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Abstract
Linear optical sampling (LOS) is one of the most powerful techniques for high-speed signal monitoring. To measure the data-rate of signal under test (SUT) in optical sampling, multi-frequency sampling (MFS) was proposed. However, the measurable data-rate range of the existing method based on MFS is limited, which makes it very difficult to measure the data-rate of high-speed signals. To solve the above problem, a range selectable data-rate measurement method based on MFS in LOS is proposed in this paper. Through this method, the measurable data-rate range can be selected to match the data-rate range of SUT and the data-rate of SUT can be measured precisely, independently of the modulation format. What’s more, the sampling order can be judged using the discriminant in the proposed method, which is key for plotting eye diagrams with correct time information. We experimentally measure the baud-rates of PDM-QPSK signal from 800 MBaud to 40.8 GBaud in different ranges and judge the sampling orders. The relative error of measured baud-rate is less than 0.17% while the error vector magnitude (EVM) is less than 0.38. Compared with the existing method, under the same sampling cost, our proposed method realizes the selectivity of the measurable data-rate range and the judgment of sampling order, greatly extends the measurable data-rate range of SUT. Hence, the data-rate measurement method with selectable range has great potential for high-speed signal data-rate monitoring.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development of information technology, the data transmission rate of optical communication system is increasing rapidly. The single channel data rate of optical transmission has reached 800 Gbit/s [1] or even higher [2,3,4], but monitoring high-speed signals is facing challenges. The traditional electrical-sampling based optical signal analyzer needs to use ultra-high speed photoelectric detectors and analog-to-digital conversion to complete high-speed signal monitoring, so this way has high requirements for hardware, high manufacturing costs, and is difficult to realize due to electronic bottleneck. Linear optical sampling (LOS) is a powerful method for high-speed optical signal monitoring [5,6]. It uses an optical pulse with low repetition frequency to sample the high-speed signal and reconstruct eye diagram and constellation diagram through software synchronization [7]. LOS realizes the linear optical process by coherently mixing the short optical pulse of the mode-locked fiber laser (MFL) with the optical signal. After linear mixing, LOS performs photoelectric conversion and sampling with low-speed balanced photodetector (BPD). Therefore, LOS obtains high-speed signal characteristics through undersampling scheme, which uses low-speed equipment to reduce hardware overhead. Many scholars have made contributions to improving the representation ability of LOS system, improving the accuracy of eye diagram reconstruction, and reducing the computational complexity [8,9,10,11]. With the wide application of wavelength division multiplexing (WDM), the fiber-optics-frequency-comb (FOFC) is proposed to replace the narrowband MFL as an optical sampling source to extend the range of operation wavelength for the LOS-based optical modulation analyzer (OMA). Since the use of BPDs introduces time bias and amplitude bias for LOS system, Liao proposed a scheme to mitigate the enhanced sampling error arising in the non-ideal response of a balanced photodetector [12]. But this procedure still introduces noise due to the constraints of the unideal response of BPDs. To overcome such issue, Zhe Yu presented a bias balance detection (BBD) scheme and demonstrated a 32 GBaud PDM-QPSK experiment for a FOFC based LOS system [13]. Meanwhile, MFL with a fixed repetition frequency has a theoretical upper limit of the measurement range [19] and the low repetition frequency of MLF increases the sampling period, which increases the constructing time of the eye diagram. To increase the repetition frequency, our team proposed a dual-pulse mixing (DPM) based LOS system using fiber delay lines with the multiplied optical pulse, which extends the measurement range and reduces error vector magnitude (EVM) bias by 9.1% [14]. All of the above studies have improved the performance of LOS in terms of measurement range, accuracy and reliability, respectively. However, in the above studies, the entire process of LOS is transparent to the data-rate, which does not provide information on the length and order of time for constructing eye diagram. In order to comprehensively monitor the optical signal, the ability of optical signal monitoring system to measure different data-rates of signal under test (SUT) is preferable.
