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In this paper, we will build up the link between parity relation design and residual evaluation via generalized likelihood ratio (GLR) test. To this end, the so-called singlesensor parity relation (SSPR) and single-actuator parity relation (SAPR) are refined for stochastic systems in frequency domain by including a whitening filter. This result is also extended to the generalized parity relations in order to detect any failure mode. The proposed design procedure not only achieves failure detection and isolation, but also generates a special residual form which can be evaluated by a GLR test. Besides, a comprehensive summary of the GLR test with rules of thumb to select the design parameters is provided. The effectiveness of the proposed procedure is illustrated by processing measured data from a pilot heat exchanger to detect sensor faults, and by performing a simulation study to detect actuator faults.

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