Our study demonstrates that a chip shaped distortion is the root cause of pseudorange biases. However, receivers with different configurations exhibit different biases to individual signal deformations. Herein, we analyze the origin of pseudorange biases from two dimensions: the satellite and user aspects. In terms of the satellite aspect, we will introduce how the chip shapes affect the tracking biases. In addition, in terms of the user aspect, we will deduce from pseudorange observation equations how receivers respond differently to individual satellite signal distortions.

### Satellite signal deformation

Taking the BDS-2 satellite signal as an example, the shapes of the BDS signal code chips are typically rectangular, as shown in Fig. 1. Satellite-generated signals must travel a long distance before they are processed at the user’s receiver for a PNT service. Herein, we combine the entire trip as a full channel \( H_{i} (f) \), including the on-board channel, atmospheric channel, and receiver channel. When \( i = 0 \), we assume there are no distortions throughout the entire channel. However, when \( i = 1 \), it indicates that the channel is experiencing distortion to a certain extent. If the bandwidth *B* is extremely wide, there will be no deformations in the signals after traveling through \( H_{0} (f) \). By contrast, a distorted channel may result in nominal signal deformations, which will probably lead to ranging errors in a BDS receiver.

The correlation peak curve between a distorted signal and a locally generated ideal signal is likely deformed to a certain extent, which may lead to ranging errors and tracking biases, and will finally influence the positioning results, as shown in Fig. 2.

To characterize the pseudorange biases from a signal deformation occurring at a satellite, we use the variation of the early-minus-late correlator (EML) tracking biases across a range of correlator spacings to convey how the users will be affected by the severity of a signal deformation. In this case, tracking biases can be obtained from the difference between an early correlator and a late correlator. From Fig. 2, we can see that, compared with the ideal signal, the correlation peak curve of the deformed signal is distorted and asymmetric, which may lead to severe tracking biases for different receiver designs.

The tracking biases can be calculated as follows: The difference in the corresponding points of the early correlator and late correlator outputs is first calculated, and then scaled by a factor of A:

$$ \varepsilon (\tau ) = \left[ {E(\tau /2) - L(\tau /2)} \right] \cdot A $$

(1)

$$ A = \frac{1}{2P} \cdot \frac{1}{{f_{B} }} \cdot c $$

(2)

where *τ* indicates the receiver correlator spacing, or the distance between an early correlator and a late correlator, *ε*(*τ*) is the tracking error, *E*(*τ*/2) is the value of the early (E) correlator, *L*(*τ*/2) is the value of the late (L) correlator, *P* is the value of the prompt (P) correlator, *f*_{B} denotes the frequency of the BDS PRN code, and *c* is the speed of light.

Figure 3 shows the tracking biases obtained from the correlation peak curve of the ideal signal and an erroneous signal, which are shown in Fig. 2. It is clear how the signal deformation influences the tracking error.

As with most common BDS user receivers, because the correlator spacings, filter characteristics, RF bandwidth, and discriminator types are extremely different for different receiver manufactories, different receivers may respond differently to an individual signal deformation. In the next section, we introduce how a receiver will respond differently based on a theoretical derivation.

### Receiver influence analysis

Our main concern regarding pseudorange biases is their impact on GNSS users, particularly the potential amplification that occurs when dual-frequency pseudoranges are applied to eliminate an ionospheric influence.

The common pseudorange observation equations can be expressed as follows [10]:

$$ P_{{B_{k} ,m}}^{i} = \rho_{m}^{i} + c\delta t_{m} - c\delta t^{i} + c \cdot IFB_{m} - c \cdot Tgd^{i} - iono_{m}^{i} - trop_{m}^{i} - rel_{m}^{i} - \alpha_{m}^{i} + \varepsilon_{m}^{i} + MP_{m}^{i} + SErro_{m}^{i} $$

(3)

$$ P_{{B_{k} ,n}}^{i} = \rho_{n}^{i} + c\delta t_{n} - c\delta t^{i} + c \cdot IFB_{n} - c \cdot Tgd^{i} - iono_{n}^{i} - trop_{n}^{i} - rel_{n}^{i} - \alpha_{n}^{i} + \varepsilon_{n}^{i} + MP_{n}^{i} + SErro_{n}^{i} $$

(4)

$$ P_{{B_{k} ,m}}^{j} = \rho_{m}^{j} + c\delta t_{m} - c\delta t^{j} + c \cdot IFB_{m} - c \cdot Tgd^{j} - iono_{m}^{j} - trop_{m}^{j} - rel_{m}^{j} - \alpha_{m}^{j} + \varepsilon_{m}^{j} + MP_{m}^{j} + SErro_{m}^{j} $$

(5)

$$ P_{{B_{k} ,n}}^{j} = \rho_{n}^{j} + c\delta t_{n} - c\delta t^{j} + c \cdot IFB_{n} - c \cdot Tgd^{j} - iono_{n}^{j} - trop_{n}^{j} - rel_{n}^{j} - \alpha_{n}^{j} + \varepsilon_{n}^{j} + MP_{n}^{j} + SErro_{n}^{j} $$

(6)

where the subscripts *i* and *j* represent satellites; *B*_{k} represents the BDS-2 frequency, with *B*_{1} = 1561.098 MHz, *B*_{2} = 1207.14 MHz, and *B*_{3} = 1268.52 MHz; subscripts *m* and *n* represent hardware receivers; \( P_{{B_{k} ,m}}^{i} \) denotes the pseudorange; \( \rho_{m}^{i} \) represents the true range; *c* is the speed of light; \( \delta t_{m} \) represents the receiver clock bias in seconds; \( \delta t^{i} \) represents the satellite clock bias in seconds; \( IFB_{m} \) is the inner-frequency biases in the receiver in seconds; \( Tgd^{i} \) is the time group delay for the inner-frequency biases on the satellite in seconds; \( iono_{m}^{i} \) is the ionosphere delay in meters; \( trop_{m}^{i} \) is the tropospheric delay in meters; \( rel_{m}^{i} \) represents the relativistic effect of the delay in meters; \( \alpha_{m}^{i} \) represents the receiver channel delay in meters; \( \varepsilon_{m}^{i} \) represents thermal noise in meters; \( MP_{m}^{i} \) is the multipath error in meters; and \( SErro_{m}^{i} \) denotes a signal deformation error in the pseudorange domain in meters.

For a characterization of the pseudorange bias difference between different receiver designs, herein we assume that the receivers are connected in a zero-baseline configuration to a high-gain dish antenna. Accordingly, there is no residual ground multipath from a large dish antenna. In this case, the results of O-C double difference processing between two receivers and two satellites can likely eliminate the effects of an ionosphere delay, a tropospheric delay, a receiver clock bias, a satellite clock bias, inner-frequency biases in the receiver, and inner-frequency biases on the satellite, among other factors. Thus, the pseudorange bias is obtained.

In Ref. [24], the authors derived in detail the signal difference and double difference among receivers in the case of common and different correlator spacings. Research results have shown that, despite the signal and double differences, the effects of a signal deformation and multiple paths cannot be eliminated if the two receiver configurations are different. However, the effects of the signal deformation and multiple paths can be eliminated when the two receiver configurations are the same.

However, if the signal distortions of all satellites are identical, or if all receiver configurations are the same, then the response of all receivers will also be identical for all satellites. As a result, there will be one shared bias for all observations. Even if the biases are different for different receiver designs, it will not have an impact on the positioning results because such an impact will be absorbed in the estimated receiver clock. However, this is not the case for the actual signals and receivers.