Adjustable Fractional-Delay FIR Filters Using the ... - IEEE Xplore

Adjustable Fractional-Delay FIR Filters Using the
Farrow Structure and Multirate Techniques
Håkan Johansson, Oscar Gustafsson, Kenny Johansson, and Lars Wanhammar
Division of Electronics Systems, Department of Electrical Engineering, SE-581 83 Linköping University, Sweden
Email: hakanj@isy.liu.se, oscarg@isy.liu.se, kennyj@isy.liu.se, larsw@isy.liu.se
Abstract — The Farrow structure can be used for efficient
realization of adjustable fractional-delay finite-length impulse
response (FIR) filters, but, nevertheless, its implementation
complexity grows rapidly as the bandwidth approaches the full
bandwidth. To reduce the complexity, a multirate approach
can be used. In this approach, the input signal is first interpolated
by a factor of two via the use of a fixed half-band linearphase
FIR filter. Then, the actual fractional-delay filtering
takes place. Finally, the so generated signal is downsampled to
retain the original input/output sampling rate. In this way, the
bandwidth of the fractional-delay filter used is halved compared
to the overall bandwidth. Because the complexity of halfband
linear-phase FIR filter interpolators is low, the overall
complexity can be reduced. In this paper, we present more
implementation details, design trade-offs, and comparisons
when the filters are implemented using multiple constant multiplication
techniques, which realize a number of constant multiplications
with a minimum number of adders and
subtracters.
x(n)
E L (z)
d
E 2 (z)
I. INTRODUCTION
This paper considers adjustable fractional delay (FD) filters
which are required in several different applications [1]–[4].
An efficient structure for adjustable FD filtering is the so
called Farrow structure [5]–[7] which makes use of a number
of fixed FIR subfilters and only one variable parameter
(d) as seen in Fig. 1. The implementation complexity is
therefore much lower for Farrow-based filtering methods
than for methods based on either on-line design or storage of
a (huge) amount of different impulse responses.
However, even if the complexity of the Farrow structure
is relatively low, it grows rapidly as the bandwidth of the FD
filter approaches the full bandwidth π [6], [7]. To reduce the
complexity, a multirate approach was proposed in [8]–[11].
In this approach, the input signal is first interpolated by a
factor of two via the use of a half-band linear-phase FIR filter.
Then, the actual FD filtering takes place. Finally, the so
generated signal is downsampled to retain the original input/
output sampling rate. In this way, the bandwidth of the FD
filter used is halved compared to the overall bandwidth.
Because the complexity of half-band linear-phase FIR filter
interpolators is low, the overall complexity can be reduced.
For a bandwidth of 0.9π, savings of some 30-50% were
reported in [10].
In this paper, we present more implementation details,
design trade-offs, and comparisons when the filters are
implemented using multiple constant multiplication (MCM)
techniques, which realize a number of constant multiplications
with a minimum number of adders and subtracters.
Following this introduction, Section II recapitulates the
basics of Farrow-based FD FIR filters and half-band linearphase
FIR filters. Sections III and IV describe the multirate
approach and the principles of MCM techniques. Section V
presents comparisons between the original Farrow structure
and the new one when the filters are implemented using
MCM techniques. Finally, Section VI concludes the paper.
II. FARROW-BASED ADJUSTABLE FD FILTERS AND
HALF-BAND FILTERS
A. Farrow-Based FD FIR Filters
Let the desired frequency response H des ( e jωT ) of an
adjustable FD filter be
H des ( e jωT ) = e – jD ( + d)ωT
(1)
= e – jDωT e – jdωT , ωT ≤ ω c T < π
where D and d are fixed and adjustable real-valued constants,
respectively. It is assumed here that D is either an
integer, or an integer plus a half, whereas d takes on values
in the interval [– 1⁄
2,
1⁄
2]
. In this way, a whole sampling
interval is covered by d, and the fractional delay equals d
(d+0.5) when D is and integer (an integer plus a half).
