Digital Signal Processing Reference
In-Depth Information
H
( )
1
Passband
Stopband
Stopband
c
0
c
(a)
H
( )
1
Passband
Stopband
Stopband
Passband
c
0
c
(b)
H
( )
1
Stop-
band
Passband
Stopband
Passband
Stopband
2
1
0
1
2
(c)
H
( )
1
Pass-
band
Stopband
Passband
Stopband
Passband
2
1
0
1
2
Figure 6.1
Frequency responses of four
types of ideal filters.
(d)
stopband
. The output of this filter consists only of those frequency components of
the input signal that are within the passband. Figure 6.2 illustrates the effect of the
ideal low-pass filter on an input signal.
Filters are used to eliminate unwanted components of signals. For example,
the high-frequency noise shown to be present in in Figure 6.2(b) is outside
the passband (in the
stopband
). Therefore, this noise is not passed through the filter
to and the desired portion of the signal is passed unaltered by the filter. This
filtering process is illustrated in Figure 6.2(c) and (d).
It should be noted that the filters described previously are called
ideal
filters.
As with most things we call ideal, they are not physically attainable. However, the
concept of the ideal filter is very helpful in the analysis of linear system operation,
because it greatly simplifies the mathematics necessary to describe the process.
That ideal filters are not possible to construct physically can be demonstrated
by reconsideration of the frequency response of the ideal low-pass filter. The trans-
fer function of this filter can be written as
V
1
(v)
V
2
(v),
H(v) = rect(v/2v
c
).
