1. science
2. mathematics
3. arithmetic

An Approach to Increase the Maximal Sample
Frequency of Lattice Wave Digital Filters Based on
Circulators and Richards’ Structures
Håkan JOHANSSON*
Abstract
This paper introduces an approach to increase the maximal sample frequency of lattice wave digital filters
based on circulators and Richards’ structures. In this
approach the allpass branches of the filters make use of
cascaded first-order and constrained third-order Richards’ allpass sections. Compared with the case in which
regular first- and second-order Richards’ allpass sections are used, the maximal sample frequency is here
two times higher. This is generally paid for by a somewhat higher computational complexity.
1 Introduction
Wave digital filters (WDFs) are known to have good
properties under finite-arithmetic conditions, in particular when stability is concerned [1]. One type of WDFs
that is suitable for implementation is the lattice WDF,
due to its modularity and regularity. However, one
drawback of WDFs, and recursive filters in general, is
that they restrict the sample frequency at which the filters can operate in an implementation [2]. This may
affect not only the speed but in fact also the power consumption, since high speed can be utilized to reduce the
power consumption via power supply voltage scaling
techniques [3]. This paper introduces an approach to
increase the maximal sample frequency of lattice
WDFs based on circulators and Richards’ structures.
can easily be computed via explicit formulas or directly
from the poles of the overall filter [4].
Maximal sample frequency
The maximal sample frequency of a recursive algorithm described by a fully specified signal-flow graph is
f max = min { N i ⁄ T opi }
where Topi denotes the total latency of the arithmetic
operations, and Ni denotes the number of delay elements in the directed loop i [2].
From Figs. 1 and 2 we see that the maximal sample
frequencies for the first- and second-order Richards’
allpass sections are
(1)
(1)
f max = 1 ⁄ T min = 1 ⁄ ( T mult + 2T add )
* Department of Electrical Engineering, Linköping University, Linköping, SE-581 83 Sweden
(2)
and
(2)
(2)
f max = 1 ⁄ T min = 1 ⁄ ( 2T mult + 4T add )
(3)
respectively. It is here assumed that all multiplier and
adder latencies are equal. For the Nth-order lattice
WDF in Fig. 3 it is obvious that recursive loops only
exist within its first- and second-order allpass sections.
Hence, from Eqs. (2) and (3) it is clear that the maximal sample frequency of the overall filter equals that of
the second-order sections.
2 Lattice wave digital filters
A lattice WDF is composed of two (WD) allpass filters
in parallel [1]. The transfer function can be written as
H ( z ) = 0.5 [ H 0 ( z ) + H 1 ( z ) ] where H0(z) and H1(z)
are stable allpass filters. Wave digital allpass filters can
be realized in many different ways. Here, we restrict
ourselves to allpass filters derived from circulators in
combination with low-order Richards’ structures. This
results in regular and modular filter structures which is
attractive from an implementation point of view. Firstand second-order Richards’ allpass sections can be
realized as shown in Figs. 1 and 2, respectively. (Here,
symmetric two-port adaptors are used. If desired, these
adaptors can be replaced with two-port series or parallel
adaptors
using
certain
equivalence
transformations [1]). In this paper we only consider
lowpass and highpass filters. In this case, the lattice
WDF is always of odd order and can for standard
approximations (Butterworth, Chebyshev, and Cauer)
be realized as shown in Fig. 3. The adaptor coefficients
(1)
i
T
Tmin
T
α0
α0
x(n)
–
y(n)
x(n)
y(n)
Figure 1. First-order Richards’ allpass section.
T
T
α2
–
α2
Tmin
T
T
α1
α1
x(n)
–
y(n)
x(n)
y(n)
Figure 2. Second-order Richards’ allpass section.
T
T
T
T
T
α4
αΝ−3
γ3
γ3
T
α0
T
α3
T
1/2
x(n)
α1
T
T
y(n)
αΝ−2
α5
T
γ2 = 0
γ2
αΝ−4
T
T
γ3
T
γ1
2T
γ1
γ1
Figure 4. Constrained third-order Richards’ allpass section.
