For vector \(\vec{x} = [x_{w-1}, x_{w-2}, \ldots, x_0]\) \[B2U_{w}(\vec{x}) \stackrel{.}= \sum_{i=0}^{w-1} x_i2^i\] \[B2T_{w}(\vec{x}) \stackrel{.}= -x_{w-1}2^{w-1} + \sum_{i=0}^{w-2} x_i2^i\] \[B2O_{w}(\vec{x}) \stackrel{.}= -x_{w-1}(2^{w-1} - 1) + \sum_{i=0}^{w-2} x_i2^i\] \[B2S_{w}(\vec{x}) \stackrel{.}= (-1)^{x_{w-1}} \cdot \left(\sum_{i=0}^{w-2} x_i2^i \right)\] \[U2T_{w}(u) = \begin{cases}u, & u\leq \text{TMax}_w\\ u-2^w, & u>\text{TMax}_w\end{cases}\]

For \(x\) such that \(\text{TMin} \leq x \leq \text{TMax}\) \[B2U_{w}(T2B_{w}(x)) = T2U_{w}(x) = x+x_{w-1}2^{w}\]

Truncation of an unsigned number \[B2U_k([x_{k-1}, x_{k-2}, \ldots, x_{0}]) = B2U_w([x_{w-1}, x_{w-2}, \ldots, x_0])\mod 2^k\] Truncation of a two’s-complement number \[B2U_k([x_{k-1}, x_{k-2}, \ldots, x_{0}]) = U2T_k(B2U_w([x_{w-1}, x_{w-2}, \ldots, x_0])\mod 2^k)\]