Accepted for presentation in International Symposium on Communications and Information Technologies (ISCIT 2012), Gold
Coast, Australia, 2-5 October 2012.
An Improved Coefficient Decimation based
Reconfigurable Low Complexity FIR Channel Filter
for Cognitive radios
Abhishek Ambede#, Smitha K. G.+ and A. P. Vinod+
School of Computer Engineering,
Nanyang Technological University,
Nanyang Avenue, Singapore 639798.
E-mail: abhishek7#@e.ntu.edu.sg, {smitha+,asvinod+}@ntu.edu.sg.
Abstract — Multi-standard channel adaptation is a critical
function in cognitive radio handsets which involves the
transmission/reception of individual frequency channels of
multiple wireless communication standards at different intervals
of time. This needs dynamically reconfigurable, low complexity
and high speed digital channel filters. In this paper, we present a
reconfigurable finite impulse response (FIR) channel filter design
technique based on the combination of the conventional
coefficient decimation method (CDM) and a modified CDM. Our
method enhances the frequency response flexibility of the filter
and doubles the center frequency resolution when compared to
the conventional CDM. The proposed channel filter has a
significantly lower multiplication complexity and achieves
superior stopband and transition band characteristics when
compared to the channel filters based on the conventional CDM.
Design example shows that a 57.1% reduction in multiplication
complexity is achieved if the proposed channel filter is designed
instead of the conventional CDM based channel filter.
Keywords — Coefficient decimation method, low complexity,
reconfigurability, FIR channel filter.
I.
INTRODUCTION
Cognitive radio (CR) provides a means for efficient
utilization of the radio electromagnetic spectrum. Its basic
principle is to sense the spectral occupancy over a wide
frequency range so as to allow unlicensed users (called
secondary users) to have opportunistic access of the vacant
frequency bands (spectrum holes) allotted to licensed users
(called primary users) [1, 2]. The channelizer used in CRs
enables them to have a time-varying adaptability to
transmit/receive signals of multiple communication standards
in the detected spectrum holes (vacant frequency bands). In
battery-powered mobile CR handsets, these functions need to
be performed with low complexity and minimum
reconfiguration overhead mainly to ensure efficient utilization
of the constrained resources. The channel adaptation can be
accomplished by using a reconfigurable finite impulse response
(FIR) filter.
Numerous methods have been proposed in literature for
implementing reconfigurable FIR filters [3]. The multiply-andaccumulate (MAC) operation based programmable filter in [3]
is an inefficient implementation resulting due to the delay and
power consumption issues of the coefficient multipliers
involved in MAC operations. In [4], a digit based
reconfigurable FIR filter implementation architecture is
proposed. In this technique, the tap (coefficient) number and
the number of non-zero digits in each tap are arbitrarily
assigned and the aim is to reduce the filter complexity by
reducing coefficient precision. A canonic signed digit based
(CSD) based reconfigurable FIR filter is proposed in [5], which
is characterized by a high speed operation. The techniques
proposed in [4, 5] have a high utilization of hardware resources
and thus a high power consumption, which makes them
infeasible for applications like cognitive radios for whom these
parameters are critical. A reconfigurable multiplier block
(ReMB) that employs graph dependence (GD) algorithms was
introduced in [6]. A time-multiplexed multiple constant
multiplication (MMCM) approach is proposed in [7], which
takes coefficients as constants and uses GD algorithms to
reduce redundancy. The methods proposed in [6, 7] have a low
speed of operation due to the sequential nature of the GD
algorithms used and the resulting longer critical path lengths
observed.
