Interspectral Error Tracking and Compensation of DSDT in Satellite Internet of Things

Abstract

With the rapid growth of satellite Internet of Things (SIoT) services, existing frequency band resources are insufficient to meet future business demands. To effectively address this issue, it is necessary to enhance the utilization of existing frequency resources. However, idle frequency resources are typically scattered across multiple bands and vary in bandwidth size. Direct Spectrum Division Transmission (DSDT), dividing a complete signal into sub-spectrum signals for transmission in idle frequency bands, can take the use of fragmented spectrum resources for satellite communication. Nevertheless, the performance of DSDT depends heavily on accurate synchronization toward multiple sub-spectrums. In this paper, an algorithm for error synchronization tracking and compensation is proposed by utilizing the focusing nature of constellation. All sub-spectrums are weighed by the minimum Euclidean distance of the constellation to compensate for amplitude–frequency–phase errors simultaneously. Simulations and experimental verification demonstrate synchronization performance and feasibility of proposed method in a multi-radio frequency channels environment.

1. Introduction

With the increasing demand for interactive data services from the satellite Internet of Things, there is a shortage of spectrum resources for satellite communications [1,2]. Recent surveys have shown that up to 80% of allocated frequency bands are under-utilized, thus resulting in a significant waste of spectrum resources [3]. Nevertheless, most of idle frequency resources are mostly scattered across multiple frequency bands, and these bandwidths of idle frequency bands are not large and not equal, which are named frequency holes in this paper.

In order to fully utilize the frequency holes scattered across multiple frequency bands, a Direct Spectrum Divsion Transmission technology has been presented in [4,5,6], which divides one complete signal into multiple sub-spectrum signals with small bandwidths for frequency holes. Combined with cognitive radio technology, DSDT can dynamically use frequency holes resources to improve communication efficiency and meet the bandwidth diversity.

The DSDT concept was initially proposed by a team of scholars led by Fumihiro Yamashita. In their work [7], the feasibility of DSDT was demonstrated through the multiple transponder transmission experiments of DVB-S2X. A simple synchronization scheme based on pilots was provided for amplitude, frequency, and phase offset in [8]. Furthermore, frequency offset can be compensated by changing the filter coefficients of the combining filters sequentially based on the estimated frequency [9]. In addition, they discussed the issues of peak-to-average power ratio and adjacent channel interference in DSDT [10]. However, in the DSDT system, its performance depends heavily on accurate synchronization toward multiple sub-spectrums.

Therefore, to remedy the synchronization error after using the pilot synchronization scheme, some free-pilot synchronization schemes for DSDT were provided in [11,12], which can be used in conjunction with other schemes based on pilots [13,14,15]. In [11], a free-pilot amplitude compensation scheme was proposed utilizing the consistency of −3 dB point at adjacent sub-spectrums, where this specific point owns identical amplitude values between neighboring sub-spectrums. Subsequently, a novel free-pilot scheme was presented in [12], which uses multiple points between adjacent sub-spectrums to estimate frequency and phase offset due to the consistency of phases of these points. As a result, its performance was better than the former [11], but it is limited by the width of transition zone between adjacent sub-spectrums. Moreover, the expansion of the width of the transition zone resulted in an increase in bandwidth. The above free-pilot synchronization schemes for DSDT only utilize the local features of adjacent spectra and do not fully exploit the correlation between all sub-spectrums; thus, there is still room for further improvement in free-pilot synchronization performance.

In this paper, a free-pilot synchronization method based on all sub-spectrums is proposed for DSDT, which can adaptively track and compensate amplitude–frequency–phase errors among sub-spectrums by utilizing the focusing nature of constellations. All sub-spectrums are weighed by the minimum Euclidean distance of the constellation to compensate for amplitude–frequency–phase errors simultaneously. This operation can be also regarded as an equalizer, as the combined spectrum with synchronization error exhibits characteristics similar to frequency selective fading. Meanwhile, the least squares iterative structure is adopted for improving its synchronization error convergence and tracking speed. Finally, some simulations are provided for analysis of the synchronization performance. Compared with the methods proposed in [11,12], our method exhibits a lower synchronization error under the same signal-to-noise ratio conditions, and it is not limited by FFT resolution. In addition, we build a DSDT test system using the Universal Software Radio Peripheral (USRP) platform to verify the feasibility of our method.

