital transmis-

sion scheme, which allows designers to employ non-linearpower amplifiers. It has been an attractive choice for time-division multiple-access (TDMA) networks where data orvoice is transmitted in a burst-mode fashion. Examples of suchsystems are the well-known GSM cellular standard [2] and thenext generation aeronautical telemetry standard [3]. Despitethe attractive features of CPM, the receiver complexity is highdue to the inherent memory of the modulation, and it requiresmaximum likelihood sequence detection (MLSD) for the bestperformance [4].

Another source of receiver complexity is the synchroniza-tion task, especially in burst-mode transmissions where the

This paper will be presented in part at the IEEE Global TelecommunicationsConference, Atlanta, Georgia, USA, December 2013.

E. Hosseini and E. Perrins are with the Department of Electrical Engineer-ing and Computer Science, University of Kansas, Lawrence, KS 66045 USA(e-mail: ehsan@ku.edu, esp@ieee.org).

Copyright 2013 IEEE. Personal use of this material is permitted. However,permission to reprint/republish this material for advertising or promotionalpurposes or for creating new collective works for resale or redistribution toservers or lists, or to reuse any copyrighted component of this work in otherworks must be obtained from the IEEE.

Published in: Ehsan Hosseini and Erik Perrins, “Timing, Carrier, and FrameSynchronization of Burst-Mode CPM,” IEEE Transactions on Communica-tions, vol.61, no.12, pp.5125-5138, December 2013

DOI: 10.1109/TCOMM.2013.111613.130667URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=

6678035&isnumber=6689285

warm-up or acquisition time must be kept as small as possible.This task has become even more challenging due to theintroduction of powerful error correction codes such as low-density parity check (LDPC) codes, which require accuratesynchronization at low signal-to-noise ratios (SNRs) in orderto achieve the full coding gain. Feedforward synchronization isa common approach in this type of application since it requiresa shorter acquisition time compared to closed-loop methods[5]. Moreover, a known synchronization preamble is usuallyappended to the beginning of each burst, which assists thesynchronization via data-aided (DA) algorithms.

The majority of works on synchronization of CPM in burst-mode transmissions have addressed minimum-shift keying(MSK)-type modulations using non-data-aided (NDA) algo-rithms, e.g. [6]–[8]. In addition to their limited application,these methods do not perform as well as DA algorithms inlow SNRs. Huang et al. [9] have proposed a feedforward DAjoint symbol timing and frequency offset estimation algorithmfor Gaussian MSK (GMSK) signals. The performance ofthis ad-hoc method relies on the amount of frequency offsetand sample timing error. A feedforward NDA symbol timingestimation is presented in [10], which can work for generalCPMs in principle. However, its performance degrades incase of partial response schemes. A few DA synchronizationalgorithms have been presented in the literature for generalCPM signals in different environments [11]–[14]. Huber andLiu [11] proposed a maximum likelihood (ML) joint tim-ing and phase synchronization algorithm for additive whiteGaussian noise (AWGN) channels. In a related work [12], theWalsh transform is used in order to derive the synchronizationalgorithm. Both of these algorithms assume the timing offsetis much smaller than the symbol duration in order to functionproperly. This limits their application in burst-mode feedfor-ward receivers as the timing offset in practice can have anyarbitrary value. Another DA joint phase and timing estimationalgorithm is proposed in [13], which is based on the minimummean-square error (MMSE) and Kalman filter criteria. Despiteits robustness in time-variant channels and short preambles,this method is implemented in a closed-loop manner whichrequires multiple initialization steps. Moreover, its mean-square error (MSE) is shown to be significantly larger thanthe Cramer-Rao bound (CRB) even at high SNRs. AnotherDA algorithm is proposed in [14] for space-time coded CPMover Rayleigh channels, which only tackles the symbol timingestimation. One important issue with all the aforementionedDA algorithms is that the carrier frequency offset has notbeen taken into account. Blind frequency estimators suchas [15], [16] can be employed prior to symbol timing andphase estimations. However, the accuracy of these frequency

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estimators is far above the CRB [15] especially in low tomoderate SNRs [16]. Residual frequency offsets result in poortiming and phase estimators as well as signal demodulation.

Another challenge in synchronization of burst-mode signalsis estimation of the burst start point, i.e. start-of-signal (SoS).This task, which will be referred to as frame synchronization,is crucial in DA algorithms where the boundaries of the knownpreamble have to be identified. Several sophisticated framesynchronization algorithms [17]–[19] have been proposed forphase shift-keying (PSK) signals in AWGN where frequencyoffset is present. The performance of the algorithm in [17]depends on the amount of frequency offset, which has to bemuch smaller than the symbol rate. Choi and Lee [18] haveassumed continuous transmissions, where the preamble is sur-rounded by random data. Although burst-mode transmissionis introduced in [19], the authors have assumed there is noguard interval between bursts and the preamble is precededby random data (similar to the continuous mode). Moreover, itassumes the tentative location of the preamble is known withinan uncertainty window. Such a knowledge might not be alwaysavailable particularly when the receiver is just powered on.

In this work, we present a feedforward DA ML algorithmfor joint estimation of frequency offset, symbol timing, andcarrier phase in burst-mode CPM signals. The proposed ap-proach takes advantage of the optimized preamble of [20],which jointly minimizes the CRBs for all three synchroniza-tion parameters. We show that the proposed algorithm iscapable of performing quite close to the CRB for variousCPMs and SNRs. Although we consider an AWGN channel,the results can be applied to time-varying channels too sincepractical wireless channels can be assumed to be static duringthe preamble period. In such environments, the estimationresults should be used in conjunction with tracking algorithmssuch as [21]. Additionally, we present a frame synchronizationalgorithm that detects the arrival of bursts and estimates theSoS within ideally one sample time. We discuss how ourapproach extends the frame synchronization algorithms in[17], [18] to our problem, i.e. CPM signals and burst-modetransmissions. We note that the order in which these twoproblems are addressed in this paper is the reverse of theirimplementation in practice where frame synchronization mustbe applied prior to timing and carrier recovery.

The remainder of this paper is organized as follows. SectionII introduces our burst-mode transmission model. In SectionIII, the joint ML timing and carrier estimation is proposed.Section IV describes the frame synchronization algorithm.Simulation results of the synchronization algorithm are re-ported in Section V, and Section VI concludes the paper.

II. BURST-MODE TRANSMISSION MODEL

In our model, we consider transmission of disjoint packetsof data, i.e. bursts. The transmitter is assumed to be turnedon at an unknown time in order to transmit a single burstafter which it is turned off again. Each burst has a knownduration and structure at the receiver, which is depicted inFig. 1 and consists of three parts. The first part is thesynchronization preamble or training sequence. It consists of

Preamble UW PayloadGuardInterval

GuardInterval

Fig. 1: The Burst-Mode Transmission Model.

L0 known and optimized data symbols which are used toestimate synchronization parameters. Although the preamblecan be used for channel estimation too, we only focus on thesynchronization task. The next section in the burst is denotedas the unique word (UW), which is utilized to identify thebursts and determine the location of data symbols within aburst. It is assumed to be a pseudo-random sequence of LUWsymbols. The last part is the payload, which carries Lpayinformation symbols.

