Digital Signal Processing Reference
In-Depth Information
pass-band error 0.003 (see Figure 6.4). The amplification factor is 0.141, or 14%
of the input noise passes through the filter. As such, the input signal bandwidth
into this filter should be no more than 0.28, if 0.003 error is acceptable, otherwise
it should be 0.25. In the case of noiseless data, full-band differentiators could be
used for signal frequencies up to the Nyquist frequency. Full-band differentiators
have been provided for this purpose.
6.5.2
Noise Amplification in Band-Pass Differentiators
Noise amplification in band-pass differentiators is found by calculating the
difference in output noise power between the low-pass differentiator at
F
c,off
and
that at
F
c,on
. Thus for
m
th-order band-pass differentiating filters
2
m
π
2
m
+
1
2
m
+
1
η
=
(
F
F
)
. (6.19)
c,
off
c,
on
2
m
+
1
For a given noise amplification factor
, and the lower cut-on frequency
F
c,on
, we
can easily show that the bandwidth
F
of the filter must be
1
/
2
m
+
1
η
(
2
m
+
1
2
m
+
1
F
=
F
+
F
, (6.20)
c,
on
c,
on
2
m
π
and that the center frequency
F
cen
is
F
=
F
+
1
F
. (6.21)
cen
c,
on
2
In the band-pass differentiating filters presented here,
F
cen
and
F
are associated
with the filter identifier (see Section 6.7.1).
6.5.3
Signal-to-Noise Ratio
The fact that we can manipulate the noise amplification on a differentiator means
there is some control over the signal-to-noise ratio (SNR) of the filtered signal.
Given a signal of strength
S
with rms noise
, then its SNR is
S
/
. If after
filtering, the output noise is changed to
y
, then the change in SNR is
/
y
.
Furthermore, if
there will be an increase in SNR. This result
assumes that the noise is white. In general, the change in SNR,
G
SNR
from (6.17)
and (6.18) is given by
y
is less than
G
(
)
=
10
log(
2
m
+
1
10
(
2
m
+
1
log
F
9
94
m
. (6.22)
SNR
dB
c