2. System Model

DSDT utilizes filter bank technology to divide the signal into multiple sub-spectra, inserts them into frequency-offset holes for transmission, and recovers the original signal through receiver-side combination filters. The schematic diagram of this transmission system is shown in Figure 1. A, B and C are the sub-spectra divided by the filter.

In practical satellite communication, not all frequency holes appear densely, and some frequency gaps have large spacing, thus requiring the use of different transceivers for transmission. The larger the spacing between channel frequency bands, the greater the difference in channel characteristics. This means that using spectrum-based transmission technology across large frequency bands or even on multiple frequency transceivers will inevitably face the problem of inconsistent amplitude, frequency, and phase between sub-spectra, as shown in Figure 2.

Since the actual frequency hole occurs in multiple frequency bands, these sub-spectrums will experience independent fading channels, thus denoting the fading coefficient of the ith sub-spectrum channel as $A_i{e}^{j{\varphi }_i}$. Meanwhile, considering a relative movement between satellite and ground station, received sub-spectrum signals in different frequency bands face different Doppler shifts, where the Doppler offset in the ith channel is denoted as $e^{j2\pi f_{di}}$. Therefore, the expression of the received signal across the ith combination filter can be written as follows:

$$s_{ri}\left(t\right)=s_i\left(t\right)\times A_i{e}^{j\left(2\pi f_{di}t+{\varphi }_i\right)}+n_i\left(t\right),$$

(1)

where $n_i\left(t\right)$ is noise in the ith channel, and $S_i\left(t\right)$ represents the ith sub-spectrum of the original signal $s_o\left(t\right)$ across division filter $h{d}_i$ and combination filter $h{c}_i$ [8] (“*” is the convolution operation).

$$s_i\left(t\right)=h{c}_i\ast h{d}_i\ast s_o\left(t\right)$$

As mentioned in [9], the original signal $s_o\left(t\right)$ could be MASK, MPSK, or MQAM and so on. Then, pilot synchronization is operated for estimations of the amplitude $\widehat{A_i}$, frequency $\widehat{f_i}$, and phase $\widehat{{\varphi }_i}$. After compensation, $s_{ri}\left(t\right)$ turns into $s_{ri}^{\prime }\left(t\right)$:

$$s_{ri}^{\prime }\left(t\right)=s_i\left(t\right)\times A_i^{\prime }{e}^{j\left(2\pi \Delta f_{i}t+\Delta {\varphi }_i\right)}+n_i\left(t\right),$$

(2)

where residual synchronization errors are

$$A_i^{\prime }=A_i/{\widehat{A}}_i$$

$$\Delta f_i=f_{di}-\widehat{f_i}$$

$$\Delta {\varphi }_i={\varphi }_i-\widehat{{\varphi }_i}$$

Finally, the combined signal with N sub-spectrums can be expressed as

$$s_r\left(t\right)=\sum _{i=1}^N{s}_{ri}^{\prime }\left(t\right),$$

(3)

When a residual synchronization error exists, the spectrum structure of Equation (3) is shown as the combined signal in Figure 2. It can be seen that the combined signal appears distorted, thus resulting in a loss of bit error rate (BER) performance. In this case, a free-pilot synchronization scheme become a good choose for reducing the synchronization error. In addition, a schematic diagram of the signal processing flow for DSDT with a free-pilot module is shown in Figure 3.

3. Adaptive Error Tracking and Equalization

As mentioned in the previous section, the distortion in the combined signal originates from inconsistent amplitude, frequency, and phase errors among the sub-spectra. Since the combined signal is formed by reconstructing the sub-spectra transmitted across a wide frequency range, the impact of the sub-spectrum amplitude and phase errors is similar to that of frequency-selective fading channels. The carrier errors introduced by multiple RF channels can be considered as time-varying selective fading caused by Doppler frequency offsets. Therefore, this paper attempts to indirectly address the distortion issue in the reconstructed signal caused by sub-spectrum amplitude, frequency, and phase errors from the perspective of reducing Inter-Symbol Interference (ISI).

