A Carrier-Based Gardner Timing Synchronization Algorithm for BPSK Signal in Maritime Communication

Abstract

Wireless communication at sea is an essential way to establish a smart ocean. In the communication system, however, signals are affected by the carrier frequency offset (CFO), which results from the Doppler effect and crystal frequency offset. The offset deteriorates the demodulation performance of the communication system. The conventional Gardner bit-synchronization algorithm performs timing synchronization on the baseband, but it fails to solve the problem of carrier frequency offset. In this paper, a carrier-based timing-synchronization Gardner algorithm was proposed. The algorithm performed error detection in the carrier signal to estimate the synchronization error in real time and the time-domain expansion in the passband signal. Based on the estimated expansion factor, the algorithm located the sampling points of the ideal signal and re-interpolated all the sampling points on the passband to recover the passband signal without Doppler. This algorithm can solve both the frequency shift and time-domain expansion caused by the Doppler effect without further CFO estimation and compensation, thus reducing the overall computational complexity of the system. The simulations showed that the proposed algorithm greatly improves the ability of the communication system to combat Doppler compared to the conventional Gardner algorithm. The algorithm is primarily designed for binary phase-shift keying (BPSK)-modulated signals under the additive white Gaussian noise (AWGN) channel and can be applied to various maritime communication scenarios.

1. Introduction

Medium-wave communication is a radio communication technique that uses electromagnetic waves with a wavelength of 1000 m~100 m (a frequency range of 300~3000 kHz) and is an important component of maritime communication. Medium-wave communication is widely used for maritime emergency rescue, aircraft and ship communication, and radio navigation. However, in digital communication systems, the received carrier signal is affected by the Doppler effect and the local crystal frequency offset, which creates frequency offset in the frequency domain and changes the length of the signal in the time domain. To synchronize the carrier signal with the local clock signal, carrier synchronization and bit synchronization are required at the receiver. Bit synchronization, also known as symbol synchronization, extracts the timing pulse sequence for sampling judgments at the receiver side and is the foundation for accurate sampling judgments.

Bit-synchronization algorithms can be divided into feedforward synchronization algorithms and feedback synchronization algorithms according to the algorithm structure. It can also be classified as data-assisted and non-data-assisted algorithms [1,2]. The classical synchronization algorithms include the early-late gate (E-LG) algorithm [3], M&M (Mueller & Muller) algorithm [4], maximum likelihood algorithm [5], wave division multiplex (WDM) algorithm [6], and Gardner algorithm [7]. Among them, refs. [3,4,6,7] belong to timing recovery synchronization algorithms. The Gardner timing recovery bit synchronization algorithm requires only two resamples in one symbol for error detection. The carrier phase does not affect the synchronization judgment performance, data assistance and judgment feedback are not required, and it is easy to implement in high-speed digital communication systems. Hence, it is widely used in engineering practice. Gardner algorithm is continuously applied to quadrature amplitude modulation (QAM) signals [8,9,10], multiple phase shift keying (MPSK) signals [11,12], Gaussian minimum frequency shift keying (GMSK) signals [13], and other modulation systems [14,15,16], with an ever-expanding range of applications.

The Gardner algorithm works by resampling the rough demodulated baseband signal. The feasibility of the algorithm depends on an assumption that the phase deviation of each sampled value is constant, which means there is no carrier frequency deviation. When the assumptions are not satisfied, the performance of the algorithm may degrade. As a result, improved algorithms for correcting frequency deviation are proposed [10]. In addition, Gardner algorithm introduces a certain amount of self-source noise when the roll-off factor (ROF) is small and adjacent symbols are the same, which affects the performance of bit synchronization.

Improved algorithms using pre-filtering method [8,17] and compensation method [18,19] have been gradually proposed for Gardner algorithm with large self-jitter in systems with small ROF. However, the complexity of these improved algorithms is relatively high. For the method applicable to MPSK modulation proposed in [11], ref. [20] proposed an improved algorithm applicable to higher-order MPSK modulation. Ref. [21] improved the Gardner phase detector (GPD) so that the algorithm still performs well in QAM modulated systems with the ROF close to 0.

When adjacent symbols are the same, the symbol difference is always zero and the Gardner algorithm cannot capture the timing error, which makes the Gardner algorithm generate large timing jitter [22] and leads to a high level of self-noise in the system. Some improved algorithms were proposed to address this issue [23]. Correction algorithms [9] and preprocessing algorithms [24] were proposed to mitigate self-noise in the Gardner algorithm. To solve the problem of long synchronization timing time, an improved method that increases the gain control processing was proposed in [25]. Other research directions on bit synchronization and the Gardner algorithm include frequency-domain bit-synchronization algorithms [26,27], parallel processing on field programmable gate array (FPGA) [28], the effect of loop delay [29] on system performance, and performance comparisons with various algorithms [30,31].

