Continuous phase modulation (CPM) has attracted much attention in recent years for its high power and spectral efficiency, and its constant signal envelope can significantly alleviate the nonlinear effects of high power amplifiers.^[1] The CPM has been widely used in satellite communications and terrestrial mobile cellular systems, ^{[2– 6]} etc. However, achieving good performance relies on precise synchronization, especially with low signal-to-noise ratios (SNRs). Thus, estimators that provide good performance and low complexity are of significant interest.

To perform synchronization, symbol timing and carrier information must be taken into consideration. Both frequency offset and phase offset estimation should be performed as part of carrier synchronization. In this study, we focused only on frequency offset estimation with CPM signals, and in subsequent analysis it was found that our estimation method is independent of timing recovery. Many researchers have proposed algorithms for frequency offset estimation.^{[7– 21]} In Refs.  [7]– [17], traditional phase-locked loop schemes are presented, which employ a feedback structure to track the precise value of the frequency offset, but such approaches require long periods of time for acquisition, and thus, they may not be feasible with burst-mode transmission. In Refs.  [10] and [11], a maximum-likelihood algorithm, the most fundamental feedforward structure, is presented for searching the frequency offset by compensating for a set of discrete points and maximizing the log-likelihood function (LLF). However, the true frequency offset point may be missed if it lies between two searching values. To avoid this situation, numerous discrete points must be used, leading to unacceptable complexity. Some other estimation methods, such as those discussed in Refs.  [12]– [14], estimate frequency offset by operating on the correlation values of the modulated signals. These approaches require high computational resources. Subsequent proposed feedforward estimation methods^{[15– 17]} are variations on the classical methods described above; they reduce the complexity to some extent, however, the level of complexity is still undesirable. In 1999, a dichotomous search frequency offset estimator was proposed by Zakharov and Tozer.^[18] This was the first time that a one-dimensional optimization search method was introduced into frequency offset estimation. This type of estimator seeks the optimum point through an iterative algorithm. Several years later, Aboutanios presented a modified dichotomous search estimator in which the initializations of the iterative algorithm are computed in a more reasonable manner.^[19] Both estimators operate in the frequency domain, meaning that a fast Fourier transform (FFT) must be carried out. To ensure the accuracy of the two types of dichotomous estimators, the order of the FFT operation must be larger than 1.5 times the number of signal points.^{[18– 20]} Subsequently, other similar estimators, such as Gaussian spectrum interpolation, parabolic interpolation, and weighted multipoint interpolation, were introduced to improve the resolution of the FFT frequency measurement.^[21] However, with these methods, practical implementation still involves unsatisfactory levels of complexity.

In this paper, we present a special pilot sequence and an improved dichotomous search frequency offset estimator that can reduce the complexity of the overall system. The proposed estimator first exploits the special pilot sequence, or training sequence, to form a sinusoidal waveform whose frequency information can be easily extracted and which requires no modulation removal. Then, the improved dichotomous search method is introduced to obtain the frequency offset estimation. All these procedures are performed in the time domain, and thus the FFT operation is eliminated. Analysis indicates that our estimator is suitable for nearly all CPM formats since it imposes no restrictions on alphabet and modulation index, except that the modulation index is rational. Analysis and simulation results demonstrate that our estimator exhibits lower complexity without accuracy degradation compared to the existing methods presented in Refs.  [18] and [19].

This paper is organized as follows: Section 2 provides a brief introduction to the burst-mode transmission model and the derivation of LLF with CPM. Section 3 provides the background information related to the traditional dichotomous search method. In Section 4, our improved dichotomous search estimator is described in detail. Simulation results are presented in Section 5 to demonstrate the benefits of our proposed method, and the paper is concluded in Section 6.

In burst-mode transmission, the modulated signal is transmitted in bursts in a deterministic manner that is known by the receiver. The transmission structure is depicted in Fig.  1. Each burst consists of three parts. The first part is the known pilot sequence used to perform parameter estimation. The next part is the unique word (UW), which is utilized to identify the bursts. The last part, referred to as the payload, transmits the unknown source information.

Fig.  1.

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	Fig.  1. Burst-mode transmission model.