Equalization is a common technique used to mitigate ISI. Among them, adaptive equalization methods update tap coefficients using training sequences to track slow time-varying errors and mitigate ISI [16,17]. However, training sequences incur additional resource overhead. On the other hand, blind equalization algorithms do not require training sequences. Among them, the Decision Feedback Equalizer (DFE) based on the Bussgang technique has good tracking performance for non-linear distortions and is expected to correct the distortion caused by sub-spectrum errors in the combined signal [18].

Therefore, referring to the DFE algorithm, two error compensation schemes are considered in the specific implementation: a conventional DFE equalization algorithm with sub-spectrum error tracking and an adaptive compensation algorithm.

3.1. Decision Feedback Equalization

The first scheme considers the problem as recovering a signal affected by frequency-selective, fading-causing Inter-Symbol Interference (ISI). After downsampling from symbol synchronization, the $l-m_b$th sampling decision symbol can be expressed as (l represents current symbol)

$$y_{l-m_b}=s_r(t-m_{b}T)=s_r(l{T}_s-m_{b}T),$$

(4)

where $T_S$ is the sampling interval, T is the symbol length, and $m_b$ is the symbol index.

Next, an adaptive equalizer can be operated to improve the detection performance by minimizing the joint error of multiple consecutive symbols. The objective function is as follows:

$$\underset{\widehat{s}\in S}{min}E\left[{\left|{\widehat{s}}_l-\left(\sum _{m_b=0}^{M_{b-1}}{w}_{m_b}{y}_{l-mb}-\sum _{n_f=1}^{N_f-1}{w}_{nf}{\widehat{s}}_{l-nf}\right)\right|}^2\right],$$

(5)

In Equation (3), $W_{mb}$ represents the weights of the inverse filter, $W_{nf}$ represents the weights of the feedforward filter,$M_b$ and $N_{f-1}$ represent the taps of the inverse filter and feedforward filter, $y_l$ represents the lth symbol of the combined signal, and ${\widehat{s}}_l$ represents the ideal constellation after the hard decision of the lth symbol. By minimizing the objective function mentioned above, the filter coefficients can be obtained through iterative optimization.

Although the DFE algorithm can mitigate the ISI caused by channel characteristics, it does not fully utilize the channel information carried by all sub-spectra. Moreover, as a traditional DFE algorithm, it has the drawbacks of slow convergence speed and high bit error rate (BER) at low signal-to-noise ratio (SNR) levels.

3.2. Sub-Spectrum Error Synchronization Tracking and Adaptive Compensation

The second scheme involves adjusting the tap weight coefficients of each sub-spectrum separately using decision feedback to achieve adaptive compensation for the combined recovered signal. Although this processing is done in the time domain, it effectively modifies the frequency components of the combined recovered signal to adaptively compensate for errors. Therefore, it can also be considered as frequency domain equalization. The objective function is as follows:

$$\underset{\widehat{s}\in S}{min}E\left[\sum _{l=0}^n{\lambda }^{n-1}{\left|{\widehat{s}}_l-\sum _{i=1}^N{g}_i{y}_{l,i}\right|}^2\right],$$

(6)

where $y_{l,i}$ is lth symbol of ith sub-spectrum after the downsampling operation

$$y_{l,i}=s_{ri}^{\prime }\left(l{T}_s\right).$$

In Equation (6), $\lambda$ represents the forgetting factor, ${\widehat{s}}_l$ is the normalized ideal constellation value of the lth symbol after combination, $g_i$ represents the equalizer weights of the (i + 1)th sub-spectrum, and $y_{l,i}$ represents the sampling point of the lth symbol in the (i + 1)th sub-spectrum under symbol synchronization. When compensating for sub-spectrum errors, ${\sum }_{i=1}^N{g}_i{y}_{l,i}$ approximates the estimated ideal constellation value under symbol estimation. Therefore, the mean square error can be used as the objective function to modify the equalizer coefficients.

Compared to the recovered signal after combination, the pre-combination sub-spectra retain more synchronization error information. Sub-spectrum level attenuation compensation can theoretically achieve better combination performance. To address the drawback of the decision feedback algorithms relying on the previous symbol’s hard decision result, a forgetting factor is introduced to enable rapid convergence of the error function and to improve the phase tracking capability, thereby reducing the risk of symbol misjudgment.

