VLSI Digital Signal Processing Systems: Design and Implementation

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⁻¹AT and b_s = T⁻¹b.

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 ...