The existing improved methods of the Gardner algorithm all detect timing errors based on the baseband signal, which is vulnerable to residual-frequency offset. In addition, the system needs to perform additional frequency-offset estimation and compensation. Based on the characteristics of the carrier signal, this paper proposed a Carrier-Based Gardner (CB-Gardner) algorithm for timing synchronization. The algorithm can directly perform error detection on the carrier signal in the passband, eliminate the carrier frequency offset caused by Doppler effect and local crystal frequency bias, and perform timing synchronous sampling. The main improvements of the algorithm include: (1) changing the error-detection point and the error-calculation formula; (2) simplifying the operation of the numerically controlled oscillator (NCO); and (3) resampling all sampling points on the passband using an interpolator. This algorithm can be applied to single-carrier signals with binary phase-shift keying (BPSK) modulation, especially in wireless communication systems.

2. Problem Formulation

2.1. Problem Statement

In this paper, the main problem we address is the effect of the Doppler on the carrier signal. In a communication system, the carrier frequency of the signal may change due to the relative motion between the transmitter and the receiver. The carrier frequency with Doppler effect is

$$f^{\prime }=(\frac{v\pm v_0}{v\mp v_s})f_c$$

(1)

where $f^{\prime }$ denotes the observed signal frequency, $v$ denotes the wave propagation speed, $v_0$ is the movement speed of the receiver, and $v_s$ denotes the movement speed of the transmitter source, $f_c$ is the original frequency of the signal. The ratio of $f^{\prime }$ and $f_c$ is defined as

$$\mathsf{\lambda }=f^{\prime }/f_c=(f_c+f_{△})/f_c=1+\mathsf{\delta }$$

(2)

where $f_{△}$ denotes the absolute Doppler shift, and the Doppler factor $\mathsf{\delta }=f_{△}/f_c$ denotes the magnitude of the Doppler shift with respect to the carrier frequency.

In the time domain, the Doppler effect causes the length of the received signal to be compressed or broadened, resulting a length ratio of $1/\mathsf{\lambda }$ between the received signal and the original signal. This length change corresponds to a change in the sampling rate at the receiver. If the original sampling rate is $F_s$, then the actual sampling rate is

$$F_s^{\prime }=F_s/\mathsf{\lambda }$$

(3)

Thus, in the time domain, the effect of Doppler is equivalent to signal resampling. The frequency shift caused by Doppler prevents the receiver from performing accurate carrier synchronization. The change in sampling rate also affects bit synchronization and thus degrades the decoding performance. Considering a single-carrier communication system with BPSK modulation, if the transmitted symbol is $x_n$, the transmitted signal can be expressed as

$$s_{\mathrm{BPSK}}(t)=\left\{\begin{array}{l}A\cos (w_{c}t) , x_n=0\\ -A\cos (w_{c}t) , x_n=1\end{array}\right.$$

(4)

The CB-Gardner algorithm proposed in this paper processes the passband signal in (4). This method can also be extended to higher-order modulations.

2.2. Solutions to Doppler

To eliminate the Doppler effects, two methods are usually used.

Method 1: Resampling the passband signal directly. This method requires estimating the Doppler factor. One approach is to calculate the Doppler factor by measuring the change of the length of signal through a synchronization method in the time domain and resample the passband signal. This method assumes a constant Doppler factor throughout the signal time, which may not hold true for time-varying Doppler. As a result, the performance of this method may degrade.

Method 2: Carrier frequency offset (CFO) estimation and compensation, and bit synchronization. The former eliminates the frequency offset of the signal, allowing for accurate bit synchronization, which eliminates the spreading effect of the signal in the time domain. The Gardner algorithm is a canonical example of bit-synchronization algorithms.

Figure 1 shows the closed-loop feedback structure of the Gardner algorithm [32]. The loop consists of an interpolator, a timing error detector (TED), a loop filter (LF) and an NCO. The input signal is the baseband signal after down-conversion, and the output is the baseband symbol after decimation.

