For a discrete LTI system, does "bounded memory" imply "rational transfer function?"
Non-trivially, the answer is no, a Bounded In Memory (BIM) LTI system do not imply the system has a rational Transfer Function (TF).
Proof (TL;DR):
Every discrete LTI system $H$ has an Impulse Response (by definition), and hence have a TF in the z-domain (by construction), which could be infinite (IIR) or finite (FIR).
A FIR LTI is trivially BIM, requiring a bounded number of operation for implement every convolution.
A IIR LTI could have a Rational TF (RTF) -such as most ARMA systems- or could have an not rational -an infinite series (IS)- TF -such as the counter-example 1 below.
If the system has an IS TF which can be decomposed as a rational TF -such as the Counter-Example 2-, then the system is bounded in memory, because the rational TF can be decomposed into a pair of finite series, which can be implemented and evaluated with finite memory and finite CPU.
Similarly, if the system has an IS TF with an Known algebraic expansion (KIS), -such as the Counter-Example 1-, then again the system is BIM, because, the general term is computed in each step, consuming an increasing amount of CPU -this is, an unbounded CPU system- but still BIM.
If this is not the case, that is, if the system has an IS TF without an unknown algebraic expansion (UIS) -which is indeed real for most LTIs- then is not BIM in practice.
The condition here for an IS to be not rational is having no poles and no zeros, that is, being bounded:
$$0&lt;H(z)=\sum_0^\infty c_nz^{-1}&lt;\infty$$ $|z|&gt;z_0$ for some $z_0$.
If the series crosses by zero through $z_i$, or diverges for some $p_i$, then it has a rational set of component $(1-z_iz^{-1})$ or $(1-p_iz^{-1})$, which is nothing more than a polynomial that can be factorized, being this condition applicable in a recurrent way.
Hence our definitions:
ALL $\leftrightarrow$ FIR $\lor$ IIR
IIR $\leftrightarrow$ TF $\lor$ IS
IS $\leftrightarrow$ KIS $\lor$ UIS
And our known relations:
FIR $\rightarrow$ BIM (By construction)
TF $\rightarrow$ BIM (By construction)
KIS $\rightarrow$ BIM (Counter-Example 1)
UIS $\rightarrow$ not BIM (Infinite Terms)
And finally, a Bounded in Memory System is a FIR, a TF or a Known Expansion Infinite Series.
BIM $\leftrightarrow$ FIR $\land$ TF $\land$ KIS
Counter Example 1:
The LTI system, with IIR defined by its Z-transform, is BIM, but has not rational TF:
$$H(z)=log(1+az^{-1}), \\ h(t)=\frac{(-1)^{t+1}a^t}{t}\Gamma(t), t&gt;0$$
We are using the exploit that:
$$log(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n}, x&lt;1$$
which in general is not valid, thus this procedure will produce unbounded LTI systems except for the trivial power series functions $sin(x)$, $cos(x)$,$e^x$...
We are also assuming that our TF must be finite in terms, which is the purpose of having a TF, provided that all Impulse Responses can be converted into series in Z space.
http://www.comm.utoronto.ca/~dkundur/course_info/362/6_Kundur_Rationalz-Transf_handouts.pdf
Counter Example 2:
The IIR AR System has both a rational and an infinite series transfer function:
$$H(z)=\frac{1}{1-az^{-1}}=\sum_{n=0}^{\infty}a^nz^{-n}, |a|&lt;|z|$$