Introduction to Filters and Filter Design Software - Practical Analog and Digital Filter Design - page 3

response in the stopband is very small. For this reason, the gain is typically
converted to decibels (dB) as indicated in (1.1). For example, a filter's passband
gain response could be specified as 0.707 or as −3.0103 dB, while the stopband
gain might be specified as 0.0001 or −80.0 dB.

gain dB = 20 ⋅ log( gain ) (1.1)

As we can see, the values in decibels are more manageable for very small
gains. Some filter designers prefer to use attenuation (or loss) values instead of
gain values. Attenuation is simply the inverse of gain. For example, a filter with a
gain of 1/2 at a particular frequency would have an attenuation of 2 at that
frequency. If we express attenuation in decibels we will find that it is simply the
negative of the gain in decibels as indicated in (1.2). Gain values expressed in
decibels will be the standard quantities used as filter specifications, although the
term attenuation (or loss) will be used occasionally when appropriate.

− 1

attn = 20 ⋅ log( gain ) = − 20 ⋅ log( gain ) = − gain

(1.2)

dB dB

1.1.1 Lowpass Filters

Figure 1.1 shows a typical lowpass filter's response using frequency and gain
specifications necessary for precision filter design. The frequency range of the
filter specification has been divided into three areas. The passband extends from
zero frequency (dc) to the passband edge frequency f pass , and the stopband extends
from the stopband edge frequency f stop to infinity. (We will see later in this text
that digital filters have a finite upper frequency limit. We will discuss that issue at
the appropriate time.) These two bands are separated by the transition band that
extends from f pass to f stop . The filter response within the passband is allowed to vary
between 0 dB and the passband gain a pass , while the gain in the stopband can vary
between the stopband gain a stop and negative infinity. (The 0 dB gain in the
passband relates to a gain of 1.0, while the gain of negative infinity in the
stopband relates to a gain of 0.0.) A lowpass filter's selectivity can now be
specified with only four parameters: the passband gain a pass , the stopband gain
a stop , the passband edge frequency f pass , and the stopband edge frequency f stop .
Lowpass filters are used whenever it is important to limit the high-frequency
content of a signal. For example, if an old audiotape has a lot of high-frequency
“hiss,” a lowpass filter with a passband edge frequency of 8 kHz could be used to
eliminate much of the hiss. Of course, it also eliminates high frequencies that were
intended to be reproduced. We should remember that any filter can differentiate
only between bands of frequencies, not between information and noise.