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VLSI Digital Signal Processing Systems: Design and Implementation by Keshab K. Parhi

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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

image

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

image

image

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

image

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 ...

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