The techniques proposed in [3-7] implement reconfigurable
filters with variable inputs along with variable coefficients. The
resulting complexity in these methods can be reduced if the
filter coefficients are kept fixed and variable frequency
responses are obtained using them, corresponding to the
different wireless communication standards. In [8], a
coefficient decimation method (CDM) is proposed to obtain
low complexity reconfigurable FIR filters. The CDM is able to
generate variable frequency responses by operating upon fixed
filter coefficients by employing two coefficient decimation
operations, one to vary the passband width of the prototype
filter (termed as CDM-II) and another to generate multi-band
frequency responses (termed as CDM-I). In [9], a
reconfigurable non-uniform channel filter is proposed based on
CDM-II, interpolation and frequency response masking
techniques. In [10], successive CDM-II operations are
performed to obtain a reconfigurable FIR filter that is used in a
spectrum sensing scheme for mobile cognitive radio terminals.
The reconfigurable filter proposed in [10] does not employ
frequency response masking filters and interpolation techniques
and thus has a lower design overhead than the channel filter
proposed in [9]. The CDM based channel filters are low
complexity alternatives to the other channelization techniques
with the primary advantage of having the ability to extract
multi-standard frequency channels.
We recently proposed a modified coefficient decimation
method [11] to obtain reconfigurable FIR filters with enhanced
frequency response flexibility and twice center frequency
resolution when compared to the conventional CDM. Our
method provides a higher degree of reconfigurability to obtain
FIR filters when compared with the CDM. The FIR filters
obtained using our method show higher stopband attenuation
when compared with those obtained using conventional CDM,
and have a lower complexity due to the requirement of a lower
order modal filter in the modified CDM.
In this paper, we propose a new channel filter design
method that is based on the combination of the modified CDM
and conventional CDM. We show that the proposed channel
filter has a significantly lower complexity when compared to
other CDM based channel filters [9, 10]. When both are
designed for the same specifications, the proposed channel
filter shows superior transition band and stopband
characteristics when compared to the CDM based channel
filters.
The rest of the paper is organized as follows: Section II
presents the proposed channel filter with its mathematical
formulation. Hardware implementation architecture for the
proposed channel filter is also presented. Section III presents a
design example along with the comparison of the proposed
channel filter with the conventional CDM-II based channel
filter. The conclusions are given in Section IV.
II.
PROPOSED CHANNEL FILTER
A. Improved Coefficient Decimation Method
In the CDM [8], the coefficients of a lowpass FIR filter
(termed as the modal filter) are decimated by M, i.e., every Mth
coefficient is retained and the others replaced by zeros, to
obtain a FIR filter with a multi-band frequency response. The
frequency response of the resulting filter has bands with center
frequencies at 2πk/M, where k is an integer ranging from 0 to
(M-1). If H (e jω ) denotes the Fourier transform of the modal
filter coefficients, then the Fourier transform of the modified
coefficients is given by
H ' (e
jω
1
)=
M
M −1
∑ H (e
k =0
2πk ⎞
⎛
j ⎜ ω−
⎟
M ⎠
⎝
)
(1)
This operation is termed as CDM-I. After performing
CDM-I by decimation factor M, if all the retained coefficients
are grouped together by discarding the zero coefficients in
between, a lowpass frequency response with its passband and
transition band widths M times that of the modal filter is
obtained. This operation is called as CDM-II.
In the new modified coefficient decimation operation
proposed by us in [11], if the modal filter is decimated by M,
every Mth coefficient is retained and the sign of every alternate
retained coefficient is reversed. All other coefficients are
replaced by zeros. As a result of this operation, an FIR filter
with a multi-band frequency response is obtained with center
frequencies at (2k+1) π/M, where k is an integer ranging from 0
to (M-1). If H (e jω ) denotes the Fourier transform of the
modal filter coefficients, then the Fourier transform of the
modified coefficients is given by
H'(e jω ) =
1
M
M −1
∑
H(e
π( 2 k +1 ) ⎞
⎛
j ⎜ ω−
⎟
M
⎠
⎝
)
(2)
k =0
We name this operation as modified coefficient decimation
method I (MCDM-I). After performing MCDM-I by
decimation factor M, if all the retained coefficients are grouped
together by discarding the zero coefficients in between, a
highpass filter response is obtained with its passband and
transition band widths M times that of the modal filter. We
name this operation as modified coefficient decimation method
II (MCDM-II).