To realize the data-rate measurement in optical sampling, multi-frequency sampling (MFS) was proposed. Xiangyu Wu proposed a bit-rate measurement method for optical performance monitoring, by alternatively tuning the repetition frequency of the optical pulse and differentially software-synchronized processing algorithm [15]. However, only several SUT with known discrete points in the low-speed range can be correctly measured like 10 Gbit/s and 40 Gbit/s in their experiment due to the existence of blind region for measurement [16]. Focus on this problem, Lin Zuo proposed a method to avoid the blind region. Through three times of optical sampling with slightly different repetition frequencies, this method calculates twice and takes the maximum result as the correct one. Almost all the bit-rate values in the whole low-speed range can be correctly measured, which achieved the monitoring of an unknown bit-rate in the low-speed range truly [17,18]. However, they found that after avoiding the blind region, the real measurable data-rate range was only a quarter of what was originally proposed in literature [15]. This makes it more difficult to monitor the data-rate of higher speed signal. There has been no major progress in linear optical sampling for the data-rate measurement of SUT in recent years. What’s more, in conventional LOS, the recovered eye diagram waveform may be the mirror image of the real waveform because sampling order cannot be judged, resulting in the error of information on time order in eye diagram. Until now, as far as we know, no method that can judge the sampling order in LOS has been proposed.
In this work, we propose a range selectable data-rate measurement method based on MFS in LOS. We derived the formula and found that the measurable range of data-rate can be greatly extended by adding a frequency selection factor. Firstly, select a measurable data-rate range, then use three laser pulses with different repetition frequencies to sample SUT for three times. Using the data obtained from each sampling, the required parameters can be obtained in software synchronization after applying a serious of DSP algorithms. Then the data-rate of SUT can be converted and the sampling order can be judged using our proposed formula, which is the key to plot the eye diagram with correct time information. We experimentally measure the baud-rates of QPSK signal from 0.8 to 40.8 GBaud and judge the sampling orders, when the EVM is less than 0.38, the relative error of the measured baud-rate is less than 0.17%. The rest of this paper is organized as follows. The theoretical mode and principle of range selectable data-rate measurement method is introduced in Section 2. The methods of avoiding blind regions and judging the sampling order are discussed in Section 3. The experiment for a triple-frequency LOS system is demonstrated and the performance of range selectable data-rate measurement method is analyzed in Section 4. Finally, the conclusion is presented.
2. Theoretical mode and principle of range selectable data-rate measurement method
LOS system generally adopts a MFL to generate a pulse with a repetition frequency of about one hundred megahertz to realize under-sampling for SUT [5]. Assuming that the data-rate of SUT is B and the repetition frequency of sampling pulse is f, the frequency reduction coefficient D can be calculated by the following formula [15]:
where $M\textrm{ = }\lfloor{{B / f}} \rfloor$ is the integer part of the frequency reduction coefficient, which is recorded as the frequency reduction factor, and X is the decimal part of the frequency reduction coefficient, recorded as the frequency reduction remainder. Define the equivalent step length of sampling as:The pulse samples the signal equivalently on the envelope with dt as the time step length [4]. Assuming that the signal period is T, the equivalent sampling points in one signal period are:
For an N-point-long sampled data stream, the number of equivalently scanned bit slots S can be obtained by the following formula:
After one sampling in LOS, without knowing B, the number of equivalently scanned bit slots S can be accurately obtained in the chirp-Z transform based software synchronization method in DSP [10]. After S is obtained, the N-point-long sampled data stream can be divided and superimposed to reconstruct the eye diagram [6].
In MFS based LOS, a mode-locked fiber laser with adjustable repetition frequency is used to generate two pulses with different repetition frequencies, recorded respectively as ${f_1},{f_2}$. For the convenience of discussion, ${f_1} > {f_2}$ is taken. The SUT with data-rate B is sampled by these two pulses, and (5) can be obtained:
The subscripts 1 and 2 of the above variables represent that the variable is obtained after the SUT was sampled by pulse with repetition frequency ${f_1}$ and ${f_2}$. Because ${f_1} > {f_2}$, (6) can be obtained from Eq. (1):
where P is the frequency selection factor, which is a non-negative integer. Then (7) is obtained:The sufficient condition for P to be a non-negative integer is:
Equivalent to:
Then the data-rate B can be converted according to (5) and (6):
Formula (10) can be obtained only when ${M_2} = {M_1} + P$ is established, the necessary condition is:
Combining (9) and (11), we can get:
It can be found that when ${f_1},{f_2}$ is determined, the measurable data-rate range can be selected by the frequency selection factor P. So, we can select a suitable P to match the data-rate range of the SUT. The reported method in [17] is actually one case of our proposed method when manually selecting P = 0. The measurable data-rate range of SUT is limited by (12) and the range that does not meet with (12) is called the first blind region for measurement, in which the data-rate cannot be calculated correctly.