Figure 4 illustrates the iterative process of the tap coefficient $\mathbf{g}\left(l\right)$ of the algorithm. $\mathbf{g}\left(l\right)$ represents the tap coefficient value corresponding to each sub-spectrum under the (l + 1)th symbol, and $\overline{e\left(l\right)}$ is the difference between the conjugate operation and the ideal normalized constellation.

In the Recursive Least Squares (RLS) algorithm, the update step for the equalizer tap coefficients is determined by calculating a weighted sum of the mean square error indices in the least squares sense. $\mathbf{k}\left(l\right)$ is the step size for updating the tap coefficients of the equalizer. The updating process is as follows:

• Calculate the combined signal and make hard decision $\widehat{s}$:

  $$\widehat{s}\left(l\right)=\mathcal{D}\left[\sum _{i=1}^N{y}_{l,i}{g}_{l-1,i}\right]$$

  (7)
• Calculate the data autocorrelation term $\mathbf{R}\left(l\right)$ and cross-correlation term $\mathbf{p}\left(l\right)$:

  $$\mathbf{R}\left(l\right)=\sum _{l=0}^N{\lambda }^{n-l}\mathbf{y}\left(l\right)\mathbf{y}{\left(l\right)}^T,$$

  (8)

  $$\mathbf{p}\left(l\right)=\sum _{l=0}^N{\lambda }^{n-l}{\widehat{s}}_l^{\ast }\mathbf{y}\left(l\right),$$

  (9)
• Calculate the tap coefficient increment $\mathbf{k}\left(l\right)$:

  $$\mathbf{T}\left(l\right)={\mathbf{R}}^{-1}\left(l\right),$$

  (10)

  $$\mathbf{k}\left(l\right)=\frac{\mathbf{T}(l-1)\mathbf{y}\left(l\right)}{\lambda +{\mathbf{y}}^T\left(l\right)\mathbf{T}(l-1)\mathbf{y}\left(l\right)},$$

  (11)
• Obtain the tap coefficient:

  $$\mathbf{g}\left(l\right)=\mathbf{g}(l-1)+\mathbf{k}\left(l\right)\overline{e}\left(l\right),$$

  (12)

  where $\overline{e}\left(l\right)=\widehat{s}\left(l\right)-{\sum }_{i=1}^N{y}_{l,i}{g}_{l-1,i}$.

Assuming the decisions in the iteration are correct, the lower bound of the convergence error for the iteration can be obtained. According to the recursive least squares theory [19], our mean square error is obtained as below

$$D\left(l\right)\approx \frac{1-\lambda }{2}{\sigma }_n^{2}tr\left[{\mathbf{R}}_y^{-1}\right]+\frac{1}{2(1-\lambda )tr\left[{\mathbf{R}}_e\right]},whenl>>1$$

(13)

where ${\mathbf{R}}_y$ is the covariance matrix of vector $\mathbf{y}$, and $\mathbf{y}=[y_{l,1},y_{l,2},y_{l,3},\dots ,y_{l,N}]$. ${\mathbf{R}}_e$ is the covariance matrix of vector $\mathbf{e}$, $\mathbf{e}=[A_1^{\prime }{e}^{j(2\pi \Delta f_{1}lT+\Delta {\varphi }_1)},\dots ,A_N^{\prime }{e}^{j(2\pi \Delta f_{N}lT+\Delta {\varphi }_N)}]$. ${\sigma }_n^2$ is the noise power, and $\lambda$ is the forgetting factor. As known from [19], its optimal forgetting factor is

$${\lambda }_{opt}\approx 1-\frac{1}{{\sigma }_n}{\left(\frac{tr{\mathbf{R}}_y}{tr{\mathbf{R}}_e}\right)}^{1/2}$$

(14)

Therefore, substituting Equation (14) into Equation (13), our minimum of mean square error is

$$D_{min}\approx {\sigma }_{n}tr{\left({\mathbf{R}}_y^{-1}tr\left[{\mathbf{R}}_e\right]\right)}^{1/2}$$

(15)

In fact, $D_{min}$ reflects the difference between the actual symbol decision and the ideal constellation in terms of the Euclidean distance.

