This paper discusses the robust fault detection design for the criteria such as ¿_-/¿_¿,¿₂/¿_¿ and¿_¿/¿_¿ in which G_d, the transfer function from disturbance to the measurement output, is a tall transfer matrix and D_d is not full column rank. It is shown that the faults in a subspace can be made arbitrarily sensitive, while the faults in the complementary subspace have bounded sensitivities that are maximized by our filter. We also discuss the decoupling and non-decoupling conditions, and their relations with image spaces of G_d and G_f (the transfer function from disturbance to the measurement output and the transfer function from fault to the measurement output). Furthermore, based on the optimal filter, a method is given to partially decouple the disturbance from the residual without impacting the rest of the fault sensitivities. An example is given to illustrate our results.