## 11.4 SCALING AND ROUNDOFF NOISE COMPUTATION

### 11.4.1 Scaling Operation

As mentioned before, the same wordlength can be assigned to all the variables of the system only if all the states have equal power. The way to achieve this is called *scaling.* The state vector is premultiplied by inverse of the scaling matrix **T**. If we denote the scaled states by **x**_{s}, we can write,

Substituting for **x** from (11.29) into the state update equation (11.10) and solving for **x**_{s} we get,

where **A**_{s} = **T**^{−1}**AT** and **b**_{s} = **T**^{−1}b.

Similarly the output equation (11.11) can be derived as follows,

The scaled **K** matrix is given by

Since it is desirable to have equal power at all states, a transformation matrix **T** is chosen such that the **K**_{s} matrix of the scaled system has all diagonal entries as 1. Further assume **T** to be diagonal, i.e.,

From (11.34) we can write the diagonal entries of **K**_{s} in terms of those of **K**

*Fig. 11.7 ...*