4. Synchronization in Multiple RF Channels

To experimentally validate the proposed schemes, a wireless transceiver platform was built based on NI-USRP, and two independent RF channels of USRP X310 were used to simulate two satellite transponders. The system environment is shown in Figure 5. NI USRP from National Instruments in Texas, USA. The NI-USRP consists of a host and wireless signal transceiver hardware. The host side programming was performed using LabVIEW, and the RIO (Reconfigurable I/O) side was processed using the Field Programmable Gate Array (FPGA) provided by LabVIEW for streaming processing.

To ensure accurate synchronization of the received signal, conventional communication systems often use the method of inserting pilots. However, for DSDT systems, the original signal recovered by aggregating multiple sub-spectra will accumulate synchronization errors from each sub-spectrum. Even if the sub-spectra are synchronized using pilots, the reconstructed signal will still have significant distortion that is difficult to eliminate further using pilots. Therefore, in the implementation of the DSDT system, the algorithm proposed in this paper is used to track and compensate for the amplitude–frequency–phase errors of the synchronized sub-spectra, thus further reducing the bit error rate.

The signal frame structure, as shown in Figure 6, includes multiple sub-spectra. In the DSDT system, independent multiple RF channels are used for transmission. Each sub-spectrum is allocated to a different RF channel. For example, sub-spectra 1 and 3 are assigned to RF channel 1, while sub-spectra 2 and 4 are assigned to RF channel 2.

Apart from the high requirements for synchronization accuracy, the hardware implementation of the DSDT system faces the following challenges:

One of the major challenges is the limitation of FPGA logic resources in the hardware devices. Due to the requirement for multi-subband filtering with arbitrary bandwidths and multi-frequency point synchronization of the sub-bands, the logic resource consumption on the RIO of the USRP is excessively high. Additionally, the processing speed of the host’s host side is slow, thus resulting in poor real-time performance.

Another challenge is how to reduce the data link load. High-speed sampling of the signal can impose a heavy load on the data transmission link, thus leading to potential data congestion and loss. To address this issue and allocate resources effectively, the system’s receiver-end structure framework is illustrated in Figure 7. The Labview implementation of the error synchronization tracking and adaptive compensation can be seen in Figure 8.

After synchronization of the signal on the RIO side, frame synchronization is achieved by utilizing the auto-correlation characteristics of Barker codes in the main sub-spectrum. Since the optimal sampling point for the main sub-spectrum is the same as that for the other sub-spectra, symbol rate sampling can be performed on all sub-spectra, thus reducing the consumption of logic resources and lowering the data link overload. Through this system architecture design, the performance degradation caused by resource limitations is to some extent avoided while ensuring the real-time performance of the system.

To assess hardware limitations and resource usage, the scheme is estimated according to the resource consumption shown in the comprehensive report generated by FPGA processing and combined with the amount of FPGA resources available in the USRP-X310 RIO.Reference is provided below in

Table 1. Comparison of resource consumption with available resources on the X310 RIO.

FPGA Resource Type	Resource Consumption	Available Resource
Slice LUTs	17,000~19,000	254,200
Slice registers	32,000~34,000	508,400
DSPs	900~1000	1540
Block RAMs	20~25	795

.

In practical applications, the resource consumption will increase with the increase in the number of sub-spectrum. The FPGA hardware model of this system is Xilinx Kintex 7-410T, and the maximum number of molecular spectra supported by this system is six. The table shows that the hardware USRP-X310 can meet the resource requirements of the synchronization scheme, and it can be predicted that the system will have better performance on the later version of USRP.

5. Simulation and Experimental Results Analysis

To validate the proposed algorithm, MATLAB simulations were conducted to compare the bit error rate (BER) performance of the DSDT under different amplitude, frequency, and phase error estimation methods. Additionally, high sampling rate satellite multi-relay transmission was simulated using NI-USRP, and the constellation diagram after sub-spectrum error compensation, as well as the measured signal BER curve, were provided.

5.1. Simulation Results

The simulation parameters of the DSDT system are listed in

Table 2. Sub-spectra synchronization experiment parameters.

Parameter	Value
Modulation	QPSK
Bandwidth/MHz	2
Sample rate/MHz	20
Number of sub-carriers	4
Sub-spectrum bandwidth/MHz	0.5
Sub-spectral roll-off factor	0.5

. To evaluate the estimation and compensation performance of the proposed algorithm for the amplitude–frequency–phase error, we first investigated the synchronization performance in the presence of amplitude error, frequency error, and phase error, respectively, and compared it with [11,12].

