## 11.2 SCALING AND ROUNDOFF NOISE

### 11.2.1 Scaling Operation

Scaling is a process of readjusting certain internal gain parameters in order to constrain internal signals to a range appropriate to the hardware with the constraint that the transfer function from input to output should not be changed.

The filter in Fig. 11.1(a) with unscaled node *x* has the transfer function

To scale the node *x*, we divide *F*(*z*) by some number *β* and multiply *G*(*z*) by the same number as in Fig. 11.1(b). Although the transfer function does not change by this operation, the signal level at node *x* has been changed. The scaling parameter *β* can be chosen to meet any specific scaling rule such as

where *f*(*i*) is the unit-sample response from input to the node *x* and the parameter *δ* can be interpreted to represent the number of standard deviations

*Fig. 11.1* (a) A filter with unscaled node *x* and (b) A filter with scaled node *x*′.

representable in the register at node *x* if the input is unit-variance white noise. If the input is bounded by |*u*(*n*)| ≤ 1, then

Equation (11.4) represents the true bound on the range of ...