Continuous phase modulation (CPM) is an attractive nonlinear modulation scheme whose signals exhibit properties of constant envelope and tight spectrum confinement . The first property will allow the transmitter to enjoy high power transmission efficiency as CPM signals have peak-to-average power ratio (PAPR) of (theoretically). Hence, traditional transmission techniques (e.g., [2, 3]) to deal with high PAPR problem in orthogonal frequency-division multiplexing (OFDM) may be avoided when high-rate transmission is not a must. The second property implies less amount of out-of-band power leakage compared to 2- and 4-ary PSK modulations , 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) , military and satellite communications , millimeter communications , and machine-type communications in 5G [7, 8]
. In dispersive channels, frequency-domain equalization (FDE) is needed to suppress the effect of intersymbol interference (ISI), followed by CPM demodulation as in flat-fading 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 .
Despite of a long history of CPM research, less has been understood on the training waveform design of CPM. According to , the 8 training binary sequences defined in GSM standard  have been found by computer search over all possible binary sequences. In 2013, Hosseini and Perrins studied the training sequence design of burst-mode CPM over additive white Gaussian noise (AWGN) channels [15, 16]. However, the CPM training sequences proposed in  may not be applicable in frequency-selective channels (as will be shown in Section IV). Motivated by this, we target at a systematic construction of CPM training waveforms for frequency-selective channels. Our main idea is to apply differential encoding to Golay complementary pair (GCP) whose aperiodic autocorrelation sums diminish to zero for all the non-zero time-shifts . Taking advantage of Laurent decomposition , we show that the resultant CPM training waveform (with modulation index of ) displays autocorrelation sidelobes close to zero.
Ii-a Introduction to CPM
Let . An equivalent lowpass -ary CPM waveform is expressed as , where () is the time-varying 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 phase-shaping waveform is defined as the integral of the frequency-shaping pulse of duration , i.e., , with for and for . is called full-response if and partial-response when . Note that can be written as
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 partial-response CPM signal can be represented as a superposition of a number of pulse-amplitude-modulated (PAM) pulses . To introduce this, we first define
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
where , for , and In general, is the most important PAM pulse as it carries more than of the total signal energy . Therefore, the CPM signal can be approximated as
For ease of presentation, are called CPM pseudo-symbols.
Ii-B Introduction to Golay Complementary Pair (GCP)
Denote by the aperiodic auto-correlation function (AACF) of which is defined as
Let be a pair of sequences with identical length of . is called a GCP  if for any . Note that compared to conventional one-dimensional sequences, the two constituent sequences in a GCP work in a cooperative way to ensure that their out-of-phase aperiodic autocorrelations sum to zero.
Let be the periodic autocorrelation function (PACF) of C at time-shift . 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).
Let and . The associated sequences of are
(Davis-Jedwab Construction of GCP ) Let
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 non-zero elements satisfying and
The PACF of the CPM approximation [cf. (4)] over is defined as
In addition, the AACF of , the most significant pulse in Laurent’s decomposition, is defined as
where if or . Hence, if . By [References, (6.45b)], we have
(12) implies that the PACF of [i.e., ] is zero for any if
Nevertheless, it is hard to find sequence satisfying (5) and (13) simultaneously. Because of this, we consider sequence pair and over . Applying the “differential-encoding”, denoted by , to , we obtain with
where and . It is clear that
Similarly, applying “differential-encoding” to , we obtain assuming . As we will see later, differential-encoding 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  using Rimoldi decomposition .
Applying as a sequence of CPM symbols, the CPM pseudo-symbols can be expressed as follows.
If , we have
If , we have
When , one can see that
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
Denote by the approximated CPM waveform (after “differential encoding”) corresponding to . Similar to (18), we have
for . Also, similar to (19), we obtain
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
Consider the CPM training waveform over and let
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.
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 differentially-encoded pairs, denoted by “Diff-GCP 1” and “Diff-GCP 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 . The resultant sequence is referred to as “Diff-GSM” 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 Diff-GCPs 1 and 2, respectively) exhibit autocorrelation sidelobes close to zero for time-shifts larger than . In contrast, the training waveform from the Diff-GSM sequence exhibits considerably large autocorrelation sidelobes when time-shift is larger than . To show the effectiveness of differential-encoding, we also depict the autocorrelation magnitudes of uncoded GCPs 1 and 2, and uncoded GSM sequence. It is shown that CPM waveforms after differential-encoding 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 16-path channel (separated by integer symbols duration) having uniform power delay profile. Specifically, we consider , where
’s are complex-valued Gaussian random variables with zero mean and. These CPM training waveforms are generated based on Diff-GCPs 1 and 2, Diff-GSM, uncoded random sequences, “Diff-Rand” sequences (i.e., differentially-encoded random “on-the-fly” sequences), uncoded Hosseini-Perrins (HP) sequence , and “Diff-HP” sequence. Using least squares (LS) estimator, comparison of channel estimation mean-squared-errors (MSEs) of different CPM training waveforms is shown in Fig. 3. The CPM training waveforms from uncoded random sequences, Diff-Rand sequences, uncoded HP sequence, and Diff-HP sequence result in relatively higher MSEs due to their autocorrelation sidelobes with larger variations and rank-deficient (sometimes) LS estimator. The GSM training waveform leads to MSE performance 8.5dB away from the Cramér-Rao lower bound (CRLB)111This CRLB is derived for perfect CPM training waveform with zero autocorrelation sidelobes for all the non-zero time-shifts .. 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 “Diff-GCP 1” (compared to 5dB distance for “Diff-GCP 2”). This is understandable as the training waveform from Diff-GCP 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 roll-off autocorrelation sidelobes at (see Figure 2).