The ideal filter response in (1) can be approximated
using the Farrow structure depicted in Fig. 1. The transfer
function is then expressed as
where E k ( z)
are FIR subfilters. Further, it is possible and
efficient (due to the coefficient symmetry) to impose linearphase
constraints on the subfilters [5]–[7]. Consequently, we
assume that each E k ( z)
is an N G th-order linear-phase FIR
filter with either a symmetric (for k even) or an anti-symmetric
(for k odd) impulse response. Furthermore, it has
been demonstrated that it is beneficial, in terms of implementation
complexity, to make use of subfilters of unequal
orders instead of equal orders [7]. To keep the notation simple,
this is here taken care of by assuming that N =
max{N k }, N k denoting the subfilter order, and introducing
d
E 1 (z)
Figure 1. Farrow structure.
L
∑
Gz ( ) = d k E k ( z)
k = 0
d
E 0 (z)
y(n)
(2)
1-4244-0387-1/06/$20.00 c○2006 IEEE
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additional delays to make all branches have the same delay.
The filter with a transfer function in the form of (2) can
approximate the ideal response (1) as close as desired by
choosing L and designing the filter appropriately [6], [7].
x(n)
2 H(z) 2
y(n)
B. Half-Band FIR Filters
For a half-band FIR filter F(z) every second impulse
response value is zero except for the centre tap that equals
1/2, i.e.,
fn ( )
=
⎧
⎪
⎨
⎪
⎩
1 ⁄ 2, n = N F ⁄ 2 = K
0, n = 13… , , , N F – 1,
n ≠ N F ⁄ 2
. (3)
where K is half the filter order and an odd number. This
means that the transfer function can be written as [12], [13]
F 0 ( z 2 ) + z – K
Fz ( ) = ------------------------------ . (4)
2
It is noted that (4) is a special case of a general polyphase
decomposition [13] Fz ( ) = F 0 ( z 2 ) + z – 1 F 1 ( z 2 ) for
F 1 ( z) = z– ( K – 1) ⁄ 2 ⁄ 2 . Furthermore, F 0 ( z)
is a Type II
linear-phase FIR filter of odd order K [12], which implies
that the overall filter Fz ( ) becomes a Type I linear-phase
FIR filter of even order N F = 2K.
III. MULTIRATE APPROACH
In the multirate approach for realizing FD filters, seen in
Fig. 2, the input signal is first upsampled by a factor of two,
then filtered through a fractional-delay filter Hz ( ) , and
finally downsampled by a factor of two. The desired overall
fractional delay D+d is obtained by letting the delay of the
FD filter between the upsampler and downsampler be
2(D+d). This is because the filter works at twice the input/
output rate.
However, in the final realization, all filtering takes place
at the lower sampling rate. In addition, although multirate
techniques are utilized, the arrangement in Fig. 2 is in fact a
linear and time-invariant system with the transfer function
being the 0th polyphase component of H(z) [13], i.e., H 0 (z)
in the polyphase representation
Hz ( ) = H 0 ( z 2 ) + z – 1 H 1 ( z 2 ) . (5)
This is because the overall transfer function of the structure
in Fig. 2 can be written as 0.5[ Hz ( 1 ⁄ 2 ) + H( – z 1 ⁄ 2 )]
which
equals H 0 (z). Here, H(z) is a cascade 1 of the FD and HB filters
considered in Section II, i.e.,
Hz ( ) = Fz ( )Gz ( )
(6)
which gives us the polyphase component H 0 (z) as
H 0 ( z) = F 0 ( z)G 0 ( z) + z – 1 F 1 ( z)G 1 ( z)
. (7)
The delay of H 0 (z) is D+d = (D H +d H )/2 where D H +d H is the
delay of H(z). The fixed part D H is given by D H = N G /2+K
1. We may also add an additional (positive or negative) delay in order to
avoid half-integer delays [10], [11] but this is not considered in this paper.
x(n)
H 0 (z)
Figure 2. Principle of multirate (two-rate) FD filtering approach.
Figure 3. Multirate-based FD filter structure.
