Causal discrete-time signals that are *linear combinations* of real or complex *exponentials* do have rational transfer functions. However, not all causal discrete-time signals are linear combinations of real or complex exponentials.

For example, consider the causal LTI system whose (infinite) impulse response is

$$h (n) = \begin{cases} \frac{1}{1+n} & \text{ if } n \geq 0\\ \,\,\,0 & \text{ if } n < 0\end{cases}$$

Taking the [Z-transform][1], we obtain the following *non*-rational transfer function

$$H (z) = \sum_{n=0}^{\infty} \frac{z^{-n}}{1+n} = - z \ln \left(\frac{z-1}{z}\right)$$

when $|z| \geq 1$ and $z \neq 1$. When $z=1$, we have the (divergent) harmonic series.

Can this LTI system be implemented? Using *finite*-precision arithmetic, $h (n)$ will eventually *underflow* at some very large $n$. Hence, we can truncate the infinite impulse response $h$, which produces an FIR filter that requires an astronomically long cascade of delays.

Of course, the same underflow would happen if we had the causal infinite impulse response $2^{-n}$. However, $2^{-n}$ is a real exponential and can be produced by the 1st order difference equation

$$y (n) - \frac 12 y (n-1) = x (n)$$ 

which requires only $1$ adder, $1$ multiplier and $1$ delay. 

Exponentials, whether real or complex, have *low complexity*, i.e., they can be generated using few adders, multipliers and delays. Using Fourier transforms, signals of interest can be *approximated* by linear combinations of complex exponentials.
 

  [1]: http://www.wolframalpha.com/input/?i=sum%20z%5E(-n)%2F(1%2Bn),%20n%3D0%20to%20infinity