The filters obtained by performing CDM-II and MCDM-II
operations on the same modal filter for identical M values are
mathematically related as
(3)
h''(n) = ( − 1 ) n h'(n)
for n = 0,1,2,..., N
where h'(n) represents the coefficients of the filter obtained
after CDM-II operation, h''(n) represents the coefficients of the
filter obtained after MCDM-II operation and N is the filter
order. From (3), it can be noted that the highpass filter obtained
after MCDM-II is the inverse of the lowpass filter obtained
after CDM-II.
In the rest of the paper, we term the combination of CDM-I
and MCDM-I operations as improved coefficient decimation
method I (ICDM-I), and the combination of CDM-II and
MCDM-II operations as improved coefficient decimation
method II (ICDM-II) respectively.
From the multiband frequency responses obtained after
ICDM-I operations, individual frequency bands with identical
BWs can be isolated by the use of frequency response masking
filters and spectral subtraction [8, 11]. However, the use of
masking filters increases the complexity involved in
implementing these methods. We obtain lowpass and highpass
frequency responses with varying passband widths after
performing ICDM-II operations. Frequency bands with
identical (uniform) as well as non-identical (non-uniform) BWs
can be obtained from these lowpass and highpass output
frequency responses by spectral subtraction / addition, without
employing masking filters.
The ICDM-II operations are further explained with the help
of an illustrative example.
If f p and f s are the desired passband and stopband edge
frequencies (normalized in the range 0 to 1, with 1
corresponding to half of sampling frequency), δ p and δs are
the desired passband and stopband peak ripple specifications,
then the order of the desired FIR filter (N) can be obtained
using the formula [12],
4 log10 (10δ p δ s )
(4)
N =−
−1
3( f s − f p )
Fig. 1(a) shows the frequency response of the modal filter
designed using (4) such that f p = 0.12, f s = 0.125, δ p =
0.1dB, δ s =-73dB and N = 1680 from (4). Figures 1(b) and 1(c)
show the frequency responses obtained using CDM-II for M=2
and M=3 respectively. Fig. 1(d) shows the frequency response
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Figure 1(c). Frequency response of filter obtained using CDM-II with M=3.
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Figure 1(d). Frequency response of filter obtained using MCDM-II with M=1.
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Figure 1(e). Frequency response of filter obtained using MCDM-II with M=4.
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Figure 1(f). Frequency response of filter obtained by spectral subtraction after
using CDM-II with M=1 and M=2.
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Magnitude (dB)
obtained from the modal filter for M=1, using MCDM-II. As
discussed above using (3), this response is the inverse of the
modal filter’s frequency response. Similarly, CDM-II with
M=1 gives the modal filter itself with the frequency response as
shown in Fig. 1(a). Fig. 1(e) shows the frequency response for
M=4 obtained using MCDM-II operation.
Figures 1(f) to 1(h) show a few of the frequency bands that
can be obtained by spectral subtraction of frequency responses
shown in figures 1(a) to 1(e). In general, uniform frequency
bands having their BWs P times that of the modal filter can be
obtained by the spectral subtraction of the frequency responses
obtained after performing ICDM-II operations successively,
with the M values separated by intervals of P (where P is an
integer). For e.g., if uniform bands with their BWs twice that of
the modal filter are to be obtained, ICDM-II operations are
performed successively with M = 2, 4, 6, 8,…, and the resulting
frequency responses are spectrally subtracted from each other.
Non-uniform frequency bands can be obtained by performing
ICDM-II operations with M values selected according to the
desired BWs and center frequency locations, and then
spectrally subtracting/adding the resulting frequency responses
based on the requirement.
