Abstract
Binary offset carrier (BOC) modulations are widely used in the modernized satellite navigation systems. But we find an interesting phenomenon that the signal composed of different BOC modulations suffers larger phase bias than the case that all the components have the same modulation due to the nonideal group delay of the channel. This issue is more obvious in navigation satellites for the use of a narrowband filter to suppress the out-of-band part. The general form of analytic expression that reveals the impact of the nonideal channel on phase bias is presented and validated with simulations. Further, by simplifying the spectrums for the signals like B1 of BeiDou and E1 of Galileo navigation systems, we obtain the concise expression of phase bias and find that the bias is mainly derived from the phase difference between the subcarrier frequencies caused by the odd order terms of group delay.
1 INTRODUCTION
The binary offset carrier (BOC) modulation, which has the advantages of splitting spectrum and better code-tracking performance due to its larger Gabor band, was proposed as a candidate modulation for satellite navigation by Betz.1 Until now, BOC modulations are widely used in various global navigation systems. For the power spectrum separation between open and authorized services, the simplification of the transmitting channel in satellite payload, and the utilization of the limited resource of the frequency band of satellite navigation, the signal components with different modulations are multiplexed and broadcasted in the same center frequency. For example, Multiplexed BOC (MBOC)(6, 1) and BOCc (15, 2.5) were chosen by Galileo in the E1 band2,3; Time-Multiplexed BOC (TMBOC)(6, 1) was chosen by Global Positioning System (GPS) in the L1 band4,5; Quadrature Multiplexed BOC (QMBOC)(6, 1)6,7 and BOC(14, 2) were chosen by BDS (BeiDou Navigation Satellite System) in the B1 band.8–10 Some signal components stay around the center frequency while some stay close to the margin of the frequency band. For these modernized signals, the phase bias, which is defined as the bias of the phase relation between the signal components from the expected one, becomes obvious to the nonideal characteristics of the channel.
Because some receivers combine multiple signal components in the tracking loop or utilize the tracking results of one component to guide the processing of the other one, it is necessary to give clear descriptions of the phase relations among the signal components and the maximum phase bias allowed. The phase relations between the signal components and the maximum phase bias are described in the interface control document. For example, the phase bias between L1C and P signals limited to the range of ±0.1 rad (5.7°) is given in IS-GPS-800.5 Therefore, we should optimize the channels in the navigation satellite payload or receiver to decrease the bias.
The conventional view about the phase bias between the signal components is that the bias is mainly affected by the modulator performance in the transmitting channel. But for modernized signals, we find that the nonideal group delay of the radio frequency (RF) channel in payload or receiver has an important impact on the phase bias. Here, the nonideal group delay means that the group delay in the bandwidth is not flat and includes higher order terms except for the zeroth-order term. Generally, the main distortion of the group delay is derived from the filters, which are used to suppress the out-of-band part of the transmitting signal. Lots of references focus on modeling the group delay and evaluating its impact on code-tracking performance. The group delay that can be presented by the form of a Taylor expansion was presented in Zhu et al.11 Instead of using the derivative of phase to angular frequency to express the group delay, they provided a new way of describing the group delay. With the Taylor expansion, the description of the group delay becomes visual. The effect of the group delay caused by ionospheric delay on the S-Curve was investigated for wideband Global Navigation Satellite System (GNSS) signals.12 They revealed that the group delay distortion from the ionospheric dispersion effect induces more distortion on wideband GNSS signals and its impact on these signals needs to be handled. The Galileo E5 signal was taken as an example to show the compensation effect of the all-pass filter on the ionospheric dispersion. The evaluations of signal distortions in the GNSS payload level and receiver were presented in Soellner et al,13 and Rebeyro et al,14 respectively. In these references, the phase bias or quadrature error between signal components is selected as an indicator to evaluate the signal quality for the signal broadcasted from the satellite. Also, the constellation diagrams of the signals distorted by the transmitting channel were evaluated. The delay of the maximum of the main peak of correlation and the analytic expression of zero crossing bias of the discriminator curve for the noncoherent Early Minus Later Power (EMLP) discriminator were presented in Liu et al.15 They explained how the group delay distorts the discriminator curve and which types of distortion of the group delay bring more distortions to the discriminator curve. Kim16 analyzed the impact of group delay variation from Controlled Reception Pattern Antenna (CRPA) hardware on the carrier phase and code biases using the simulation method. The code and phase biases from GNSS hardware became important and deteriorated the performance of integer ambiguity resolution if not handled properly.17 By using measurements from receivers, Hauschild et al18 performed a detailed analysis of the impact of correlator design on code bias and showed that the correlator spacing of the receiver plays an important role in code bias. Methods of estimating the amplitude and group delay of the channel from the received signal were provided and compared in Yan et al.19 They showed that the correlation-domain method can provide an accurate estimate of the amplitude and group delay except for the zero value area of the spectrum. These previous works presented that the evaluation and constraint of the code and phase biases from GNSS hardware are important for high-accuracy positioning. Most of the works focus on modeling the impact of the RF channel on code bias or evaluating the biases based on the measurements of the receiver. Few references analyze the impact of the nonideal group delay on the phase bias between the signal components with different spectrums and reveal how the group delay affects the phase bias.
