# A GPS Phase-Locked Loop Performance Metric Based on the Phase Discriminator Output
## 3 Augmented PLL Linear Model
\[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]8 \[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]9 (20) \[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]0 (21) \[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]1  
Since the phase error, \(\phi\), and additive noise is small, the 2 nd-order small terms are neglected, resulting in,  
\[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]2Figure 2 shows the linear block diagram drawn from Equation (23), where the transfer functions \(C(s)\), \(F(s)\), and \(G(s)\) correspond to the coherent integration, loop filter and NCO, respectively. This figure helps to clarify the definition of phase error, \(\phi\), and tracking error, \(\epsilon\), by visual representation. The AWGN input is represented by \(n^{\prime}_{Q^{\prime}}\), and since our PLL is implemented in software, there is no phase-noise contribution from the NCO. The phase-noise of both the receiver and satellite clocks will only enter the PLL through the input phase process, \(\psi\), due to mixing in the receiver front-end. When working with traditional receivers, the added phase-noise contribution from the NCO can easily be included. Its absence in our particular PLL implementation does not affect the analysis techniques or conclusions reached using this linear model. It is worth noting that the only difference between this augmented linear model and the conventional linear model ([9], pp. 127-128) is the additional effect of coherent averaging, \(C(s)\). It is included to obtain a more accurate transfer function from AWGN input to tracking error. We will see the impact this additional block has on transfer function computations in Section 6. We ultimately want to express the variance of the tracking error, \(\sigma_{\epsilon}^{2}\), by using the Wiener-Khinchin theorem. To do so, we need the transfer function from the noise and phase inputs to the tracking error, which we will call the \"tracking error transfer function\". Using \(C(s)\) to denote the transfer function for the coherent averaging, the tracking error transfer function, \(H_{TE}\), is,  
\[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]3  
This transfer function describes the following input-error relation,  
\[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]4  
Then, the variance of the _tracking error_, \(\epsilon\), can be computed as,  
\[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]5  
where \(S_{n_{i}}(f)\) is the single-sided power spectral density (PSD) of the noise input, \(n^{\prime}_{Q^{\prime}}\), and \(S_{clk}(f)\) is the single-sided PSD of clock phase noise. It can be shown that the relationship between the input noise PSD and carrier to noise ratio is,  
\[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]6  
when \(A=1\) and \(K=1\); these values are chosen for convenience in our simulations. The modeling of oscillator phase noise PSDs is discussed in Section 4. \[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]7 Augmented PLL linear model. In addition, we would like to write the variance of the _phase error_, \(\sigma_{\phi}^{2}\), since this is the principal component of the phase jitter metric. The jitter metric will be formally introduced in Section 5. The variance of the phase error, \(\phi\), can be written as,  
\[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]8  
where, \(\sigma_{\phi,n}^{2}\) represents the variance of \(\phi\) due to AWGN, and \(\sigma_{\phi,\text{clk}}^{2}\) represents the variance due to clock phase-noise.