In ICDM-II, the order of the resulting filters after
coefficient decimation decreases as the decimation factor value
is increased, due to elimination of filter coefficients that are not
retained. As the group delays of filters with different filter
orders are unequal, the corresponding group delays of the
concerned filters are to be compensated before performing
spectral subtraction. This is done by adding k delay elements
where k is the difference between the group delays of the filters
whose frequency responses are subtracted. Note that group
delay compensation is a requirement in conventional CDM also
whenever spectral subtraction needs to be performed. The
outputs of these filters then correspond to the same time
instances and thus can be spectrally subtracted. Also, note that
for performing this compensation, the group delay values of the
corresponding filters have to be integer values. Thus, to ensure
that the frequency responses obtained after performing ICDMII operations can be spectrally subtracted, the order of the
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Figure 1(g). Frequency response of filter obtained by spectral subtraction after
using CDM-II with M=2 and M=3.
Normalized Frequency (×π rad/sample)
Figure 1(a). Frequency response of the modal filter.
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Figure 1(b). Frequency response of filter obtained using CDM-II with M=2.
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1
Figure 1(h). Frequency response of filter obtained by spectral subtraction after
using MCDM-II with M=1 and M=4.
Figure 2. ICDM-II based reconfigurable channel filter architecture.
modal filter should be chosen as a multiple of the least
common multiple (LCM) of all the decimation factor values
involved. This condition can be represented as
N = m × [ LCM {M 1 , M 2 , M 3 ,...}]
(5)
where m = 2, 3, 4,… and {M 1 , M 2 , M 3 ...} represent the various
decimation factor values used.
We use the ICDM-II operations to obtain the proposed
reconfigurable channel filter. The proposed channel filter is
henceforth termed as ICDM-II based channel filter in this
paper.
B. Modal Filter Overdesign
From Figures 1 and 2, it can be noted that the stopband
attenuation (SA) of the filters obtained after coefficient
decimation deteriorates as the value of M increases. This is an
inherent drawback of CDM-II and is present in ICDM-II as
well. The deterioration in SA can be mathematically given by
δ s (final)
δ s (modal) =
(6)
M
where δ s (modal) is the SA of the modal filter and δ s (final) is the
SA of the filter obtained after coefficient decimation by M [9].
This SA deterioration problem can be addressed by over
designing the modal filter. If the desired SA of the resulting
coefficient decimated filter is δ s (final) , then the modal filter has
to be designed such that its SA is greater than or equal to
δs (final) / M .
From (4) and (6), we derive that if a filter is to be
coefficient decimated by M and the SA of the resulting filter is
to be kept within the desired value δ s , the minimum order of
the overdesigned modal filter is given by
N =−
4 log10 [10δ p (δ s / M )]
3( f s − f p )
−1
By rearranging the terms, we obtain
(7)
⎡ 4 log10 (10δ p δ s ) ⎤ 4 log10 M
N = ⎢−
− 1⎥ +
3( f s − f p )
⎥⎦ 3( f s − f p )
⎣⎢
(8)
The second term on the right hand side of (8) is the increase
in the order of the overdesigned modal filter required to
compensate the deterioration in SA after coefficient decimation
by M.
C. ICDM-II based channel filter architecture
Fig. 2 shows the hardware implementation architecture for
the proposed ICDM-II based channel filter. It consists of 2:1
multiplexers (mux) that are used to select the filter coefficients
according to the different M values being used for the ICDMII operations being performed. The decimation selector logic
drives the select lines (S) of the 2:1 multiplexers. If S=1 for a
multiplexer corresponding to a particular filter coefficient, that
coefficient is retained and when S=0, that coefficient is
bypassed. The adder/subtractor (add/sub) blocks are used to
perform the sign reversal of the alternate retained coefficients
in the ICDM-II operations. This is valid because
(−h) ⊗ x = −(h ⊗ x) , where h is the filter coefficient, x is
the input and ⊗ represents the convolution operation. The
select signal (sel) of the add/sub blocks is to be set to 1 (for
subtraction) or 0 (for addition) according to the desired
operation. This function is also performed by the decimation
selector which drives the add/sub blocks accordingly. From
(1) and (2), we observe that the resulting outputs after the
ICDM-II operations need to be scaled by M to obtain the
original magnitude response. This can be done by multiplying
the output of the channel filter by M as shown in Fig. 2 using
the associative property of convolution [13], i.e.,
( M × h) ⊗ x = M × ( h ⊗ x ) .