First, we present and model the phenomenon of the phase relation between the signal components with different types of spectrums affected by the nonideal group delay of the RF channel in general form in Section 2. Then, in Section 3, we validate the theoretical results. Third, we simplify the presented expression and analyze the effect of different order terms of the group delay on phase bias in Section 4. Finally, we further analyze the impact of other factors on phase bias in Section 5 and make the conclusions in Section 6.
2 MODEL OF PHASE BIAS
The payloads of navigation satellites contain linear and nonlinear distortions. The linear distortions are mainly from the nonideal amplitude and group delay of the filters in the RF channel, while the nonlinear distortions are mainly from the high-power amplifier (HPA) in the transmitting channel. The nonideal properties of HPA and filters in the payload both distort the phase relation between the signal components with different spectrums. The effect of the cascade connection of the HPA and filter on phase bias can be approximately treated as the superposition of the biases from the HPA and filter, which is illuminated by the example of using a simulation method in Section 5. With including the nonlinear distortions of HPA in the model, it is difficult to obtain concise and analytic expressions, and simulation is an efficient way to evaluate the impacts. For evaluating and improving the phase bias from the nonideal group delay efficiently, it is meaningful to give a clear description between the nonideal group delay and the phase bias using an analytic method.
The signal consisting of BOC(1, 1) and BOC(14, 2) is chosen as an example. The signal is filtered with the group delay shown in Equation (9). Due to the impact of the nonideal group delay on the phase relation between BOC(1, 1) and BOC(14, 2), the phase orthogonality between the components is distorted. After normalizing and rotating the filtered baseband signal, the IQ constellation diagram is obtained by choosing the real and imaginary parts of the processed baseband signal as the x-axis and y-axis of the diagram, respectively. Figure 1 shows the well properties of the baseband signal without distortion. The IQ constellation diagram illuminated in Figure 2 reveals that the nonideal group delay distorts the phase relation between BOC(1, 1) and BOC(14, 2).
IQ constellation diagram of the signal composed of BOC(14, 2) and BOC(1, 1) for the ideal case without distortion of group delay
IQ constellation diagram of the signal composed of BOC(14, 2) and BOC(1, 1) affected by the nonideal group delay of the filter shown in Equation (9). The phase orthogonality between the components is distorted
In satellite navigation systems, several signal components with BOC or Binary Phase Shift Keying(BPSK) modulations are multiplexed to form a constant envelope signal with the same center frequency. The multiplexed signal components may have different modulations, spectrums, and power allocations. Considering that the constant envelope multiplex modulation and data modulation have limited impacts on the evaluation of the phase bias from the nonideal group delay, we ignore them in the model for simplifying the analysis process. The similar assumptions of omitting data modulation or constant envelope modulation can be found in Huo et al,20 Yoo et al,21 Yao and Lu,22 and Julien et al23 for the sake of clarity when analyzing the interference mitigation or tracking performance. In this paper, we mainly focus on analyzing the impact of the nonideal group delay; the other factors such as nonlinear distortion form HPA are not included in the model, but we extend the analyses of other factors on phase bias using the simulation method in Section 5.
The signal s(t) that consists of two components, that is, c1(t) and c2(t), can be modeled as
1
where c1(t) and c2(t) are low-pass signals that depend on two different pseudorandom code sequences; 
and 
are the filtered versions of c1(t) and c2(t), respectively; A1 and A2 are the amplitude of the corresponding components; α is the amplitude imbalance of the modulator; φΔ is the quadrature error of the modulator; f0 is the carrier frequency; and φ0 is the initial phase. Here, the phase relation between c1(t) and c2(t) is assumed to be 90°.
For convenience, we assume the carrier frequency is accurately estimated. The baseband form of the transmitting or receiving signal is expressed as
2
where 
and 
can be further written as
3
4
where the symbol ⊗ means convolution and h(t) is the equivalent low-pass filter. Here, h(t) can be treated as the equivalent filter for the transmitting channel, the receiving channel, or the cascade connection of transmitting and receiving channels.
