Digital Signal Processing Reference
In-Depth Information
The transfer function of digital filters
In principle, the same conventions are chosen as for analog filters (see Chapter 6, "System
Analysis"):
• Transfer function, as shown in
two
images: amplitude - and phase response.
• Transfer function in
one
representation: Local curve in the GAUSSian plane.
In Figure 220, the transfer function of a bandpass filter is shown in the two types of
presentation:
(a)
The stop band of the bandpass filter is characterized by many "irregular" phase
jumps of 180
0
or
rad at the zeros of the filter graph. Because of the tiny amplitude
the entire blocking area appears concentrated at the center of the locus (see "zoom
range").
π
(b)
Within the ripple pass band a completely regular, linear phase response is present,
an important advantage of digital FIR filter. Within the passband, the phase curve
rotates about 24 times around the wheel full angle 360
0
and 2
. The "jumps" in the
upper right phase response are therefore caused by the representation and are not
real in a physical sense, because the values are identical -180
0
or -
π
π
and
π
rad or
180
0
rad!
(c)
The distance between the measurement points or measurement crosses in the locus
below is exactly 1 Hz because the values for sampling rate and block length were
chosen to be identical (8192).
(d)
The circular orbits of the locus does not overlap exactly, because the amplitude re-
sponse in the passband is wavy. The width of the "tube" is therefore a measure of
the ripple.
(e)
In the passband the distance between the individual measuring points is greatest,
little smaller at the filter edges, and finally tiny in the stopband. It is typical of loci,
that they represent the area of interest, as if increased in size by a magnifying glass!
The comparison of the transfer function of "poor" analog filter with "high-quality" digital
filters provide overall the following results:
Inevitably quality filters have a long-lasting impulse response
h (t) (Uncertainty principle!). This "time-stretching" in the lan-
guage of the sine waves implies a "giant" phase shift of n
<
2
π
rad.
Excellent digital filter - steep edge and without ripple - has there-
fore a markedly narrow annular locus for the passage area and a
small center point range (for the blocking region). The filter
edges form the relatively short, symmetrical lines connecting ring
and center.