D. Design Procedure
The steps involved in the realization of the proposed
ICDM-II based channel filter are given below.
to M=5.
The frequency responses shown in Fig. 3 are spectrally
subtracted from each other to obtain uniform frequency bands
for extracting 10 non-overlapping Bluetooth channels
occurring at different time instances in the input signal. Zigbee
channels having any of the frequency locations {(0MHz4MHz), (1MHz-5MHz), (2MHz-6MHz), (3MHz-7MHz),
(4MHz-8MHz), (5MHz-9MHz), (6MHz-10MHz)}, are
extracted by performing spectral subtraction/addition of the
ICDM-II output frequency responses for different
combinations of M values. For example, output frequency
responses with M=1 and M=5 can be used for extracting two
non-simultaneously occurring Zigbee channels located at
1MHz-5MHz and 6MHz-10MHz in the input signal.
The designed channel filter is functionally verified by
inputting a signal containing Bluetooth and Zigbee channels as
shown in Fig. 4. Note that our assumption is that Zigbee and
Bluetooth channels do not exist simultaneously; the channels of
both these standards are shown together in Fig. 4 due to space
constraints. Assume that during the time interval t1-t2, when the
receiver is operating in Zigbee mode, one Zigbee channel is
located between 0MHz-4MHz and at another time
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Step-1: Fix the passband width of the lowpass modal filter
with respect to the greatest common divisor (GCD) of
the channel BWs of the different communication
standards involved.
Step-2: Identify the decimation factor (M) values required to
obtain the frequency bands corresponding to the
different standards and their channel distribution in a
wideband frequency range, using ICDM-II
operations. Compute the LCM of all the identified M
values. Let the maximum decimation factor identified
be M max .
Step-3: Fix the transition BW (TBW) of the modal filter to be
1 / M max times that of the minimum TBW
specification amongst those of the different standards.
Step-4: Using (5) and (8), compute the order (N) of the modal
filter such that it is a multiple of the LCM calculated
in Step-2. This ensures that the group delays of all the
decimated filters are integer values. Also, to meet the
desired SA specifications, compute the value of N
with respect to M max .
Step-5: Design the modal filter for the specifications obtained
in steps 1, 3, and 4.
Step-6: Perform ICDM-II operations on the modal filter using
the M values identified in Step-2 and obtain the
corresponding frequency responses.
Step-7: Obtain desired frequency bands corresponding to the
different standards by spectral subtraction/addition of
the frequency responses obtained in Step-6.
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III.
RESULTS AND ANALYSIS
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0
f s = 0.1, δ p = 0.1dB, δ s = -65dB. Using ICDM-II operations,
the decimation factors required to obtain different frequency
bands to extract frequency channels of the two standards are
{1, 2, 3, 4, 5}. The maximum decimation factor involved is
thus 5 and the LCM of the identified M values is 60. Using (5)
and (8), the order of the modal filter is calculated as 2160. Fig.
3 shows the frequency responses that are obtained after
performing ICDM-II operations on the modal filter with M=1
1
2
3
4
5
6
7
8
9
10
Frequency (MHz)
Figure 3. ICDM-II frequency responses for M=1 to M=4.
100
4MHz bandwidth Zigbee channel
1MHz Bandwidth Bluetooth channel
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Figure 4. Input signal spectrum with Zigbee and Bluetooth Channels.
