Consider a classical phase detector implemented with
analog multiplier and low-pass filter. Here f^1(\theta^1(t)) and f^2(\theta^2(t)) denote high-frequency signals,
piecewise differentiable functions f^1(\theta), f^2(\theta) represent
waveforms of input signals, \theta^{1,2}(t) denote phases, and g(t) denotes the output of the filter. If f^{1,2}(\theta) and \theta^{1,2}(t) satisfy the high frequency conditions (see ) then phase detector characteristic \varphi(\theta) is calculated in such a way that time-domain model filter output : g(t) = \int\limits_0^t f^1(\theta^1(\tau))f^2(\theta^2(\tau))d \tau and filter output for phase-frequency domain model : G(t) = \int\limits_0^t \varphi(\theta^1(\tau) - \theta^2(\tau))d \tau are almost equal: :g(t) - G(t) \approx 0 :
Sine waveforms case Consider a simple case of harmonic waveforms f^1(\theta)=\sin(\theta), f^2(\theta)=\cos(\theta) and integration filter. :\sin(\theta^1(t))\cos(\theta^2(t)) = \frac{1}{2}\sin(\theta^1(t) + \theta^2(t)) + \frac{1}{2}\sin(\theta^1(t) - \theta^2(t)) Standard engineering assumption is that the filter removes the upper sideband \sin(\theta^1(t) + \theta^2(t)) from the input but leaves the lower sideband \sin(\theta^1(t) - \theta^2(t)) without change. Consequently, the PD characteristic in the case of sinusoidal waveforms is : \varphi(\theta) = \frac{1}{2}\sin(\theta).
Square waveforms case Consider high-frequency square-wave signals f^1(t) = \sgn(\sin(\theta^1(t))) and f^2(t) = \sgn(\cos(\theta^2(t))). For this signals it was found that similar thing takes place. The characteristic for the case of square waveforms is : \varphi(\theta) = \begin{cases} 1+\frac{2\theta}{\pi}, & \text{if }\theta \in [-\pi,0],\\ 1-\frac{2\theta}{\pi}, & \text{if }\theta \in [0,\pi].\\ \end{cases}
General waveforms case Consider general case of piecewise-differentiable waveforms f^{1}(\theta), f^2(\theta). This class of functions can be expanded in Fourier series. Denote by : a^p_i=\frac{1}{\pi}\int\limits_{-\pi}^{\pi} f^p(x)\sin(ix)dx, b^p_i=\frac{1}{\pi}\int\limits_{-\pi}^{\pi} f^p(x)\cos(ix)dx, c^p_i=\frac{1}{\pi}\int\limits_{-\pi}^{\pi} f^p(x)dx, p = 1,2 the Fourier coefficients of f^1(\theta) and f^2(\theta). Then the phase detector characteristic is
Examples == References ==