Page 33 - Analog and Digital Filter Design

30 Analog and Digital Filter Design

All analog filters are designed from a normalized lowpass model. This model is
a set of component values that are normalized for a w = 1 rad/s at the passband
edge. Passive filter models have component values that are normalized for a 1 R
load. Normalization allows the use of a table or set of component values, in
conjunction with a single graph, to determine any filter design. This is a very
powerful method, but transforming and scaling are necessary for each filter
design undertaken.

Component values are scaled to produce an analog lowpass filter with a more
practical passband and, in the case of passive filters, a more practical load resist-
ance. The scaling process requires simple arithmetic to multiply and divide by
certain factors. The result of scaling is that the cutoff point is changed from
1 rad/s to the required frequency and the load impedance is changed from 1 R
to the required value.

Highpass filters can be produced from a lowpass model. The frequency response
is the reciprocal of the lowpass response; so the attenuation of a lowpass filter
model at w = 2 is the same as the equivalent highpass at o = 0.5. Passive high-
pass filter components are the reciprocal of the normalized lowpass filter. This
means that where there are capacitors in the lowpass model, they are replaced
by inductors in the highpass model. Similarly, where there are inductors in the
lowpass model, they are replaced by capacitors in the highpass model.

Bandpass and bandstop filters are more complex but can still be derived from
a normalized lowpass model. As an illustration, I will consider a bandpass filter
and describe how to find out whether the filter specification is demanding, and
hence I will be able to determine the filter order required to achieve it. First, I
need to find out the bandwidth of the passband. Second, I need to find out the
stopband attenuation and the width of the passband skirt.

If the desired passband (between the points where the filter provides less than
3dB attenuation) extends from 2OkHz to 24kHz, the passband bandwidth is
4kHz. Suppose a 40dB stopband attenuation is required at frequencies below
10 kHz and above 40 kHz. The width of the passband skirt is thus 30 kHz, being
the difference between the two. The ratio of skirt width to bandwidth is 30 + 4
= 7.5. In terms of the lowpass model, the passband width is 1 rad/s, and hence
the skirt response at 7.5 rad/s must provide the desired 40dB attenuation. This
is not very demanding, so a simple filter will do.

Bandstop filters have the inverse response of the bandpass filters described
above. The normalized frequency of attenuation is given by the 3 dB bandwidth
divided by the width of the stopband.

Active filters do not use normalized component value tables. Instead, they use
something called pole and zero locations. (Do not worry too much about this