NOTCH FILTERS

8.2.2 NOTCH FILTERS

A notch filter is a filter that contains one or more deep notches or, ideally, perfect nulls in its frequency response. Figure 8.5 illustrates the frequency response of a notch filter with a null at the frequency ω = ω 0 . Notch filters are useful in many applications where specific frequency components must

be eliminated. For example, instrumentation systems require that the power line frequency of 60 Hz and its harmonics be eliminated. To create a null in the frequency response of a filter at a frequency ω 0 , we simply introduce a pair of complex-conjugate zeros on the unit

circle at the angle ω 0 . Hence, the zeros are selected as

z ± 1,2 =e jω 0 (8.19)

Then, the system function for the notch filter is

H(z) = b

jω 0 z − 1 − jω 0 z 0 − (1 − e )(1 − e 1 )

(8.20) where b 0 is a gain factor. Figure 8.6 illustrates the magnitude response of

=b 0 (1 − (2 cos ω 0 )z − 1 +z − 2 )

a notch filter having a null at ω = π/4.

The major problem with this notch filter is that the notch has a rela- tively large bandwidth, which means that other frequency components

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Chapter 8

IIR FILTER DESIGN

Pole–zero Plot

Magnitude Response

0.4 Decibel

Part 0.2 –50 –1

Phase Response

Imaginary

Radians /

Real Part

ω in π units

FIGURE 8.5 Frequency response of a typical notch filter

around the desired null are severely attenuated. To reduce the band- width of the null, we may resort to the more sophisticated, longer FIR filter designed according to the optimum equiripple design method de- scribed in Chapter 7. Alternatively, we could attempt to improve the fre- quency response of the filter by introducing poles in the system function.

Pole–zero Plot

Magnitude Response

0.4 Decibel Part 0.2 –50

Phase Response

Imaginary –0.4

Radians /

Real Part

ω in π units

FIGURE 8.6 Frequency response of a notch filter with ω 0 = π/4

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Some Special Filter Types 395

Pole–zero Plot

Magnitude Response

rt

Decibel

a 0.4

P 0.2 –50 –1

ry 1

a 0 in

g –0.2 Phase Response

Im

Radians /

Real Part

ω in π units

FIGURE 8.7

Magnitude and phase responses of notch filter with poles (solid lines) and without poles (dotted lines) for ω 0 = π/4 and r = 0.85

In particular, suppose that we select the poles at

Hence, the system function becomes

1 − (2 cos ω 0 )z − 1 +z 2

H(z) = b 0

1 − (2r cos ω 0 )z − 1 +r 2 z − 2

The magnitude of the frequency response #

# $ H e jω %# # of this filter is illus- trated in Figure 8.7 for ω 0 = π/4 and r = 0.85. Also plotted in this figure is the frequency response without the poles. We observe that the effect of

the pole is to introduce a resonance in the vicinity of the null and, thus, to reduce the bandwidth of the notch. In addition to reducing the bandwidth of the notch, the introduction of a pole in the vicinity of the null may re- sult in a small ripple in the passband of the filter due to the resonance created by the pole.

8.2.3 COMB FILTERS In its simplest form, a comb filter may be viewed as a notch filter in which the nulls occur periodically across the frequency band, hence the analogy to an ordinary comb that has periodically spaced teeth. Comb filters are used in many practical systems, including the rejections of power-line

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Chapter 8

IIR FILTER DESIGN

harmonics, and the suppression of clutter from fixed objects in moving- target indicator (MTI) radars.

We can create a comb filter by taking our FIR filter with system function * M

H(z) =

h(k)z − k

k=0

and replacing z by z L , where L is a positive integer. Thus, the new FIR filter has the system function

H L (z) =

h(k)z − kL

k=0

If the frequency response of the original FIR filter is H $ e jω % , the frequency

response of the filter given by (8.24) is

L e jω % *

= % h(k)e jkLω =H e jLω

k=0

Consequently, the frequency response characteristic H $

L e jω % is an L-order repetition of H $ e jω % in the range 0 ≤ ω ≤ 2π. Figure 8.8 illustrates the

relationship between H $

L e jω %

and H $ e jω % for L = 4. The introduction of

a pole at each notch may be used to narrow the bandwidth of each notch, as just described.

8.2.4 ALLPASS FILTERS An allpass filter is characterized by a system function that has a constant magnitude response for all frequencies, i.e.,

A simple example of an allpass system is a system that introduces a pure delay to an input signal, i.e.,

(8.27) This system passes all frequency components of an input signal without

H(z) = z − k

any frequency dependent attenuation. It simply delays all frequency com- ponents by k samples.

A more general characterization of an allpass filter is one having a system function of the form

a N − +a N −1 z 1 +···+a 1 z − N +1 +z − N

H(z) =

(8.28) 1+a 1 z − 1 +···+a N −1 z − N +1 +a N z − N

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Some Special Filter Types 397

(a)

(b)

FIGURE 8.8

$ Comb filters with frequency response H % L e jω obtained from

$ H % e jω for L = 4

which may be expressed in the compact form as

N A(z − H(z) = z )

A(z) where

A(z) = − a z k

We observe that

H − e jω # = H(z)H(z 1 )|

z=e jω =1

for all frequencies. Hence, the system is all-pass.

From the form of H(z) given by (8.28), we observe that if z 0 is a pole of H(z), then 1/z 0 is a zero of H(z). That is, the poles and zeros are reciprocals of one another. Figure 8.9 illustrates the typical pole-zero pat- tern for a single-pole, single-zero filter and a 2-pole, 2-zero filter. Graphs of the magnitude and phase characteristics of these two filters are shown

in Figure 8.10 for a = 0.6 and r = 0.9, ω 0 = π/4, where A(z) for the two

filters is, respectively, given as

A(z) = 1 + az − 1 (8.32a) )z − A(z) = 1 − (2r cos ω 1 0 +r 2 z − 2 (8.32b)

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Chapter 8

IIR FILTER DESIGN