3. Blind region avoiding and sampling order judgement
3.1 Blind region avoiding
In the actual simulation, it is found that even if ${M_2} = {M_1} + P$ is met, there will be wrong results caused by the fact that the frequency reduction remainder ${X_1},{X_2}$ are in (0,0.5) and (0.5,1) respectively, or both are in (0.5,1), which are the second and third blind regions. In order to explain the formation for the second and third blind regions, it is necessary to explore the characteristics of fast Fourier transform (FFT) in software synchronization.
In coarse synchronization, due to the conjugate symmetry of FFT, there will be two completely symmetrical peaks in the spectrum of the sampled signal after variance transform, as shown in Fig. 1.
In fact, it is feasible to take the abscissa of any of these two peaks as the number of equivalently scanned bit slots S for eye diagram reconstruction, and eye diagrams with equal Q value can be obtained, as shown in Fig. 2. This is because the down frequency remainder $X = {S / N}$. When ${X_1},{X_2}$ are symmetrical about 0.5, it takes the same number of steps to sample a period of SUT. But the abscissa point of first peak is always taken as the number of equivalently scanned bit slots S in the software synchronization process. This means that when processing two optical sampled data streams with equal length and symmetrical down frequency remainders about 0.5, they will have the same S after software synchronization, rather than the original two S. Then it is impossible to distinguish which is the correct number of equivalently scanned bit slots from software synchronization.
Fig. 2. Eye diagram of 8GBaud QPSK signal. (a) Eye diagram constructed by taking the abscissa of first peak as S. (b) Eye diagram constructed by taking the abscissa of second peak as S. (a) and (b) share the equal Q value, Q = 8.162 dB.
In practice, there are four situations as shown in (13) for the distribution of the down frequency remainder after two times of sampling,
Only when $0 < {X_1} < 0.5$, $0 < {X_2} < 0.5$, the measured data-rate is correct by taking the abscissa of the first peak as S. In the other three cases, due to the wrong selection of the abscissa, the measured data-rate will be less than or equal to the correct one, which is proved in detail in Appendix A.
To resolve the above issue, a method to avoid the blind region based on three times of sampling is proposed in this paper. Take ${f_1} > {f_2} > {f_3}$, ${f_1} - {f_2} = {f_2} - {f_3}\textrm{ = }\Delta f$, and the data-rate B of SUT should meet the condition in (14) at the same time:
Then take the intersection of the ranges defined by formulas in (15) as:
When the data-rate B of SUT meets the condition of (16), ${S_1}$, ${S_2}$ and ${S_3}$ are calculated respectively in software synchronization after three times of sampling. Then use the data obtained from the first and second sampling, and the data obtained from the second and third sampling to calculate the data-rate respectively
The data-rate calculated according to (17) is recorded as ${B_{12}}$, and the data-rate calculated according to (18) is recorded as ${B_{21}}$. Similarly, ${B_{23}}$ and ${B_{32}}$ can be obtained. These four results will finally be used to determine the correct data-rate and judge the sampling sequence.
To analyze how to avoid the blind region, the six different distributions of frequency reduction coefficient D on the number axis are plotted as shown in Fig. 3.
Fig. 3. Distributions of frequency reduction coefficient D on the number axis. In each distribution, the results of B are different. (a) B12 wrong, B21 correct, B23 wrong, B32 correct. (b) B12 wrong, B21 correct, B23 wrong, B32 wrong. (c) B12 wrong, B21 wrong, B23 correct, B32 wrong. (d) B12 correct, B21 wrong, B23 correct, B32 wrong. (e) B12 correct, B21 wrong, B23 wrong, B32 wrong. (f) B12 wrong, B21 wrong, B23wrong, B32 correct.
In the six cases shown in Fig. 3, at least one correct B can be found from ${B_{12}},{B_{21}},{B_{23}}$ and ${B_{32}}$. As it is proved in Appendix A that when in the blind region, the measured result is always less than or equal to the correct one, we only need to take the maximum of ${B_{12}},{B_{21}},{B_{23}}$ and ${B_{32}}$ as the correct data-rate B, that is:
3.2 Sampling order judgement
When $X \in ({0, 0.5} )$, the sampling order is sequential, and the recovered waveform in eye diagram is the same as the real waveform. But when $X \in ({0.5,1} )$, the sampling order is reverse, which will cause the recovered waveform to be the mirror image of the real waveform, resulting in the error of time order, as shown in the following Fig. 4.
