# A GPS Phase-Locked Loop Performance Metric Based on the Phase Discriminator Output
## 3 Augmented PLL Linear Model  
In this section, we derive a linear model for the PLL shown in Figure 1. The advantage of a linear model is that it may be analyzed using conventional control system techniques. The derivation that follows assumes \"small\" levels of additive white Gaussian noise (AWGN), or \"high\" \(C/N_{0}\). We anticipate that the model will likely break down for low carrier-to-noise ratios we are interested in. However, in Section 8, we will show that the linear model produces accurate results at much lower carrier-to-noise ratios than expected. The PLL input signal \(y(t)\), following code wipe-off, may be written as,  
\[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]  
where \(A\) is the RMS amplitude, and \(n(t)\) is assumed AWGN (it is actually band-passed in the front-end, but the PLL bandwidth is much smaller, so we will assume AWGN for simplicity). The navigation data \(D(t-\tau)\) are not included in Equation (5) because we are assuming the data bit value does not change during a coherent averaging period. This can be ensured by using coherent averaging times of less than the duration of one bit. Note that we can also write the additive noise as,  
\[n(t)=\sqrt{2}n_{I}(t)\cos(\omega_{IF}t)-\sqrt{2}n_{Q}(t)\sin(\omega_{IF}t) \tag{6}\]  
where \(n_{I}(t)\) and \(n_{Q}(t)\) are independent WGN processes ([15], p. 64). The in-phase signal after mixing with replica from Equation (3) may be written as,  
\[I(t) =y(t)x_{I}(t) \tag{7}\] \[=2 AK\cos(\omega_{IF}t+\psi(t))\cos(\omega_{IF}t+\hat{\psi}(t))\] \[\quad+n(t)\sqrt{2}K\cos(\omega_{IF}t+\hat{\psi}(t)) \tag{8}\]  
Substituting Equation (6) for \(n(t)\), and using product-to-sum trigonometric identities, we have,  
\[I(t) =AK\cos(\psi(t)-\hat{\psi}(t))\] \[\quad+K[n_{I}(t)\cos\hat{\psi}(t)+n_{Q}(t)\sin\hat{\psi}(t)] \tag{9}\]  
where the resultant higher-frequency components are neglected. They will be filtered out by coherent averaging, which follows the mixing operation. Finally, defining the phase error as \(\phi(t):=\psi(t)-\hat{\psi}(t)\), the in-phase signal is,  
\[I(t)=AK\big{[}\cos\phi(t)+n^{\prime}_{I}(t)\big{]} \tag{10}\]  
with,  
\[n^{\prime}_{I}(t):=\frac{1}{A}[n_{I}(t)\cos\hat{\psi}(t)+n_{Q}(t)\sin\hat{ \psi}(t)] \tag{11}\]Similarly, we can write the quadrature signal as,  
\[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]0 \[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]1 \[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]2 (13) \[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]3  
with,  
\[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]4  
The IQ signals are then filtered in the coherent averaging operation (denoted using angled brackets), and the tracking error is computed as,  
\[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]5  
Substituting Equations (10) and (14) for \(I(t)\) and \(Q(t)\), respectively,  
\[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]6  
To proceed further, we consider a large carrier-to-noise ratio, \(C/N_{0}\), which implies that the additive noise is small. Under this condition, it is assumed that the PLL will be tracking properly, and therefore the phase error, \(\phi(t)\), will be small. Linearizing the sine and cosine terms, we may write Equation (17) as,  
\[y(t)=\sqrt{2}A\cos(\omega_{IF}t+\psi(t))+n(t) \tag{5}\]7  
Now, the arctangent is linearized, and we focus on the denominator. First we note that the averaging applies only to the in-phase noise component. In addition, the binomial theorem can be used because the noise is assumed to be small.