Databases Reference
In-Depth Information
There are several things we can see from this example. First, the number of different
values that we transmit is the same, whether we send the original sequence
{
x
n
}
or the two
subsequences
sequence into subsequences did not result
in any increase in the number of values that we need to transmit. Second, the two subsequences
had distinctly different characteristics, which led to our use of different techniques to encode
the different sequences.
{
y
n
}
and
{
z
n
}
. Decomposing the
{
x
n
}
sequence, we would have been using
essentially the same approach to compress both subsequences. Finally, we could have used
the same decomposition approach to decompose the two constituent sequences, which then
could be decomposed further still.
While this example was specific to a particular set of values, we can see that decomposing
a signal can lead to different ways of looking at the problem of compression. This added
flexibility can lead to improved compression performance.
Before we leave this example let us formalize the process of decomposing or
analysis
, and
recomposing or
synthesis
. In our example, we decomposed the input sequence
If we had not split the
{
x
n
}
{
x
n
}
into two
subsequences
{
y
n
}
and
{
z
n
}
by the operations
x
n
+
x
n
−
1
2
y
n
=
(6)
x
n
−
x
n
−
1
2
z
n
=
(7)
We can implement these operations using discrete time filters. We briefly considered discrete
time filters in Chapter 11. We take a slightly more detailed look at filters in the next section.
14.3 Filters
A system that isolates certain frequency components is called a
filter
. The analogy here with
mechanical filters such as coffee filters is obvious. Acoffee filter or a filter in awater purification
systemblocks coarse particles and allows only the finer-grained components of the input to pass
through. The analogy is not complete, however, because mechanical filters always block the
coarser components of the input, while the filters we are discussing can selectively let through
or block any range of frequencies. Filters that only let through components below a certain
frequency
f
0
are called low-pass filters; filters that block all frequency components below a
certain value
f
0
are called high-pass filters. The frequency
f
0
is called the
cutoff frequency
.
Filters that let through components that have frequency content above some frequency
f
1
but
below frequency
f
2
are called band-pass filters.
One way to characterize filters is by their
magnitude transfer function
—the ratio of the
magnitude of the input and output of the filter as a function of frequency. In Figure
14.4
we
show the magnitude transfer function for an ideal low-pass filter and a more realistic low-pass
filter, both with a cutoff frequency of
f
0
. In the ideal case, all components of the input signal
with frequencies below
f
0
are unaffected except for a constant amount of amplification. All
frequencies above
f
0
are blocked. In other words, the cutoff is sharp. In the case of the more
realistic filter, the cutoff is more gradual. Also, the amplification for the components with
