I Introduction
Continuous phase modulation (CPM) is an attractive nonlinear modulation scheme whose signals exhibit properties of constant envelope and tight spectrum confinement [1]. The first property will allow the transmitter to enjoy high power transmission efficiency as CPM signals have peaktoaverage power ratio (PAPR) of (theoretically). Hence, traditional transmission techniques (e.g., [2, 3]) to deal with high PAPR problem in orthogonal frequencydivision multiplexing (OFDM) may be avoided when highrate transmission is not a must. The second property implies less amount of outofband power leakage compared to 2 and 4ary PSK modulations [1], therefore, causing less interference to other applications (operated over adjacent spectral bands) and leading to relatively higher spectral efficiency.
Nowadays, CPM has been used in many areas such as global system for mobile communications (GSM) [4], military and satellite communications [5], millimeter communications [6], and machinetype communications in 5G [7, 8]
. In dispersive channels, frequencydomain equalization (FDE) is needed to suppress the effect of intersymbol interference (ISI), followed by CPM demodulation as in flatfading channels. FDE based CPM receiver design can be found in
[9, 10, 11], in which the channel fading coefficients are assumed to be perfectly known at the receiver. Recently, random sequence based channel estimation and equalization has been investigated in [12].Despite of a long history of CPM research, less has been understood on the training waveform design of CPM. According to [13], the 8 training binary sequences defined in GSM standard [14] have been found by computer search over all possible binary sequences. In 2013, Hosseini and Perrins studied the training sequence design of burstmode CPM over additive white Gaussian noise (AWGN) channels [15, 16]. However, the CPM training sequences proposed in [15] may not be applicable in frequencyselective channels (as will be shown in Section IV). Motivated by this, we target at a systematic construction of CPM training waveforms for frequencyselective channels. Our main idea is to apply differential encoding to Golay complementary pair (GCP) whose aperiodic autocorrelation sums diminish to zero for all the nonzero timeshifts [17]. Taking advantage of Laurent decomposition [18], we show that the resultant CPM training waveform (with modulation index of ) displays autocorrelation sidelobes close to zero.
Ii Preliminaries
Iia Introduction to CPM
Let . An equivalent lowpass ary CPM waveform is expressed as , where () is the timevarying phase depending on the information sequence with being the th CPM symbol drawn from the set of , is the modulation index, and is the symbol duration. The phaseshaping waveform is defined as the integral of the frequencyshaping pulse of duration , i.e., , with for and for . is called fullresponse if and partialresponse when . Note that can be written as
(1) 
Let and . It is easy to see that the phase depends on the modulator state , where and are called the phase state and the correlative state, respectively.
Laurent’s Decomposition: Laurent showed that a binary partialresponse CPM signal can be represented as a superposition of a number of pulseamplitudemodulated (PAM) pulses [18]. To introduce this, we first define
(2) 
Also, denote by the coefficients of the binary representation of integer in the set of , i.e., Then, the CPM signal can be written as
(3) 
where , for , and In general, is the most important PAM pulse as it carries more than of the total signal energy [18]. Therefore, the CPM signal can be approximated as
(4) 
with
(5) 
For ease of presentation, are called CPM pseudosymbols.
IiB Introduction to Golay Complementary Pair (GCP)
Denote by the aperiodic autocorrelation function (AACF) of which is defined as
(6) 
Let be a pair of sequences with identical length of . is called a GCP [17] if for any . Note that compared to conventional onedimensional sequences, the two constituent sequences in a GCP work in a cooperative way to ensure that their outofphase aperiodic autocorrelations sum to zero.
Let be the periodic autocorrelation function (PACF) of C at timeshift . Clearly, for any if is a GCP.
Denote by the set of integers modulo . For , a generalized Boolean function (GBF) (or ) is defined as a mapping . Let be the binary representation of the integer , with denoting the most significant bit. Given (or ), define , and
We present the example below to illustrate GBFs defined above. One can find it useful in understanding the GCP construction in Lemma 1 (which is formed by summation of a series of quadratic and linear terms of GBFs).
Example 1
Let and . The associated sequences of are
respectively.
Lemma 1
(DavisJedwab Construction of GCP [20]) Let
(7) 
where is a permutation of the set , and ( even integer). Then, for any , form a GCP over of length .
Iii Proposed CPM Training Waveform Design
In this section, we will propose a training waveform design for CPM signal [or the approximation ] with periodic autocorrelation sidelobes close to zero. Throughout the proposed design, we consider binary CPM with . Therefore, . To get started, we first consider a sequence of with nonzero elements satisfying and
(8) 
The PACF of the CPM approximation [cf. (4)] over is defined as
(9) 
where
(10) 
In addition, the AACF of , the most significant pulse in Laurent’s decomposition, is defined as
(11) 
where if or . Hence, if . By [References, (6.45b)], we have
(12) 
(12) implies that the PACF of [i.e., ] is zero for any if
(13) 
Nevertheless, it is hard to find sequence satisfying (5) and (13) simultaneously. Because of this, we consider sequence pair and over . Applying the “differentialencoding”, denoted by , to , we obtain with
(14) 
where and . It is clear that
(15) 
Similarly, applying “differentialencoding” to , we obtain assuming . As we will see later, differentialencoding helps to mitigate the correlations among CPM symbols which facilitates the design of CPM training waveforms with autocorrelation sidelobes close to zero. Then, send and for CPM modulation following the transmission structure shown in Fig. 1. The “ tail bits”, placed before (or ), are used to return the CPM modulator phase state to zero. Systematic method for the generation of tail bits can be found in [9] using Rimoldi decomposition [21].