Fig. 4. SUT is drawn as a blue solid line, while the two optical sampling pulses are drawn as a red solid line and an orange solid line. Because their down frequency remainders are located at (0,0.5) and (0.5, 1) respectively, their reconstructed eye images are mirror images of each other.
Now we analyze how to judge the sampling order. As it can be seen from Fig. 3 that the correct B can be obtained only when the both ${X_1}$ and ${X_2}$ are in $({0,0.5} )$ or $({0.5,1} )$, and at most one in ${B_{ij}}$ and ${B_{ji}}$ is correct. According to this feature, a method to judge the sampling order can be provided. When ${M_2} = {M_1}\textrm{ + }P$ and ${X_1}, {X_2} \in ({0.5,1} )$ are met at the same time, ${X_2} > {X_1}$ can be obtained according to ${B / {{f_2}}} = {M_1} + P + {X_2}$, ${B / {{f_1}}} = {M_1} + {X_1}$ and ${B / {{f_2}}} - {B / {{f_1}}} > P$. Since in software synchronization we take the abscissa of the first peak point as S, the data-rate converted from formula (18) is correct.
Remove the absolute value symbol in (18) to obtain ${B_{Jud}}$, which is called sampling order discriminant:
When ${X_1}, {X_2} \in ({0.5,1} )$ and P > 0, ${B_{Jud}}$ is negative.
When ${X_1}, {X_2} \in ({0, 0.5} )$, the correct measured data-rate is ${B_{12}}$, and remove the absolute value symbol in (17) to obtain:
When ${X_1}, {X_2} \in ({0, 0.5} )$, P > 0, ${B_{Jud}}$ is positive.
Then it can be judged that when ${B_{Jud}}$ is positive, the sampling order is sequential, and correspondingly when ${B_{Jud}}$ is negative, the sampling order is reverse. This method is effective only when P > 0 because the result of (10) after exchanging subscripts 1 and 2 are equal when P = 0.
As we known, the frequency selection factor P should be selected manually according to the prior knowledge of the data-rate range of SUT. For example, when f1 = 98.53 MHz, f2= 97.33 MHz, f3 = 96.13 MHz is set, the measurable data-rate range is shown in Table 1. When the data-rate of SUT is more than 7.991 GBaud and less than 9.746 GBaud, P = 1 should be selected during calculating the data-rate. But it should be noted that the width of the range determined by (16) gradually decreases with the increase of P. When the width of the range is equal to 0, there is:
Table 1. measurable range corresponding to P
Simplify (22) to obtain:
which shows that there is an upper limit of P,Practically, ${P_{\max }}$ must not be selected to ensure that the width of the measurable range matches the data-rate range of SUT.
In the actual processing of data streams, in addition to data-rate measurement, some DSP algorithms for conventional coherent optical communication are also applied, such as normalization orthogonalization, polarization division demultiplexing, frequency offset estimation, carrier phase recovery, etc. The flow chart of all DSP algorithms applied in data-rate measurement is shown in Fig. 5.
The range selectable data-rate measurement method is actually a calculation of the transmission frequency of the time-domain waveform, which means that it is transparent to the modulation format. Thus, it is also feasible to measure the data-rate of signals of different modulation formats such as 16QAM, 64QAM, etc.
4. Experimental analysis
We conducted an experiment to verify the data-rate measurement method with selectable range described in Section 2. A custom MFL pump source, with operating frequency of 95.00 MHz to 100.00 MHz and adjustable accuracy of 0.01 MHz (gain spectrum width 20 nm), is used as a seed source. We set the pulse repetition frequency as f1 = 98.53 MHz, f2 = 97.33 MHz, f3 = 96.13 MHz. A fiber laser (EXFO IQS-636) is used to output a continuous wavelength laser with a center wavelength of 1550 nm and a linewidth of 100kHz as an optical carrier. The optical carrier is modulated by IQ Mach Zehnder modulator (IQ MZM), which is driven by the PRBS signal trace from the arbitrary waveform generator (Keysight M8192). The signal generated by the arbitrary waveform generator has been shaped by Raised Cosine filter with the roll-off coefficient α= 0.38. In the LOS system, we use a 2 × 8-wavelength hybrid (Kylia, COH24-X) to realize the mixing of SUT and optical pulse. The data-rates of SUT are set from 800 MBaud to 40.8 GBaud as in Table 2, and each SUT is sampled by pulses with different repetition frequencies for three times. The signal trace after the wavelength hybrid is shown in Fig. 6. After the photoelectric conversion in four BPDs (Thorlabs, BPD470C), a 4-channel data acquisition card (NI5160) is applied to collect the time waveform at a sampling rate of 1.25 G samples/s. After collecting the data, we use the DSP process in Fig. 5 to measure the data-rate and characterize SUT. When processing, the P is selected from 0 to 5 corresponding to the measurable data-rate range shown in Table 1. The baud-rate setting and the corresponding P are shown in Table 2. Since the baud-rate of a QPSK signal is equal to the bit-rate of channel I or channel Q, channel I of SUT is taken for data-rate measurement in experiment. For each SUT, it is feasible to use the data collected from any of the three samples for eye diagram reconstruction. In the experiment we use the data collected by the sampling with the repetition frequency of 96.13 MHz.
