Filter Design and Implementation - Digital Filter Design Solutions - page 45

fact, if G i ( F ) is the i th filter spectrum, then the output noise power for M filters is
given by

1

M

∏ =

x ∫

2 2 2

G ( F ) dF (2.39)

i

i 1

0

Note that if all the filters are identical, then the first term on the right-hand side
will result in the same pass band but with absolute attenuation raised to the M th
power, or correspondingly, the original decibel attenuation of the filter scaled up
by M . If the filters are different, the output noise power will depend on the profile
of the resultant cascaded filter. High-order polynomial Butterworth filter models
can be used to estimate the right-hand side of (2.39) for various combinations of
filters [8]. Figure 2.31 shows the SNR gain for several filter types. Note that for
differentiators, which will be presented in Chapter 6, there is a 0 dB crossover.
When the SNR gain is negative, the input noise is amplified through the filter.
Table 2.2 summarizes the noise output and SNR gain for common filter types.

2.11 THE CENTER COEFFICIENT

Before closing, we want to take a brief look at what happens to the frequency
response of an odd-length filter when its center coefficient (i.e., at ( L /2+1)), is
changed slightly. Take for example a 99-point, 0.4 low-pass filter. If the middle
coefficient alone were changed by 0.5% and 5% we would observe the responses

100

80

2nd Order
Differentiator

60

1st Order
Differentiator

40

High Pass
Filter Low Pass

Filter

20

0

0dB Cross Over

-20

0 0.2 0.4 0.6 0.8 1

Normalized Cut-off Frequency, F c

Figure 2.31 SNR gain for typical filters.