Figure 2 shows the principle of the Gardner algorithm for TED on the baseband signal. The figure depicts a baseband signal containing two symbols, where $y\left(n-1\right)$, $y\left(n-1/2\right)$ and $y\left(n\right)$ are the three detection points required to detect timing error. $y\left(n-1\right)$ and $y\left(n\right)$ are the optimal sampling instants for the two adjacent symbols in the baseband, and $y\left(n-1/2\right)$ is the midpoint of these two adjacent symbols. Therefore, the Gardner algorithm performs two resampling points per symbol on average.

Let $x\left(n-1\right)$ and $x\left(n\right)$ be the symbols adjudicated based on $y\left(n-1\right)$ and $y\left(n\right)$. The error detection equation of Gardner algorithm is

$$er{r}_n=y(n-\frac{1}{2})\ast [x(n-1)-x(n)]$$

(5)

where $er{r}_n$ contains the information that the signal is extended or compressed and therefore Doppler can be detected.

As seen in Figure 2, the shape of the baseband signal bears some similarity to the cosine signal. This similarity serves for timing error detection in CB-Gardner, which can eliminate the impact of Doppler directly on the passband while maintaining the real-time iterative property of the Gardner algorithm.

3. Materials and Methods

The proposed CB-Gardner in this paper is a feedback loop consisting mainly of two interpolators, a timing error detector, a loop filter, and a numerically controlled oscillator. This feedback loop is executed once per symbol to counteract the time-varying Doppler. Figure 3 illustrates the structure and model of the proposed CB-Gardner.

The fundamental difference between CB-Gardner and Gardner algorithm lies in the error detector, which is mainly reflected in the difference of the location of error detection point and the change of the error detection equation. Error detection in the CB-Gardner can be divided into two steps: finding the error-detection point and calculating the timing error. An interpolator is also needed before calculating the timing error. Next, we will describe the detailed process of the algorithm.

3.1. Error Detection Point

For a BPSK signal, (4) shows that the signal can be represented as a cosine waveform. However, the signal amplitude around the symbol instant decreases gradually due to the forming filter. Nevertheless, since the symbol period is usually much larger than the carrier period for medium wave, the amplitude of the filter can be approximated as constant when considering only the waveform of the first carrier period after the symbol instant. At this point, the passband waveform can be approximated as a cosine function.

Observing the baseband waveform in Figure 2, we find that its shape is similar to that of a cosine waveform. Therefore, the three points used for error detection in the CB-Gardner are chosen with a phase difference of $\mathsf{\pi }/2$, where the phase of the first detection point belongs to $[0,\frac{\mathsf{\pi }}{2},\mathsf{\pi },\frac{3\mathsf{\pi }}{2}]$, and all detection points are located after the symbol instant. Specifically, the phase values of the three error-detection points ($y_1,y_2,y_3$) are

$$\left\{\begin{array}{l}{\mathsf{\phi }}_{y_1}\in [0,\frac{\mathsf{\pi }}{2},\mathsf{\pi },\frac{3\mathsf{\pi }}{2}]\\ {\mathsf{\phi }}_{y_2}={\mathsf{\phi }}_{y_1}+\frac{\mathsf{\pi }}{2}\\ {\mathsf{\phi }}_{y_3}={\mathsf{\phi }}_{y_1}+\mathsf{\pi }\end{array}\right.$$

(6)

To minimize the decay of the cosine waveform, we choose the optimal sampling instant for each symbol as the reference and position of $y_1$ as close to the symbol instant as possible. Figure 4 shows the positions of the three detection points ($y_1,y_2,y_3$) in the improved algorithm when ${\mathsf{\phi }}_{y_1}=0$. $S_1$ is the optimal judgment instant for one symbol and $S_2$ is the optimal judgment instant for another symbol after $S_1$. For symbol $S_1$, the nearest $y_1$ is located at the instant with a phase of 0. The phases of $y_2, y_3$ are then $\mathsf{\pi }/2$ and $\mathsf{\pi }$, respectively.

Suppose the symbol rate is $F_b$, and the symbol period is $T=1/F_b$. We set the first symbol instant to $t_1=0$, its sampling point to $P_1=1$. For the nth symbol instant, we have $t_n=\left(n-1\right)T$ and its sampling points are $P_n$. The phase of the nth symbol instant is

$${\mathsf{\phi }}_n=w_c{t}_n+{\mathsf{\phi }}_{x_n}=\left\{\begin{array}{l}(n-1)w_{c}T , \hspace{1em}{x}_n=0\\ (n-1)w_{c}T+\mathsf{\pi }, x_n=1\end{array}\right.$$