We consider CPM signaling for transmission of burstsin our model. The complex baseband CPM signal duringtransmission of each burst can be expressed as

s(t) =

√EsTs

exp{jφ(t;α)} (1)

where Es is the energy per transmitted symbol. The phase ofthe signal φ(t;α) is represented as

φ(t;α) = 2πh

Lb−1∑i=0

αiq(t− iTs) (2)

where αi is the sequence of M -ary data symbols selected fromthe set of {±1,±3, . . . ,±(M − 1)}. Lb is the total numberof such symbols in a burst, that is Lb = L0 + Luw + Lpay.The variable h is the modulation index, which can varyfrom symbol to symbol in the case of multi-h CPM. Thewaveform q(t) is the phase response of CPM and in generalis represented as the integral of the frequency pulse g(t) witha duration of LTs. If L = 1, the signal is called full responseCPM, and for L > 1, it is called partial response CPM. InCPM literature, there are two well-known frequency pulsesdenoted by length L rectangular (LREC) and length L raisedcosine (LRC) [22]. Another commonly-used frequency pulseis the Gaussian minimum-shift keying (GMSK) pulse withbandwidth parameter BTs. In our discussion, we will useBTs = 0.3, which is the bandwidth parameter in the GSMstandard.

Assuming transmission over an AWGN channel, the com-plex baseband representation of the received signal is

r(t) = ej(2πfdt+θ)s(t− τ) + w(t) (3)

where θ is the unknown carrier phase, fd is the frequencyoffset, τ is the timing offset, and w(t) is complex basebandAWGN with zero mean and power spectral density N0. Wedenote the transmitted data symbols during the preamble byα = [α0, α1, · · · , αL0−1]. Our goal is to determine the syn-chronization parameters, i.e. u = [fd, θ, τ ]T , by observing thepreamble portion of the burst, which corresponds to α. Here,it is assumed that u is a vector of unknown but deterministicparameters which are to be jointly estimated at the receiver.Note that α is implicit in the definition of s(t).

Since data arrives in bursts at the receiver, τ can assumeany value. However, a DA estimator requires the approximate

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Fig. 2: The optimum synchronization preamble (training se-quence) for M -ary CPM signals containing L0 symbols.

knowledge of τ in order to perform the estimation algorithmon the received preamble. Therefore, we decompose τ intotwo parts based on

τ = µTs + εTs (4)

where µ ≥ 0 is an integer that represents the integer delayand −0.5 < ε < 0.5 represents the fractional delay. In thiswork, we address these two components separately. First weassume µ is known and the goal is to estimate ε, fd and θ.Later in Section IV, we consider estimation of µ, i.e. the SoSlocation, regardless of fd and θ values.

The last item we need to specify is the synchronizationpreamble. In our recent work [20], we proposed the optimumtraining sequence for joint estimation of u based on the CRBcriterion. This sequence, which is depicted in Fig. 2, minimizesthe CRBs for fd, θ and ε simultaneously. It also has a similarpattern for the entire CPM family. We exploit the structure ofthe preamble in order to facilitate the algorithm design processand then to reduce its complexity.

III. MAXIMUM LIKELIHOOD TIMING AND CARRIERSYNCHRONIZATION

A. Derivation of the Algorithm

Reliable detection of CPM signals depends on accuratetiming and carrier synchronization, which requires knowledgeof fd, θ and τ . These parameters can be estimated via varioustechniques. In this work, we apply joint ML estimation inwhich α is known to the receiver. The likelihood functionfor the estimation of a set of parameters from a waveform inAWGN is given in [22]. It can be easily shown that in our prob-lem, i.e. when the signal is complex and constant envelope,the joint log-likelihood function (LLF) for the synchronizationparameters is expressed within a constant factor of

Λ[r(t); fd, θ, ε]=Re

[∫ T0+εTs

εTs

e−j(2πfdt+θ)r(t)s∗(t− εTs) dt

](5)

where fd, θ and ε are hypothetical values for fd, θ and εrespectively, and T0 = L0Ts is the preamble duration. Notethat we disregard µ in this section for the sake of clarity.According to the ML criterion, we choose the trial values thatmaximize (5) as the best estimates for the unknown parametersu. We denote the ML estimates as u = [fd, θ, τ ]T .

In practice, r(t) is sampled N times per symbol. This resultsin a discrete-time version of the LLF as

Λ(r; ν, θ, ε) ≈ Re

[NL0−1∑n=0

e−j(2πnν+θ)r[n]s∗ε[n]

](6)

0 4 8 12 16

0

1RC (M = 2, h = 1/2)

t/Ts

Unwrapped

Phase

0 4 8 12 16

0

t/Ts

2RC (M = 4, h = 1/4)

Unwrapped

Phase

0 4 8 12 16

0

t/Ts

Unwrapped

Phase

GMSK (BTs = 0.3,M = 2, h = 1/2)

2π

−2π

2π

−2π

2π

−2π

Fig. 3: The phase response of different CPMs to the optimumtraining sequence (shown in solid lines). The dashed linesshow the response of the same sequence to the 1REC CPMwith the same h.

where ν = fdTs/N , i.e. the normalized frequency offsetwith respect to the sampling frequency. r[n] and sε[n] arethe sampled versions of r(t) and s(t − εTs) at t = nTs/Nrespectively. Note that ε is assumed to be zero in the integrallimits of (5) in order to derive (6). This is the main contributorto the approximation in the above given that the samplingfrequency is large enough to avoid aliasing.

Based on (6), the maximization of the LLF requires at least atwo-dimensional grid search on (ν, ε) in general because bothof these parameters are embedded inside the above summation.Therefore, we are interested in a method that decouples εand ν. We note that the preamble of Fig. 2, regardless of itsunderlying CPM, can be divided into three parts, each of whichhaving the same data symbols. This distinct pattern causes theCPM phase to change with a uniform rate of approximatelyπh(M − 1) radians per symbol in the same direction foreach part. We have illustrated this fact in Fig. 3 by plottingthe unwrapped phase response of three different CPMs whenpreamble of Fig. 2 with L0 = 16 is utilized. The first signalphase corresponds to the 1RC frequency pulse with binary datasymbols and h = 1/2. Additionally, the partial-response 4-ary2RC CPM is provided in which h = 1/4. The GMSK schemewith BTs = 0.3 is also included, which is binary, L = 4and h = 1/2. We have compared each case with the phaseresponse of 1REC frequency pulse to the same α and h. Itis observed that despite the fundamental differences betweentheir frequency pulses, the overall phase response of all CPMsignals are approximately similar. More detailed observationscan be made as the following:

1) GMSK and 2RC phase responses follow a straight line

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within each part similar to the 1REC pulse shape in spiteof their bell-shaped pulses. This is due to the overlap ofthe frequency pulses when the subsequent data symbolsare the same, which leads to uniform phase variations.

2) The overall phase response is delayed when partial-response CPMs such as 2RC and GMSK are employed.We denote this lag time by Tl which is equivalent to Nlsamples.

3) 1RC CPM shows the largest deviations from the 1RECphase response because its frequency pulse is full re-sponse (non-overlapping) and has the highest peak.