T
α2
α6
αΝ−1
T
T
T
Figure 3. Conventional Nth-order lattice WDF.
T
T
T
β0
βm–1
γ0
T
T
γ4
γΝ−3
2T
2T
3 Proposed approach
To increase the maximal sample frequency we will
make use of constrained third-order allpass sections
instead of second-order sections. A constrained thirdorder section is obtained by fixing one of the adaptor
coefficients in a third-order Richards’ allpass section to
zero, as illustrated in Fig. 4. In the constrained thirdorder section, there are two delay elements between the
two adaptors. By again making use of Fig. 2 it is clear
that the maximal sample frequency of the constrained
third-order section equals that of the first-order section
in Fig. 1, i.e.,
(3)
(1)
f max = f max
(4)
assuming again that all multiplier and adder latencies
are equal. Therefore, by using constrained third-order
sections instead of regular second-order sections, the
maximal sample frequency of the overall filter can be
increased by two.
However, if we substitute a third-order section for a
second-order section in one of the allpass branches, it is
inevitable that an additional first-order section is introduced into the same branch. Therefore, a first-order
section must be inserted into the other branch to compensate for this. If we do like this for each of the second-order sections in the filter in Fig. 3, we end up with
the filter in Fig. 5. The overall transfer function of this
filter can be written as
m
H (z) = H (z)
∏
(1)
H api ( z )
(5)
i=0
where m = (N–3)/2, H(z) is the transfer function of the
( 1 ) ( z ) denotes the extra firstconventional filter, and H api
order allpass section i. The magnitude response is thus
retained in this approach, whereas the phase response is
changed.
γ3
γΝ−4
x(n)
y(n)
1/2
β1
βm
T
T
γ1
2T
γΝ−2
γ5
2T
2T
γ2
γ6
T
T
γΝ−1
T
Figure 5. Proposed lattice WDF with m = (N–3)/2.
Adaptor coefficients
The adaptor coefficients of the proposed filter in Fig. 5
can easily be computed from the coefficients of the
conventional filter in Fig. 3. To see how, we first study
the transfer functions of the first-, second-, and constrained third-order sections alone.
The transfer functions of the first-order, secondorder, and constrained third-order allpass sections in
Figs. 1, 2, and 4 are given by Eqs. (6), (7), and (8),
respectively.
– α0 z + 1
(1)
H ap ( z ) = ----------------------z –α0
(6)
– α 1 z 2 – α 2 ( 1 – α 1 )z + 1
(2)
H ap ( z ) = ------------------------------------------------------------z 2 – α 2 ( 1 – α 1 )z – α 1
(7)
– γ 1z3 + γ 1γ 3z2 – γ 3z + 1
(3)
H ap ( z ) = -------------------------------------------------------------z3 – γ 3z2 + γ 1γ 3z – γ 1
(8)
It is desired that the constrained third-order section and
the corresponding second-order section realize the
same complex conjugated pole pair. Therefore, we
rewrite Eq. (8) as
which the delay element has been replaced with two
cascaded delay elements (cf. bireciprocal lattice
WDFs [1]), which means that a higher fmax is obtained
automatically for the conventional filter.
 – α 1 z 2 – α 2 ( 1 – α 1 )z + 1  – β 0 z + 1
(3)
- ---------------------- (9)
H ap ( z ) =  ----------------------------------------------------------- z 2 – α 2 ( 1 – α 1 )z – α 1   z – β 0 
4 Some properties
where β0 is a function of α1 and α2. From Eqs. (8)
and (9) we get the following relation between α1, α2,
β0, γ1, and γ3:
γ 1 = – α 1 β 0,
γ 1 γ 3 = α2 β0 ( 1 – α1 ) – α1
γ 3 = α2 ( 1 – α1 ) + β0
(10)
By solving Eq. (10) we can express β0 as a function of
α1 and α2:
2
α 2 ( 1 – α 12 )
 α 2 ( 1 – α 12 )
β 0 = – --------------------------- ±  --------------------------- + 1
2α 1 
2α 1

(11)
+
–
Let the two roots in Eq. (11) be denoted as β 0 and β 0
(respectively associated with + and – in front of
). It
is evident that we have
+ –
β0 β0 = –1
(12)
i.e., the product of the two roots is always equal to –1.
