An iterative least squares method with modified residuals is presented which is dedicated to the robust coefficient estimation of Walsh functions for time series contaminated with noise, some of which may even be outliers. Instead of the mean-square approximation error (MSE), a robust criterion is proposed for estimating the coefficients of the time series. It is minimised by applying the ordinary iterative Gauss-Newton approach so that an arbitrary function, which is absolutely integrable in the interval [0,T), can be properly approximated by the firstMWalsh functions. A proof of convergence of the proposed method is provided. Results of simulation confirming robustness and convergence of the robust estimates are included. This method should be of great value in real-life situations.