Based on the above discussion, we approximate the phaseresponse of any given CPM signal to the optimum preambleα∗ with a delayed version of 1REC CPM to α∗ and the sameh . In fact, the optimum preamble enables us to accuratelyapply a piecewise linear approximation to the phase of CPM.Therefore, the approximated phase response can be mathemat-ically expressed as

φ(t,α∗)≈

−(M − 1)πh t−Tl

TsTl < t ≤ T0

4 + Tl

(M − 1)πh t−Tl−T0/2Ts

T0

4 + Tl < t ≤ 3T0

4 + Tl

−(M − 1)πh t−Tl−T0

Ts

3T0

4 + Tl < t ≤ T0 + Tl

0 otherwise(7)

where Tl is fixed for a given CPM and is known to thereceiver. In the Appendix, it is shown that Tl = (L−1)

2 Ts forsymmetric g(t), which is the case for rectangular, raised-cosineand Gaussian pulse shapes. In the rest of our discussion, weassume the channel observation starts from t = Tl, and hence,we ignore Tl. In practice, we can append dTl/Tse “−(M −1)symbols” to the end of the preamble for partial-responseCPMs in order to avoid unwanted variations at the end ofthe observation interval, which is now shifted by Tl. Thus, weuse (7) to express sε[n] during the preamble transmission as

sε[n]≈

exp[−j(M−1)πh( nN −ε)] 0 < n ≤ NL0

4

exp[+j(M−1)πh( nN −L0

2 −ε)]NL0

4 < n ≤ 3NL0

4

exp[−j(M−1)πh( nN −L0−ε)] 3NL0

4 < n ≤ NL0.(8)

We take advantage of the above approximation in order tosimplify the LLF and its maximization algorithm. Using (8)in (6) results in

Λ∗(r; ν, θ, ε)≈Re

{e−jθ

[NL0/4−1∑n=0

e−j2πνnr[n]ej(M−1)πh(n/N−ε)

+

3NL0/4−1∑n=NL0/4

e−j2πνnr[n]e−j(M−1)πh(n/N−L0/2−ε)

+

NL0−1∑n=3NL0/4

e−j2πνnr[n]ej(M−1)πh(n/N−L0−ε)]}(9)

where Λ∗(·) represents the joint LLF given α∗. It is evidentfrom (9) that the symbol timing is now decoupled from thefrequency offset and can be moved outside the summations of

the LLF. Hence, (9) can be simplified as

Λ∗(r; ν, θ, ε) ≈

Re

{e−jθ

[e−j(M−1)πhελ1(ν) + ej(M−1)πhελ2(ν)

]} (10)

where

λ1(ν) =

NL0/4−1∑n=0

e−j2πνnr[n]ej(M−1)πhn/N

+ e−j(M−1)πhL0

NL0−1∑n=3NL0/4

e−j2πνnr[n]ej(M−1)πhn/N

(11)

and

λ2(ν)= ej(M−1)πhL0/2

3NL0/4−1∑n=NL0/4

e−j2πνnr[n]e−j(M−1)πhn/N .

(12)As the estimation parameters are now decoupled, the maxi-

mization of the LLF becomes straightforward. Let us proceedby denoting the term in (10) which corresponds to symboltiming and frequency offset as

Γ(ν, ε) = e−j(M−1)πhελ1(ν) + ej(M−1)πhελ2(ν). (13)

It is easily seen that for any value of (ν, ε), Λ∗(·) is maximizedby choosing θ such that it rotates Γ(ν, ε) towards the real axis,i.e.,

θ = arg{Γ(ν, ε)}. (14)

which reduces the LLF to |Γ(ν, ε)|. Thus, the ML estimatesof ν and ε are found by maximizing

|Γ(ν, ε)|2 = |λ1(ν)|2+|λ2(ν)|2+2Re[e−j2(M−1)πhελ1(ν)λ∗2(ν)](15)

with respect to (ν, ε). The first two terms on the right-handside of (15) do not depend on ε. Using a similar argument asθ, the third term is maximized by selecting ε according to

ε =arg{λ1(ν)λ∗2(ν)}

2(M − 1)πh(16)

so that the term inside the real part operator in (15) be-comes purely real and equal to |λ1(ν)λ∗2(ν)|. Therefore, themaximization of the LLF is now a one-dimensional problemthat results in the ML estimate of ν. This can be expressedmathematically in the form of

ν = argmaxν

{X(ν) = |λ1(ν)|+ |λ2(ν)|} (17)

which in turn leads to the ML estimates of the normalizedsymbol timing and phase offset via

ε =arg{λ1(ν)λ∗2(ν)}

2(M − 1)πh(18)

and

θ = arg{e−j(M−1)πhελ1(ν) + ej(M−1)πhελ2(ν)

}. (19)

respectively.

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B. Implementation of the Frequency Offset Estimator

In the previous section, we derived simple closed-formexpressions for estimation of phase and symbol timing. How-ever, the frequency offset estimation requires computing themaximum of a one-dimensional function as defined in (17).λ1(ν) and λ2(ν) have the form of Fourier transforms of r(t)and should be expected to have fluctuations due to the presenceof noise, which results in several local maxima. Thus, a gridsearch is inevitable in order to find the correct frequency offsetwith confidence.

According to (11) and (12), computations of λ1(ν) andλ2(ν) require a different number of summations with differentlimits. In order to make both of them consistent, we define twonew signals, i.e. r1[n] and r2[n], such that

r1[n] =

r[n] 0 ≤ n < NL0/4

exp[−j(M − 1)πhL0]r[n] 3NL0/4 ≤ n < NL0

0 otherwise(20)

and

r2[n]=

{exp[j(M − 1)πhL0/2]r[n] NL0/4 ≤n< 3NL0/4

0 otherwise.(21)

The above modifications to r[n] leads to similar forms forλ1(ν) and λ2(ν), where each one requires computation of onesummation with NL0 terms, i.e.,

λ1(ν) =

NL0−1∑n=0

r1[n]ej(M−1)πhn/Ne−j2πnν (22)

and

λ2(ν) =

NL0−1∑n=0

r2[n]e−j(M−1)πhn/Ne−j2πnν. (23)

The computations of (22) and (23) for different ν valuesresemble the discrete Fourier transform (DFT) operation,where ν is replaced by trial discrete frequencies. These op-erations can be performed efficiently using the fast Fouriertransform (FFT). The FFT size will be equal to the summa-tion length assuming NL0 is a power of two. This processresults in trial values for λ1(ν) and λ2(ν) such that ν ∈[0, 1/NL0, . . . , (NL0 − 1)/NL0], which are then inserted in(17) in order to find ν. Therefore, the frequency offset estimaterequires two FFTs of the same size.

The frequency estimation performance is limited by theresolution of the FFT operations, i.e. the distance between thediscrete frequency components. A low resolution estimate maycause a ripple effect on the estimation performance of otherparameters. In order to increase the accuracy of the frequencyestimate, two approaches are considered. The first approach isto zero pad the FFT operands in (22) and (23) such that bothFFTs have a size of Nf = KfNL0 where Kf is a powerof two. This procedure results in a frequency resolution of1/KfL0 with respect to the symbol rate. The second approachis to employ an interpolator in order to estimate the truemaximum of (17) between the discrete frequency values. In[23], it was shown that the Gaussian interpolator is superior to

X

X

FFT

FFT

+ MAX Interpolator

X

X

X

X

X

+

Symbol Timing ( )

Carrier Phase ( )

Frequency Offset ( )

Fig. 4: Block diagram of the feedforward joint frequencyoffset, symbol timing and carrier phase estimator.

a parabolic one in terms of improving FFT resolution. The onlyadded complexity is an extra look-up table for computationof the logarithm function. The Gaussian interpolation can beexpressed as

ν = ν0 +1

2KfNL0

logX(ν−1)− logX(ν1)

logX(ν−1) + logX(ν1)− 2 logX(ν0)(24)

where ν0 represents the maximizing frequency resulting from(17). ν−1 and ν1 denote the discrete frequency componentsimmediately before and after ν0 respectively in terms of theFFT operation. The above operation can be regarded as a finesearch while FFTs perform a coarse search on the frequencyoffset.

