Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 49

0.1 200

Linear phase

0.05 100

0 0

-0.05 -100

-0.1 -200

0 5 10 15 0 0.1 0.2 0.3 0.4 0.5

Time

Frequency

(a) (b)

Figure 3.3 Coefficients of a moving-average filter

The coefficient vector , comprising the b i stacked together, is of dimension

2N 1; it is shown in Figure 3.3(a). Notice that the coefficients are symmetric;
we have drawn an arrow at the symmetric point.
From the theory of polynomials, we recollect that such a symmetric property
produces a linear phase characteristic. The same is shown in Figure 3.3(b), which
shows the linear phase of the MA filter. In the same figure, we see the corresponding
non-linear phase of the original (parent) IIR narrowband filter. The choice of these
coefficients is deliberate and is intended to describe the linear phase property. In
addition, we demonstrate conversion of an IIR filter to an MA filter using the
truncated impulse response of the IIR filter.
Figure 3.4 shows the frequency response of the IIR filter and the MA filter
(truncated IIR). The performance of the IIR filter is far superior to the performance
of the MA filter. Computationally, an IIR filter needs less resources, but stability
and phase are the compromises. In addition, a careful examination reveals a
compromise in the filter's skirt sharpness and its sidelobe levels are high.

b

1

0.8

O< ---- Original IIR filter

0.6

O< --- 15 Coeffcients MA filter

0.4

0.2

0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Figure 3.4 Moving-average filter