To estimate the phase relation between the signal components, we need to make despread first. The local replica signals of 
and 
are 
and 
, where τ1 and τ2 are the estimated delays of 
and 
, respectively. The output of the integration and dump for c1(t) is
5
where P1(f) is the power spectrum of c1(t), Bf is the frontend bandwidth, and H(f) is the response of the filter. In a similar way, the output of the integration and dump for c2(t) is
6
where P2(f) is the power spectrum of c2(t).
Thus, the general form of the carrier phase difference between c1(t) and c2(t) is calculated by
7
Substituting Equation (5) and Equation (6) into Equation (7) yields
8
Equation (8) shows the phase relation between c1(t) and c2(t). Here, we can see that the amplitude imbalance does not impact the phase relation and that the quadrature error of the modulator causes an additional phase bias φΔ to the case of only existing nonideal group delay. Thus, in the following analysis, the errors from the modulator are omitted for clarity. Here, we note that the phase relation between c1(t) and c2(t) depends on the components’ spectrums and the filter response.
The moment of generating c1(t) and c2(t) is usually synchronized, but the estimated delays τ1 and τ2 may show little difference due to the nonideal group delay of the channel. Here, the delay denotes the time offset of the local replica code related to the analyzed signal. The steps of accurate calculation of the delay are given in Liu et al,15 and the accurate phase bias can be obtained by using the expression in Equation (8) together with the accurately calculated delay. The error in the estimated delay mainly causes the decrease of the amplitude of the correlator output, as long as the estimated delay is still limited in the scope of the main peak of the correlation function. Further, the estimated delay that exceeds the range of the main peak of the correlation function for BOC can cause an unexpected 180° flip of the phase. For convenience in the analysis of phase bias, the delays of the two components can be set to the zeroth-order group delay due to the limited impact on the results in most cases. This is reasonable—the estimated delay is mainly from the zeroth-order group delay of the channel, and higher order terms of group delays cause little distortion to the delay of the main peak of the correlation function. The error from the approximation will be evaluated in the section on validation.
3 VALIDATION
In this section, we first introduce the model of the output filter used in the validation. Then, three types of signals belonging to different navigation systems are selected as examples to validate the analysis results. Finally, we perform further validation of the theoretical results by extracting the phase bias from the signal generated in the test bench.
To reveal why and how the group delay induces phase bias, we expand the group delay with the Taylor form.11 Generally, the group delay of the filter can be properly fitted by third-order and below order terms with little residual errors. The curve of the group delay within the bandwidth in Figure 3, which has the typical feature of a parabola, is the cubic fitting result of the measurement of an output filter from a vector network analyzer. If necessary, similar analysis and discussion can be performed for other filters with higher order terms of the group delay involved.
Cubic fitting group delay within the bandwidth
The model of group delay is given by
9
where f is the frequency variable and τg, τg1, τg2, and τg3 are the zeroth-order, first-order, second-order, and third-order group delays, respectively. The values of these coefficients used for the description of the curve in Figure 3 are given as follows: τg is 11.2E−9 (s); τg1 is 2.06E−17 (s2/rad); τg2 is 1.73E−24 (s3/rad); τg3 is 3.58E−33 (s4/rad). Meanwhile, the phase response of the filter is written as
10
where θ0 is the initial phase.
The impact of the amplitude of the RF channel on phase bias can be evaluated by using the general form in Equation (8). For its limited impact on phase bias, the amplitude of the channel is assumed to be ideal for focusing on group delay. In the validation, H(f) is given by H(f)= ei * θ(f).
In this section, three types of typical signals belonging to different satellite navigation systems are taken as examples. The signal in Case 1 denotes the signal types in L5 of GPS, E5a or E5b of Galileo, and B2a of BDS; the signal in Case 2 presents the signal types using different BOC modulations in B1 of BDS and E1 of Galileo; and the signal in Case 3 accords with the signal types in L1 of GPS. Besides for validation, the results also show that the signals containing different BOC modulations suffer larger phase bias in the presence of the nonideal group delay than the other cases. The group delay model in the validation contains the zeroth-order, first-order, and third-order terms of group delay, which covers the typical characteristics of filters in practice. If needed, higher order terms of group delay can be included in the model using the similar way.
First, the signals with specific modulations are generated. Then the generated signals are filtered using a Fast Fourier Transform (FFT) algorithm to minimize the error of group delay induced in the filter process.