Based on DFT properties, FFT operations are periodic witha period of NL0. Therefore, values of 1/2 ≤ ν < 1 representnegative frequency offsets, and hence, ν is estimated over[−1/2, 1/2). This limits the frequency estimation range to

− N

2Ts≤ fd <

N

2Ts(25)

which can be increased by increasing the sampling frequency.Therefore, the proposed algorithm can easily handle applica-tions in which the frequency offset is greater than the symbolrate.

The final design for our feedforward joint frequency offset,symbol timing and carrier phase estimator is illustrated in Fig.4. Based on (20) and (21), r1[n] and r2[n] should be multi-plied by exp[−j(M − 1)πhL0] and exp[j(M − 1)πhL0/2]respectively. However, we have not shown this in Fig. 4 forthe sake of clarity, and because the aforementioned factors arebasically equal to one in our examples. Based on the aboveblock diagram, the joint estimator requires 2NL0+3 complexmultiplications, NL0 real multiplications, NL0 + 1 complexadditions and KfNL0 real additions. These exclude blockssuch as FFT, interpolation, | · |, exp(·), and arg(·) as theircomplexity depends on their implementation method.

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Noise Only Preamble Random Data

Fig. 5: The observation window for the SoS estimation algo-rithm.

IV. FRAME SYNCHRONIZATION

So far, we have assumed the carrier and timing synchroniza-tion algorithm have the knowledge of the SoS within ±Ts/2,which has to be carried out by the frame synchronizationalgorithm. In this work, we decompose the frame synchro-nization into two tasks: SoS detection and SoS Estimation. TheSoS detector determines the arrival of a new burst such thatthe preamble is located within an observation or uncertaintywindow. The SoS estimation algorithm then tries to find theexact location of the SoS within that window. Using a reverseapproach, we initially derive the SoS estimation algorithm.Based on its results, we propose a simple SoS detectionalgorithm. It should be mentioned that if the observationwindow does not contain the whole preamble due to SoSdetection failures, the SoS estimation results are no longerreliable, and hence, an entire burst might be missed.

A. SoS Estimation Algorithm

The framework for the SoS estimation algorithm is depictedin Fig. 5 where an observation window of Nw samples isconsidered. The first δ samples contain only WGN whichcorrespond to the guard interval prior to the beginning ofsignal transmission. Note that δ differs from µ in (4) wherethe latter one represents the overall delay from the transmitterto the receiver. It is followed by Np samples of the preamble.Finally, there are Nw− δ−Np samples which are assumed tobe generated from a random CPM signal, and are associatedwith the UW and/or payload portion of the burst. The SoSestimation algorithm attempts to find the best estimate of δaccording to the above observation window.

The received and sampled signal within the observationwindow can be expressed as

r[n] =

{w[n] 0 ≤ n < δ

ej(2πνn+θ)s[n− δ] + w[n] δ ≤ n < Nw(26)

where w[n] is complex white Gaussian random sequence witha variance of σ2 = N(Es/N0)−1. Additionally, we haveassumed Ts = 1 and |s[n]| = 1. It should be noted that θin (26) is different from its value in (3) due to the frequencyoffset and different reference points. Finally, we denote thevalues of r[n] within the observation window by r.

Based on the ML rules, the best estimate of δ is the valuewhich maximizes the likelihood function p(r; δ). However, letus first consider the likelihood function as a function of all

unknown parameters, i.e.,

p(r; δ, ν, θ,αd)=1

(πσ2)Nwexp

(− 1

σ2

δ−1∑n=0

|r[n]|2)

.exp

(− 1

σ2

Nw−1∑n=δ

|r[n]−s[n− δ]ej(2πνn+θ)|2)

(27)

where αd represents the random data sequence (uniformlydistributed among all sequences of that length) in the non-preamble portion of s[n] and r is the received signal vector. Ifwe omit constant factors in the likelihood function, it becomes

p(r; δ, ν, θ,αd)= exp

(δ −Nwσ2

).exp

(2

σ2

Nw−1∑n=δ

Re{r∗[n]s[n−δ]ej(2πνn+θ)

}).

(28)

In order to compute p(r; δ) from (28), we must either estimateor average out the nuisance parameters, i.e. ν, θ and αd, whichis not trivial due to the form of the above function. Instead, weinitially approximate the exponential function with its seconddegree Taylor’s series in the neighborhood of zero, i.e.,

p(r; δ, ν, θ,αd)

≈ C(δ)

(1+

2

σ2

Nw−1∑n=δ

Re{r∗[n]s[n− δ]ej(2πνn+θ)

}+

1

σ4

Nw−1∑n=δ

Nw−1∑m=δ

Re{r∗[n]r∗[m]s[n−δ]s[m−δ]ej(2πν(m+n)+2θ)

}+

1

σ4

Nw−1∑n=δ

Nw−1∑m=δ

Re{r∗[n]r[m]s[n−δ]s∗[m−δ]ej(2πν(n−m))

})(29)

where C(δ) represents exp( δ−Nw

σ2 ) in (28), which is nota function of the nuisance parameters. However, we avoidusing it in its original form because it can be very smalland adversely affect the approximated likelihood function.Nevertheless, we will propose an approximation for C(δ) oncethe final form of the likelihood function becomes available.

Assuming θ is uniformly distributed over [−π, π), it can beeliminated from the likelihood function by averaging (29) overθ, i.e.,

p(r; δ, ν,αd) =1

2π

∫ π

−πp(r; δ, ν, θ,αd) dθ

≈ C(δ)1

σ4

Nw−1∑n=δ

Nw−1∑m=δ

Re

{r∗[n]r[m]s[n− δ]

.s∗[m− δ]ej(2πν(n−m))

}.

(30)

Note that we have neglected 1 in (29) because it is muchsmaller than the fourth term especially when noise variance issmall. We also omit 1/σ4 from the above as it is a constant

7

factor. If we denote d = m − n in (30), it can be rearrangedas

p(r; δ, ν,αd) ≈ C(δ)

(Nw−1∑n=δ

|r[n]|2

+2

Nw−δ−1∑d=1

Re

{e−j2πdν

Nw−d−1∑n=δ

r∗[n]r[n+d]s[n−δ]s∗[n+d−δ]

})(31)

which allows us to investigate signal correlation dueto the presence of random αd. The computation ofEαd{p(r; δ, ν,αd)} leads us to compute Eαd

{s[n− δ]s∗[n+d− δ]} which, in our problem, is

Eαd{s[n− δ]s∗[n+ d− δ]}

=

s[n− δ]s∗[n+ d− δ] δ ≤ n < Np + δ − d0 Np + δ − d ≤ n < Np + δ

Rss(d) n ≥ Np + δ

(32)

where Rss(d) is the autocorrelation function of the CPMsignal normalized to the sample duration. Rss(d) can becomputed numerically as described in [22, p. 208]. The firstcase in (32) corresponds to the preamble, which has norandomness. The second case is zero because s[n − δ] isdeterministic whereas s∗[n+d−δ] is generated by the randomdata and its expected value is zero. Therefore, taking theexpected value of (31) with respect to αd results in

p(r; δ, ν) ≈ C(δ)

(Nw−1∑n=δ

|r[n]|2 + 2

Np−1∑d=1

Re

{e−j2πdν

.