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A. Design Example
In this section, reconfigurability of the proposed channel
filter to extract frequency channels of multiple wireless
communication standards is illustrated with the help of a design
example.
Consider a Bluetooth/ Zigbee dual standard scenario where
frequency bands corresponding to these two standards are to be
extracted. The channel BWs of Bluetooth and Zigbee standards
are 1MHz and 4 MHz respectively. The desired passband and
stopband peak ripple specifications are 0.1dB and -65dB for
Bluetooth channels, 0.1dB and -50dB for Zigbee channels. The
TBW specifications are 200KHz for Bluetooth and 300 KHz
for the Zigbee. The sampling frequency is 20MHz.
The passband width of the modal filter is chosen as 0.1 for
the normalized frequency range 0 to 1, where 1 corresponds to
half of sampling frequency. Considering the worst case (most
stringent) SA and TBW specifications from the ones desired,
the specifications of the modal filter are chosen as f p = 0.096,
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Zigbee
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Frequency (MHz)
Figure 5. ICDM-II frequency responses required for the Zigbee/Bluetooth
channel distribution.
interval t2-t3, when the receiver is operating in Bluetooth mode,
one Bluetooth channel is located between 9MHz-10MHz as
shown in Fig. 4. The ICDM-II output responses that are used to
obtain the frequency bands corresponding to this channel
distribution are shown in Fig. 5. The frequency band used to
extract the Zigbee channel is the CDM-II output response for
M=4. The Bluetooth channel is extracted using the frequency
band obtained as the output frequency response using MCDMII for M=1.
B. Complexity analysis
The complexity of a channel filter implementation is
mainly dependent on the order of the filter used in it and the
corresponding number of coefficient multiplications required.
For the ICDM-II based channel filter implementation in the
design example in Section III.A, the order of the modal filter
used is 2160. Exploiting its symmetric property, the total
number of coefficient multiplications required is
⎡2161 / 2⎤ = 1081 . This modal filter is reconfigured to extract
channels of the different communication standards and thus,
only the modal filter’s implementation accounts for the total
multiplication complexity involved.
In contrast to just 5 decimation factor values (M=1 to M=5)
required in the proposed ICDM-II based channel filter design
for the Bluetooth/Zigbee design example, decimation factors
from M=1 to M=10 are required in the conventional CDM-II
based implementation. The maximum decimation factor
involved is 10 and the corresponding LCM of all the required
M values is 2520. Using (5) and (8), the order of the modal
filter for the same SA and TBW specifications is calculated as
5040. Using transposed direct form FIR filter structure for the
modal filter implementation, the total number of coefficient
multiplications involved in the CDM-II based approach is
⎡5041 / 2⎤ = 2521 . Thus, a 57.12% reduction in multiplication
complexity is achieved using our ICDM-II based channel filter
when compared with a CDM-II based channel filter designed
for the same frequency response specifications.
The maximum decimation factor involved in the ICDM-II
based channel filter design is half of that required in the CDMII based approach. Thus, for a given multi-standard scenario, if
the proposed ICDM-II based channel filter and the CDM-II
based channel filter are designed using the same modal filter ,
the worst SA and maximum TBW values observed in ICDMII based design are half of those observed in CDM-II based
approach.
IV.
When compared with a CDM-II based channel filter designed
for the same specifications, the ICDM-II based channel filter
has a significant reduction in multiplication complexity. Also,
if the same modal filter is used to obtain CDM-II based and
ICDM-II based channel filters, the worst case stopband
attenuation and transition bandwidth values observed in the
ICDM-II based design are half of those observed in CDM-II
based approach. Design example showed that the proposed
channel filter achieves a 57.12% reduction in multiplication
complexity when compared with the CDM-II based channel
filter designed for the same specifications.
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CONCLUSION
In this paper, we have proposed a reconfigurable channel
filter based on an improved coefficient decimation method. The
proposed ICDM-II based channel filter is a low complexity
alternative to the conventional CDM based channel filters.
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