It is assumed that the CPM signal is transmitted according to a burst-mode model. The complex baseband CPM signal during the transmission of each burst can be expressed as

where E_s denotes the energy per transmitted symbol, T is the symbol period, and ψ (t, α ) is the phase function expressed as follows:

Here, h is the rational modulation index, and α _i is the data sequence transmitted up to time nT. The data symbols are independent, equally likely sequences belonging to the M – ary bipolar alphabet {± 1, ± 3, … , ± (M − 1)}. The phase response pulse q(t) is a continuous, monotonic function defined as follows:

The frequency response pulse g(t) = dq(t)/dt is nonzero over the interval [0, LT], where L is the correlation length. The schemes most commonly used for g(t) are L-rectangular frequency pulse (LREC), L-raised cosine pulse (LRC), and L-interval Gaussian-minimum shift keying pulse (L-GMSK). The corresponding mathematical expressions are listed in Table  1. In the Appendix A, we demonstrate that the LLF has the same expression for all of these frequency response pulses, and this indicates that our algorithm is suitable for most types of CPM signals.

Table 1.

Table 1.

Table 1. Commonly used schemes for g(t). g(t) Mathematical expression LREC LRC L-GMSK

Table 1. Commonly used schemes for g(t).

In Table  1, B represents the − 3  dB bandwidth of the Gaussian pulse shape, and Q(x) is the Q-function expressed as follows:

The received signal has the form

where

In Eqs.  (5) and (6), v, τ , and φ are the frequency offset, the timing delay, and the channel phase, respectively. Both τ ∈ (− T/2, T/2) and φ ∈ (− π , π ) obey uniform distribution, w(t) is a zero-mean complex white Gaussian noise term with two-sided power spectral density N₀/2. In Eq.  (6), the signal component s(t) is a function of time combined with a set of parameters γ . This set can consist of v,   τ ,   φ , and data symbols; however, it does not necessarily contain all of them at once. In our system, the data symbols are known and are not included in γ because a training sequence is transmitted; thus, only v,   τ , and φ are included. To emphasize the dependence of the signal on unknown parameters, we adopt the notation s(t, γ ) in place of s(t) and rewrite Eq.  (5) as

If we assume that γ ̃ is a hypothetical value set of γ , the joint LLF for the synchronization parameters takes the following form:^[9]

where N is the length of the pilot sequence and Re is the real operator. CPM has a constant envelope, and thus the second term in Eq.  (8) becomes a constant and can be omitted for convenience. It is shown in the appendix that s_CE(t) is a sinusoidal waveform with frequency f₁ = h/(2T) if an all-one training sequence is transmitted. Suppose the initial phase of the sinusoidal waveform is θ , which is deterministic for a given CPM format, and s_CE(t) can be rewritten as

Substituting Eq.  (9) into Eq.  (8) and ignoring the immaterial factor, we can obtain

where φ ̃ ₁ = − 2π f₁τ ̃ + φ ̃ + θ is a joint phase related to τ ̃ , φ ̃ , and θ .

The parameters are decoupled now, and thus the maximization of the LLF becomes straightforward. The term that corresponds to the frequency offset in Eq.  (10) is defined as

and thus, maximizing the LLF is equivalent to maximizing the objective function

Discrete-time estimation can be accomplished by sampling the continuous-time function. The number of samples per symbol interval is denoted by N_s, and the discrete-time format of Eq.  (12) can be written as

where the parameters ṽ and f₁ in Eq.  (13) have been normalized to the symbol interval.

From Eqs.  (10)– (13) it is clear that parameter τ ̃ can be incorporated into the joint phase φ ̃ ₁ and does not appear in the objective function; therefore, our algorithm for frequency offset estimation is independent of timing recovery. Furthermore, because the objective function has the form of a sinusoid, the operation of modulation removal can be bypassed. This represents one aspect of the reduced complexity provided by our estimator. In Section 4, this property will be discussed in detail.

To provide supporting information for our later analysis, a brief introduction to the dichotomous and modified dichotomous search frequency offset estimator is presented; readers who are interested in a more detailed discussion can refer to Refs.  [18] and [19]. This estimator first obtains the maximum bin of a K-point FFT, then a dichotomous (or modified dichotomous) search method is used to derive the true frequency offset value. The index of the maximum bin is denoted by m, and the procedure of the algorithm is summarized below.

