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The problem of relaxed state estimation design of multi-instant fuzzy switching observer for discrete-time nonlinear systems is studied by proposing a deep division approach. Firstly,both the enhancement factor and the attenuation factor for each normalized fuzzy weighting function are for the first time introduced synchronously in order to derive much finer subdivisions of the whole spanning space constituted by all the normalized fuzzy weighting functions. Consequently, a more advanced ranking-based switching mechanism can be proposed over the previous results. Secondly, a new multi-instant fuzzy switching observer with different gain matrices for each finer subdivision is designed with the help of the given switching mechanism and thus much more freedom can be brought for reducing the conservatism of existing designs of fuzzy observers. More importantly, it is worth noting that the calculation of all the involved gain matrices belongs to a feasible off-line process. Finally, two simulation examples including tunnel diode circuits are provided to validate the progressiveness of the proposed deep division approach. [ABSTRACT FROM AUTHOR]