(Np+δ−d−1∑

n=δ

r∗[n]r[n+ d]s[n− δ]s∗[n+ d− δ]

+Rss(d)

Nw−d−1∑n=Np+δ

r∗[n]r[n+ d]

)}).

(33)

In general, CPM autocorrelation function becomes zero for lagtimes greater than LTs. Therefore, Rss(d) is zero except forits first few values.

The last step to obtain p(r; δ) is removing ν from (33). Itcan be verified that averaging (33) with respect to ν, uniformlydistributed over [-0.5,0.5), completely eliminates the secondsummation. This indeed results in a poor ML estimate for δbecause it ignores the knowledge of the known preamble. Abetter approach is to estimate ν by maximizing the secondsummation in the above. However, a closed-form solutionseems to be unavailable due to the range of d. Instead, wecan derive different estimates νd based on single terms insidethe summation via

νd =1

2πdarg

{Np+δ−d−1∑

n=δ

r∗[n]r[n+ d]s[n− δ]s∗[n+ d− δ]

+Rss(d)

Nw−d−1∑n=Np+δ

r∗[n]r[n+ d]

}.

(34)

The above method is the basis for some well-known carrierfrequency estimation algorithms, such as [24]. If we use νdvalues and plug them back into (33), the likelihood functionbecomes independent of the frequency offset. Thus,

p(r; δ) ≈ C(δ)

(Nw−1∑n=δ

|r[n]|2

+ 2

Np−1∑d=1

∣∣∣∣∣Np+δ−d−1∑

n=δ

r∗[n]r[n+ d]s[n− δ]s∗[n+ d− δ]

+Rss(d)

Nw−d−1∑n=Np+δ

r∗[n]r[n+ d]

∣∣∣∣∣)

(35)

which must be maximized with respect to δ in order to deriveδ.

The computational complexity of (35) can be reduced bytruncating the summation over d. This results in a sub-optimum, reduced-complexity estimator, i.e.,

δ = argmaxδ

{C(δ)

.

(Nw−1∑n=δ

|r[n]|2 + 2

D∑d=1

∣∣∣∣∣Np+δ−d−1∑

n=δ

r∗[n]r[n+d]s[n−δ]s∗[n+d−δ]

+Rss(d)

Nw−d−1∑n=Np+δ

r∗[n]r[n+ d]

∣∣∣∣∣)}

(36)

where 1 ≤ D < Np is a design parameter, which allows atrade-off between complexity and performance. In fact, if weassume Rss(d) = 0 for d 6= 0, it is observed that computationof the argument of (36) requires D(2Np − D − 1) + Nw −Np complex multiplications, 2 real multiplications, D(Np −D+12 ) +Nw−Np complex additions, D real additions, and D

computations of the absolute value. Therefore, the complexityof proposed algorithm is approximately a linear function ofD.

As mentioned earlier, C(δ) needs to be adjusted based onthe final form of the likelihood function. We note that (35)is dominated by the summation over d. If we ignore Rss(d)due to its short length, the computations inside the absolutevalue are performed over a sliding window that covers thehypothetical preamble. If this window is shifted to the left byone sample, one signal-plus-noise sample will be replace byone noise-only sample, which has a smaller energy comparedto the former one. However, shifting the window to the rightreplaces it with a different signal-plus-noise sample. Therefore,we expect p(r; δ+ 1) > p(r; δ− 1), if δ is its true value. Thismakes the likelihood test biased, i.e. δ is more likely to tendtowards δ + 1 than δ − 1. We introduce a simple solution tothis issue by proposing

C(δ) , (Nw − δ)q (37)

where q ≥ 0 is another design parameter which has to bechosen according to D. As we will see in the simulation

8

results, q = 1 is a good choice for the full-complexityestimator, i.e. D = Np − 1, while it has to be reduced forsmaller values of D.

Choi and Lee [18] have presented a ML frame synchro-nization algorithm through a different path for PSK signalswhere the preamble is surrounded by random data. Despitesimilarities to (36), our estimator addresses a different scenarioin which the preamble is preceded by the noise-only samplesso that C(δ) was introduced. Additionally, the memory ofCPM signals is handled via the presence of Rss(d). Fi-nally, it should be mentioned that each of the summations∑n r∗[n]r[n + d]s[n − δ]s∗[n + d − δ] in (36) is referred to

as a double-correlation in [18].

B. SoS Detection Algorithm

As the last piece of our synchronization algorithm, wepresent a simple ML detection algorithm, which is closelyrelated to our previous discussion. Let us assume a receiverwhich collects vectors of Np samples using a sliding window.We denote this vector by rp. Additionally, consider twohypotheses H0 and H1. H0 is the hypothesis where the entirevector of samples in rp are noise-only samples, which happenswhen no burst is received. On the other hand, H1 is thehypothesis where rp is perfectly aligned with the preamble.We can distinguish these two hypotheses by performing alikelihood ratio test (LRT) according to

L(rp) =p(rp;H1)

p(rp,H0)

H1

≷H0

γ (38)

where p(rp;Hi) is the likelihood function under Hi. Basedon the above test, H1 is selected when L(rp) is greater thana threshold γ. Otherwise, we select H0, i.e. no preamble ispresent.

Obviously, several other hypotheses also occur in betweenthese two in which rp contains only a fraction of the preamble,i.e. a mixed-signal scenario. However, we note that the LRT isperformed on a sliding window, which makes it a generalizedLRT (GLRT). In this manner, the unknown time delay is beingestimated at the same time as the preamble is detected. Thepoint at which the above test exceeds the threshold indicatesthe presence of the preamble and an estimate of the SoS. Later,the SoS estimator improves this estimate by going over anadditional Np samples to find the peak when it considers thefull structure of a burst, i.e. the preamble is followed by CPMsignal rather than noise.

The likelihood ratio of (38) can be easily obtained from(35). In fact, p(rp;H1) becomes equal to p(r; δ) when δ = 0and Nw = Np. We also note that we can multiply p(rp; δ = 0)

by exp(− 1σ2

∑Np−1n=0 |r[n]|2) because it is not a function of δ.

The latter factor is basically equal to p(rp;H0). Thus, thelikelihood test can be approximated by

L(rp) =p(rp; δ = 0)p(rp;H0)

p(rp;H0)

≈Np−1∑n=0

|r[n]|2+2

Np−1∑d=1

∣∣∣∣∣Np−d−1∑n=0

r∗[n]r[n+ d]s[n]s∗[n+ d]

∣∣∣∣∣ ≷ γ′.

(39)

Similar to (36), we propose a reduced-complexity test, i.e.

LD′(rp) ,D′∑d=1

∣∣∣∣∣Np−d−1∑n=0

r∗[n]r[n+ d]s[n]s∗[n+ d]

∣∣∣∣∣ ≷ γD′ .

(40)where 1 ≤ D′ < Np is a design parameter and γD′ representsthe test threshold for a given D′. It can be verified thatcomputation of (40) requires D′(2Np − D′ − 1) complexmultiplications, D′(Np − D′+1

2 ) complex additions, D′ realadditions, and D′ absolute value functions. Moreover, themajority of computations in SoS detection and SoS estimationare the same, which allows a high degree of resource sharingbetween these two blocks.