Figure 9 shows a comparison of [11] and the proposed algorithm with respect to the performance of amplitude compensation. It can be seen that the proposed algorithm exhibited a lower bit error rate after amplitude compensation. Meanwhile, the performance comparisons of the phase and frequency compensation with [12] are illustrated in Figure 10 and Figure 11. As a result, the proposed algorithm demonstrates superior performance compared to the methods in [11,12], and it is capable of simultaneously handling amplitude, phase, and frequency offset.

Figure 12 shows a BER comparison of various error compensation methods in the presence of amplitude–frequency–phase errors. It can be seen that the proposed algorithm had a significantly lower BER compared to the traditional decision feedback equalization (DFE) algorithm and [12]. On the other hand, although the performance of the proposed algorithm was not as good as that of the pilot synchronization scheme, it could achieve better performance when used together with the pilot algorithm, as shown by purple line in Figure 12.

Figure 13 shows a BER performance for various modulation types after synchronization. Compared to the ideal synchronization, the proposed algorithm could all effectively avoid performance loss for these modulation types. It also indicates that the proposed algorithm is not limited by the modulation type of the original signal.

5.2. Experimental Results and Analysis of Multiple RF Channels

To validate the performance of the algorithm, experiments were conducted using the NI-USRP wireless transceiver platform. The experimental parameters are shown in

Table 3. Experimental parameters under multiple RF channels.

Parameter	Value
Modulation	QPSK
bandwidth/MHz	2
Sample rate/MHz	100
Number of sub-carriers	4
Sub-spectrum bandwidth/MHz	0.5
Subspectral roll-off factor	0.5
Antenna carrier frequency/GHz	2.3

.

The sub-spectrum of the signal after four segments is shown in Figure 14. Figure 15 shows the constellation diagrams before and after using the sub-spectrum error tracking and adaptive compensation algorithm. It can be observed that the constellation diagram after pilot synchronization and the direct combination of sub-spectrums is more dispersed and exhibits some rotation, thus indicating the presence of residual frequency and amplitude phase errors in the sub-spectrum signals. After processing with the proposed algorithm, the combined signal’s constellation diagram is greatly focused with almost no scatter points, thus demonstrating the high-precision error synchronization and tracking performance of the algorithm. Figure 16 shows the error convergence curve, thus indicating that the signal achieved error convergence within approximately 20 symbol periods, and the mean square error ended up being generally within 0.05 after convergence, thus reflecting the algorithm’s fast error convergence and tracking performance.

Finally, Gaussian white noise was introduced using the built-in controls of LabVIEW, and the BER curves before and after applying the proposed algorithm were compared at different SNRs, as shown in Figure 11.

Figure 17 shows the measured BER of the DSDT; it can be observed that at an SNR of 10 dB, the reconstructed signal’s BER is close to 1 × 10⁻⁵.

After applying the proposed algorithm, the BER further decreased compared to direct reconstruction after pilot synchronization, thus validating the conclusions of the simulation experiments.

6. Conclusions

This paper addresses the issue of amplitude–frequency–phase distortion in sub-carriers in DSDT systems during satellite cross-transponder transmission, along with the high synchronization accuracy requirements. It combines the decision feedback algorithm in equalization and introduces the error vector of each sub-carrier based on the least squares criterion. A data assistant-free algorithm for inter-subcarrier error synchronization tracking and adaptive compensation has been proposed. Experimental simulation analysis and measurements based on the NI-USRP wireless transceiver platform demonstrate that the error feedback mechanism of multiple sub-carriers has significant performance advantages over traditional decision feedback equalization of the combined signal, thus effectively correcting amplitude, frequency, and phase errors. Compared to the blind estimation algorithm in [12], the proposed algorithm avoids the contradiction between FFT points and estimation accuracy, thus resulting in improved estimation accuracy. By combining it with pilot synchronization, the algorithm can further track and compensate for subtle residual errors after synchronization without introducing additional pilot overhead. This approach maximizes the demodulation performance of the DSDT system and performs well in practical engineering applications. The simulations and measurements in this paper have been conducted under limited error conditions, but the analysis of the algorithm’s effective tracking range for maximum errors is lacking. Further exploration of this issue will be undertaken in future research.