Soft-Output MIMO Detectors - Introduction
The fundamentals of soft demapping are explained in the Soft-Demapping webdemo. The described approach can be implemented in receivers for multiple antenna systems as well.
$$ \mathbf{y} = \mathbf{H} \mathbf{s} + \mathbf{n} $$ It is $\mathbf{y}$ the receive vector, $\mathbf{H}$ the channel matrix, $\mathbf{s}$ the transmit vector, that results from the mapping of a bit vector $\mathbf{x}$, and $\mathbf{n}$ a random noise vector. Each element of $\mathbf{s}$ emanates the set of possible QAM-symbols $\mathcal{S}$ and therefore $\mathbf{s} \in \mathcal{S}^{N_{\mathrm{tx}}}$.
The maximum A Posteriori Probability receiver (APP) computes the L-value of the m-th bit $x_{\mathrm{m}}$ according to $$ {L}_{\mathrm{APP}} \left(x_{\mathrm{m}}|\mathbf{y}\right) = \mathrm{ln} \frac{\sum_{\mathbf{s} \in \mathcal{S}^{N_{\mathrm{tx}}}_{\mathrm{m},1}} \exp \left(- \|\mathbf{y}-\mathbf{H}\mathbf{s} \|^2 / {\sigma^2_{\mathrm{n}}} \right)} {\sum_{\mathbf{s} \in \mathcal{S}^{N_{\mathrm{tx}}}_{\mathrm{m},0}} \exp \left(- \|\mathbf{y}-\mathbf{H}\mathbf{s} \|^2 / {\sigma^2_{\mathrm{n}}} \right)} $$ Hereby, $\mathcal{S}^{N_{\mathrm{tx}}}_{\mathrm{m},b}$ denotes the set of transmit vectors that result from the mapping of a bit vector whose m-th bit has the value $b \in \{ 0,1 \}$. The use of $N_{\mathrm{tx}}$ transmit antennas and $M_{\mathrm{b}}$ bit per symbol, leads to $\left| \mathcal{S}^{N_{\mathrm{tx}}}_{\mathrm{m},b} \right| = 2^{M_{\mathrm{b}} · N_{\mathrm{tx}} - 1}$. An exhaustive calculation according to this formula is not feasible for large numbers of possible vectors.
The APP receiver features BERs similar to the ML receiver. However, the measure of mutual information shows an improved performance of the soft-output receiver. An illustration of these measures can be found in the Physical Layer Performance Measures webdemo.
Zero Forcing with Per-Antenna A Posteriori Probability calculation (ZF+PA²P²) equips the Zero Forcing receiver by a soft-output back end, that replaces the element-wise quantization step. The linear front end performes the matrix computation $\mathbf{z}_{\mathrm{ZF}} = \left( \mathbf{H}^H \mathbf{H} \right)^{\!-1} \mathbf{H}^H \mathbf{y}$ $$ {L}_{\mathrm{ZF+PA\kern-0.1em^2P^2}} \left(x_{\mathrm{m}}|\mathbf{y}\right) = \mathrm{ln} \frac{\sum_{s \in \mathcal{S}_{\mathrm{m},1}} \exp \left(- \left| \left[\mathbf{z}_{\mathrm{ZF}}\right]_k - s \right|^2 / {\sigma^2_{kk}} \right)} {\sum_{s \in \mathcal{S}_{\mathrm{m},0}} \exp \left(- \left| \left[\mathbf{z}_{\mathrm{ZF}}\right]_k - s \right|^2 / {\sigma^2_{kk}} \right)} $$ The colored noise in $\mathbf{z}_{\mathrm{ZF}}$ is incorporated in $\sigma^2_{kk} = \sigma^2_{\mathrm{n}} \left[ \left( \mathbf{H}^H \mathbf{H} \right)^{\!-1} \right]_{kk}$. Hereby, the transmit antenna index of the m-th bit is given by $k = \big\lfloor \frac{m-1}{M_{\mathrm{b}}} \big\rfloor + 1$. An equal noise weight (EQU) according to $\sigma^2_{kk} = \sigma^2_{\mathrm{n}}$ yields a reduced computational complexity.
