Unlike deterministic testing, it is unnecessary for scan-based BIST to apply a complete test vector into the circuit via the scan chains. A new scan-based BIST scheme is proposed by properly controlling the test signals of the scan chains. Different weighted random signals are assigned to the test signals of different scan chains. In the proposed test scheme, capture cycles can be inserted at any clock cycle. Testability calculation procedure according to the proposed testing scheme is presented. Techniques for selecting different weights on the test signals of the scan chains are also proposed. Experimental results show that the proposed method can improve the test effectiveness of scan-based BIST greatly, and most circuits can reach complete fault coverage or very close to complete fault coverage.