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## 11.9    PROBLEMS

1. Consider the SFG in Fig. 11.25 which corresponds to the all-pole transfer function

where − 1 < a < 1 for stability.

(a)  Calculate the output roundoff noise. Show that the noise variance gets amplified through the feedback section, i.e., even though e(n) may be small, its effect on y(n) can be large.

(b)  Compute roundoff noise for (i) a = 0.1, (ii) a = 0.5, and (iii) a = 0.998. Comment on the relation between roundoff noise and pole locations.

Fig. 11.25    Signal flow graph of an all-pole IIR filter.

Fig. 11.26    Signal flow graph of a scaled all-pole IIR filter.

(c)  For narrowband filters, |a| is typically very close to 1. For example, a = 0.998 is not uncommon. For such filters, the noise variance gets amplified enormously. (In that case, low-noise/sensitivity filter structures such as lattice filters and wave digital filters need to be considered.) Explain conceptually why poles are located very close to the unit circle for narrowband filters.

2. Assume that the ...

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