This article generalises the Kalman-Yakubovic-Popov (KYP) lemma in delta domain to treat some conditions on different restricted frequency ranges. The frequency ranges are characterised by two quadratic forms. This characterisation encompasses low/middle/high frequency conditions for delta operator systems. By utilising the obtained delta domain generalised KYP lemma, a fault detection filter design problem is dealt with for delta operator systems in finite-frequency domain. The fault detection filter design problem is formulated as a twoobjective optimisation algorithm with H_ and H-infinity performance indexes. A numerical example is given to illustrate the effectiveness and potential for the developed techniques.