( 3 ) ( z ) it must obviously be
Since β0 is a root of H ap
smaller than one in magnitude in order to make this fil+
ter stable. From Eq. (12), we see that if one of β 0 and
–
β 0 is larger than one, then the other is always smaller
+
than one. Thus, if β 0 ≠ 1 , then it is always possible to
(
3
)
make H ap ( z ) stable by choosing β0 as
+
+
β 0 = β 0 , β 0 < 1,
+
+
β0 = –1 ⁄ β0 , β0 > 1
(13)
Based on the results above it is clear that, given all
adaptor coefficients of the filter in Fig. 3, all adaptor
coefficients of the filter in Fig. 5 can be computed
according to
+
+
β i = β i , β i < 1,
γ 0 = α 0,
+
+
βi = –1 ⁄ βi , βi > 1
γ 2i + 1 = – α 2i + 1 β i
(14)
(15)
γ 2i + 2 = α 2i + 2 ( 1 – α 2i + 1 ) + β i
for i = 0, 1, ..., m = (N–3)/2, where
2
α 2i + 2 ( 1 – α 2i
+
+ 1)
β i = – --------------------------------------------2α 2i + 1
2
2
 α 2i + 2 ( 1 – α 2i
+ 1 )
+  --------------------------------------------- + 1
2α 2i + 1


+
–
(16)
From above it is clear that if β 0 = β 0 = 1 , then
( 3 ) ( z ) stable. It can easily
it is not possible to make H ap
be concluded that this occurs when the desired complex
conjugated poles are purely imaginary. In this case it is
however possible to use instead a first-order section in
The proposed approach can be used for all cases of
interest, but it is more attractive for some specifications
than for others. This will be discussed in this section.
Poles of first-order sections
In the proposed approach, a number of extra first-order
allpass sections are introduced. The real pole, p, of one
such section is determined by a desired complex conjugated pole pair at re ± jθ . Figure 6 plots p versus θ and
r. We see that as the complex-conjugated pole pair
approaches the imaginary axis (θ approaches π/2), the
real pole approaches +1 or –1, regardless the value of r.
Round-off noise and sensitivity
The poles of the extra first-order sections approach +1
or –1 as the bandwidth (θ) gets closer to π/2. One can
therefore expect that the round-off noise and sensitivity
of the new filters increase as the bandwidth approaches
π/2, because they tend to increase when the poles move
towards the unit circle. To confirm this, we have in
Fig. 7 plotted the total noise gains (after L2-norm scaling) at the output of the overall filter that emanate from
a constrained third-order section and its associated
first-order section, and the corresponding conventional
second-order section, respectively. Since the total noise
gain at the output of the overall filter is the sum of the
noise gains of each allpass section, the same conclusions can be drawn for the overall filter as for these allpass sections. As expected, we see that the noise
increases rapidly as the bandwidth approaches π/2. We
also see that for very narrow and wide bandwidths, the
new filters can in fact have lower noise. As the bandwidth increases, the noise increases to finally become
higher than for the conventional filter. However, when
the bandwidth is narrower than about 0.45π, or wider
than about 0.55π, the difference between the two cases
is still small. The sensitivity has not been investigated
thoroughly, but it seems that the same conclusions can
be drawn for the sensitivity as for the round-off noise.
Phase linearity
To assess the phase linearity, we first study the group
delay response of the extra first-order allpass sections.
The group delay response of one such section is plotted
in the top figure of Fig. 8 for some different pole locations (p). We see that the variation is largest near 0 or π,
and that the variation increases with p . In the remaining region the response is rather flat. If the original lattice WDF is a lowpass filter with poles in the right halfplane (narrow-band), the extra first-order allpass sections have poles close to p = –1. In this case, the group
delay of these sections has large variations near π, but
is rather flat for small values of ωT. The group delay
variation in the passband is therefore not much
affected, as illustrated in the middle figure of Fig. 8.
