The paper describes the problem of identification of a linear discrete-time system in l(1), from noisy impulse response data of the system. Many tuned algorithms using window functions are proposed in the literature for the problem of H-infinity identification. The study concerns the use of suitable window functions and tuned and untuned algorithms for the problem of identification in l(1). The properties of window functions suitable for l(1) identification are analysed and it is shown that the use of a parameterised exponential window function leads to a convergent worst-case error. The optimal value of the window parameter which results in the least worst-case model error is given in terms of the a priori assumptions on the system and the noise. The tuned algorithm using the optimal parameter is proved to be robustly convergent.