(7)

where $w_c=2\mathsf{\pi }{f}_c$. At this point, the instant of the nearest $y_1$ after $t_n$ is

$$\begin{array}{l}{t}_{y_1}=t_n+\{\frac{\mathsf{\pi }}{2}-[{\mathsf{\phi }}_n\mathrm{mod}(\frac{\mathsf{\pi }}{2})]\}/w_c\hfill \\ \hspace{1em} =t_n+\{\frac{\mathsf{\pi }}{2}-[(n-1)w_{c}T]\mathrm{mod}(\frac{\mathsf{\pi }}{2})\}/w_c\hfill \end{array}$$

(8)

where the time difference $t_{y_1}$ with $t_n$ is $△t_{y_1,n}=\{\frac{\mathsf{\pi }}{2}-[(n-1)w_{c}T]\mathrm{mod}(\frac{\mathsf{\pi }}{2})\}/w_c$. For the current nth symbol, we can use the estimated ${\mathsf{\lambda }}_{n-1}$ from the previous symbol as a priori information. Considering the variation of the signal length due to Doppler, the number of sampling points corresponding to $△t_{y_1,n}$ is

$$\begin{array}{l}△p=△t_{y_1,n}{F}_s/{\mathsf{\lambda }}_{n-1}\\ \hspace{1em} =\{\frac{\mathsf{\pi }}{2}-[(\mathrm{n}-1)w_{c}T]\mathrm{mod}(\frac{\mathsf{\pi }}{2})\}/w_c\ast F_s/{\mathsf{\lambda }}_{n-1}\\ \hspace{1em} =1/{\mathsf{\lambda }}_{n-1}\ast \{\frac{\mathsf{\pi }}{2}-[(\mathrm{n}-1)w_{c}T]\mathrm{mod}(\frac{\mathsf{\pi }}{2})\}/(2\mathsf{\pi })\ast \frac{F_s}{f_c}\\ \hspace{1em} =1/{\mathsf{\lambda }}_{n-1}\ast spc\ast \{\frac{\mathsf{\pi }}{2}-[(\mathrm{n}-1)w_{c}T]\mathrm{mod}(\frac{\mathsf{\pi }}{2})\}/(2\mathsf{\pi })\end{array}$$

(9)

where $spc=F_s/f_c$ denotes the average number of samples per carrier cycle. Therefore, the positions of the error-detection points ($y_1,y_2,y_3$) are

$$\left\{\begin{array}{l}{p}_{y_1,n}=P_n+1/{\mathsf{\lambda }}_{n-1}\ast spc\ast \{\frac{\mathsf{\pi }}{2}-[(n-1)w_{c}T]\mathrm{mod}(\frac{\mathsf{\pi }}{2})\}/(2\mathsf{\pi })\\ p_{y_2,n}=p_{y_1,n}+1/{\mathsf{\lambda }}_{n-1}\ast spc/4\\ p_{y_3,n}=p_{y_1,n}+1/{\mathsf{\lambda }}_{n-1}\ast spc/2\end{array}\right.$$

(10)

Once we get the positions of all three error detection points, we can obtain their corresponding values using the interpolation method, which will be described later.

3.2. Timing Error Detection

Unlike (5), the CB-Gardner performs error detection by utilizing three sampled values on the passband instead of determined symbols. When ${\mathsf{\phi }}_{y_1}\in [0,\mathsf{\pi }]$, for example ${\mathsf{\phi }}_{y_1}=0$, as shown in Figure 5, and assuming that the position of $y_1$ is relatively accurate, the signal may broaden due to the effect of Doppler (i.e., $\mathsf{\lambda }<1$), and the detection points $y_2$ and $y_3$ may shift.

Referring to the error detection Equation (5) of the Gardner algorithm, we define the detection error as

$$er{r}_n=y_2\ast (y_1-y_3)$$

(11)

It can be inferred that $er{r}_n>0$, which leads to a decrease in the control word in the NCO. Consequently, the NCO estimates that ${\mathsf{\lambda }}_n<1$. Conversely, when $\mathsf{\lambda }>1$, the NCO estimates that ${\mathsf{\lambda }}_n>1$. Thus, $er{r}_n$ accurately indicates the directional information of Doppler and can be used as timing synchronization error. But when ${\mathsf{\phi }}_{y_1}\in [\frac{\mathsf{\pi }}{2},\frac{3\mathsf{\pi }}{2}]$, for example, ${\mathsf{\phi }}_{y_1}=\frac{\mathsf{\pi }}{2}$, if $\mathsf{\lambda }<1$, as shown in Figure 6, according to (11), we can see that $er{r}_n<0$ at this point. This will give the NCO incorrect information of ${\mathsf{\lambda }}_n>1$.