Table 2. Set baud-rate and corresponding P
The measurement results for each SUT are compared with the data-rate set in Table 2, and the accuracy performance of the data-rate measurement method is tested by relative error.
A comparison of the measured baud-rate and the real baud-rate is shown in Fig. 7(a). The data-rate set according to Table 2 is drawn as a red solid line in Fig. 7(a) while the measured data-rate is drawn as a blue hollow circle. In LOS, the quality of optical signal is characterized by EVM value. In order to test the performance of our proposed method under different signal quality, the EVM values of SUT at each data-rate are also applied for comparison.
Fig. 7. Experimental data analysis. (a) Real baud-rate B and measured baud-rate B. (b) EVM and relative error.
In Fig. 7(b), the relative error of the measured data-rate is plotted as a red line, while the EVM value is plotted as a blue line. It can be seen that the quality of SUT is best at about 8 GBaud, and the EVM value gradually increases with the increase of the baud-rate. However, the relative error is not sensitive to the deterioration of signal quality, which shows that the proposed method has better applicability. Figure 7(b) shows that when the EVM is less than 0.38, the relative error of the measured baud-rate is less than 0.17%. This proves that our proposed method has a high accuracy even when the signal quality is very poor, which shows the reliability of the method. The reconstructed eye diagram with measured time information and constellation diagram of 8 GBaud QPSK signal using the repetition frequency of 96.13 MHz is shown in Fig. 8.
Fig. 8. Eye diagram with measured time information and constellation diagram of 8 GBaud QPSK signal, Q = 8.1620 dB, EVM = 0.1503.
5. Conclusion
This paper proposes a range selectable data-rate measurement method based on MFS in LOS. In our proposed method, the measurement range can be selected according to the data-rate range of the SUT. In LOS system, a MFL is used to generate pulses with different repetition frequencies to sample the SUT for three times, and then the data acquired by the ADC is used to calculate the data-rate of the SUT in our proposed method. The data-rate of SUT can be measured without blind region in the selected range, and the sampling order can be judged, which is key for plotting eye diagrams with correct time information. Compared with the existing method, under the same sampling cost, our proposed method realizes the selectivity of the measurable data-rate range and the judgment of the sampling order, greatly extend the measurable data-rate range of SUT. We experimentally measure the data-rates of QPSK signal as high as 40.8 GBaud in a triple-frequency LOS system and judge the sampling order. When EVM is lower than 0.38, the relative error of measured baud-rate is less than 0.17%. Thus, the range selectable data-rate measurement method has great potential for future high-speed signal monitoring.
Appendix A
Since ${f_1}$ and ${f_2}$ are equal in status in multi-frequency sampling, for the convenience of discussion, ${f_1} > {f_2}$ is taken, the calculation formula of the measured data-rate ${B^{\prime}}$ is written as follows:
In the first case, when ${M_2} = {M_1}\textrm{ + }P$ is satisfied but not $({X_1} - 0.5)({X_2} - 0.5) > 0$, because of ${f_1} > {f_2}$, ${X_2} > {X_1}$ can be obtained according to ${B / {{f_2}}} = {M_1} + P + {X_2}$, ${B / {{f_1}}} = {M_1} + {X_1}$ and ${B / {{f_2}}} - {B / {{f_1}}} > P$, there must be ${X_2} \in ({0.5,1} )$ and ${X_1} \in ({0,0.5} )$. Since the abscissa of the first peak point is taken as the number of equivalently scanned bit slots S in software synchronization, ${X_2}$ should be replaced by $1 - {X_2}$ in (26), and ${B^{\prime}}$ is:
Using $\frac{B}{{{f_1}}} = {M_1} + {X_1}$, $\frac{B}{{{f_2}}} = {M_1} + P + {X_2}$ to eliminate ${X_2},{X_1}$ in (27):
In order to prove the measured data-rate ${B^{\prime}}$ is less than or equal to the real data-rate B, it is only necessary to prove:
Considering ${f_1} - {f_2} > 0$, (29) can be written as:
The proof of (30) is equivalent to the proof of (31) and (32):
For (31), substitute ${M_2} = {M_1}\textrm{ + }P$ and simplify it to obtain:
Formula (33) divides $2{f_1}$ by both sides and sort out:
Substitute ${B / {{f_1}}} = {M_1} + {X_1}$ into (34) to obtain:
Substitute ${M_2} = {M_1}\textrm{ + }P$ into (35) and simplify it to obtain:
According to ${X_1} \in ({0,0.5} )$, when P takes any natural number, (36) is obviously true, so (31) is true.
