11.6    ROUNDOFF NOISE COMPUTATION USING STATE VARIABLE DESCRIPTION

Parseval’s relation and Cauchy’s residue theorem are useful for finding signal power or roundoff noise of digital filters. However, it is not easy to apply Parseval’s relation to a filter with complex structure. (For example, see the 3rd-order scaled-normalized lattice filter in Fig. 11.18.)

However, due to simple computing algorithms available [3], the power at each internal node and the output roundoff noise of a digital filter can be easily computed once the digital filter is described in state variable form. Using (11.24), K can be computed efficiently by the following algorithm [3].

Algorithm for Computing K:

  • Initialize:

    image

  • Loop:

    image

  • Continue until F = 0.

After 1st loop iteration,

image

After 2nd loop iteration,

image

Thus, each execution of the loop doubles the number of terms in the sum of (11.24). The above algorithm converges as long as the filter is stable since the eigenvalues of the matrix A are the poles of the transfer function. (It is well-known from linear algebra theory that the matrix powers Ai tend to the zero ...

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