Therefore, we need to correct (11). To differentiate between the two cases of ${\mathsf{\phi }}_{y_1}\in [0,\mathsf{\pi }]$ and ${\mathsf{\phi }}_{y_1}\in [\frac{\mathsf{\pi }}{2},\frac{3\mathsf{\pi }}{2}]$, a simple method is to compare the absolute values of $y_1$ and $y_2$ when the signal-to-noise ratio (SNR) is not very low. Thus, the final equation for CB-Gardner’s error detection is

$$er{r}_n=\left\{\begin{array}{l}{y}_2\ast (y_1-y_3) , \left|y_1\right|>\left|y_2\right|\\ y_2\ast (y_3-y_1) , \left|y_1\right|<\left|y_2\right|\end{array}\right.$$

(12)

In this case, the sign of $er{r}_n$ will correctly indicate the directional information of Doppler. In addition, $er{r}_n$ also contains information about the magnitude of the Doppler, enabling the NCO to accurately detect both the magnitude and direction of the Doppler. Since the error detection is carrier-based and independent of the specific symbol, the CB-Gardner does not fail even when there are a large number of consecutive identical symbols.

For BPSK signals, (11) is used for error detection, while for quadrature phase shift keying (QPSK) and higher-order modulation methods, (12) is required.

3.3. Loop Filter

For the nth symbol, once the synchronization error $er{r}_n$ is detected by the error detector, it is fed into the loop filter. The loop filter here is identical to the one used in the Gardner algorithm, which is a second-order proportional integral filter. The output of the filter can be obtained by

$$E_{LF,n}=c_{1}er{r}_n+c_2{\displaystyle \sum _{i=1}^{n}er{r}_n}$$

(13)

where $c_1$ and $c_2$ are scaling and integration coefficients, respectively. The filter output $E_{LP,n}$ is then fed to the NCO.

3.4. Numerically Controlled Oscillator

The NCO is used to determine the position of the optimal sampling instant for the next symbol, i.e., $P_{n+1}$. Figure 7 shows the flowchart of NCO in the CB-Gardner algorithm.

In the CB-Gardner, we modified the expression of the control word in the NCO. Since the NCO only needs to obtain the position of the next symbol and does not require knowledge of the midpoint position between two symbols, the NCO only needs to overflow once per symbol period, i.e., the NCO overflows with a period of $T_i=T$. Therefore, the initial value of the control word is

$$W_0=T_s/T_i=\frac{1}{F_s}/T=\frac{1}{F_s}/\frac{1}{F_{\mathrm{b}}}==1/(\frac{F_s}{F_{\mathrm{b}}})=1/sps$$

(14)

where $T_s$ is the sampling period, and $sps=\frac{F_s}{F_{\mathrm{b}}}$ denotes the average number of samples per symbol. For the nth symbol, the control word within the NCO is

$$W_n=W_0-E_{LP,n}=1/sps-E_{LF,n}$$

(15)

Below, we provide a brief derivation of the register values in the NCO after each overflow. We define the location of each NCO overflow as the best sampling instant for the symbols. For the nth symbol, we assume that the NCO overflows at the position of $P_n=m_n+u_n$, where $m_n$ is the integer part of $P_n$ and $u_n$ is the fractional part. The register value after the overflow is ${\mathsf{\eta }}_n$, which is

$${\mathsf{\eta }}_n=(m_k+u_k)W_n$$

(16)

where $m_k$ is an integer and $u_k$ is a fractional number. From $P_n$, ${\mathsf{\eta }}_n$ and $W_n$, we can directly obtain the position and register value of the next NCO overflow, as given by

$$P_{n+1}=m_n+1+m_k+u_k=\mathrm{int}(P_n)+1+\frac{{\mathsf{\eta }}_n}{W_n}$$

(17)

$${\mathsf{\eta }}_{n+1}={\mathsf{\eta }}_n-(m_k+1)W_n+1={\mathsf{\eta }}_n-[\mathrm{int}(\frac{{\mathsf{\eta }}_n}{W_n})+1]W_n+1$$

(18)

Then, from the nth symbol to the (n + 1)th symbol, the NCO estimates the signal expansion factor as

$${\mathsf{\lambda }}_n=sps/(P_{n+1}-P_n)=sps/[\mathrm{int}(P_n)+1+\frac{{\mathsf{\eta }}_n}{W_n}-P_n]$$

(19)

In this way, we estimate the magnitude of the signal expansion factor in real time. Subsequently, we again search for the nearest three error detection points after the (n + 1)th symbol to form a feedback loop.

