To model IQ imbalance in the analog parts of the sender we write the sender signal $x_{m}=\Re\left\{ x_{m}\right\} +\mathrm{j}\cdot\Im\left\{ x_{m}\right\}$ in vector representation: $$ \mathbf{x}_{m}=\begin{pmatrix}\Re\left\{ x_{m}\right\} \\ \Im\left\{ x_{m}\right\} \end{pmatrix} $$ Now we can define the phase offset $\varphi_{\mathrm{imb}}$ and amplitude offset $a_{\mathrm{imb}}$ of the Q channel with the transformation matrix $\mathbf{T}_{\mathrm{imb}}$ $$ \mathbf{T}_{\mathrm{imb}}=\begin{pmatrix}1 & -a_{\mathrm{imb}}\cdot\sin\left(\varphi_{\mathrm{imb}}\right)\\ 0 & a_{\mathrm{imb}}\cdot\cos\left(\varphi_{\mathrm{imb}}\right) \end{pmatrix} $$ The DC offset $d_{\mathrm{imb}}$ of the Q channel can be modeled by the following vector $$ \mathbf{d}_{\mathrm{imb}}=\begin{pmatrix}0\\ d_{\mathrm{imb}} \end{pmatrix} $$ combining $\mathbf{T}_{\mathrm{imb}}$ and $d_{\mathrm{imb}}$ we get the sender signal with applied IQ imbalance $x_{\mathrm{imb},m}=\Re\left\{ x_{\mathrm{imb,}m}\right\} +\mathrm{j}\cdot\Im\left\{ x_{\mathrm{imb,}m}\right\}$ in vector form $$ \mathbf{x}_{\mathrm{imb},m}=\begin{pmatrix}\Re\left\{ x_{\mathrm{imb,}m}\right\} \\ \Im\left\{ x_{\mathrm{imb,}m}\right\} \end{pmatrix}=\mathbf{T}_{\mathrm{imb}}\cdot\mathbf{x}_{m}+\mathbf{d}_{\mathrm{imb}} $$