Consider the 1-pole IIR filter transfer function
We are interested in the total roundoff noise of this filter for different levels of pipelining (i.e., different values of M) with and without l2 scaling. The SFG of the filter is shown in Fig. 11.9.
M = 1 Unscaled system:
As we can see in Fig. 11.9 there is just one rounding operation in the filter (at node 1); therefore, there is just one error contribution e(n) and just one state. As we have shown in (11.44), the roundoff noise is Σng2(n), where g(n) is the unit-sample response from the state to the output y(n). From the SFG of Fig. 11.9 we can write the transfer function G(z), from the roundoff input to the output:
The roundoff noise can be computed as
M = 1 Scaled system:
To compute the ...