Second, the tracking and measurement processes are performed in a software receiver. The receiver uses a classical code-tracking loop using EMLP discriminator with the side-peak lock detection algorithm and a Costas carrier tracking loop. We apply two independent carrier tracking loops for each signal component. After the lock of the loops, we extract the phase information from the two carrier loops at the same time and calculate the phase difference between the two components. Then, the phase bias is obtained by subtracting the expected phase relation between the two components from the calculated phase difference.
3.1 Case 1
The first case is that the signal contains two traditional BPSK(10) signals filtered by the model shown in Equation (9). The expression of the spectrum for BPSK(10) signals is given by24
11
where Tc is Pseudo-Random Noise (PRN) code duration time and sinc(x) = sin(x)/x.
For the two components with the same modulation, the spectrums of them are equal; that is, P1(f)= P2(f)= PBPSK(f). Thus, the phase difference calculated from Equation (8) is
12
The phase difference is the same as expected; therefore, no bias exists.
In the validation of simulation, we generate two BSPK(10) signals with the phase relation shown in Equation (2). The simulated result of phase bias is 0.0015°, and the non-zero of the phase bias is mainly from the nonideal feature of the PRN codes. The simulated result agrees well with the theoretical result. Thus, we can conclude that no phase bias is induced for the components with the same spectrum even if the group delay is nonideal.
3.2 Case 2
Here, the signal composed of BOC(1, 1) and BOC(14, 2) is taken as an example. c1(t) is set to BOC(1, 1), and c2(t) is set to BOC(14, 2). It is noted that the BOC in the validation stands for the sine-phase binary offset carrier modulation. The spectrum of BOC(1, 1) is given by Kaplan et al24
13
where 
is the PRN code duration time of BOC(1, 1) and the subcarrier frequency of BOC(1, 1) fse1 is equal to 1.023 MHz. The spectrum of BOC(14, 2) is given by Kaplan et al24
14
where 
is the PRN code duration time of BOC(14, 2) and the subcarrier frequency of BOC(14, 2) fse2 is equal to 14.322 MHz. The main spectrums of the components concentrate in their own subcarrier frequencies, as shown in Figure 4. Substituting Equation (9), Equation (13), and Equation (14) into Equation (8), we obtain the theoretical phase bias of 4.93°.
Theoretical spectrums of BOC(1, 1) and BOC(14, 2)
Simulation is performed to validate the analysis. In the simulation, BOC(1, 1) and BOC(14, 2) are generated and filtered by the filter with the group delay shown in Figure 3. After the convergences of the tracking loops, the simulated result of the phase bias is 4.92°. Thus, it can be concluded that the phase bias from the simulation agrees well with the theoretical result.
The effect of the approximated delay on phase bias is further discussed here. Using the model of the group delay in Figure 3 and the spectrums in Equation (13) and Equation (14), we obtain the accurate delays of the maximum main peak of the correlation functions of BOC(1, 1) and BOC(14, 2). The zeroth-order group delay is 11.2 ns, while the accurate delays τ1 and τ2 for BOC(1, 1) and BOC(14, 2) are 12.8 and 13.7 ns, respectively. The phases of the correlator output versus delays are shown in Figure 5. We can see that the phase of the correlator output changes slowly around τ = 13.7 ns for BOC(14, 2) and the phase of the correlator output almost keeps constant in the range of [3–25] ns for BOC(1, 1) in Figure 5B. The phases calculated by using τ1 = τ2 = 11.2 ns are approximately equal to the phases from τ1 = 12.8 ns and τ2 = 13.7 ns. This can be explained as follows: the signal with BOC modulation has multiple peaks. The range of the delay of the replica code corresponding to the main peak of BOC(14, 2) is about –3 to 31 ns, and the range for BOC(1, 1) is much larger than that for BOC(14, 2). Thus, we can see that the phases of correlation output for BOC(1, 1) and BOC(14, 2) almost keep constant around the delay of the maximum main peak of the correlation function of BOC(14, 2); that is, τ = 13.7 ns. When the delay is approaching the margin of the main peak, the correlation output tends to be zero and further becomes opposite, which results in that the phase for BOC(14, 2) shows the inverse of 180°. Thus, we can understand that the calculation of the phase bias using the zeroth-order group delay instead of the accurate delay of the maximum main peak of the correlation function is acceptable.
Phase of the correlation output versus delays for BOC(1, 1) and BOC(14, 2). A, Full view. B, Zoom in of (A)
For comparison, the phase bias between BOC(1, 1) and BOCc(15, 2.5) in E1 of Galileo is also calculated. Using a similar process, we obtain that the phase bias between BOC(1, 1) and BOCc(15, 2.5) in E1 of Galileo is 6.3°. Here, we can see that the phase bias between BOC(1, 1) and BOCc(15, 2.5) is larger than the phase bias between BOC(1, 1) and BOC(14, 2). This is because the subcarrier frequency of BOCc(15, 2.5) is farther away from the center frequency than that for BOC(14, 2).
