# An Enhanced Non-Coherent Pre-Filter Design for Tracking Error Estimation in GNSS Receivers
## 2 Tracking Error Estimation
### Relevant Baseband Signal Model  
Figure 1 illustrates the schematic block diagram of the baseband processing of GNSS receivers. The mixer, correlator, tracking error estimation model, numerically-controlled oscillator (NCO) command generation model, local carrier generator, and local PRN code generator constitute the basic GNSS  
tracking loop. The GNSS tracking loop, together with Bit/Frame synchronization model, navigation data demodulation model, and GNSS measurements formulation model constitute the intact baseband processing module. The whole baseband processing flow is described as follows. After the GNSS signal passes through the radio frequency (RF) front end and is sampled, the center frequency of the digitalized signal is located in intermediate frequency (IF). As mentioned above, signal acquisition is first done to roughly estimate the GNSS signal parameters of all visible satellites. After that, GNSS tracking loops are employed to process the IF signal and accurately estimate GNSS signal parameters. The tracking loops include the carrier tracking loop and code tracking loop. The code tracking loop is often implemented as a delay lock loop (DLL), while the carrier tracking loop implemented as a phase lock loop (PLL), frequency lock loop (FLL), or FLL-assisted-PLL. In order to accomplish the GNSS signal tracking, the GNSS IF signal is first mixed with local generated sin/cos carrier signals with the aim of converting the signal center frequency from IF to baseband, and then correlated with local generated Early/Prompt/Late (E/P/L) PRN codes whose phase have certain interval to remove the PRN code modulated in GNSS signals. This correlation process can also be considered as coherent integration, which often contains the in-phase (I) component and quadrature (Q) component. Non-coherent integration can be further done by accumulating the auto-correlation amplitude or power which will be discussed later. The coherent/non-coherent integration contain the real tracking error. Thus, tracking error can be extracted from the coherent/non-coherent integration through traditional discriminator or pre-filter. After that, the NCO command generation model generates the carrier NCO and code NCO feedback command according to tracking error estimation, aiming at decreasing the tracking error. There are tracking loop algorithms in the NCO command generation model. Loop filter is the traditional method of generating the NCO feedback command [9; 10; 11]. The PLL, FLL, and DLL in this case all have similar structure in terms of loop filter design. Moreover, the Kalman filter-based tracking loop, the linear-quadratic-Gaussian (LQG)-based control method, and intelligent control method can also be used in the NCO command generation [20; 21; 22; 23; 24; 25]. The other parts of baseband processing are Bit/Frame synchronization, navigation data demodulation, and GNSS measurements formulation, which are also shown in Figure 1, and are not discussed in detail in this paper. Based on the description above, it is coherent/non-coherent integration in baseband signal processing that are relevant to tracking error estimation. So, we will make a brief review of the mathematical model of coherent and non-coherent integration in the following part. The analogy IF signal for all visible GNSS satellites is modeled as:  
\[s(t)=\sum_{i=1}^{N}a_{i}D_{i}(t)x_{i}(t)\cos[2\pi\cdot(f_{\rm IF}+f_{\rm d,i})t +\varphi_{0,i}]+n_{0}(t) \tag{1}\]  
where \(i\) indicates the \(i\)-th visible satellite, \(t\) is the signal reception time, \(a\) is signal amplitude, \(D\) is binary navigation data bits, \(x\) is the PRN code sequence, \(f_{\rm IF}\) is analogy IF [Hz], \(f_{\rm d}\) is the Doppler frequency (Hz) caused by relative motion, \(\varphi_{0}\) is the initial carrier phase (rad), \(n_{0}(t)\) is additive band-limited white Gaussian noise. The one-sided power spectral density (PSD) of \(n_{0}(t)\) is \(N_{0}\) (Hz). The IF signal  
Figure 1: Baseband signal processing flow of GNSS receivers.