VLSI Digital Signal Processing Systems: Design and Implementation

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 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 Σ_ng²(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 ...