3.3 Case 3
In the third case, the signal containing BOC(1, 1) and BPSK(10) is taken as an example to be analyzed in the validation. Their power spectrums are shown in Figure 6. Substituting Equation (10), Equation (11), and Equation (13) into Equation (8), we obtain that the theoretical phase bias is −0.39°.
Theoretical power spectrums of BOC(1, 1) and BPSK(10)
Similar to Case 2, the simulation is performed, and the simulated phase bias is −0.40°. Thus, we can obtain that the theoretical result agrees well with the simulated result and the phase bias is slighter compared with the result in Case 2.
3.4 Validation with test bench
We perform a validation based on our test bench. The signal composed of BOC(1, 1) and BOC(14, 2) is generated in a signal generator. The model of signal generation is based on Equation (1) with equal power allocation. The modulation of the signal is implemented in a digital modulator; thus, the errors from the modulator are ignorable. The generated signal goes through a cavity filter with the group delay shown in Figure 7. The signal at the output of the filter is directly sampled with a high fidelity signal recorder with a rate of 1 GHz. Then the obtained phase bias between BOC(1, 1) and BOC(14, 2) from the sampled signal is 3.35°. Compared with the theoretical phase bias of 3.42° calculated by Equation (8), we can obtain that the theoretical result agrees with the result from the sample data.
Group delay of the cavity filter in the validation
4 FURTHER ANALYSIS AND SIMPLIFIED MODELS
In this section, we first analyze the phase bias with the components having the same spectrum. Then, the impacts of zeroth-order, first-order, second-order, and third-order terms of group delay on phase bias are discussed. By simplifying the expressions of the spectrums, we analyze how the group delay deteriorates the phase bias and which terms of group delay have a greater impact on the bias. Lastly, the steps of calculating the phase bias under different conditions are summarized and given in detail.
From the results presented in the above section, we understand that the signals in B1 of BDS and E1 of Galileo suffer larger phase bias than the other types of signals. Thus, the group delays of the transmitting and receiving channels for those signals should be optimized for this indicator. The general form expression of the phase bias is complicated, and the relationship between the phase bias and group delay is not distinct and clear. Simplification should be performed to reveal the essential factors that cause the main bias and provide a quick way of evaluating the bias.
Observing Figure 8, we find that the spectrums of BOC(1, 1) and BOC(14, 2) concentrate in their subcarrier frequencies and the spectrums are symmetrical about the center frequency. Therefore, we can use the Dirac delta function with the frequency located in the subcarrier frequency instead of the original signal spectrum for simplification, as shown in Figure 8.
Simplified spectrums of BOC(1, 1) and BOC(14, 2)
Then, the simplified spectrums become
15
16
In the following analysis, we select several typical cases for presenting the influence of group delay on phase bias. The simplified expressions will be used for discussing the effect of higher order terms of group delay on phase bias.
4.1 Signal components with the same spectrum
It is noted that the spectrum mentioned in this paper means the spectrum envelope. In this case, the spectrums of the components are the same; that is, P1(f) = P2(f). By using the presented expressions, we obtain that the phase difference is 90°, and the phase bias between the components is immune to the nonideal group delay. This is due to the fact that the components with the same spectrum suffer the same level of distortion. In fact, the signals with the same envelope spectrum may have different spectrum details with different PRN codes, which would lead to the existence of a small bias. In this case, the detailed spectrum can also be substituted into Equation (8) to reveal its impact on phase bias.
4.2 Effect of zeroth-order group delay on phase bias
In this case, the phase response of the filter is
17
The estimated delays are equal to the zeroth-order group delay, that is, τ1 = τ2 = τg. Substituting Equation (17) into Equation (8), we have
18
It is noted that P1(f) and P2(f) in Equation (18) are nonnegative real. Then, the phase difference is the same as expected. Thus, we can see that only the zeroth-order group delay does not introduce bias regardless of the signal components. The zeroth-order group delay is an ideal case, and it delays all the frequency components with the same amount. However, it is hard to design an RF channel having only zeroth-order group delay without using channel equalization. In practice, the group delay of the channel is more complex and usually contains the zeroth-order, first-order, or higher order terms.
