## 11.5 ROUNDOFF NOISE IN PIPELINED IIR FILTERS

In this section, roundoff noise properties in pipelined 1st-order and 2nd-order direct-form IIR filters are discussed [4],[5]. Roundoff noise of the scaled pipelined versions are also considered.

*Fig. 11.9* Signal flow graph of 1st-order nonpipelined IIR filter.

### 11.5.1 First-Order IIR Filter

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 *l*_{2} 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 Σ_{n}*g*^{2}(*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 ...