The threshold γD′ can be chosen based on the Neyman-Pearson criterion [25] in which the probability of false alarmis fixed. Here, the probability of false alarm is defined asPFA = Pr{LD′(rp) > γD′ |H0}. Once the threshold is chosen,the probability of missed detection can be calculated viaPMD = Pr{LD′(rp) < γD′ |H1}. The probability of correctdetection is PD = 1 − PMD. Exact closed-form expressionsfor PFA and PD may not be realized due to the magnitudeoperators and multiplications in (40). For instance, if wedenote the output of each double-correlation as a randomvariable, i.e., Xd =

∑Np−d−1n=0 r∗[n]r[n + d]s[n]s∗[n + d], a

simple yet acceptable (for large Np − d) approximation is toconsider Xd as a complex Gaussian random variable (RV).This forces |Xd| to become a Rayleigh RV under H0 andRician RV under H1 due to the presence of signal. Thus,LD′(rp) can be approximated as sum of Rayleigh or RicianRVs depending on the hypothesis. In [26], [27], approximatecumulative distribution functions (CDFs) are provided for suchRVs. However, our investigations show that the approximationerror is considerable because we are interested in regionswhere PFA and PMD are very low. Therefore, we resort toMonte-Carlo simulations with a large sample size in order tocompute these probabilities, γD′ , and the receiver operatingcharacteristic (ROC).

V. RESULTS AND DISCUSSION

A. Approximation Error

In this section, we study the error in representing theCPM phase during preamble transmission using (7). For thesake of clarity, we denote the original CPM phase by φ(t),and its approximated value in (7) by φ′(t). Therefore, therepresentation error is e(t) = φ(t) − φ′(t). We define theapproximation error as the ratio of the energy in the error(Ee) to the signal energy during the preamble transmission,i.e.,

ea ,

∫ T0

0|ejφ(t) − ejφ′(t)|2 dt∫ T0

0|ejφ(t)|2 dt

=

∫ T0

0|1− eje(t)|2 dtL0Ts

=EeL0Ts

.

(41)In the above, |1 − eje(t)|2 can be approximated by e2(t) forsmall values of |e(t)|, which gives us an approximate value ofEe ≈

∫ T0

0e2(t) dt. The computation of Ee for full response

9

CPMs is

Ee =

L0−1∑k=0

∫ (k+1)Ts

kTs

e2(t) dt

= 4L0π2h2(M − 1)2

∫ Ts

0

(q(t)− t

2Ts

)2

dt

(42)

which is basically proportional to the difference between q(t)and the 1REC phase response over a single symbol interval.

The computation of Ee for partial response signals can bedivided into four parts based on the preamble structure asfollows. For each part, we denote Ee and e(t) by Ei andei(t) respectively. e1(t) corresponds to the first L−1 symbolswhere CPM modulator does not have any memory of t < 0.Moreover, we have introduced a shift of Tl for partial responsesignals. Therefore, e1(t) during 0 ≤ t < (L − 1)Ts/2 isexpressed as

e1(t) = 2πh(M − 1)

{t

2Ts−L−2∑i=0

q

(t+ (

L− 1

2− i)Ts

)}(43)

which is used to compute E1.E2 and E3 correspond to the two intervals, i.e. T2 and T3,

where there are transitions in the preamble. Each of theseintervals has a duration of (L − 1)Ts in which there are atleast two symbols with different signs. Due to the symmetry,E2 = E3, we need only to derive e2(t). At the beginning ofT2, the responses of the L − 1 previous symbols are still ineffect, and hence, we must consider 2(L− 1) symbols, wherethe first half have a value of −(M − 1), and the second halfhave a value of M − 1. Additionally, we assume that the timeorigin is moved to the start of T2 such that φ(t) is replacedby φ2(t). Therefore, φ2(t) is expressed as

φ2(t) = 2πh(M − 1)

2(L−1)∑i=L

q(t+ (L− i)Ts)

−L−1∑i=1

q(t+ (L− i)Ts)

}.

(44)

We also note that φ2(t) is symmetric with respect to t =(L−1)/2Ts. Thus, it is sufficient to compare it with the 1RECphase response only for 0 ≤ t < (L− 1)/2Ts, i.e.,

|e2(t)| = |e3(t)| =∣∣∣∣φ2(t) +

πh(M − 1)t

Ts+πh(L− 1)

4

∣∣∣∣ .(45)

Finally, we need to investigate time intervals where the past(L−1) symbols are similar to the current one. These intervalsmake up the majority of the preamble with an overall durationof [L0− 2(L− 1)]Ts seconds. The absolute value of the errorwithin each such symbol interval is

|e(t)| = 2πh(M − 1)

∣∣∣∣p(t)− t

2Ts− (L− 1)

4

∣∣∣∣ (46)

where

p(t) =

(L−1)∑i=0

q(t+ iTs) (47)

for 0 ≤ t < Ts. p(t) can be simplified based on g(t), i.e.,

p(t)=

(L−1)∑i=0

∫ iTs+t

0

g(τ) dτ=

(L−1)∑i=0

∫ iTs

0

g(τ) dτ +

(L−1)∑i=0

∫ iTs+t

iTs

g(τ) dτ

(48a)

=L−1

4+

(L−1)∑i=0

∫ iTs+t

iTs

g(τ) dτ (48b)

=L−1

4+

∫ t

0

L−1∑i=0

g(τ + iTs) dτ (48c)

where (48b) is due to (52). It is straightforward to show thatthe second term in (48c) is equal to t/2Ts for partial responseLREC, LRC, and Gaussian pulse shapes. Using LRC as anexample, the integrand in (48c) can be written as

L−1∑i=0

g(τ + iTs)dτ =

L−1∑i=0

1

2LTs

(1− cos

2π(t+ iTs)

LTs

)

=1

2Ts−L−1∑i=0

cos2π(t+ iTs)

LTs=

1

2Ts

(49)

where the last equality holds because the summation ofcomplex points that are uniformly distributed over the unitcircle is equal to zero. Hence, e(t) = 0 for our partial responseexamples except for T2, T3, and the start of the preamble. Thisresults in

ea ≈1

L0Ts

[∫ (L−1)/2Ts

0

e21(t)dt+ 4

∫ (L−1)/2Ts

0

e22(t)dt

](50)

for partial response CPMs of our interest. We note that theintegrals in (50) are independent of L0. Therefore, the approx-imation error decreases as L0 increases. We also observe thatthe approximation error occurs whenever there is a transitionin the preamble. Since the optimized preamble has only twotransitions, it conveniently limits the approximation error inour approach. On the other hand, if we use (42) in (41), weobserve that ea for full response CPMs is constant with respectto L0.

We have computed ea for different CPMs based on theabove relations and have plotted them in Fig. 6 with respectto L0. We have normalized the curves by h2(M − 1)2 inorder to isolate the effect of L0 and q(t) on the approximationerror. We observe that longer frequency pulses result in largerapproximation errors because the interval for which theyexhibit deviations from 1REC is proportional to L. It is alsoseen that the approximation error for 1RC is much larger thanother examples. Its effect on the estimation performance willbe seen in the next section.

