I saw these formula in this paper,and i don't know why can the $p_k$ be written like the formula ,in the photo, below
I derive the formula,and i think $p_k$ should be like this ,according to the first equation
$p_k=\frac{\sum\limits_{j \neq k}(\mathbf h^H_k \bar{\mathbf f_j})^2+\sigma^2_{a_k}+\frac{\sigma^2_{d_k}}{\rho_k}}{(\mathbf h^H_k \bar{\mathbf f_k})^2}\times \bar \gamma_k$ .Is my $p_k$ the same as the paper shown?
By the way,if i want to write the K+1 equations to solve the $\rho_k$,
how do i rewrite the formula? because $\rho$ should be the variable,but there are $\rho_k$ and $\rho_j$,i want to rewrite the K+1 equations as this form
$[\mathbf A ] \begin{bmatrix} \rho_1 \\ \rho_2 \\ \rho_3 \\ \rho_4 \\ \end{bmatrix} $=P
Yes, the $p$ vector that you calculated is the same as the paper's but it's in component form, not in standard matrix-vector form.
Instead of putting the restriction $(j\ne k)$ on the summation, simply add the $k^{th}$ component to the sum and explicitly subtract it outside the sum. This allows the sum to be replaced with a matrix-vector product (specifically $\,Lp\,$ below).
Let $L$ be a full matrix and $G$ be a diagonal matrix (note the "$1+{\rm term}$" that comes from moving the term discussed in the previous paragraph) whose elements are given by
$$\eqalign{ L_{ij} &= \big|h_i^Hf_j\big|^2, \quad G_{kk} &= \bigg(1+\frac{1}{\gamma_k}\bigg)L_{kk} \cr }$$ also define a vector with components $$\eqalign{ v_k &= (\sigma_a^2)_k + \frac{(\sigma_d^2)_k}{\rho_k} \cr }$$ Now the formula can be written and solved in matrix form. $$\eqalign{ Gp-Lp &= v \cr (G-L)p &= v \cr Mp &= v \cr p &= M^{-1}v \cr }$$