Compensated CIC-Cosine Decimation Filter
8
ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL.4, NO.1 MAY 2010
Gordana Jovanovic Dolecek and Fernando Javier Trejo Torres
Compensated
CIC-Cosine
Decimation
Institute INAOE,
Department of Electronics,
Puebla, Mexico Filter
Gordana Jovanovic Dolecek and Fernando Javier Trejo Torres, Non-members
structure of (1) can be used [2], [3]. More details on a
of the performances of the recursive and nonThis
paper presents efficient modification of the CIC-datacomparison
rate resulting
in a larger chip
higher
ABSTRACT
recursive
implementations
are area
givenand
in [2].
Additionally,
cosine decimation filter. The second order compensatorpower consumption especially when
the
decimation
This paper presents efficient modification of the
the
magnitude
response
of
the
CIC
filter
has
filter is introduced at low rate in order to improve thefactor is high. In order to resolve this problem thea high droop
CICcosine decimation filter. The second order comin the desired passband and low stopband attenuation.
passband
of interest
of the overall
filter.
The
structure of (1) can be used [2], [3].
pensator
filter is introduced
at low
rate
in coefficients
order to non-recursive
The
decimation
factor Rof of
next decimation
stage
of the
compensator
filter
are
presented
in
a
canonical
More
details
on
a comparison
the the
performances
of
improve the passband of interest of the overall fildetermines
the
frequency
ω
,
where
the
worst
passband
c
are
signed
form,ofand
be implemented
usingthe recursive and nonrecursive implementations
ter. digits
The (CSD)
coefficients
thecan
compensator
filter are
ω
,
where
distortion
occurs,
and
the
frequency
given
in
[2].
Additionally,
the
magnitude
response
of the worst
A
onlypresented
adders and
shifts.
Consequently,
the
resulting
filter
in a canonical signed digits (CSD) form,
the
CIC
filter
has
a
high
droop
in
the
desired
passωc/π
case
aliasing
occurs
[4].
The
normalized
frequencies
can be implemented
onlyaadders
and shifts. in
is a and
multiplierfree
filter andusing
exhibits
high attenuation
band
and
low
stopband
attenuation.
The
decimation
Consequently,
the as
resulting
filter is adroop.
multiplierfree
and ωA/π are at
the stopband,
as well
a low passband
decimation stage determines the
filter and exhibits a high attenuation in the stopband, factor Rωofc the next
ω Al
1
2
1
2R − 1
π 1 the worst
frequency ωc=, where
passband
. (2)
⋅
=
= distortion
−
=
as
well
as
a
low
passband
droop.
Keywords: Decimation filter, Compensator, CIC filter, occurs, and
π theMR
π MR
π theMworst
MRcase MR
frequency
ωA , where
Cosine filter.
occursstructures
[4]. The normalized
Several
have beenfrequencies
proposed ωto
c /πdesign CIC
Keywords: Decimation filter, Compensator, CIC aliasing
and filters
ωA /π are
at improved frequency response as for example
with
filter, Cosine filter.
1. INTRODUCTION
[4-11]. Usually methods deal to improve either passband
1.
INTRODUCTION
The process of converting the given rate of a signal ωc [5-6],π or1stopband
1 ω[7-10].
2
1
2R-1
Al
=
.
=
= work
− is =
. (2) a simple
The
goal
of
this
to
propose
into a The
different
is called the
sampling
rateof conversion
processrate
of converting
given rate
a signal
π
MR π
MR π
M MR
MR
(SRC).
reduction
of a sampling
sampling
is called multiplierless decimation filter with no filtering at high
into aThe
different
rate is called
raterate
conversion
(SRC). The
of a sampling
is called
decimation,
and reduction
is accomplished
in two rate
stages,
filtering input rate, and with a desired high stopband attenuation
decimation,
and
is
accomplished
in
two
stages,
fil- Several
andstructures
a desired
low
passband
ToCIC
this end we
have
been
proposeddroop.
to design
and downsampling.
tering
and
downsampling.
