Databases Reference
In-Depth Information
Magnitude
Magnitude
f
0
f
0
Frequency
Frequency
F I GU R E 14 . 4
Ideal and realistic low-pass filter characteristics.
frequency less than
f
0
is not constant, and components with frequencies above
f
0
are not
totally blocked. This phenomenon is referred to as
ripple
in the passband and stopband.
The filters we will discuss are digital filters, which operate on a sequence of numbers that
are usually samples of a continuously varying signal. We discussed sampling in Chapter 12.
For those of you who skipped that chapter, let us take a brief look at the sampling operation.
How often does a signal have to be sampled in order to reconstruct the signal from the
samples? If one signal changes more rapidly than another, it is reasonable to assume that we
would need to sample the more rapidly varying signal more often than the slowly varying signal
in order to achieve an accurate representation. In fact, it can be shown mathematically that if
the highest frequency component of a signal is
f
0
, then we need to sample the signal at more
than 2
f
0
times per second. This result is known as the
Nyquist theorem
or
Nyquist rule
after
Harry Nyquist, a famous mathematician from Bell Laboratories. His pioneering work laid the
groundwork for much of digital communication. The Nyquist rule can also be extended to
signals that only have frequency components between two frequencies
f
1
and
f
2
.If
f
1
and
f
2
satisfy certain criteria, then we can show that in order to recover the signal exactly, we need
to sample the signal at a rate of at least 2
samples per second [
134
].
What would happen if we violated the Nyquist rule and sampled at less than twice the
highest frequency? In Chapter 12 we showed that it would be impossible to recover the
original signal from the sample. Components with frequencies higher than half the sampling
rate show up at lower frequencies. This process is called
aliasing
. In order to prevent aliasing,
most systems that require sampling will contain an “anti-aliasing filter” that restricts the input
to the sampler to be less than half the sampling frequency. If the signal contains components
at more than half the sampling frequency, we will introduce distortion by filtering out these
components. However, the distortion due to aliasing is generallymore severe than the distortion
we introduce due to filtering.
Digital filtering involves taking a weighted sum of current and past inputs to the filter and,
in some cases, the past outputs of the filter. The general form of the input-output relationships
of the filter is given by
(
f
2
−
f
1
)
N
M
y
n
=
a
i
x
n
−
i
+
b
i
y
n
−
i
(8)
i
=
0
i
=
1