4.3 Effect of zeroth-order and first-order group delays on phase bias
In the presence of zeroth-order and first-order terms of group delay, the phase response becomes
19
In this case, the delay is set to the zeroth-order group delay as discussed in Section 2. Substituting Equation (19) into Equation (8) yields
20
The expression in Equation (20) is for all kinds of signals, but we find that the effect of group delay on phase bias is complicated. To obtain more concise and plain expressions for optimizing the group delay of the channel for the sensitive signals, we substitute the simplified expressions of the spectrum given in Equation (15) and Equation (16) into Equation (8) and obtain
21
Equation (21) can be further expressed as
22
where 
is the phase response of the filter from the first-order group delay τg1. Thus, we obtain the phase bias between the signal components, which mainly depends on the phase difference from the first-order group delay between fse1 and fse2 in this case.
The signal components in B1 of BDS are taken as an example to show the efficiency of the spectrum simplification. Substituting the value of τg1 into Equation (22), we obtain that the phase bias between BOC(1, 1) and BOC(14, 2) is 4.74°. The phase bias obtained from the original expression in Equation (8) is 4.402°. Therefore, we can conclude that the error caused by the spectrum simplification is acceptable.
The phase response of the filter from the first-order group delay is shown in Figure 9. By further observing Equation (22) and Figure 9, we can see that the phase bias is related to the difference of the phase response of the filter between fse1 and fse2 and that the phase between the signal components with larger subcarrier frequency suffers more bias.
Phase response of the filter caused by the first-order group delay
4.4 Effect of zeroth-order, first-order, and second-order group delay on phase bias
With the addition of the second-order group delay in Equation (19), we have
23
Using the similar process for the estimated delay, the general form of the phase difference for such a case is given by
24
Using the simplified spectrum model, we obtain
25
Here, we first analyze the terms in the two cosines affected by the second-order term of group delay. Take the group delay of the filter shown in Figure 3 for example. The second-order term of the group delay is plotted in Figure 10. We can see that even if the fluctuation of the second-order term of the group delay reaches 14 ns in the bandwidth, the phase response of the filter as shown in Figure 11 due to the second-order term of group delay is still limited to 13° in the range of [fse1fse2]. It is still easy to satisfy the condition in which the two cosines take the same sign. If the filter is so terrible that the fluctuation of the second-order term of group delay causes the opposite sign of the two cosines, the inverse of 180° of phase relationship would happen.
Group delay fluctuations caused by the second-order terms in the bandwidth
Phase response of filter due to the second-order term of group delay
Here, we focus on the case that the two cos(·) functions have the same signs; then Equation (25) can be further simplified as
26
It is noted that the phase difference in Equation (26) is the same as the case of the only existence of the zeroth-order and first-order terms of group delay in Equation (22).
The phase bias in this case, calculated from the original expression in Equation (8), is 4.403°. By comparing the results in Section 4.3, we find that the addition of the second-order group delay slightly changes the phase bias. This can be found in Equation (24) where the impact of the first-order and second-order terms on the bias is mixed. Both the second-order group delay and the omitted spectrum components in the simplification cause the difference.
4.5 Effect of zeroth-order, first-order, second-order, and third-order group delay on phase bias
Substituting Equation (9) into Equation (8) and performing the similar operation to the equation, we obtain
27
Using the simplified spectrum model, we obtain
28
With the same assumption, the distortions caused by the second-order group delay in cos(·) functions have the same signs, and then Equation (28) can be further simplified as
29
Thus, the phase difference, in this case, is given by
30
where 
and 
. θg1(f) and θg3(f) are the phases caused by the first-order group delay and the third-order group delay, respectively. It is noted that the impact of the third-order group delay on the phase difference is directly added to the bias caused by the first-order group delay.
In this condition, the phase biases calculated by the original expression and the simplified expression are 4.93° and 5.31°, respectively. The results show that the simplified model agrees with the original one. Comparing this with the phase bias of 4.40° from the original model in the presence of zeroth-order and first-order terms of group delay, we find that the more the degradation of 0.5° is from the addition of the third-order group delay. The phase response of the filter caused by the first-order and third-order terms of group delay is shown in Figure 12. By comparing Figure 9 with Figure 12, we find that the curves in the two cases are similar and the phase at the subcarrier frequency of BOC(14, 2) is 0.5° larger than the phase of the only existence of the first-order group delay. It is reasonable that the third-order group delay is small and the phase caused by the third-order group delay at the frequency far from the center is larger than the phase near to the center frequency.
Phase of the filter from the first-order and third-order terms of group delay
The impact of group delay of high-order terms on phase bias can be analyzed by using the similar way. Based on the analysis presented above, we can conclude that the phase difference between fse1 and fse2 from the group delays of odd order terms plays an important role in the phase bias.