Hogenauer
[1]
introduced
filters
with
improved
frequency
response
as
for
examHogenauer [1] introduced a very simple decimation propose to compensate the passband droop of the
very simple
decimation
filtercomb
called(CIC)
cascaded
Usually
methods
deal to improve
either
cascaded
CIC-cosine
decimation
filter [10],
introducing a
filtera called
cascaded
integrator
filter,intewhichple [4-11].
grator
comb
(CIC)
filter,
which
consists
of
cascaded
passband
[5-6],
or
stopband
[7-10].
consists of cascaded integrators and differentiators simple compensator at the low rate. The rest of the paper
integrators and differentiators separated by a downgoal of thisaswork
is to In
propose
a simple
mulis organized
follows.
Section
2 we propose
the new
separated by a downsampler, as shown in Fig. 1. The The
sampler, as shown in Fig. 1. The transfer function of tiplierless decimation filter with no filtering at high
second order compensator filter which is illustrated with
transfer
function decimation
of the resulting
the resulting
filter decimation
is given as filter is giveninput rate, and with a desired high stopband attenutwo examples. Section 3 briefly introduces the cascaded
as
ation and a desired low passband droop. To this end
CIC-cosine
decimation
from
[10].
The proposed
we propose
to compensate
the filter
passband
droop
of the
KK
-M
modified
compensated
CIC-cosine
decimation
filter is
1 1 − z −M ⎤
cascaded CIC-cosine decimation filter [10], introduc⎞ .
(1)
H (z) = ⎡ 1 ⎛ 1 − z−1
in section 3,
illustrated
with
one
simple compensator
at and
the low
rate. The
rest
of example.
(1)ing adescribed
H (z ) = ⎢M ⎜1 − z −1 ⎟⎥ ,
⎜
⎟
M
the
paper
is
organized
as
follows.
In
Section
2
we
pro1
−
z
⎠⎦⎥
⎣⎢ ⎝
ABSTRACT
where M is the decimation factor and K represents
the number of cascaded filters.
pose the new second order compensator filter which
is illustrated
with two
examples. Section FILTER
3 briefly
2. PROPOSED
COMPENSATION
where M is the decimation factor and K represents theintroduces the cascaded CIC-cosine decimation filter
TheThe
transfer
function
of compensated
the proposedCICcompensation
number of cascaded filters.
from [10].
proposed
modified
filter
is
given
by
cosine decimation filter is described in section 3, and
M
illustrated with H
one
example.
= a + bz − M + az −2 M ,
(3)
COMP z
( )
where a and b are real valued constants, and M is
decimation factor. Using a multirate identity this filter
The
function
therate
proposed
cantransfer
be moved
to a of
low
which compensais M times less than
Fig.1:
CIC
decimation
filter.
Fig.1: CIC decimation filter.
tion high
filter input
is given
by
rate becoming a second order filter.
response of (3) is expressed as
The above decimation filter is attractive in many
The magnitude
The
above decimation filter is attractive in many
HCOMP zM = a + bz-M + az-2M .
(3)
applications because of its very low complexity. It
applications
because
of
its
very
low
complexity.
It
is
is usually used at the first stage of decimation [1].
H COMP e jMω = 2a cos(Mω ) + b . (4)
usually
used at
thedecimation
first stagefilter
of decimation
[1]. However,
However,
this
exhibits several
drawthis backs.
decimation
filter exhibits
several
drawbacks.
The integrator
section works
at the
high inputThewhere a and b are real valued constants, and M is
factor.
a multirate
identity this
fil- b. To this
integrator section works at the high input data ratedecimation
Next issue
is Using
to define
the coefficients
a and
The authors are with the Institute INAOE, Deter
can
be
moved
to
a
low
rate
which
is
M
times
less
resulting
in a larger chip Puebla,
area and
higher power
end we consider magnitude response of Eq. (4) in two
partment of Electronics,
Mexico.
Email:gordana.dolecek@gmail.com,
gordana@ieee.org,
and isthan high input rate becoming a second order filter.
consumption
especially when the
decimation factor
points ωof=0,
and
ω=ωc,aswhere the worst
gordana@inaoep.mx
The frequency
magnitude response
(3) is
expressed
high. In order to resolve this problem the non-recursive
passband distortion occurs, defined in Eq. (2). In ω=0 the
2. PROPOSED COMPENSATION FILTER
(
)
Compensated CIC-Cosine Decimation Filter
HCOMP ejMω
9
= |2acos (Mω) + b| .
2aq + bq = 1.
(4)
(11)
Next issue is to define the coefficients a and b. To
this end we consider magnitude response of Eq. (4)
in two frequency points ω=0, and ω = ωc , where the
worst passband distortion occurs, defined in Eq. (2).