With G(z) and F(z) in the form of (2) and (4), Structure 1 can
be implemented as shown in Fig. 3 where E k0 ( z)
and
E k1 ( z) are the polyphase components of E k ( z)
.
The advantage of this structure is that it is fixed. This
makes it suitable for applications where the fractional delay
d changes from sample to sample (or at least frequently). The
(minor) disadvantage is that the variable fractional delay at
the higher rate, i.e., d H , must cover the interval [– 1,
1]
when
the desired overall variable fractional delay d H covers
[– 1⁄
2,
1 ⁄ 2]
. The wider this range is, the higher will the filter
order of the fractional delay filter G(z) be. But the overall
complexity can despite of this fact be lower than that of a
regular Farrow-based FD filter [10], [11].
Is is worth pointing out that the complexity can be
reduced even further by also utilizing the two optional structures
obtained by adding a positive and negative delay to
H(z) in (6) [10], [11]. By switching between the three different
structures appropriately, one can make the interval for d H
become [– 1⁄
2,
1 ⁄ 2]
, which reduces the complexity of the
Farrow-based FD filter. The disadvantage is that, due to the
switching between the two different structures, the implementation
becomes more complex if one wants to avoid transient
problems. This case is therefore more attractive for
applications where the fractional delay d does not change too
often. Due to a lack of space, this alternative will not be discussed
further in this paper.
IV. MULTIPLE CONSTANT MULTIPLICATION TECHNIQUES
For dedicated hardware implementations, one can take
advantage of multiple constant multiplication (MCM) techniques
to reduce the implementation cost. Multiplications by
constant coefficients can be performed using adders, subtracters,
and shifts. As adders and subtracters have approximately
the same implementation complexity we will
henceforth refer to both as adders. Efficient realization of
constant multiplications is an active research area and much
effort has been laid on the case where one input data is multiplied
by several constant coefficients. This problem has
mainly been motivated by single-rate FIR filters, where for a
transposed direct form FIR filter the input is multiplied by
several coefficients [14]–[18]. The resulting implementation
y(n)
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Figure 4. Realization of Farrow filter using transposed direct form
subfilters resulting in a multiplier block and several sets of registers.
of several multiplications is denoted multiplier block [14].
Work has also been done for sampling rate change with
an integer factor in [19], where it was shown that FIR filters
in parallel can be implemented either using one multiplier
block or by using a constant matrix multiplication block
[21]–[23], with the first approach requiring more delay elements
than the latter. As a Farrow filter also is composed of
several FIR filters in parallel we have the same implementation
alternatives here, not only the single multiplier block
case as reported in [20]. In Fig. 4 the approach proposed in
[20] is shown. This approach typically requires few additions
for the multiplier block. However, a separate set of registers
is required for each subfilter and the number of structural
adders is high. On the other hand, in Fig. 5 an approach
based on the observation in [19] is shown. Here, only one set
of registers is required and the structural adders of the subfilters
are merged into the matrix-vector multiplication.
The Farrow filter part of the proposed filter structure can
be implemented similarly to what is shown in Figs. 4 or 5.
For transposed direct form subfilters, as in Fig. 4, the corresponding
structure would have two inputs, and, hence, result
in a constant matrix multiplication. Using direct form subfilters,
as in Fig. 5, requires two sets of registers, one for each
input. For the half-band filter it is convenient to use a direct
form subfilter as the delayed input values are easily obtained
from the registers. Now, note that the input to the lower
branch subfilters in Fig. 3 is just delayed version of the input,
which are available from the first stage filter. Therefore, it is
possible to use the registers of the half-band filter as registers
for direct form subfilters. The resulting structure is illustrated
in Fig. 6.
Naturally, it is also possible to use a transposed direct
form half-band filter and/or transposed direct form subfilters
Figure 5. Realization of Farrow filter using direct form subfilters resulting
in a constant matrix multiplication and a single set of registers.
in the Farrow filter part. From a complexity point of view, a
transposed direct form half-band filter will have the same
number of adders and registers. However, it will not be possible
to share registers as shown in Fig. 6.
