The design of a minimal order observer estimating a given function \(Kx(t)\) of the state vector \(x(t)\) for given observer poles and \(K\) is considered. The design problem is reduced to a set of linear equations \(K= K_z\text{diag}(c_1,\dots, c_r)D\), where \(r\) is the observer order, \(D\) is determined and the other parameters are free. This set of equations provides the upper bound of \(r\), \(\min\{n,v_1+\cdots+ v_p\}\) and \(\min\{n- m,(v_1-1)+\cdots+ (v_p- 1)\}\) for strictly proper and proper observers, respectively, where \(n\), \(m\), \(p\) and \(v_{i0}\) \((i= 1,\dots, p)\) are the plant order, number of outputs, number of inputs and observability indices in decreasing order, respectively.