Furthermore, under the same frequency-selective channel model, Fig. 4 compares the uncoded bit-error-rates (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 , where single-carrier frequency-domain equalization (SC-FDE) with minimum MSE is adopted. It is seen that the BER curves corresponding to the two Diff-GCPs are very close, and are about 1.4dB to the BER curve with perfect CSI. On the other hand, the BER curve corresponding to Diff-GSM displays 5dB distance to that with perfect CSI.
A systematic construction of CPM training waveform displaying autocorrelation sidelobes close to zero has been proposed. Our idea is to apply differential-encoding 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., non-binary modulation orders, rational/integer modulation indices) using CPM decompositions reported in  and .
-  J. B. Anderson, T. Aulin, and C. E. Sundberg, Digital Phase Modulation, New York: Plenum, 1986.
-  T. Jiang and G. Zhu, “Complement block coding for reduction in peak-to-average power ratio of OFDM signals,” IEEE Communi. Mag., vol. 43, no. 9, S17-S22, Sept. 2005.
-  T. Jiang and Y. Wu, “An Overview: Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals,” IEEE Trans. Broadcast., vol. 54, no. 2, pp. 257-268, Jun. 2008.
-  Digital Cellular Telecommunications System, Modulation (GSM 05.04), ETSI standard ETS 300959, pp. 1-10, May. 1997.
-  A. Emmanuele, A. Zanier, F. Boccolini, et al., “Spread-spectrum continuous-phase-modulated signals for satellite navigation,” IEEE Trans. Aeros. Electro. Systems, vol. 48, no. 4, pp. 3234-3249, Oct. 2012.
-  R. C. Daniels and R.W. Heath, Jr., “60 GHz wireless communications: emerging requirements and design recommendations,” IEEE Veh. Technol. Mag., vol. 2, no. 3, pp. 41-50, Sept. 2007.
-  M. Wylie-Green, E. Perrins and T. Svensson, “Introduction to CPM-SCFDMA - a novel multiple-access power-efficient transmission scheme,” IEEE Trans. Commun., vol. 7, no. 59, pp. 1904-1915, Jul. 2011.
-  C. Bockelmann et al, “Massive machine-type communications in 5G: physical and MAC-layer solutions,” IEEE Commun. Magazine, pp. 59-65, Sep. 2016.
-  J. Tan and G. L. Stuber, “Frequency-domain equalization for continuous phase modulation,” IEEE Trans. Wireless Commun., vol. 4, pp. 2479-2490, 2005.
-  F. Pancaldi and G. M. Vitetta, “Equalization algorithms in the frequency domain for continuous phase modulations,” IEEE Trans. Commun., vol. 54, no. 4, pp. 648-658, 2006.
-  W. Van Thillo, F. Horlin, J. Nsenga, V. Ramon, A. Bourdoux, and R. Lauwereins, “Low-complexity linear frequency domain equalization for continuous phase modulation,” IEEE Trans. Wireless Commun., vol. 8, no. 3, pp. 1435-1445, 2009.
-  R. Chayot, N. Thomas, C. Poulliat, M.-L. Boucheret, N. V. Wambeke and G. Lesthievent, “Channel estimation and equalization for CPM with appliation for aeronautical communications via a satellite link,” in IEEE Military Communications Conference (MILCOM), Baltimore, MD, pp. 888-893, Oct. 2017.
-  M. Drutarovsky, “GSM channel equalization algorithm - modern DSP coprocessor approach,” Radioengineering, vol. 8, no. 4, pp. 26-31, Dec. 1999.
-  Digital cellular telecommunications system (Phase 2), Multiplexing and multiple access on the radio path (GSM 05.02), ETSI standard ETS 300574, pp. 1-39, Apr. 1997.
-  E. Hosseini and E. Perrins, “The Cramér-Rao bound for training sequence design for burst-mode CPM,” IEEE Trans. Commun., vol. 61, no. 6, pp. 2396-2407, Jun. 2013.
-  E. Hosseini and E. Perrins, “Timing, carrier, and frame synchronization of burst-mode CPM,” IEEE Trans. Commun., vol. 61, no. 12, pp. 5125-5138, Dec. 2013.
-  M. J. E. Golay, “Complementary series,” IRE Trans. Inf. Theory, vol. IT-7, pp. 82-87, Apr. 1961.
-  P. A. Laurent, “Exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses,” IEEE Trans. Commun., vol. COM-34, no. 2, pp. 150-160, Feb. 1986.
-  J. S. Lee and L. E. Miller, CDMA Systems Engineering Handbook, Mobile Communications series, Artech House Publishers, 1998.
-  J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inf. Theory, vol. 45, pp. 2397-2417, Nov. 1999.
-  B. E. Rimoldi, “A decomposition approach to CPM,” IEEE Trans. Inf. Theory, vol. 34, no. 2, pp. 260-270, Mar. 1988.
-  A. Milewski, “Periodic sequences with optimal properties for channel estimation and fast start-up equalziation,” IBM J. Res. Develop., vol. 27, no. 5, pp. 426-439, 1983.
-  U. Mengali and M. Morelli. “Decomposition of M-ary CPM signals into PAM waveforms,” IEEE Trans. Inf. Theory, vol. 41, no. 5, pp. 1265-1275, Sept. 1995.
-  X. Huang and Y. Li, “The PAM decomposition of CPM signals with integer modulation index,” IEEE Trans. Communi., vol. 51, no. 4, pp. 543-546, Apr. 2003.