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

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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.

image

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

image

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 imageΣng2(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:

image

The roundoff noise can be computed as

M = 1 Scaled system:

To compute the ...

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