Filter Design and Implementation - Digital Filter Design Solutions - page 43

where min means the smaller of the two ratios in the brackets. The scheme is
shown in Figure 2.29. The symbols (↑ M ) and (↓ P ) refer to the insertion of ( M - 1)
zeros and the discarding of ( P - 1) data respectively. The sampling period is
( P / M ) t .

In the next section we will give some general rules for governing the passage
of white noise through FIR filters. This gives a fairly reasonable reference point
even in circumstances where the noise is colored.

2.10 NOISE CONSIDERATION

In many situations, we are interested in estimating the extent of noise attenuation
through an FIR filter. Noise attenuation is often expressed in terms of the input
noise power , this being the variance of the input noise fluctuations. In this
section, a brief review of the noise propagation through low-pass, high-pass, band-
pass, and band-stop filters will be given. Noise propagation through differentiating
filters and Hilbert transformers will be presented in Chapters 6 and 7, respectively.
Further details on noise propagation in various filter types can be obtained from
[3,5,6,8]

2

2.10.1 Low-Pass and High-Pass Filters

2 2

x

Given a white noise input of power

σ
c is given by

, the ideal output noise power for a

low-pass filter with cut-off at F

2

F

2

(2.34)

x c

whereas for the high-pass filter with cut-on at F c , the ideal output noise power
becomes

2

( F )

2

(2.35)

x c

These results assume an ideal brick-wall type filter with zero transition width.
However, in practice, there are deviations that can be linked to the shape of the
filter in the frequency domain, and to the fact that at very low cut-off frequencies,
such filters are constrained by the shape of the window [8]. Figure 2.30 shows the
output noise power from a 33- and a 55-tap low-pass filter for successive cut-off
frequencies. The input noise was taken from a CCD spectrometer. It is clear that
apart from very low frequencies, (2.34) is good working model of noise
attenuation through a low-pass filter. A similar observation could be made for
white noise propagation through high pass filters as modeled by (2.35). The gain
in signal-to-noise ratio G SNR , as a result of the smoothing effect (or decrease