# Is a filter/control transfer function with positive phase “causal”?

In control we often use transfer functions with positive phase, i.e., a "lead filter" has transfer function

$$G_c(s) = \frac{\alpha \tau s+1}{\tau s+1}$$

(with $\alpha>1$). Since the zero occurs at a lower frequency than the pole, the phase is positive approximately beteen $\omega = 1/\alpha \tau$ (the zero) and $\omega = 1/\tau$ (the pole), with the maximum phase occuring at the midpoint $\omega = 1/\tau \sqrt{\alpha}$. Typically it's used to increase phase at a particular frequency $\omega_c$ for stability.

• But how can a system with positive phase actually exist?
• Does the positive phase at certain frequencies mean that if I remove the lead filter from the control loop and input a sinusoid, the output will occur before the input?
• How is this related to causality and realizability?

## 2 Answers

In general, from the positiveness of the phase nothing can be concluded about the causality of the corresponding system. Note that the phase of the given system is

$$\phi(\omega)=\arctan(\alpha\omega\tau)-\arctan(\omega\tau)\tag{1}$$

which is positive for all $\omega>0$ if $\alpha>1$. Yet the system is causal, which can be easily seen by calculating its impulse response:

\begin{align}G(s)&=\frac{\alpha s+\frac{1}{\tau}}{s+\frac{1}{\tau}}=\alpha\frac{s+\frac{1}{\tau}-\frac{1}{\tau}}{s+\frac{1}{\tau}}+\frac{1}{\tau}\frac{1}{s+\frac{1}{\tau}}=\alpha+\frac{1-\alpha}{\tau}\frac{1}{s+\frac{1}{\tau}}\\\Longleftrightarrow g(t)&=\alpha\delta(t)+\frac{1-\alpha}{\tau}e^{-t/\tau}u(t)\tag{2}\end{align}

where $u(t)$ is the unit step function. Obviously, we have $g(t)=0$ for $t<0$, i.e., the system is causal.

Note that the system's phase is not only positive, but it also has a positive slope for $\omega<1/(\tau\sqrt{\alpha})$. This means that its group delay (the negative derivative of the phase) is negative in that frequency range, as shown in the figure ($\alpha=2$, $\tau=1$): So even a negative group delay in some frequency range says nothing about the causality of a system.

A rational transfer function with all its poles in the left half-plane (as is the case in your example) and a region of convergence $\text{Re}\{s\}>0$ always corresponds to a causal and stable system, even if its phase is positive or its group delay is negative in a certain frequency range.

I like this question, and although I can not give an intuitive answer I can offer this example implementation to show how this case can exist (and perhaps this will help you or others provide more insight):

I tried an example case where the zero would occur before the pole in a typical lead filter implemented digitally. The pole zero map would appear as in the figure below, and to use actual values I set the zero at z=0.8 and the pole at z=0.5: The transfer function for this case is given as:

$$H(z) = \frac{z-0.8}{z-0.5}$$ $$= \frac{1-0.8z^{-1}}{1-0.5z^{-1}}$$

Given $H(z) = Y(z)/X(z)$, this leads to the implementation from:

$$Y(z) = X(z)-0.8X(z)z^{-1}+0.5Y(z)z^{-1}$$

Which I show in the block diagram below: And using freqz([1 -.8], [1 -.5]) we can view the frequency response confirming the leading phase characteristic, and from the block diagram implementation, for a causal system. Also from the analog world consider the case of a series inductor with impedance $X_L= sL$, which is therefore also leading in phase (voltage leads current in an inductor, while voltage lags current in a capacitor). That too is for a causal system but has a leading phase.