Many attempts have been made on finding a frequency response estimator which minimizes the bias error in cases where both the input and output signals of a linear system are contaminated by extraneous noise, for example, Hv and Hs. It is wellknown that these estimators only minimize the bias error if the input and output extraneous noise spectra are known, which they are normally not. This paper describes how time domain averaging (cyclic averaging) of periodic excitation signals can be used to eliminate the bias due to both input and output extraneous noise. It is demonstrated by simulation results that asymptotically unbiased estimates of frequency response functions can be obtained by using time domain averaging and periodic random noise. Examples are given of both single-input/single-output (SISO) and of multiple-input/multiple-output (MIMO) systems. The fact that periodic excitation signals in this way can be used to eliminate the bias error in FRF estimates does not seem to have been recognized previously.