3.5. Interpolator

After performing the aforementioned operation, we can obtain the position of optimal sampling instant $P_n$ for each symbol. However, as we are dealing with a passband carrier signal, we need to determine the position of each sampling point and resample them in order to recover the passband signal without Doppler effects. This necessitates the use of interpolation algorithms.

3.5.1. Location of Sampling Points

There should be $sps$ sampling points within each symbol. Assuming that Doppler is approximately constant within a symbol period, these sampling points should be equidistantly distributed. Therefore, the position interval of each sampling point between the nth symbol and the (n + 1)th symbol is

$$P_{△n}=(P_{n+1}-P_n)/sps=1/{\mathsf{\lambda }}_n$$

(20)

The location of each sampling point is:

$$P_{n,i}=P_n+i\ast P_{△n}=P_n+i/{\mathsf{\lambda }}_n,i=1,2,\dots ,sps$$

(21)

After that, we can get the values of all sampling points again according to $P_{n,i}$. For $n=1$, the position of the first symbol is $P_1=1$.

3.5.2. Interpolation Algorithm

The interpolation method of CB-Gardner uses the four-point Lagrange cubic interpolation method with Farrow structure [33]. Once the positions of the error detection points ($p_{y,n}$) and each sampling point ($P_{n,i}$) are obtained, an interpolation algorithm is applied to resample them and obtain their true values without the influence of Doppler. For any resampling position $P=m+u$, where $m$ is the integer part of $P$ and $u$ is the fractional part of $P$, let $s$ be the received original passband signal, the interpolated value after resampling according to $P$ is

$$y_{\mathrm{in}}=X^{\mathrm{T}}{B}^{\mathrm{T}}U$$

(22)

where $X$ consists of four sampling points in $s$:

$$X=\left[\begin{array}{l}s(m+2)\\ s(m+1)\\ s(m)\\ s(m-1)\end{array}\right]$$

(23)

$B$ is the Farrow cubic interpolation structure matrix:

$$B=\left[\begin{array}{cccc}0& 0& 1& 0\\ -\frac{1}{6}& 1& -\frac{1}{2}& -\frac{1}{3}\\ 0& \frac{1}{2}& -1& \frac{1}{2}\\ \frac{1}{6}& -\frac{1}{2}& \frac{1}{2}& -\frac{1}{6}\end{array}\right]$$

(24)

and $U$ is the matrix determined by $u$:

$$U=\left[\begin{array}{l}1\\ u\\ u^2\\ u^3\end{array}\right]$$

(25)

The resampling value for the sampling point location $P_{n,i}$ is $y_{\mathrm{in}\_n,i}$, and the combination of $y_{\mathrm{in}\_n,i}$ forms the final passband signal $s_{CB}$ after Doppler elimination, i.e.,

$$s_{CB}[(n-1)\ast sps+i]=y_{\mathrm{in}\_n,i} , n=1,2,\dots ;i=1,2,\dots ,sps$$

(26)

and $s_{CB}(1)=s(1)$. The algorithm terminates when the number of symbols ($n$) in a packet reaches its maximum value ($N$). In the subsequent demodulation process, we only need to perform down-conversion, matched filtering, and equalization operations on $s_{CB}$ to decode the transmitted symbols $x$.

Algorithm 1 summarizes the proposed CB-Gardner algorithm.

Algorithm 1: CB-Gardner algorithm.

1: $s,sps,N$  2: Initialization of ${\mathsf{\lambda }}_0=1,P_1=1,{\mathsf{\eta }}_1=1,s_{CB}(1)=s(1)$;

