The Impact of Finite Wordlength Implementation - Digital Signal Processing: An Introduction with MATLAB and Applications - page 207

(a) (b)

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0

−0.1 −0.1

−0.2 −0.2

−0.3 −0.3

−0.4 −0.4

5 10 15 20 5 10 15 20

n n

Fig. 4.16 Zero-input response of a first-order IIR filter with a =-0.815, a infinite-precision,
b fixed-point 1 ? 4 bits

as x(n) = 0. Since the rounding error is bounded by D = 2, then

j Q ½ ay q ð n 1 Þ ay q ð n 1 Þj D

2 :

ð 4 : 40 Þ

Substituting ( 4.39 ) into ( 4.40 ), yields

D
2 ð 1 j a jÞ :

j y q ð n 1 Þj ð 4 : 41 Þ

or, for a steady-state output y q

D
2 ð 1 j a jÞ :

j y q j ð 4 : 42 Þ

This relation imposes an upper limit on the magnitude of granular limit cycles.
Recall that in Example (4), y q (n - 1) B 0.1689. This result satisfies the condition
given in ( 4.42 ). Notice that granular limit cycles occur due to the small error
introduced by rounding quantization. According to ( 4.42 ), the magnitude of this
oscillation is proportional to the quantization step size D, and this magnitude can
therefore be reduced by increasing the number of precision bits.