4.6 Summaries of calculating phase bias
Here, the steps of calculating the phase bias for some typical cases are summarized and shown in detail. The calculations of phase bias can be divided into three cases:
a. For the signal components with the same or similar spectrum envelope, the impact of nonideal group delay on phase bias between signal components is limited and ignorable. These types of signals include L5 of GPS, B2a of BDS, and E5a of Galileo.
b. For the signal containing BPSK and BOC components, we need to adopt the original expression of phase bias for calculating. It is noted that the following steps are also suitable for all types of signal components. The detailed steps are given by:
1. Extract the phase response of the channel in the bandwidth noted as θ(f);
2. Calculate the group delay from the phase response of the filter using
;3. Polyfit the group delay, and set the obtained zeroth-order term of group delay as the delays for both of the signal components. If needed, accurate delays can also be calculated using the method shown in Liu et al15;
4. Substitute the spectrum expression and phase response of the filter into Equations (5) and (8);
5. Calculate the phase difference between the components using Equation (8), then the bias is obtained by subtracting the ideal phase difference between the signal components from the calculated phase difference.
c. For the signal composed of different BOC components, such as the signal in B1 of BDS, the phase bias between the signal components becomes obvious. The calculation of phase bias can be rapidly performed using the simplified method. The steps are given as follows:
1. Extract the phase response of the channel in the bandwidth noted as θ(f);
2. Calculate the group delay from the phase response of the filter using
;3. Polyfit the group delay with third and below order terms, then we obtain the coefficients of the first-order and third-order terms. If needed, the coefficients of higher odd order terms can be obtained using the similar way;
4. Calculate the phase difference between the frequencies of the subcarriers of BOC components using the coefficients obtained in Step 3 by using Equation (30);
5. The bias is obtained by subtracting the ideal phase difference between the signal components from the calculated phase difference.
5 IMPACTS OF OTHER FACTORS ON PHASE BIAS
In the previous sections, we have analyzed the impact of group delay on phase bias between the signal components. To evaluate other factors in the channel affecting the phase bias, we have performed the analyses of the impacts of HPA and ionosphere on the phase bias in this section. Finally, the tracking biases induced in different carrier tracking loops have been analyzed in the presence of phase bias.
5.1 Impact of nonlinear HPA on phase bias
We use a simulation method to present the effect of nonlinear HPA on phase bias. The link of signal generation is shown in Figure 13. The group delay of the filter used in the simulation is described by Equation (9). The AM/AM and AM/PM characteristics of the HPA are shown in Figure 14.
Signal flow in the simulation to show the impact of HPA on phase bias
Nonlinear model of the HPA used in the simulation. A, Mode of AM/AM for the HPA. B, Mode of AM/PM for the HPA
The signal has an ideal constant envelope under the condition of infinite bandwidth. In practice, the signal generated in the payload is bandlimited, and the finite bandwidth actually deteriorates the envelope properties of the signal even if it is multiplexed with a constant envelope. The constant envelope modulated signal with wider bandwidth usually has better envelope properties and would suffer less nonlinear distortion of HPA. Thus, in the design and implementation of payload, the bandwidth of the filter before HPA is much wider than that of the output filter. The detailed analyses of the effect of different bandwidths of the filter before HPA on phase bias using simulation have been performed. The phase bias at point B contains the phase bias from the nonlinear HPA and the nonideal group delay of the filter, while the phase bias at point A only contains the distortion from the nonideal HPA. The phase biases at both points A and B are calculated under different bandwidths to show that the phase bias at point B can be treated as the phase bias distorted by the group delay of the filter plus the phase bias affected by the nonlinear distortion of the HPA, as shown in Table 1. From the results in the table, we can see that the phase bias from the HPA also needs to be considered and evaluated, except for the phase bias from the nonideal group delay. Further, it also indicates that the wider bandwidth of the filter before HPA leads to less nonlinear distortions on phase bias at point A. The results for the bandwidth less than or equal to ±40 MHz show larger phase bias at point A than those for the bandwidth larger than ±40 MHz. This can be explained as that the envelope of the signal becomes much better when the third harmonic of the subcarrier (3 * 14 * 1.023 MHz) is preserved, as shown in Figure 15.
Impact of the bandwidth of the filter before the HPA on phase bias
Spectrum of BOC(14, 2) under different bandwidths
5.2 Impact of ionosphere on phase bias
For the dispersive effect of ionosphere, the model of ionosphere can be treated as a special type of nonideal group delay and is given by Guo et al12
31
where c is the vacuum speed of light and total electron content (TEC) is the line integral of free electron density along the direct ray path.