3: for $n=1:N$  4: Step 1: Finding detection points  5:   getting $p_{y_1,\mathrm{n}},p_{y_2,\mathrm{n}},p_{y_3,\mathrm{n}}$  6:   interpolating to get $y_1,y_2,y_3$  7: Step 2: Timing Error detection  8:   $er{r}_n=y_2\ast (y_1-y_3) \mathrm{or} er{r}_n=y_2\ast (y_3-y_1)$  9: Step 3: Loop filter  10:  $E_{LF,n}=c_{1}er{r}_n+c_2{\displaystyle \sum _{i=1}^{n}er{r}_n}$  11: Step 4: Numerically controlled oscillator  12:  $W_n=1/sps-E_{LF,n}$  13:  $P_{n+1}=\mathrm{int}(P_n)+1+{\mathsf{\eta }}_n/W_n$  14:  ${\mathsf{\eta }}_{n+1}={\mathsf{\eta }}_n-[\mathrm{int}({\mathsf{\eta }}_n/W_n)+1]W_n+1$  15:  ${\mathsf{\lambda }}_n=sps/(P_{n+1}-P_n)$  16: Step 5: Calculating $s_{CB}$  17:   for $i=1:sps$  18:    $P_{n,i}=P_n+i/{\mathsf{\lambda }}_n$  19:    interpolating to get $y_{\mathrm{in}\_n,i}$  20:    $s_{CB}[(n-1)\ast sps+i]=y_{\mathrm{in}\_n,i}$  21:   end for  22: end for

4. Simulation

To evaluate the effectiveness of the CB-Gardner algorithm, a simulation was performed using Matlab R2017b. The transmitted signal was a BPSK single-carrier signal that passed through an additive white Gaussian noise (AWGN) channel and had artificially Doppler added to simulate the signal distortion caused by the Doppler effect and local crystal frequency offset. In our method, the received signal was processed by the CB-Gardner algorithm, after which the output signal underwent down-conversion and matched filtering. The method differed from the traditional Gardner method where these two procedures were inverted. Finally, equalization was performed, and the bit error rate (BER) was calculated. The transmitted signal consisted of both pilot and data. The pilot was an m-sequence of length 511, and the data consisted of $N$ symbols, with both the pilot and data modulated with BPSK.

For the Gardner algorithm, the receiver needs to perform CFO estimation and compensation before applying the Gardner algorithm. The CFO estimation method used in this paper is as follows: assume that the maximum CFO is $\pm f_{△max}$, all frequency offsets between $f_c-f_{△max}$ and $f_c+f_{△max}$ are traversed with an interval of $f_d$, and then frequency compensation is performed. After that, the m-sequence part of the received signal is correlated with the local m-sequence signal, and the CFO is estimated as the frequency offset resulting in the maximum correlation peak. When the CB-Gardner is used, the passband signal is directly fed into it, and the receiver-side process is much simpler because CB-Gardner eliminates both the frequency offset and the sampling bias.

Although the carrier signal is not affected by multipaths in the AWGN channel, residual frequency offset still exists when using the Gardner algorithm due to the inaccuracy of the estimated CFO, which can lead to an increasing phase shift of the symbols. The phase-locked loop of the equalizer is utilized to eliminate the residual phase distortion of the symbols. Therefore, it is necessary to use an equalization method when implementing the Gardner algorithm. For the sake of fairness, the CB-Gardener also employs equalization, which uses a decision-feedback equalizer based on root recursive least squares [34].

Since the CB-Gardner requires multiple sampling points in each carrier cycle of the carrier signal for error detection, this method is suitable for medium-wave communication scenarios with relatively low frequencies. To simulate the medium-wave communication system at sea, the simulation parameters were set as follows: the number of symbols $N$ is 10,000, the sampling rate $F_s$ is 2.4 MHz, the symbol rate $F_b$ is 100 k, the carrier frequency $f_c$ is 0.4 MHz, the roll-off factor of the forming filter α is 0.2, the filter length is 40, and the feedforward and feedback filter lengths of the equalizer are 5 and 4. The number of simulations is set to 4000. The added frequency offset is measured in $PPM$ (parts per million), i.e.,

$$\frac{PPM}{{10}^6}=\mathsf{\delta }=f_{△}/f_c$$

(27)

First, we set the SNR of the passband signal to 25 dB and varied the PPM from −32,000 to 32,000 with an interval of 4000. Figure 8 shows the simulated BER variation curves for different PPM when using the Gardner algorithm and the CB-Gardner algorithm. As shown in the figure, the BER is almost 0.5 when PPM ≥ 8000 or PPM ≤ −12,000 when using the Gardner algorithm, indicating that the communication system cannot work at all. For the CB-Gardner, however, the communication system fails when PPM ≥ 28,000 or PPM ≤ −32000. Therefore, the CB-Gardner can resist a maximum PPM of about $\pm \mathrm{28,000}$, which is much larger than the corresponding $\pm 8000$ value for the Gardner. For the Gardner algorithm, its BER is much less than 10⁻⁴ when −8000 ≤ PPM ≤ 4000. Correspondingly, the BER for the CB-Gardner is much less than 10⁻⁴ when −28,000 ≤ PPM ≤ 20,000. The use of the CB-Gardner significantly enhanced the ability of the communication system against Doppler.

