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Detecting small delay defects (SDDs) has become increasingly important to address the quality and reliability concerns of integrated circuits. Without considering functional constraints in the circuits under test, however, existing techniques may generate test patterns that are functionally-unreachable. Such SDD patterns may incur excessive (or limited) power supply noise (PSN) on sensitized paths in test mode, thus leading to over-testing or under-testing of the circuits. In this paper, we propose novel pseudo-functional testing techniques to tackle the above problem. Firstly, by taking the circuit layout information into account, functional constraints related to critical paths are extracted. Then, we generate functionally-reachable test cubes for SDD faults in the circuit. Finally, we use ATPG-like algorithm to justify transitions that pose the maximized PSN effects on sensitized critical paths under the consideration of functional constraints. The effectiveness of the proposed methodology is verified with large ISCAS'89 and ILWS'05 benchmark circuits.