Functional magnetic resonance imaging (fMRI) is used to study brain activity due

to stimuli presented to subjects in a scanner. It is important to conduct statistical

inference on such time series fMRI data obtained. It is also important to select optimal designs for practical experiments. Design selection under autoregressive models

have not been thoroughly discussed before. This paper derives general information

matrices for orthogonal designs under autoregressive model with an arbitrary number

of correlation coefficients. We further provide the minimum trace of orthogonal circulant designs under AR(1) model, which is used as a criterion to compare practical

designs such as M-sequence designs and circulant (almost) orthogonal array designs.

We also explore optimal designs under AR(2) model. In practice, types of stimuli can

be more than one, but in this paper we only consider the simplest situation with only

one type of stimuli.