Causality of a discrete recursive system

Question

I am new in this site and this my first question.

How can I mathematically prove that the system with a transfer function like the below one is causal? $$H(z)=\frac{B(z)}{A(z)}=\frac{b_{0}+b_{1} z^{-1}+b_{2} z^{-2}+\cdots+b_{N} z^{-N}}{1+a_{1} z^{-1}+a_{2} z^{-2}+\cdots+a_{M} z^{-M}}$$

P.S:

I know that for a system to be causal there has to be past inputs in the equation of difference and that the inpulse resoponse has to be zero for negative $n$ ($h[n]=0 , n<0$) but the thing is that when i try to prove the second one by putting values into $n$ and writing down the result of $H[n]$, I always get a reslut that depents on a past value of $H[n]$

Answer 1

Just write down the corresponding difference equation. Then it is obvious that the system is causal. There is no need to compute the pulse response. The difference equation reads

$y(n) + a_1 y(n-1) + \ldots = b_0 x(n) + b_1 x(n-1) + \ldots$

Clearly, if you solve for $y(n)$, the right hand side only depends on previous $y(n)$ and previous $x(n)$ so it is causal.

It would not be causal, if any exponent in the numerator or denominator of the transfer function is positive (e.g. there is some term with $z^2$, instead of $z^{-2}$ for instance). The difference equation would then contain $x(n+2)$ in the right hand side for $y(n)$. That is clearly not causal.

In an abstract way, all systems are causal that have transfer functions which is rational polynomial in $z^{-1}$ with a nonzero lowest order coefficient in the denominator.

Answer 2

How can I mathematically prove that the system with a transfer function like the below one is causal?

You can't.

The transfer function $H(z)$ in itself doesn't tell you if the filter is causal or not.

More precisely. An LTI is causal if $h[n] =0$ for $n<0$. But $H(z)$ is not enough to determine $h[n]$. You need also the ROC (region of convergence of the Z transform).

For example, $H(z)=\frac{1}{1+z^{-1}}$ (a particular case of your transfer function) is the Z transform both of a causal response function and an anti-causal one (details here).

Specifically: for a LTI to be causal, the ROC of the $Z$ transform must extend "to the outside", ie., must be of the form $|z| > R_0$ or $|z| \ge R_0$. For a given rational funtion, $R_0$ would be the largest absolute value of all poles.