Zero-forcing precoding: Difference between revisions

Revision as of 05:37, 8 November 2020

Revision as of 06:03, 8 November 2020

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In a multiple antenna downlink system which comprises an <math>N_t</math> transmit antenna access point and <math>K</math> single receive antenna users, such that <math>K \leq N_t</math>, the received signal of user <math>k</math> is described as

:<math>y_k = \mathbf{h}_k^TH \mathbf{x} + n_k, \quad k=1,2, \ldots, K</math>

where <math>\mathbf{x} = \sum_{i=1}^K \sqrt{P_i} s_i \mathbf{w}_i</math> is the <math>N_t \times 1</math> vector of transmitted symbols, <math>n_k</math> is the noise signal, <math>\mathbf{h}_k</math> is the <math>N_t \times 1</math> channel vector and <math>\mathbf{w}_i</math> is some <math>N_t \times 1</math> linear precoding vector. Here <math>(\cdot)^TH</math> is the matrixHermitian transpose, <math>\sqrt{P_i}</math> is the square root of transmit power, and <math>s_i</math> is the message signal with zero mean and variance <math>\mathbf{E}(|s_i|^2) = 1</math>.

The above signal model can be more compactly re-written as

:<math> \mathbf{y} = \mathbf{H}^TH \mathbf{W} \mathbf{D} \mathbf{s} + \mathbf{n}. </math>

where

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From the fact that each beam generated by zero-forcing precoder is orthogonal to all the other user channel vectors, one can rewrite the received signal as

:<math>y_k = \mathbf{h}_k^TH \sum_{i=1}^K \sqrt{P_i} s_i \mathbf{w}_i + n_k = \mathbf{h}_k^TH \mathbf{w}_k \sqrt{P_i} s_k + n_k, \quad k=1,2, \ldots, K</math>

The orthogonality condition can be expressed in matrix form as

:<math>\mathbf{H}^TH \mathbf{W} = \mathbf{Q}</math>

where <math>\mathbf{Q}</math> is some <math>K \times K</math> diagonal matrix. Typically, <math>\mathbf{Q}</math> is selected to be an identity matrix. This makes <math>\mathbf{W}</math> the right [[Moore–Penrose inverse|Moore-Penrose pseudo-inverse]] of <math>\mathbf{H}^TH</math> given by

:<math>\mathbf{W} = \left( \mathbf{H}^TH \right)^+ = \mathbf{H} (\mathbf{H}^TH \mathbf{H})^{-1}</math>

Given this zero-forcing precoder design, the received signal at each user is decoupled from each other as