I have read MMSE equalizer for ISI affected channels from three books:
- Digital Communication by Proakis
- Principles of Mobile communication by Stuber
- Wireless Communications by Andrea Goldsmith
In all of the books the analysis is more or less same, however there is one question that constantly bugs me, it is explained as follows. In MMSE we need to minimize $\bf{J}$:
Let $\bf{w}$ and $\bf{v}$ be a column vector of $N$ complex entries. I have to minimize the quantity: $$\bf{J}=\mathbb{E}[\bf{w}^T \bf{v} \bf{v}^H \bf{w}^*-2Re\{\bf{v}^H\bf{w}^*d_k\}+|d_k|^2]$$ Define $\bf{M}_v=\mathbb{E}[\bf{vv}^H]$ and $\bf{v}^d=\mathbb{E}[\bf{v}^Hd_k]$. Here $()^H$ is complex conjugate transpose, $T$ is transpose, $()^*$ is complex conjugate, $d_k$ is the symbol.
Then $v_k=\sum_{n=0}^{L}g_nd_{k-n}+\eta_k$ a filter with $L-$tap coefficients. It is always given that $$v^d=\mathbb{E}[d_{n-m}v^*_{k-j}]=\sigma^2_{d}g^*_{m-i}$$ where $\sigma_k^2=\mathbb{E}[|d_k|^2]$ $$\mathbb{E}[v_{k-i}v^*_{k-j}]=\sigma_d^2f_{j-i}+N_o\delta_{ij}$$ where $f_m=\sum_{k=0}^{L-m} g_k^*g_{k+m}$ Both of these derivations require that $$\mathbb{E}[d_kd_l]=0$$ when $k\neq l$.
For BPSK it seems true but what for other schemes, how one should ensure this condition? None of the book elaborates or even mention this condition, and I am not sure if I am correct. Please help!