The presented expression of calculating phase bias is also suitable for evaluating the phase bias from the ionosphere. Here, we take a typical value of TEC to demonstrate its impact on phase bias. The group delay for TEC = 50TECU due to the ionosphere is shown in Figure 16. The group delay in the bandwidth of 40 MHz is approximately linear and can be well fitted by the zeroth-order and first-order terms of group delay. Then, the model of group delay and the corresponding phase response are given by
Group delay due to the ionosphere with TEC = 50 TECU
32
33
where τg = 27 E − 9 (s) and τg1 = − 5.4E − 17 (s2/rad).
Using Equations (8), (13), (14), and (33), we obtain that the phase bias due to the ionosphere is −1.16°, which is opposite to the phase bias from the filter with the group delay shown in Figure 3. Further, if the transmitted signal from the payload affected by the filter goes through the ionosphere with the model shown in Figure 16, the final phase bias affected by the cascade connection of the filter and ionosphere is 3.77° and decreases −1.16° compared with the case of the only existence of the filter depicted in Equation (9).
5.3 Carrier tracking loop in the presence of phase bias
The existence of the nonideal group delay causes the phase bias between the signal components with different modulations. At the same time, the nonideal group delay also affects limited tracking performance variation under noise. Compared with the tracking performance under noise, the constant phase bias may bring more trouble to high-accuracy positioning. Thus, we should analyze the bias in different configurations of the carrier tracking loop in the presence of phase bias. For the reason that the impact of group delay on the phase is just as likely to add a constant phase rotation on the phase of the signal component, the analysis of bias in the carrier tracking loops can be equivalent to analyze the phase after the correlator outputs. There are two cases for different configurations of the carrier tracking loop.
The first case is that two independent carrier tracking loops are used for tracking the signal components, respectively. The phase difference between the two loops accords with the phase relation between the signal components. The distortions of the group delay on the signal would result in the phase relation deviating from the expected one. Thus, the phase bias, in this case, agrees with the results obtained by using the model in Equation (8).
For the second case, a coherently combined carrier tracking algorithm is used for tracking both of the signal components in the same loop. Thus, a fixed phase bias may exist in the tracking loop due to the fact that the local replica signal composed of two signal components for despreading is still generated according to the phase relation defined in the interface control document. The local replica signal for this case is given by
34
Then, the correlator output becomes
35
Further, 
can be approximately expressed as
36
where θ1 and θ2 are the phase rotations caused by 
and 
, respectively. In the tracking loop, the common phase rotation can be accurately removed. If we choose the quadrature branch as the reference, then we have
37
where θΔ is equal to θ1 − θ2 − φΔ and it contains the phase error from the modulator and the phase bias between the signal components affected by the group delay analyzed in the previous section. Thus, the tracking bias in the coherently combined carrier tracing loop is given by
38
The tracking bias depends on both the phase bias between the signal components and the errors from the modulator. We can see that in the ideal case, that is, θΔ = 0 and 
becomes 
and ΔθT = 0. However, considering the development of the performance of the digital signal process, most of the modulations are implemented in the digital domain, and the modulation errors become easy to be controlled. Thus, we assume φΔ = 0 and α = 1 for convenience to show the impact of the phase bias due to the nonideal group delay on the tracking bias in the coherently combined carrier tracking loop. With the assumptions of equal power allocation for the signal components, we obtain
39
Observing Equation (39), we find that the bias for the coherently combined carrier tracking loop is equal to half of the phase bias caused by the nonideal group delay calculated from Equation (8) in this case.
6 CONCLUSIONS
The nonideal group delay is usually inevitable in the transmitting or receiving channels, and the phase bias between the signal components caused by the nonideal group delay becomes serious for the modernized signals with different BOC modulations. We have established the general model for this phenomenon and derived the analytic expression by simplifying the signal spectrum for the signals like B1 of BDS and E1 of Galileo. The impacts of nonideal modulator, nonlinear HPA, ionosphere on phase bias, and phase bias on different configurations of carrier tracking loops are further analyzed and discussed. The obtained conclusions and expressions are useful for improving the phase bias in the design of the RF channel in the navigation payload or receiver. The simplified model provides a quick way to evaluate the phase bias caused by the nonideal group delay of the channel. Meanwhile, the corresponding results can be applied to analyze the phase relation between the signal components for other systems.
HOW TO CITE THIS ARTICLE
Liu Y, Yang Y, Chen L, Pan H, Ran Y. Analysis of phase bias between GNSS signal components caused by nonideal group delay. NAVIGATION-US. 2020;67:291–305. https://doi.org/10.1002/navi.363
- Received May 2, 2019.
- Revision received December 26, 2019.
- Accepted February 24, 2020.
- © 2020 Institute of Navigation
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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