Concerned is residual generation in systems with mild nonlinearities, that is, nonlinearities which involve the inputs, outputs and faults but not the state. It is shown how structured residuals can be designed by the parity space method, in discrete time, if mild nonlinearities are present. Next a number of fundamental links between the parity space and the input-output representations are explored. Utilizing these results, a new design method is developed which finds structured residuals for the mildly nonlinear system entirely in the input-output framework. It is also shown that the two approaches, parity space and input-output, lead to identical residuals.