Noise Rejection Performance

In this section the ability of a phaselock loop to cope with noise is examined. Consider the input signal of a phaselock loop and assume that as well as a sinusoidal signal, additive noise of n(t)volts with one sided spectral density S_n(f) is also admitted. Therefore the input voltage to the phase detector is given by

$$v_i(t)=V_s\sin(\omega_it+\varphi_i)+n(t)\quad [V].$$

The other input to the phase detector is due to the VCO. Because it is typical that VCO's lock in quadrature to the incoming signal, the VCO signal is typically $\frac{\pi}{2}$ out of phase with v_i. In anticipation of this fact, it is useful to write

$$v_o(t)=V_o\cos(\omega_it+\varphi_o).$$

The additive noise n(t) can be expanded into two quadrature components

$$n(t)=n_c(t)\cos\omega_it-n_s(t)\sin\omega_it$$

One may then define the dimensionless quantity n' as

$$n'(t)=\frac{n_c(t)}{V_s}\cos\varphi_0\frac{n(t)}{V_s}$$

It can be shown [11] that the variance, $\overline{\varphi_{no}}$ of the output noise n(t) is given by

$$\overline{\varphi_{no}}=\int_0^{\infty}S_{n'}(f)\vert H(j\omega)\vert^2df.$$

where $H(j\omega)$ is the filter transfer function defined by equation (3.13), S_n'(f) is the one-sided spectral density of n' and $\omega=2\pi f$. In general, the form of S_n(f) impedes the evaluation of the above integral. If one considers white noise where $S_n(f)=\frac{1}{2}\eta$, the above integral simplifies to

$$\overline{ \varphi_{no}}=\frac{\eta}{V_s^2}\int_0^{\infty}\vert H(j\omega)\vert^2df$$

where

$$B_L=\int_0^{\infty}\vert H(j\omega)\vert^2df$$

is the noise bandwidth . Hence for input white noise, the phase variance is simply  

$$\overline{ \varphi_{no}}=\frac{\eta B_L}{V_s^2}.$$ (21)

Gardner [11] details the noise bandwidths of various loop types and they are reproduced in table 3.3.

 

Loop Description Noise Bandwidth, B_L(Hz))

$$\frac{1}{4}K$$ First Order

$$\frac{1}{4}K\frac{K+\frac{1}{\tau_2}}{K+\frac{1}{\tau_1}}$$ Second order passive

$$\frac{1}{4}K\left(1+\frac{1}{\tau_2K}\right)$$ Second order active

where K is the open loop gain defined for various loops in table 3.2. It can be seen that K plays a central role in establishing the noise bandwidth and hence the output phase variance. It is necessary for K and hence K_oK_d, to be small in order to minimise the output jitter due to external noise.