Step 1 Perform modulation removal to obtain a sinusoid as

Step 2 Let K = lNN_s, then calculate S = FFT(x, K) and Y = | S(n)| ².

Step 3 Find .

Step 4 (i) For dichotomous search method, set Δ = 1, Y_{− 1} = Y(m − 1), Y₀ = Y(m), and Y₁ = Y(m + 1). (ii) For modified dichotomous search method, set Δ = 0.75, Y_{− 1} = Y(m − 1), Y₀ = Y(m), and Y₁ = Y(m + 1). If Y₁ > Y_{− 1}, then

else

Step 5 For Q iterations do

If Y₁ > Y_{− 1}, then

else

Update such that

Step 6 Obtain the normalized frequency offset value

The dichotomous search frequency offset estimator was first proposed by Zakharov and Tozer.^[18] A binary search method is used to locate the true FFT peak (Steps 1– 3), then the FFT coefficients on both sides of the maximum are compared to halve the frequency resolution (Step 4). The procedure is repeated for Q iterations in order to ensure the accuracy of the estimation (Step 5). Later, Aboutanios modified the method simply by making it possible to calculate the initialization of the iterations more reasonably. However, to ensure that the variance of the estimator achieves the Cramer– Rao bound (CRB), l ≥ 1.5 and Q ≥ 10 are necessary, and this leads to high computational load. We will now present an improved algorithm that reduces the complexity of the dichotomous search method. For the sake of brevity in the forthcoming comparisons, we will refer to the conventional method as FFT & dichotomous and our proposed method as improved dichotomous.

To clarify the relationship between our algorithm and the conventional algorithms described in the previous section, we will use the same format when presenting the procedure of our improved estimator. To begin with, an initialized range must be given for use in subsequent processing. We specify the range as (v_{− 1}  ,   v₁) with v_{− 1} < v₁.

Step 1 Initialize the search range (v_{− 1}  ,   v₁).

Step 2 Calculate Y_{− 1} = f (v_{− 1}) and Y₁ = f (v₁) using Eq.  (13).

Step 3 For Q iterations, do

If Y_{− 1} < Y₁, then

else

Step 4 Obtain the normalized frequency offset value

If our estimator is compared to the conventional estimators described in Section 3, it is clear that the complexity is greatly reduced owing to the elimination of modulation removal and FFT operations. However, the reduced complexity is achieved at the expense of a small estimation range. Based on one-dimensional optimization search theory, the dichotomous search method is available only if the objective function is unimodal; ^{[22, 23]} thus, the estimation range is confined to the unimodal interval of the objective function. Substituting Eqs.  (5) and (6) into Eq.  (12) and rearranging the expression, eventually we obtain

where w̃ (t) is Gaussian noise. If the noise term is ignored, the objective function can be simplified as

The objective function reaches the maximum at ṽ = v. However, because the noise is always present, the maximum bin is more accurately written as ṽ = v + v_w, where v_w is a random term representing the noise. Thus, we have

where E represents the expectation operator. Equation  (16) suggests that our estimator is unbiased. In Section 5, the simulation results will be presented to corroborate this conclusion.

The curve of the normalized objective function is shown in Fig.  2; based on this curve, we know that in order to ensure the unimodality of the function, the estimation range must be confined to the main lobe, that is

Fig.  2.

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	Fig.  2. Curve of the normalized objective function.

From Eq.  (17), it is clear that the estimation range is small compared to that of the FFT & dichotomous method, whose estimation range is (− 1/2T,   1/2T). However, this is a desirable trade-off because in most cases the residual offset is much smaller than ± 1/2T.

We proposed an improved dichotomous search frequency offset estimator for burst-mode CPM. The estimator can operate independently of timing recovery and exhibits relatively low outlier. Most importantly, it offers reduced implementation complexity. Our proposed estimator eliminates the modulation removal and FFT operations, and it offers faster convergence rate compared to the dichotomous and modified dichotomous methods.