In exercise [1, p. 34 exercise 2.5] we are asked to show that the matrix $\mathbf{R}=\sigma^2 \mathbf{I}+P_1\mathbf{e_1}\mathbf{e_1}^H+P_2\mathbf{e_2}\mathbf{e_2}^H$ is circulant when $\mathbf{e_i}^H=\left[ {\begin{array}{*{20}c} 1 & {e^{-j2\pi f_i } } & {e^{-j2\pi (2f_i) } } & \cdots & {e^{-j2\pi (n - 1)f_i } } \end{array}} \right], i=1,2$. read the conclusion >