It is well known that the weighted least squares (WLS) identification algorithm provides estimates that are in general not in the membership set and in this sense are falsified estimates. This paper shows that: (1) if the noise bound is known, the WLS estimates can be made to lie in or converge to the membership set by choosing the weights properly and (2) if the noise bound is unknown, the same results can still be achieved by using white input signals for finite impulse response systems (FIR).