In ω=0 the desired magnitude characteristic has the
value 1, resulting in the following equation
Starting with k=2 the values of k are increased
until the condition (11) is satisfied. If the passband
droop is within the desired limit, the corresponding quantized coefficients aq and bq are presented in
canonical signed digit (CSD) form [12-15].
2a + b = 1.
HCOMP-CSD zM = aCSD + bCSD z-M + aCSD z-2M . (12)
(5)
desired magnitude characteristic has the value 1, resulting
in the following equation
START
In order to compensate the passband droop δc at the
frequency ωc , we have from Eq. (4)
where aCSD , and bCSD are the CSD representations
of the quantized coefficients aq , bq of the proposed
2a + b = 1 .
(5)
compensation filter (3).
Compute a and b
Otherwise, the procedure is continued
(Eq. (9))until the deIn order to compensate
at the
2acos (Mωc )the
+ bpassband
= 1/δc . droop δc(6)
sired passband compensation is obtained. This prob the correa
frequency ωc, we have from Eq. (4)
cedure is programmed in MATLAB,
and
sponding block diagram is shown in Fig. 2.
2a cos( Mωc ) + b = 1 / δ c .
Denoting
k=2
(6)
desired magnitude characteristic has the value 1, resulting
Denoting
in the following equation
δcomp = 1/δc .
k=k +1
(7)
δ comp2a=+1 /bδ=c 1 .
(7)
(5)
order to compensate the passband droop δc at the
we In
have
frequency ωc, we have from Eq. (4)
1 a ⎡ 1 ⎤
⎡ 22 2a cos(
⎤ ⎡ 1)⎤+ b = 1 / δ .
1
δ⎢comp
c ⎥ =. ⎢
⎥c . (8) (6)(8)
⎢ 2cos (Mc )1 =M⎥ω
⎣2 cos(Mωc ) 1⎦ ⎣b ⎦
Compute
a and bq
START q
(Eq. (10))
⎣δ comp ⎦
Denoting
No
k=2
No
k=k +1
and bb as
Solving
(8) we
get the
Solving (8)
we get
the coefficients
coefficients aa and
as
δ comp = 1 / δ c
(7)
1
1
⎤
⎡
−
we ⎢have
2(cos(Mω c ) − 1) 2(cos(Mω c ) − 1) ⎥ ⎡ 1 ⎤
⎥
⎡a ⎤ ⎢
⎥ . (9)
⎥⎢
⎢b ⎥ = ⎢
1⎤ ⎡a ⎤1 ⎡ 1⎥ ⎣δ⎤ comp ⎦
⎣ ⎦ ⎢
cos⎡(Mω c )2
(8)
⎢2 cos(Mω ) − 1⎥ ⎢b ⎥ = ⎢δ ⎥ ⎥ .
⎢ cos(M
⎣ ω c ) − 1 c cos
⎦ ⎣(M⎦ω c )⎣− 1comp
⎦ ⎦
⎣
and b asfrom (9) are
(8) we
the coefficients
a aand
In Solving
following,
thegetcoefficients
(9)
quantized as shown in the next relations
Compute a and b
(Eq.(11)
(9)) satisfied?
Is Eq.
b
a
No
Yes
Is the passband
compensation
Compute aOK?
q and bq
(Eq. (10))
Yes
CSD
Is Eq. (11) satisfied?
END
Yes
No
Is the passband
compensation
Fig.2: Block diagram of OK?
the compensator filter design.
Yes
1
1
The method is illustratedCSD
in the following example.
⎤
⎡
−
⎢ 2(cos− k(Mω ) −−k1) 2(cos(Mω
− k ) − 1) ⎥− k
2 a /c2
b = 2 c b / 2 ⎡ 1, ⎤ (10)
⎡a ⎤ a⎢ q = the
In following,
coefficients aq and b from⎥ ⎢(9) are
Example 1: Consider the
design of compensator for
END
⎥ . (9)
⎥ δ
⎢b ⎥ = ⎢
quantized
⎣ ⎦ ⎢ as shown
cascaded CIC decimation filter with M=4, K=4, and
cos(Mωinc )the next relations
1
⎥ ⎣ comp ⎦
− factor, aq and
where k ⎢is cos
the(Mquantization
decimation
factor of
stage is R=filter
8. The
ω c ) −-k1
ω c )-k−1 ⎥⎦ bq are the
cos-k(M
Fig.2:
Block diagram
of second
the compensator
de- passband
⎣
-k
aq = 2quantized
a/2
bq = 2 b/2and .⎣x⎦ represents
(10) sign. droop of CIC filter at normalized frequency ωc/π is
corresponding
coefficients,
Fig.2: Block diagram of the compensator filter design.
