Page 427 - Analog and Digital Filter Design

424 Analog and Digital Filter Design

As the filter order increases, the noise equivalent bandwidth approaches the
3 dB bandwidth. To find the actual bandwidth, simply multiply the figure given
by the 3dB bandwidth of the filter that you wish to assess.

Chebyshev Noise Bandwidth
The noise bandwidth of Chebyshev filters having a 3 dB cutoff point is given by:

Applying this formula gives a noise bandwidth of less than the 3 dB bandwidth
for high-order filters. This can be explained by remembering that Chebyshev
filters have ripple in the passband. This means that, on average, the gain in
the passband will be less than one. Also, Chebyshev filters have a steeper skirt
response; the attenuation beyond the filter cutoff rises sharply with frequency.
Therefore the noise equivalent bandwidth of Chebyshev filters can be expected
to be less than a Butterworth filter of the same order.

In fact the amplitude of the voltage ripple is:

In terms of power this becomes:

1
g=-
I+&'
The term E is equal to dx

The average power gain within the ripple part of the passband can therefore be
approximated by assunling that the ripple is symmetrical. In that case the
average power gain is halfway between the gain at the peaks and the gain in the
troughs, that is, between one andg. The equation for this is -. Table A.4 shows
2
the passband power gain for Chebyshev filters that have a 3 dB cutoff point.