We propose a new algorithm to identify a Wiener-Hammerstein system. This model represents a communication channel where two linear filters are separated by a non-linear function modelling an amplifier. The algorithm enables to recover each parameter of the model, namely the two linear filters and the non-linear function. This is to be opposed with estimation algorithms which identify the equivalent Volterra system. The algorithm is composed of three main steps and uses three distinct pilot sequences. The estimation of the parameters is done in the time domain via several instances of the least-square algorithm. However, arguments based on the spectral representation of the signals and filters are used to design the pilot sequences. We also provide an analysis of the proposed algorithm. We estimate, via the theory and simulations, the minimum required size of the pilot sequences to achieve a target mean squared error between the output of the true channel and the output of the estimated model. We obtain that the new method requires reduced-size pilot sequences: The sum of the length of the pilot sequences is approximately the one needed to estimate the convolutional product of the two linear filters with a back-off. A comparison with the Volterra approach is also provided.