the integer part of x. The relation (5) must also hold for 0.2095792dB.
a and(5)b must
from also
(9) are
In following,
theof coefficients
the
integer part
x. The relation
athe
and
b resulting
in the condition
the quantized
quantized values
However, we want to have a resulting passband droop
shownofinvalues
next
hold for the as
quantized
ofrelations
a and b resulting in
less
Thethan
method
is illustrated
in the
following
example.
The method
is0.01dB.
illustrated
in the
following
example.
the condition
⎣
⎦
⎣
⎦
a q = 2 −k 2
aa/ q2 −+k bq =b1q .= 2 − k b / 2 − k ,
⎣
⎦
⎣
⎦
(11)
(10)
Starting
k=2quantization
the values of
k are aincreased
until the
where kwith
is the
factor,
q and bq are the
condition
(11)
is
satisfied.
If
the
passband
droop
is
within
corresponding quantized coefficients, and ⎣x⎦ represents
the the
desired
limit,
the
corresponding
quantized
coefficients
integer part of x. The relation (5) must also hold for
From Eq. (7), it follows
Example 1: Consider
the design of compensator for (13)
δcomp=1.02442214823189.
cascaded CIC decimation filter with M=4, K=4, and
decimation
factor
of second stage is R= 8. The passband
From (9)
we have
droop of CIC filter at normalized frequency ωc/π is
Fig.2: Block diagram of the compensator filter design.
11)
the
hin
ents
SD)
12)
the
sed
red
e is
ock
The method is illustrated in the following example.
Example
1: Consider
the the
design
compensatorforfor
Example
1: Consider
designofof compensator
cascaded
decimation
filterwith
with M=4,
M=4, K=4,
K=4,and
and
cascaded
CIC CIC
decimation
filter
decimation
factor
of
second
stage
is
R=
8.
The
passdecimation factor of second stage is R= 8. The passband
droopfilter
of CIC
filter at normalized
frequency
droop band
of CIC
at normalized
frequency
ωc/π is
ωc /π is 0.2095792dB.
0.2095792dB.
However, we want to have a resulting passband
However,
we want to have a resulting passband droop
droop less than 0.01dB.
less thanFrom
0.01dB.
Eq. (7), it follows
From Eq. (7), it follows
δcomp=1.02442214823189.
(13)
Next example compares the proposed compensation
filter with that recently proposed in literature [5].
Example 2: Consider the design of a CIC decimation filter for the design parameters M=11, K=5,
[5].
From [5], the normalized passband and stopband frequencies of interest are ωc /π=0.035455 and
ωAl /π=0.146364. The corresponding passband droop
δcomp = 1.02442214823189.
(13) and stopband attenuation of the CIC filter are 2.7288dB and AAl =-63.9441dB, respectively.
From (9) we have
The compensation filter from [5] has the correFrom (9) we have
sponding passband droop 0.01597dB, and stopband
attenuation - 61.199338dB, respectively.
⎡a ⎤ ⎡- 0.16041774990093⎤
Using the proposed compensator for the quantized
.
(14)
⎢b ⎥ = ⎢ 1.32083549980185 ⎥
factor k=8, we got the passband droop of -0.01475dB
⎣ ⎦ ⎣
⎦
the stopband
attenuation
of -61.215311dB.
corresponding CSD terms for quantized coefficients and and
compensation
filter
with the corresponding
CSD The
terms of
compensation
filter with
the by
corresponding CSD
the
transfer
function
is
given
by
quantized
coefficients
is
given
corresponding
terms
quantized
coefficients
and
filter with
the corresponding
CSD terms of
Starting
with kCSD
=2 we
findforthe
quantized
values shown
(14) compensation
terms of quantized
coefficients
is given by
the
transfer
function
is
given
by
quantized
coefficients
is
given
by
in Table
I.
