Digital Signal Processing Reference
In-Depth Information
(don't-care band)
∆
=
ω
S
−
ω
C
, where
ω
C
and
ω
S
denote the passband and stopband edges,
respectively, see [20, 21]. Hence, when the don't-care band decreases towards zero, the order
increases rapidly. Then, using a direct-form realization, the computational complexity may
become intolerable as it follows the filter order. The same trend exists also for other functions
that are not frequency selective filters, like differentiation and integration, as seen in [22].
2.2. TWO-RATE BASED STRUCTURE
To reduce the complexity, we consider here a structure that is derived via a two-rate
approach, seen in Fig. 1. This structure is efficient for left-band systems (like a differentiator)
targeting the frequency region
ω
∈ [
0,
ω
C
]
, 0
<
ω
C
<
π
. The same structure can also be used
for right-band systems targeting the band
ω
∈ [
ω
C
,
π
]
, 0
<
ω
C
<
π
. The only difference will
appear in the design, and we will therefore focus on the left-band case in this chapter, and
only comment upon the right-band case in the design section.
For a left-band specification, the basic idea is to first interpolate the input signal
X
(
N
)
by
two through upsampling by two followed by a lowpass filter with transfer function
F
(
Z
)
2
. Then, a subsequent filter with transfer function
G
(
Z
)
follows that performs the actual
function. Finally, downsampling by two takes place to retain the original sampling rate.
Using multi-rate theory, see [23], it is readily shown that this scheme corresponds to a linear
and time-invariant (LTI) system with a transfer function
H
(
Z
)
that equals the 0th polyphase
component of the cascaded filter
F
(
Z
)
G
(
Z
)
, i.e.,
H
(
Z
) =
F
0
(
Z
)
G
0
(
Z
) +
Z
−
1
F
1
(
Z
)
G
1
(
Z
)
(4)
where
F
(
Z
) =
F
0
(
Z
2
) +
Z
−
1
F
1
(
Z
2
)
(5)
and
G(Z) = G
0
(Z
2
) + Z
−
1
G
1
(Z
2
)
.
(6)
The final realization is thus a single-rate structure. A two-rate technique is only used to
derive efficient structures. It is noted here that the order and delay of the overall filter
H(Z)
is
N
H
= (N
F
+ N
G
)
/2 and
D
H
= (D
F
+ D
G
)
/2, respectively. This can be understood by
noting that
F(Z)
and
G(Z)
can be viewed as operating (in principle) at two times the input
rate, because the structure is derived by sandwiching
F(Z)G(Z)
between upsampling and
downsampling by two.
2
The same function can be achieved by sampling the underlying analog signal with a higher sampling rate instead of
sampling it slower and then use interpolation in the digital domain. However, this also increases the requirements
on the analog-to-digital converters which are power-hungry components and in many cases one of the bottlenecks
in overall systems. It is therefore often preferred to perform interpolation in the digital domain.
