# Optimal Particle Filter Weight for Bayesian Direct Position Estimation in a GNSS Receiver
## 2 BDPE Receiver Architecture  
The concept of BDPE has been implemented into the SX3 [28], a commercial software-based GNSS receiver (IFEN GmbH, Poing, Germany). The implementation follows the idea to make maximum reuse of existing, well proven and computationally optimized GNSS receiver elements. The implemented receiver design is shown in Figure 1 and it consists of four blocks. The first block (**1**) refers to the first stage of the GNSS signal processing. The Radio Frequency (RF) signal is gathered with a GNSS antenna. Afterwards, the RF signal is down-converted to an Intermediate Frequency (IF). Then, the analogue IF signal is converted into a digital signal by an Analogue to Digital Converter (ADC). Finally, the digital and to IF down-converted sample stream is transmitted via USB to a PC, where the whole GNSS receiver processing is performed. The second step in the processing sequence is the signal correlation, indicated with (**2**) in Figure 1. At this stage, the receiver is configured to act in the same way as a vector tracking receiver. For each of the \(N\) GNSS signals, a complex-valued replica signal is generated using the best estimate \(\hat{f}_{d}\) and \(\hat{\tau}\), at the Doppler and code phase. These estimates come from the latest PVT estimate (cf. step **4**) after applying a LOS projection. Thus, there are Doppler and code phase estimates for each of the \(N\) channels. The signal replica is then correlated with the incoming signal to produce a new time series of correlation values. This is done for a wide range of code phase offsets, not only for Early, Prompt and Late, as it is typically done in GNSS receivers. Thereafter, a post-correlation Fast Fourier Transform (FFT) is applied to this complex valued correlation values in order to obtain synthetically generated correlation values in the Doppler direction. The resulting grid of correlation values in code phase and Doppler direction are called synthetically generated Multi-Correlation (MC) values (cf. step **3**). These maps contain now correlation values for a range of Doppler bins and code phase offsets. The MC processing performs a navigation data bit wipeoff which also allows for long coherent integration times. The MC maps are centered at the old fed back Doppler \(\hat{f}_{d}\) and code phase \(\hat{\tau}\), symbolically indicated with the black square. In the next step, each green PVT particle of a particle filter in the PVT domain (**4**) is mapped to its corresponding correlation value in (**3**). This is done for each channel \(N\) and using a _sinc_ interpolation in both directions, code phase and Doppler. Based on the fact that the correlation values are coupled via the geometric relationship of its PVT particle, it is allowed to sum them up, the basic idea of DPE. The correlation values are used as input to determine the weight change of a particle, a crucial step in a Bayesian filtering approach, which is discussed in detail within this work. Using synthetically generated correlation values as input to the Bayes filter is a valid approach for DPE because it was proven in [26] that these synthetically generated correlation values represent mathematically the exact value of a real correlator. This approach mimics the direct correlation and it is computationally very efficient. Figure 1: Bayesian Direct Position Estimation (BDPE) processing scheme as implemented in the software based Global Navigation Satellite System (GNSS) receiver; (**1**) refers to the analogue to digital conversion, the first stage of the receiver; (**2**) refers to the GNSS signal processing, which produces correlation values for each tracked channel as symbolically shown in (**3**). The correlation values are mapped and weighted to a particle cloud of a particle filter as shown in (**4**). The last block **(4)** in Figure 1 refers to the navigation processor and thus to the implemented particle filter. There exist various types of particle filters--for example, Sampling Importance Resampling (SIR), Auxiliary SIR (ASIR) or Regularized Particle Filter (RPF).