Similarly we conclude that, if the original filter is a
lowpass filter with poles in the left half-plane (wideband) then the group delay variation is more affected
since the large group delay variations of the first-order
sections then occur in the passband, as illustrated in the
bottom figure of Fig. 8. The same conclusions can also
be drawn for highpass filters. Hence, for lowpass and
highpass filters with bandwidths wider than π/2, there
may be problems with phase linearity deterioration.
100
0
Noise gain
One price to pay for the increased maximal sample frequency is a number of extra first-order allpass sections.
However, it is possible to further optimize the proposed
filter in order to improve the frequency response (in
some sense) compared with that of the corresponding
conventional filter. This is possible since we have an
additional number of degrees of freedom (adaptor coefficients). It seems though that, due the restrictions that
we have imposed, large improvements are not allowed.
Nevertheless, it may in some cases be worth to optimize the filter. To illustrate this, we start with a 5thorder Cauer filter with passband and stopband edges at
0.2π and 0.3π rad, and passband and stopband ripples
of –0.1 and –46.0 dB. We then optimize the resulting
proposed filter using the program minimax.m in MATLAB straightforwardly, aiming at minimizing the stopband ripple. For the optimized filter this ripple becomes
–48.1 dB, and thus a reduction of 2.1 dB is achieved.
The overall proposed filter requires 7 adaptors. If a conventional lattice WDF is to meet the same requirements, its order must be increased to 7, implying that
this filter also requires 7 adaptors. Thus, in this case,
the complexity of the proposed filter does not exceed
that of the conventional filter. The magnitude responses
of the 5th-order Cauer filter and the corresponding optimized proposed filter are shown in Fig. 9.
[2]
[3]
[4]
20
40
60
80
100
120
140
60
160
r = 0.7
40
20
0
20
40
60
80
100
120
140
160
40
r = 0.5
20
0
20
40
60
80
100
θ in degrees
120
140
160
Figure 7. Total noise gain of a constrained third-order section
and its associated extra first-order section (dashed line), and
the corresponding conventional second-order section (solid
line), as a function of θ and r of a desired complex conjugated
pole pair at re ± jθ .
20
p = 0.9
p = -0.9
10
p = 0.5
References
[1]
r = 0.9
200
Group delay in samples
5 Optimization
300
0
0
40
p = -0.5
20
40
60
80
100
120
140
160
180
20
40
60
80
100
120
140
160
180
20
40
60
80
100
120
140
160
180
20
0
0
40
20
A. Fettweis, “Wave digital filters: Theory and practice,” Proc.
IEEE, vol. 74, no. 2, pp. 270-327, Feb. 1986.
M. Renfors and Y. Neuvo, “The maximum sampling rate of digital filters under hardware speed constraints,” IEEE Trans. Circuits Syst., vol. CAS-28, no. 3, pp. 196-202, Mar. 1981.
A. P. Chandrakasan and R. W. Brodersen, Low Power Digital
CMOS Design. Norwell, MA: Kluwer, 1995.
L. Gazsi, “Explicit formulas for lattice wave digital filters,”
IEEE Trans. Circuits Syst., vol. CAS-32, no. 1, pp. 68-88, Jan.
1985.
0
0
ωT in degrees
Figure 8. Illustration of group delay responses. Top: group
delay response of a first-order allpass section with a pole at p.
Middle and bottom: group delay responses of the conventional (solid line) and proposed (dashed line) filters for a narrow-band (middle) and wideband (bottom) Cauer filter
10
r
1
0.95
0
0.80
-10
0.65
-20
0.8
0
0.6
0.50
0.35
0.20
0.05
p
0.2
0
-0.2
Magnitude in dB
-0.05
0.4
-0.1
-30
0
10
100
120
20
30
-40
-50
-0.4
-60
-0.6
-70
-0.8
-1
0
20
40
60
80
100
θ in degrees
120
140
160
180
Figure 6. Pole location (p) of the extra first-order allpass section as a function of θ and r of a desired complex conjugated
pole pair at re ± jθ .
-80
0
20
40
60
80
ωT in degrees
140
160
180
Figure 9. Magnitude response of a 5th-order Cauer filter
(solid line) and the corresponding optimized proposed filter
(dashed line).