Applying as a sequence of CPM symbols, the CPM pseudosymbols can be expressed as follows.

If , we have
(16) 
If , we have
(17)
When , one can see that
(18) 
In this case, (8) holds. This implies that is a periodic transmission of two identical length sequences, which in turn allows the calculation of PACF at the local receiver. The approximated CPM waveform in (4) can be written as
(19) 
Denote by the approximated CPM waveform (after “differential encoding”) corresponding to . Similar to (18), we have
(20) 
for . Also, similar to (19), we obtain
(21) 
for .
Next, we will show and form a quaternary GCP provided that is a binary GCP generated by the DavisJedwab construction (see Lemma 1). In the context of Lemma 1, let
(22) 
and be the GBFs of C and D, respectively, where ( as should be divisible by 4). Lifting these two GBFs from to and noting that , the corresponding GBFs of and can be expressed as
(23) 
and , respectively. It is clear that satisfy the GBF forms in Lemma 1. Thus, and are a quaternary GCP. Applying (12) to (19) and (21), we assert that
(24) 
Consider the CPM training waveform over and let
(25) 
At the receiver, and are taken as two local reference waveforms for correlation with and , respectively. This is because (and ) will be spread into the time window of [and ] owing to the multipath propagation. Finally, we have the following assertion.
(26) 
Iv Simulation Results
In the context of Lemma 1, let (i.e., ) and . Consider two GCPs, with and for GCP 1, and and for GCP 2. The resultant GCPs are given below.
Applying differential encoding to GCPs 1 and 2, we obtain differentiallyencoded pairs, denoted by “DiffGCP 1” and “DiffGCP 2”, respectively. Each pair will be sent as and (see Section III) for CPM modulation following the transmission structure in Fig. 1. For comparison, differential encoding is also applied to GSM sequence [14]. The resultant sequence is referred to as “DiffGSM” and will be sent as only for CPM training. For simulation, we consider the GMSK frequency pulse below.
where and . Considering GMSK frequency pulse with and truncated frequency pulse length of , we obtain CPM training waveform , where in this example. The (normalized) autocorrelation magnitudes of CPM waveforms, i.e., , are shown in Fig. 2. One can see that the proposed training waveforms 1 and 2 (corresponding to DiffGCPs 1 and 2, respectively) exhibit autocorrelation sidelobes close to zero for timeshifts larger than . In contrast, the training waveform from the DiffGSM sequence exhibits considerably large autocorrelation sidelobes when timeshift is larger than . To show the effectiveness of differentialencoding, we also depict the autocorrelation magnitudes of uncoded GCPs 1 and 2, and uncoded GSM sequence. It is shown that CPM waveforms after differentialencoding exhibit lower autocorrelation sidelobes compared to the uncoded ones.
Next, we apply a set of CPM training waveforms, which are normalized to have identical energy in the transmission, for estimation of a 16path channel (separated by integer symbols duration) having uniform power delay profile. Specifically, we consider , where
’s are complexvalued Gaussian random variables with zero mean and
. These CPM training waveforms are generated based on DiffGCPs 1 and 2, DiffGSM, uncoded random sequences, “DiffRand” sequences (i.e., differentiallyencoded random “onthefly” sequences), uncoded HosseiniPerrins (HP) sequence [15], and “DiffHP” sequence. Using least squares (LS) estimator, comparison of channel estimation meansquarederrors (MSEs) of different CPM training waveforms is shown in Fig. 3. The CPM training waveforms from uncoded random sequences, DiffRand sequences, uncoded HP sequence, and DiffHP sequence result in relatively higher MSEs due to their autocorrelation sidelobes with larger variations and rankdeficient (sometimes) LS estimator. The GSM training waveform leads to MSE performance 8.5dB away from the CramérRao lower bound (CRLB)^{1}^{1}1This CRLB is derived for perfect CPM training waveform with zero autocorrelation sidelobes for all the nonzero timeshifts [22].. The MSEs of using the proposed training waveforms (based on differentially encoded GCPs) exhibit MSEs much closer to CRLB, as close as 4dB from the CRLB for the case of “DiffGCP 1” (compared to 5dB distance for “DiffGCP 2”). This is understandable as the training waveform from DiffGCP 1 displays the best autocorrelation performance with uniformly low sidelobes, as shown in Figure 2. It should be noted that the estimates of and are more likely to suffer from larger MSEs due to the rolloff autocorrelation sidelobes at (see Figure 2).Furthermore, under the same frequencyselective channel model, Fig. 4 compares the uncoded biterrorrates (BERs) of GMSK systems using perfect channel state information (CSI) and estimated CSI from the proposed CPM training waveforms. Here, we follow the CPM receiver design developed in [9], where singlecarrier frequencydomain equalization (SCFDE) with minimum MSE is adopted. It is seen that the BER curves corresponding to the two DiffGCPs are very close, and are about 1.4dB to the BER curve with perfect CSI. On the other hand, the BER curve corresponding to DiffGSM displays 5dB distance to that with perfect CSI.
V Conclusions
A systematic construction of CPM training waveform displaying autocorrelation sidelobes close to zero has been proposed. Our idea is to apply differentialencoding to a GCP and then send the encoded component sequences one after another, separated by tail bits, to the CPM modulator. Note that this work is focused on binary CPM with modulation index of . It would be interesting to extend the presented training waveform design to generic CPM schemes (e.g., nonbinary modulation orders, rational/integer modulation indices) using CPM decompositions reported in [23] and [24].
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