−2
5
5
Starting
withz 4k=2
H COMP _ CSD11z11 = − 2−2
− 2 −−5
+ 2 −8 −8
1 + z −22 −22 H COMP _ CSD
= −we
2 −3find
− 2 −the
1 +quantized
z −8 + 20 values
+ 2 −2 + shown
2 − 4 z −4 .
Thein
condition
(11)
is
satisfied
only
for
the
values
of
(16)
H
z
=
−2
−
2
+
2
1 + .z
COMP-CSD 11
Table I. 4
−5
5
−8
1
−1z
H COMP
=− 4 −− 22−−27 −z2−11
+ 2 −8 1 + z −22 .
H COMP
z = bold
− 2 −3 −in2 −Table
1 + z I.
+ 20 + 2 −2 + 2 −4 z −4(15)
+
2
2
+
2
CSD
_−
_ CSD
k=3 and
kThe
=5,
shown
condition (11) is satisfied only for the values
(16)
+ 21 − 2−1 + 2−4 − 2−7 . z−11 .(16)
=3 the
passband
not (15)
satisfy + 21 − 2−1 + 2− 4 − 2− 7 z −11
However,
for kk=5,
of k=3 and
shown
bold indroop
Tabledoes
I.
The passband details for CIC filter, the proposed
the desired
specifications.
For
k =5 the
desired
However,
k=3 the
passband
droop
does passband
not satTable Ifor
: Quantized Factors and Coefficients.
compensated
CIC,
and
compensated
from
[5] are
isfy the is
desired
specifications.
For k k=5
the
desired
=5
we
find
the
specification
satisfied.
Therefore
using
The
passband
details
forfor
CIC
the proposed
proposed
The
passband
details
CIC filter,
filter, CIC
the
k I: Quantized
aq
bq and Coefficients.
Ac(dB)
Table
given
in
Fig.4.
passband
specificationFactors
is satisfied.
Therefore using k compensated
CIC,
and
compensated
CIC
from
[5]
are
compensated CIC, and compensated CIC from [5] are
k 2 find athe
b1.25
Ac(terms
dB) for quan- given in Fig.4.
q 0 corresponding
q
=5 we
CSD
given in Fig.4.
-0.0458388
3
-0.125
1.25
0 and the
1.25
tized2 coefficients
transfer function
is given by
0.25
-0. 125 1.25
1.3125 -0.0458388
3 4 -0.125
4
−3
−5
−8
0.25
=1.3125
−2 − 2-0.005381495
1+z
4H5COMP-CSD
-0.
125 z 1.3125
-0.15625
0.2
−4
0
−2
−4
-0.005381495
5 -0.15625
1.3125
0.2
+ 2 +2 +2
z . (15)
( ) (
( ) (
)(
)(
) (
) (
)
)
( ) (
( ) ( )
)
(
(
)(
)(
)
)
0.15
The passband
passband details
details of the
The
the CIC
CIC and
andthe
theproposed
proposed
compensated
CIC
filters
are
shown
in Fig.
3.
compensated
filters
shown
inand
Fig.
3. proposed
The
passband CIC
details
of are
the
CIC
the
There exists
a trade-off
between
the
compensated
CIC filters
are shown
in Fig.
3. desired pass0.01
0.15
Gain (dB)
the
ents
for
on
ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL.4, NO.1 MAY 2010
Gain (dB)
0)
10
0.1
0.1
0.05
0.05
0
0.01
0
-0.05
0
0
-0.01
-0.01
Gain(dB)
Gain(dB)
are
-0.02
-0.02
CIC
CICCompensated CIC, k=5
Compensated CIC, k=5
CIC Filter
CIC Filter+Roll-Off Compensator, [5]
CIC Filter
CIC Filter+Proposed
Compensator,
-0.05
CIC Filter+Roll-Off
Compensator,
[5] k=8
-0.1
CIC Filter+Proposed Compensator, k=8
0
0.005
0.01
0.015
0.02
0.025
0.03 0.035
-0.1
Frequency
(ω/π )0.03 0.035
0
0.005
0.01 Normalized
0.015
0.02
0.025
Normalized Frequency (ω/π )
Fig. 4: Example 2.
Fig.4:
Example2.2.
Fig. 4: Example
Note that the proposed compensator results in a slightly
Note
that
proposed
compensator
in a
Note
that
thethe
proposed
compensator
resultsresults
in a slightly
-0.04
better
passband
characteristic.