Then prove (32), substitute ${M_2} = {M_1}\textrm{ + }P$ into (32) and simplify it to obtain (37):
Formula (37) divides $2{f_2}$ by both sides to get (38):
Substitute ${B / {{f_2}}} = {M_2} + {X_2}$ into (38) and simplify it to:
Similarly, according to ${X_2} \in ({0.5,1} )$, easy to know (39) is obviously true, so (32) is true. Then ${B^{\prime}} \le B$ is certified in the first case.
In the second case, when neither ${M_2} = {M_1}\textrm{ + }P$ nor $({X_1} - 0.5)({X_2} - 0.5) > 0$ is satisfied, ${f_1} > {f_2}$, ${B / {{f_2}}} = {M_1} + P + {X_2}$, ${B / {{f_1}}} = {M_1} + {X_1}$ and ${B / {{f_2}}} - {B / {{f_1}}} > P$, so ${X_2} > {X_1}$. It is easy to get ${X_1} \in ({0.5,1} )$, ${X_2} \in ({0,0.5} )$, ${M_2} = {M_1}\textrm{ + }P\textrm{ + }1$. Because the software synchronization takes the abscissa of the first peak point as S, ${X_1}$ should be replaced with $1 - {X_1}$ in (26), and the measured data-rate ${B^{\prime}}$ is:
Similarly, use ${B / {{f_1}}} = {M_1} + {X_1}$, ${B / {{f_2}}} = {M_1} + P + {X_2}$ to eliminate ${X_2},{X_1}$ from (40):
To prove ${B^{\prime}} \le B$, just prove:
Using simplification steps similar to the first case, (42) can be equivalent to (43) and (44).
Formula (43) is equivalent to ${X_1} \le 1$, which is obviously true from ${X_1} \in ({0.5,1} )$. Formula (44) is established because ${X_2} \in ({0,0.5} )$ and $P \ge 0$. Then ${B^{\prime}} \le B$ in the second case is proved.
In the third case, when both ${M_2} = {M_1}\textrm{ + }P$ and $({X_1} - 0.5)({X_2} - 0.5) > 0$ are satisfied, but ${X_1}, {X_2} \in ({0.5,1} )$, because ${f_1} > {f_2}$, ${B / {{f_2}}} = {M_1} + P + {X_2}$, ${B / {{f_1}}} = {M_1} + {X_1}$ and ${B / {{f_2}}} - {B / {{f_1}}} > P$, there is still ${X_2} > {X_1}$. Since the abscissa of the first peak point is taken as S in software synchronization, ${X_1} - {X_2}$ should be replaced by ${X_2} - {X_1}$ in (26), and the measured data-rate ${B^{\prime}}$ is:
Similarly, use ${B / {{f_1}}} = {M_1} + {X_1}$, ${B / {{f_2}}} = {M_1} + P + {X_2}$ to eliminate ${X_2},{X_1}$ in (45):
To prove ${B^{\prime}} \le B$, just prove:
Using simplification steps similar to the first case, (47) can be simplified to (48) and (49).
Using simplification steps as those in the first and second cases, (48) can be simplified to:
Easy to get that (50) must be established.
Similarly, formula (49) can be reduced to:
It can be seen from ${X_1} < {X_2}$ that (51) must be established.
To sum up, ${B^{\prime}} \le B$ is certified in all cases.
Funding
National Key Research and Development Program of China (2018YFB1801705); National Natural Science Foundation of China (61727817, 62021005).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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