B. Timing and Carrier Recovery Performance

In this section, we compute the error variances of frequencyoffset, carrier phase, and symbol timing for the proposed MLestimation algorithm using simulations. We have considered

10

0 20 40 60 80 100 120 14010

−4

10−3

10−2

10−1

100

L0

Approxim

ationError(e

a)

2RC

3RC

2REC

3RECGaussian (BT

s=0.3)

Gaussian (BTs=0.5)

1RC

Fig. 6: The approximation error for different phase responses.These values should be scaled by h2(M − 1)2 for a specificCPM.

the three examples of Fig. 3 along with MSK, which is abinary CPM with h = 1/2 and 1REC frequency pulse. In allexamples, the optimum preamble with L0 = 64 is deployed. Inaddition to AWGN, we apply ν, θ and ε that are uniformly dis-tributed over [−0.5, 0.5), [0, 2π) and [−0.5, 0.5) respectively.Additionally, we have considered N = 2, Kf = 2, and theGaussian interpolator (24) for the estimation. Our simulationsshow that both interpolation and zero padding must be presentin order to have reliable estimates [28, Fig. 3].

The estimation error variances corresponding to the nor-malized frequency offset and carrier phase are depicted inFigs. 7 (a) and 7 (b) respectively for different CPM schemes.The frequency estimation plots demonstrate that the proposedestimator performs with almost the same accuracy for all theschemes and is less than 0.5 dB away from the CRB for lowto moderate SNRs. As it was shown in [20], the frequency andphase estimation CRBs for the optimum training sequence areindependent of the particular CPM scheme. Hence, only oneCRB plot is shown in each figure. Moreover, it is observedthat the 1RC scheme performs slightly worse than the otherschemes because it has the largest deviations from the 1RECtemplate (refer to Fig. 3). For the remaining schemes, the gapbetween the error variances and the CRB is mostly due tothe FFT precision and can be reduced by increasing Kf . Thisdivergence is the beginning of a floor at high SNRs becauseerrors caused by the thermal noise become less significant thanthe FFT and interpolation precision, which are constant at allSNR.

The normalized timing error variances are plotted in Fig.8. It reveals that the proposed estimator reaches the CRB forthe majority of the examples. The only exception is again the1RC scheme as discussed above. For all other examples, theML estimator attains the lower limit of the CRB despite thevisible loss in the frequency estimation. This is because theoptimum training sequence decouples timing from frequencyin terms of the Fisher information matrix (FIM) [20], whichmeans that small errors in the frequency estimate do not affectthe symbol timing estimate. The optimum training sequence

0 1 2 3 4 5 6 7 8 9 1010

−8

10−7

10−6

Es/N0 (dB)

Normalized

Frequency

Error

Variance

MSK

1RC (M=2,h=1/2)

GMSK

2RC (M=4,h=1/4)

CRB

(a)

0 1 2 3 4 5 6 7 8 9 1010

−3

10−2

10−1

Es/N0 (dB)

PhaseError

Variance

MSK

1RC (M=2,h=1/2)

GMSK

2RC (M=4,h=1/4)

CRB

(b)

Fig. 7: The error variance of frequency offset (a) and carrierphase (b) estimations for different CPM schemes when L0 =64. The frequency is normalized with respect to the symbolrate.

does not decouple frequency and phase, and hence, errors inthe frequency estimate leak into the phase estimator, whichresults in a slight performance degradation that is visible inFig. 7 (b).

It should be mentioned that the FFT operations will bereplaced by simple correlations when fd = 0. In such applica-tions, (22) and (23) are computed for ν = 0 without any needto perform the maximization of (17) and the interpolation. Thisleads to a joint symbol timing and carrier phase estimator,which is efficient yet less complex compared to other DAworks such as [11]–[13]. This simplicity is a direct result ofunique structure of the optimized preamble.

C. Frame Synchronization Performance

The performance of the SoS estimation algorithm is charac-terized by the probability of false lock, which is PFL = Pr{δ 6=δ}. This probability is computed given that the preamble iscorrectly detected and fully resides within the observationwindow.

11

0 1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

Es/N0 (dB)

Normalized

Tim

ingError

Variance

MSK

CRB (MSK)

1RC (M=2,h=1/2)

CRB (1RC)

GMSK

CRB (GMSK)

2RC (M=4,h=1/4)

CRB (2RC)

Fig. 8: The variance of symbol timing estimation for differentCPM schemes when L0 = 64. The symbol timing is normal-ized with respect to the symbol duration.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

−3

10−2

10−1

100

q

Probabilityof

False

Lock(P

FL)

D=2

D=4

D = 8

D=63

Fig. 9: The effect of correction term C(δ) (Equation (37)) andits exponent q on PFL. GMSK signaling is used when Np = 64and Es/N0 = 1 dB.

The effect of the C(δ) as a function of q on PFL is studiedin Fig. 9 using simulations. The GMSK scheme is used whereL0 = Np = 64, Nw = 96 and Es/N0 = 1 dB. Additionally,we have varied q over the range of [0, 2] and computed PFL forseveral values of D. It is observed that the introduction of C(δ)reduces PFL given that q is carefully selected. We observe thata value of q = 1 is suitable for D = 63, while it needs to bedecreased for smaller values of D. In fact, C(δ) becomes lessimportant for small values of D such as D = 2 and can simplybe ignored, i.e. q = 0. Nevertheless, it visibly improves theperformance for D = 63 such that it becomes superior toD = 8 only in the presence of C(δ). Our simulations alsoconfirm that the SoS estimator becomes unbiased only for theoptimized q, which was the main motivation for introductionof C(δ) as in (37).

The SoS estimator’s performance with respect to SNR isshown in Fig. 10 for two different preamble lengths andmultiple values of D. Our simulations show that variationsof the optimum q with respect to SNR is less than 0.1 for

0 1 2 3 410

−5

10−4

10−3

10−2

10−1

100

Es/N0 (dB)

Probabilityof

False

Lock(P

FL)

L0=64, D=63, q=1

L0=64, D=8, q=0.5

L0=64, D=4, q=0.4

L0=64, D=2, q=0.3

L0=32, D=31, q=1

L0=32, D=8, q=0.5

L0=32, D=4, q=0.4

L0=32, D=2, q=0.3

Fig. 10: The probability of false lock versus SNR for differentpreamble lengths. The values of q are optimized for each caseusing simulations at Es/N0 = 1 dB. The signal is sampled atN = 1, which results in Np = L0.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10−4

0.99

0.991

0.992

0.993

0.994

0.995

0.996

0.997

0.998

0.999

1

Probability of False Alarm (PFA)

Probabilityof

Detection

(PD)

D=2, L0=32

D=3, L0=32

D=4, L0=32

D=2, L0=64

Fig. 11: Receiver operating characteristics for the proposeddetector. The optimum preamble is transmitted over an AWGNchannel when Es/N0 = 1 dB and GMSK modulation is used.

the plotted range. Therefore, we use a fixed q for each plot.We note that the proposed parameter of D allows us to avoidunwieldy complexity of D = Np − 1. For instance, choosingD = 4 results in only a loss of 0.7 dB for L0 = 64 incomparison with D = 63. Yet, the computational complexityis reduced by a factor of approximately 16. Another importantobservation that can be made is that increasing L0 from 32 to64 yields a gain of only a fraction of dB in terms of the SNR.