Based on Figure 8, it can be seen that the BER of the system experiences a sharp increase when the PPM exceeds a certain value, irrespective of whether the Gardner or CB-Gardner algorithm is employed. The result indicates that the stability of the communication system undergoes an abrupt change instead of a smooth change as the PPM increases. Additionally, the results showed that the system can resist a greater Doppler when PPM < 0 in comparison with PPM > 0 for both Gardner and CB-Gardner algorithm. This could be attributed to the smaller jitter of the Doppler factor estimated by the NCO of the algorithm when PPM < 0, with the current loop filter parameters used in the simulation.

We conducted further simulations to verify the performance of CB-Gardner on SNR. The PPM was set as −2000, and the SNR of the passband signal ranged from −10 dB to 50 dB with an interval of 5 dB. Figure 9 shows the BER variation curves for different SNRs when using the Gardner and CB-Gardner algorithms for BPSK- and QPSK-modulated signals. Figure 9 demonstrates that the BER using the Gardner algorithm for BPSK signals is less than 10⁻⁶ when SNR > 10 dB. In contrast, when using CB-Gardner, the BER for BPSK signals is less than 10⁻⁶ when SNR ≥ 20 dB. However, for QPSK signals, the BER remains greater than 10⁻² when SNR ≥ 50 dB. The simulation results indicate that CB-Gardner is currently only suitable for BPSK signals, and the required SNR is less than 10 dB larger than that of the Gardner. This is because the error-detection points of the Gardner algorithm are located in the symbol envelope, whereas the error-detection point of the CB-Gardner algorithm are located within one carrier cycle. As a result, it can be inferred that the error detection in the CB-Gardner is more susceptible to noise than that of the Gardner. Although the error-detection equation (12) of the CB-Gardner can theoretically be used for QPSK, the BER is relatively high. Thus, the CB-Gardner is currently less applicable to QPSK signals.

5. Discussion

The CB-Gardner algorithm proposed in this paper opens up several avenues for future research and discussion, as described below.

(1) In this paper, the first sampling point was set as the exact first symbol instant. However, in a practical communication system, it is challenging to locate the first symbol instant with such precision. One potential solution is to use pilot signals to detect the arrival time of the signal;

(2) CB-Gardner requires knowledge of the phase of each symbol instant to locate the error-detection point. However, in a multipath environment, the phase of the symbol instant may be random, making it impossible for the algorithm to locate the error-detection point. Although we have explored alternative methods to estimate phase, they also eliminate the phase variation caused by Doppler, leading to error-detection failure. Therefore, CB-Gardner is currently unsuitable for multipath environments, especially complex underwater acoustic environments, but it may be useful in wireless communication with fewer channel taps;

(3) In our simulations, CB-Gardner had a high BER for QPSK modulated signals. Several factors may have contributed to this result. First, the phase variations in QPSK are more diverse, resulting in a waveform at the symbol instant that is less like a cosine waveform. Second, the error detection Equation (12) may fail, causing the localized symbol instant to drift. Third, the constellations of the QPSK signal are relatively dense and require a higher SNR. These factors can increase the self-jitter of error detection, resulting in a higher BER.

6. Conclusions

The Gardner algorithm and various improved algorithms are based on baseband signals for error detection. In this paper, we proposed an improved Gardner algorithm applied to the carrier signals based on the similarity between the carrier signal and the baseband symbols in the waveform. Compared to the Gardner algorithm, the CB-Gardner algorithm redefined the position of the error detection point, improved the error-detection equation, simplified the working process of the NCO, and resampled all passband sampling points using an interpolator. The improved algorithm performed error detection directly on the passband while eliminating the frequency offset and sampling-point bias caused by Doppler; thus, the estimation and compensation of CFO at the receiver side are no longer required, simplifying the processing flow. The error detection remained valid when adjacent symbols were the same, reducing the self-noise to a certain extent. CB-Gardner retains the real-time iterative property of the Gardner algorithm and can resist time-varying Doppler.

The CB-Gardner algorithm was primarily designed for single-carrier BPSK signals under AWGN channels, particularly for wireless communication systems located in the medium-wave band, making it suitable for maritime wireless communication. Simulations showed that the performance of CB-Gardner against Doppler is about three times better than that of the Gardner algorithm. However, CB-Gardner requires a higher SNR for the system. The main improvement direction for CB-Gardner may be to continue reducing self-noise, making it suitable for QPSK and other modulation systems.