The increased performance is again only visible in terms of mutual information.
$$ \mathbf{y} = \mathbf{H} \mathbf{s} + \mathbf{n} $$ It is $\mathbf{y}$ the receive vector, $\mathbf{H}$ the channel matrix, $\mathbf{s}$ the transmit vector, that results from the mapping of a bit vector $\mathbf{x}$, and $\mathbf{n}$ a random noise vector. Each element of $\mathbf{s}$ emanates the set of possible QAM-symbols $\mathcal{S}$ and therefore $\mathbf{s} \in \mathcal{S}^{N_{\mathrm{tx}}}$.
The maximum A Posteriori Probability receiver (APP) computes the L-value of the m-th bit $x_{\mathrm{m}}$ according to $$ {L}_{\mathrm{APP}} \left(x_{\mathrm{m}}|\mathbf{y}\right) = \mathrm{ln} \frac{\sum_{\mathbf{s} \in \mathcal{S}^{N_{\mathrm{tx}}}_{\mathrm{m},1}} \exp \left(- \|\mathbf{y}-\mathbf{H}\mathbf{s} \|^2 / {\sigma^2_{\mathrm{n}}} \right)} {\sum_{\mathbf{s} \in \mathcal{S}^{N_{\mathrm{tx}}}_{\mathrm{m},0}} \exp \left(- \|\mathbf{y}-\mathbf{H}\mathbf{s} \|^2 / {\sigma^2_{\mathrm{n}}} \right)} $$ Hereby, $\mathcal{S}^{N_{\mathrm{tx}}}_{\mathrm{m},b}$ denotes the set of transmit vectors that result from the mapping of a bit vector whose m-th bit has the value $b \in \{ 0,1 \}$. The use of $N_{\mathrm{tx}}$ transmit antennas and $M_{\mathrm{b}}$ bit per symbol, leads to $\left| \mathcal{S}^{N_{\mathrm{tx}}}_{\mathrm{m},b} \right| = 2^{M_{\mathrm{b}} · N_{\mathrm{tx}} - 1}$. An exhaustive calculation according to this formula is not feasible for large numbers of possible vectors.
The APP receiver features BERs similar to the ML receiver. However, the measure of mutual information shows an improved performance of the soft-output receiver. An illustration of these measures can be found in the Physical Layer Performance Measures webdemo.
Zero Forcing with Per-Antenna A Posteriori Probability calculation (ZF+PA²P²) equips the Zero Forcing receiver by a soft-output back end, that replaces the element-wise quantization step. The linear front end performes the matrix computation $\mathbf{z}_{\mathrm{ZF}} = \left( \mathbf{H}^H \mathbf{H} \right)^{\!-1} \mathbf{H}^H \mathbf{y}$ $$ {L}_{\mathrm{ZF+PA\kern-0.1em^2P^2}} \left(x_{\mathrm{m}}|\mathbf{y}\right) = \mathrm{ln} \frac{\sum_{s \in \mathcal{S}_{\mathrm{m},1}} \exp \left(- \left| \left[\mathbf{z}_{\mathrm{ZF}}\right]_k - s \right|^2 / {\sigma^2_{kk}} \right)} {\sum_{s \in \mathcal{S}_{\mathrm{m},0}} \exp \left(- \left| \left[\mathbf{z}_{\mathrm{ZF}}\right]_k - s \right|^2 / {\sigma^2_{kk}} \right)} $$ The colored noise in $\mathbf{z}_{\mathrm{ZF}}$ is incorporated in $\sigma^2_{kk} = \sigma^2_{\mathrm{n}} \left[ \left( \mathbf{H}^H \mathbf{H} \right)^{\!-1} \right]_{kk}$. Hereby, the transmit antenna index of the m-th bit is given by $k = \big\lfloor \frac{m-1}{M_{\mathrm{b}}} \big\rfloor + 1$. An equal noise weight (EQU) according to $\sigma^2_{kk} = \sigma^2_{\mathrm{n}}$ yields a reduced computational complexity.
The increased performance is again only visible in terms of mutual information.