Previous applications of the enclosure method with a finite set of observation data to a mathematical model of electrical impedance tomography are based on the assumption that the conductivity of the background body is homogeneous and known. This paper considers the case when the conductivity is homogeneous and unknown. It is shown that, in two dimensions if the domain occupied by the background body is enclosed by an ellipse, then it is still possible to extract some information about the location of unknown cavities or inclusions embedded in the body without knowing the background conductivity provided the Fourier series expansion of the voltage on the boundary does not contain high frequency parts (band limited) and satisfies a non vanishing condition of a quantity involving the Fourier coefficients.