The authors study the realization and approximation of linear systems whose impulse response is an element of \(L_ 1\cap L_ 2(0,\infty,{\mathbb{C}}^{P^*m})\), and whose Hankel operator is nuclear. First a state space realization is made for these impulse responses. The realization is constructed on \(\ell_ 2\), and has an unbounded input and output operator, but it is well-defined as an input-state-output map. The truncation of these realizations converges in various norms to the original impulse response, and error bounds for the truncated transfer function as well as for the truncated impulse responses are given. These truncations also generate an approximating sequence to the optimal Hankel-norm approximations to the original system, and various error bounds of these approximations are deduced. In the last section an example of an impulse response with compact suppot is presented.