In this contribution we consider sparse linear regression problems. It is well known that the mutual coherence, i.e. the maximum correlation of the regressors, is important for the ability of any algorithm to recover the sparsity pattern of an unknown parameter vector from data. A low mutual coherence improves the ability of recovery. In optimal experiment design this requirement may be in conflict with other objectives encoded by the desired Fisher matrix. In this contribution we alleviate this issue by combining optimal input design with a recently proposed approach to achieve low mutual coherence by way of a linear coordinate transformation. The resulting optimization problem is solved using cyclic minimization. Via simulations we demonstrate that the resulting algorithm is able to achieve a Fisher matrix which results in a performance close to the performance if the sparsity would have been known, while at the same time being able to recover the sparsity pattern.