Digital Signal Processing Reference
In-Depth Information
N
H
N
G
N
F
D
H
DE Mult Add
Regular
124
-
-
62
124
125
124
Two-rate, regular
F
(
z
) =
P
(
z
2
)
126
10
372
63
126
73
134
Two-rate, FRM
F(z) = P(z
2
)
140
10
412
70
140
41
59
TABLE2.
Results of Example 3.
3.1. Complexity savings
As opposed to the case of linear-phase overall filters considered in Section 2, we can
here achieve complexity savings without using additional FRM techniques. The reason is
two-fold. First, as seen above,
F
(
z
) =
P
(
z
2
)
is already sparse. Second, as a mid-band
system is often a nonlinear-phase system, the filter coefficient are not symmetric. By using
the two-rate based structure, symmetry can partially be utilized as
F
(
z
)
is a symmetric filter
whereas only the low-order
G
(
z
)
is unsymmetric. As to the sparsity, the degree of sparseness
can be increased by realizing
P
(
z
)
as an FRM third-band filter. Details are given in [18].
3.2. Examples
Example 3:
Consider the approximation of a fractional-degree differentiator with the desired
function
D
(
j
ω
) =
e
−
j
ω
(
N
G
+
N
F
)
/4
(
j
ω
)
0.5
in the frequency band
ω
∈ [
0.02
π
, 0.98
π
]
and for an
approximation error of
δ
e
=
0.01. Figure 5 shows the frequency response and approximation
error of the two-rate based design. The filter has been designed using essentially the same
three-step procedure described earlier, but after minor appropriate modifications, as detailed
in [18]. Table 2 gives the results for the conventional direct-form realization and for the
two-rate based realizations, both with a sparse regular bandpass filter and a sparse FRM
bandpass filter. The quantity
D
H
denotes the integer part of the group delay whereas DE
denotes the number of delay elements. As seen from the table, substantial savings are
achieved using the two-rate based structures, especially when the FRM technique is also
utilized. As usual when using the FRM technique, one has to pay a price in a somewhat
increased delay. It is also noted that the savings increase/decrease with increased/decreased
bandwidth (decreased/increased width of the don't-care bands). This is in line with the basic
two-rate based scheme and it was exemplified earlier in Example 2.
4. Multi-function systems
In this section, we will discuss the extension to the realization of multifunction systems. The
two-rate based approach is even more efficient for such systems as the same
F
(
z
)
, and thus
the same
F
0
(
z
)
, is shared between all functions. We will illustrate this for Farrow-structure
based (see [34]) variable fractional-delay (VFD) filters. As an example will reveal, the two-rate
based
structure
offers
dramatic
complexity
reductions
in
this
application,
even
without
using the additional FRM approach.
However, incorporating the FRM approach, further
complexity savings are obtained.
4.1. Variable fractional-delay filters
Variable fractional-delay filters find applications in many different contexts like interpolation,
resampling, delay estimation, and signal reconstruction, see [35-40].