The performance of the SoS detection algorithm can beexamined through the ROC plots. A few examples of theROC are plotted in Fig. 11 where PFA and PD are calculatedusing simulations by varying γD′ . These ROCs are obtainedfor the GMSK scheme when Es/N0 = 1 dB, and N = 1(Np = L0). It is observed that we are able to attain a verylow PFA at low SNR even with a relatively short preambleof L0 = 32. It is also seen the improvement becomes lesssignificant when D′ is changed from 4 to 8. Therefore, a

12

0 2 4 6 8 1010

−5

10−4

10−3

10−2

10−1

100

Eb/N0 (dB)

BitError

Rate(B

ER)

L0=32

L0=64

Perfect Synchronization

(a) GMSK (BTs = 0.3)

0 2 4 6 8 1010

−4

10−3

10−2

10−1

100

Eb/N0 (dB)

BitError

Rate(B

ER)

L0=32

L0=64

Perfect Synchronization

(b) 2RC (h = 1/4,M = 4)

Fig. 12: BER for the burst-mode CPM receiver. L0 is thepreamble length in terms of data symbols.

small value of D′ looks sufficient to achieve a PD that isclose to the full-complexity detector, i.e. D′ = Np−1. This issimilar to the performance improvement of the SoS estimationalgorithm versus D (Figs. 9 and 10). On the other hand, theperformance is improved substantially when L0 = 64. Forinstance, PD = 1− 5× 10−7 ≈ 1 and PFA = 4.86× 10−6 forγ2 = 40. Comparing these metrics with PFL in Fig. 10 revealsthat the performance of the frame synchronization algorithmis limited by the false locks rather than false alarms or misseddetections.

D. BER Performance

In this section, we evaluate the overall BER performanceof the proposed synchronization scheme, including SoS de-tection, SoS estimation, and timing and carrier recovery, usingsimulations. We have considered two examples of GMSK and4-ary, h = 1/4 CPM with 2RC frequency pulse. Each burst

consists of a preamble of L0 symbols, a UW of 64 random butknown bits and 4096 information bits. The UW is used in orderto adjust the beginning of each burst by correlating it with thedemodulated bits. In our simulations, the transmitter sendsindividual bursts that are preceded by a fixed but unknownamount of guard time. The AWGN is then added to thewaveform along with random frequency and phase offsets. Thereceived signal is sampled at N = 2 samples per symbol. TheMLSD CPM demodulator is designed according to [4], whichuses the Viterbi algorithm (VA). We have also employed adecision-directed (DD) phase and timing tracking loop [21] inwhich phase and timing error signals are generated accordingthe decisions made inside the VA. The phase tracking loop isessential because even very small residual frequency offsets,after the DA synchronization, result in large phase rotationsas the burst is being demodulated. The phase and timing loopbandwidths are both set to 10−3/Ts. Finally, we have setD′ = D = 4 and Nw = 2NL0 for the frame synchronization.

The BER performance of the burst-mode receiver for theGMSK scheme with two different preamble lengths is depictedin Fig. 12 (a). It is observed that the receiver operates withinless than 0.1 dB of the ideal synchronization for L0 = 64over the whole range of Eb/N0. However, there is a substantialBER degradation at the low SNR region for the short preambleof L0 = 32. Our simulation results show that this is mainlydue to the SoS false locks that are more likely to happen atlow SNRs and short preamble lengths. False locks reduce theaccuracy of the timing and carrier recovery algorithm, whichimpact the BER. At higher SNRs, there is about 0.2 dB gapthat is caused by estimation errors, which are increased whenL0 is reduced.

The BER performance for the 2RC scheme is reported inFig. 12 (b). Similar to GMSK, L0 = 64 performs almost idealand within about 0.1 dB of perfect synchronization. However,the preamble of L0 = 32 shows slightly different behaviorthan that of GMSK, where no BER degradation at low SNRsis visible. This is because Es = 2Eb for the 4-ary schemeand both Figs. 12 (a) and 12 (b) are expressed in Eb/N0. Inother words, Es/N0 for GMSK is 3 dB less than for the 2RC,and hence, PFL becomes larger. In fact, 2RC with L0 = 32should be compared to GMSK with L0 = 64 in order tohave a fair comparison where both preambles contain 64 bits.We also note there is no visible difference between the twopreambles in terms of the BER, and hence, L0 = 32 is anadequate length in practice. Finally, this scheme, i.e. non-binary and partial response, is known to be prone to falselocks when DD timing estimation algorithms such as [11] or[21] are implemented. Here, we showed that our proposedDA algorithm with a short preamble can be another methodto solve the false lock problem while it significantly reducesthe acquisition time.

VI. CONCLUSION

In this paper, we addressed the synchronization problemfor CPM signals in burst-mode transmissions. Thanks to theunique structure of the optimized synchronization preamble,we developed a DA ML algorithm, which jointly estimates

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the frequency offset, carrier phase and symbol timing. Theproposed algorithm, which is implemented in a feedforwardmanner, estimates the frequency offset via two FFT operations.Once the frequency estimate is available, the carrier phaseand symbol timing are easily computed via simple closed-form expressions. Our method can be applied to the wholerange of CPM signals. The computed MSEs demonstrate thatits performance is within 0.5 dB of the CRB for all threesynchronization parameters for various examples. Moreover, itoperates at frequency offsets as large as half of the samplingfrequency without sacrificing the estimation accuracy.

In the second part of this paper, we addressed the framesynchronization issue in burst-mode CPM transmissions usingML principles. We developed a simple test for detection of theSoS after which the exact location of the SoS is estimated via aone-dimensional search. We numerically computed the ROCsfor the SoS detector along with the false lock probabilities forthe SoS estimator. The frame synchronization allowed us toimplement a realistic burst-mode CPM receiver. The simulatedBER curves demonstrated an almost ideal performance forpreambles as short as 64 bits and SNRs as low as 0 dB.

APPENDIX

DERIVATION OF Tl

We start by assuming transmission of K “M − 1” symbolswhen the phase response length is L. The CPM phase at t =KTs when K > L can be written as

φ(KTs) = 2πh

K−1∑i=0

(M − 1)q(KTs − iTs)

= πh(M − 1)(K − L+ 1) +2πh(M − 1)

L−1∑l=1

q(lTs)

(51)

where the second equality holds since q(mTs) = 1/2 form ≥ L. Without loss of generality we assume L is odd.Additionally, we consider frequency pulses which have evensymmetry around LTs/2. Therefore, the second term on theright-hand side of (51) can be expressed as

L−1∑l=1

q(lTs) =

(L−1)/2∑k=1

q(kTs) + q((L− k)Ts)

=

(L−1)/2∑k=1

∫ (L/2)Ts

0

g(t)dt−∫ (L/2)Ts

kTs

g(t)dt

+

∫ (L/2)Ts

0

g(t)dt+

∫ (L−k)Ts

(L/2)Ts

g(t)dt

=

(L−1)/2∑k=1

1

2=L− 1

4.

(52)

The last equality is true due to the following equalities forsymmetric g(t),∫ (L/2)Ts

0

g(t)dt =1

2

∫ LTs

0

g(t)dt =1

4(53)∫ (L/2)Ts

kTs

g(t)dt =

∫ (L−k)Ts

(L/2)Ts

g(t)dt. (54)

where k < L/2. Thus, (51) is simplified to

φ(KTs) = πh(M − 1)[K − L− 1

2]. (55)

It can be shown that the above results hold for even values ofL as well. It is observed that the signal phase in (55) is equalto the phase of a CPM signal with 1REC pulse shape, sameh and data sequence at t = (K − L−1

2 )Ts. The latter signal isbasically the approximated phase response, and hence,

Tl = KTs − (K − L− 1

2)Ts =

L− 1

2Ts. (56)

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