V. EXAMPLE AND COMPARISONS
We assume a specification that was considered in [7], [10],
[11] where the bandwidth is 0.9π and the modulus of the
complex error should be below 0.0042. To meet this specification,
the Farrow structure, with subfilters jointly optimized
as outlined in [7], requires 45 fixed multipliers, 88 fixed
adders, and 5 variable multipliers. The structure in Fig. 3
requires 30 fixed multipliers, 53 fixed adders, and 5 variable
multipliers. Thus, in terms of number of multiplications and
additions, the multirate approach is superior.
Now, to refine the comparison when constant multiplication
techniques are applied we must naturally quantize the
filter coefficients. For a relative comparison, one can use
simple rounding. We found that the original Farrow structure
requires 11 bits to fulfill the requirements whereas the structure
in Fig. 3 requires 13 fractional bits. The slightly larger
number of bits for the multirate approach is explained by the
fact that a cascaded filter must meet the requirements which
leads to a somewhat more stringent requirement on the subfilters,
at least when simple rounding is used.
The number of adders and registers may serve as a rough
estimation of the chip area required. However, for power
consumption the number of cascaded adders are also important
as the amount of glitches increases with the number of
cascaded adders [24]. Obviously, the number of cascaded
adders will also affect the sample rate. Furthermore, the
wordlength required for adders and registers vary depending
on where they are positioned [25].
For the structure in Fig. 4 a total of 33 adders are required
for the multiplier block using the RAG-n algorithm [14].
This is an optimal result since there are 33 different (odd)
coefficients. Furthermore, 80 structural adders and 118 registers
2 are required for the FIR subfilters. As for all structures
five general multipliers and four additional adders are
required.
Instead, using the proposed structure in Fig. 5 the constant
matrix multiplication can be realized using 96 adders
2. These numbers are different since the subfilters are of different length
which increases the number of registers but not the structural adders.
Figure 6. Realization of the proposed filter structure using direct
form subfilters resulting in a sum-of-products block and a constant
matrix multiplication.
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using the algorithm in [22]. In addition, 26 structural adders
are required. One observation is that separating the symmetric
and anti-symmetric subfilters may reduce the complexity
as some algorithms work better for fewer columns. If this is
utilized, the number of adders can be reduced to 107 by
applying the algorithm in [21] to the resulting two matrices
and adding and subtracting the results. The number of registers
is now decreased to 30, whereas the number of general
multipliers and additional adders is constant.
Using the multirate-based structure in Fig. 6 the halfband
filter requires 43 adders and 29 registers. The sum of
products are realized by computing the corresponding multiplier
block with RAG-n and transposing it. For the constant
matrix multiplication 38 adders are required. This number is
not reduced by separating the symmetric and anti-symmetric
subfilters. A total of six additional registers are required as
well as the five general multipliers and four additional
adders.
The results are summarized in Table 1. From this it is
obvious that the proposed multirate realization has the lowest
complexity, for most implementation technologies five registers
will be less complex than 26 adders. Furthermore,
whereas the use of transposed direct form subfilters for the
Farrow filter, as proposed in [20], reduces the number of
adders related to the multiplication, it is in this case still
more efficient to use direct form subfilters. Also, the number
of registers decreases dramatically.
VI. CONCLUSIONS
This paper has considered a multirate approach for adjustable
FD filtering. In this approach, the signal is first interpolated
by a factor of two before the actual fractional filtering
takes place via the Farrow structure. Finally, downsampling
by two takes place to retain the original sampling rate. The
paper presented comparisons between the original Farrow
structure and the new one when the filters are implemented
using MCM techniques. The results indicated that the multirate
approach can still be advantageous but also that the two
options are closer to each other regarding the implementation
complexity compared to the case where the multipliers are
implemented separately.
Finally, we also point out that there exist alternatives to
the multirate approach to reduce the complexity of adjustable
fractional delay filters [26], [27]. It is however beyond the
scope of this paper to make a detailed comparison between
these approaches. It is a topic for future research.
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