-0.04
slightly
better
passband
characteristic.
better passband characteristic.
-0.05
3. CIC-COSINE DECIMATION FILTER
0.005
0.01
0.015
0.02
0.025
0.03
-0.05 0
.
CIC-COSINE
DECIMATION FILTER
3
Normalized
)
0
0.005
0.01
0.015Frequency
0.02 (ω /π0.025
0.03
3. CIC-COSINE
FILTER factor
Normalized Frequency (ω /π)
Considering the DECIMATION
case when the down-sampling
Considering
the
case
when
the
down-sampling
can
be expressed
Fig.3:
Considering
the as
case when the down-samplingfactor
facFig. 3:Example
Example1.
1.
can
be
expressed
as
Fig. 3: Example 1.
M = M1M 2 M 3...M N
tor can be expressed as
(17)
M = M1M 2 M 3...M N
band
droop
the number
of thethe
CSD
termspassband
of co(17)
There
existsand
a trade-off
between
desired
we can rewrite Eq. (1) as
There
exists
anumber
trade-off
desired
passband a we can rewrite M
efficients
atheand
b. of between
(17)
Eq.=(1)M
as1 M2 M3 ...MN . K
droop
and
the CSDthe
terms
of coefficients
i −1
droop
and
the
number
of
the
CSD
terms
of
coefficients
a
⎡
and b.
i −1 ∏ M jK ⎤
⎡ N⎢ N
and b.
Mj =j 0 ⎤ ⎥
H ( z )⎢=
H i∏
)
j(
= 0z
H
(
z
)
H
(
z
)⎥ ⎥
=
⎢ i =1i
Next example compares the proposed compensation filter
⎥
⎢
Next
compares
the proposed
compensation
filter
⎥⎦
⎣1
i =⎢
withexample
that recently
proposed
in literature
[5].
(18)
⎦⎥
⎣⎢
with that recently proposed in literature [5].
(18)
where
-0.03
-0.03
∏∏
5
better passband characteristic.
0.03
3. CIC-COSINE DECIMATION FILTER
Compensated CIC-Cosine Decimation Filter
d passband
efficients a
ation filter
decimation
stopband
455 and
droop and
288dB and
esponding
enuation -
quantized
475dB and
dB. The
11
Considering the case when the down-sampling factor
can be expressed as
M = M1M 2 M 3...M N
(17)
we can rewrite Eq. (1) as
we can rewrite Eq. (1) as
K
i −1
⎡" N
∏ M j #⎤
N
Q j = 0 ⎥K
Y
⎢
HH((z)
z) =
Hi i (z z M ) .
= ⎢∏H
⎥
i =1
⎥⎦
⎢⎣ i=1
At first we design the modified CIC-cosine filter
H
(z) [10] in order to satisfy the desired stopmCCOS
i-1
j
j=0
(18) band attenuation. Next, we find the passband droop
δ in the frequency of interest ω = ωc of the designed
(18) c
modified CIC-cosine filter. Using (7) we compute
where
δcomp and solving equations (8) we get the coefficients
i
⎛
a and b of the compensator. Finally, the coefficients
i −1
−∏ M j ⎞
⎟
⎜
j =1
∏M j
a and b are expressed in CSD form as explained in
1 ⎜1− z
⎟; M = 1
H i ( z j =0 ) =
Section
2.
0
i −1
. (19)
Mi ⎜
−∏M j ⎟
The method is illustrated in Example 3.
⎜ 1 − z j =0 ⎟
Example 3: We design decimation filter for
⎠
⎝
M=32, which has passband droop at the frequency of
interest less than 0.002dB, and stopband attenuation
Using Eqs. (18) and (19) we express the modified
(19)CICin the frequency of interest at least of 110 dB. First we
filter H m (z ) as, [10]
design CIC-cosine filter choosing M1 =2, M2 =4 and
Using Eqs. (18) and (19) we express the modified M3 =4 yielding
k2
M1
H1k1 ( zas,
) ⋅ H[10]
) ⋅ ⋅ ⋅ H Nk N ( z M 1 ⋅⋅⋅M N −1 ),
(20)
CICHfilter
Hm(z)
m ( z) =
2 (z
1 1 − z−8
1 1 − z−2
2
,
H
,
z
=
H
(z)
=
where ki is the number of the cascaded filters H i.
2
1
2 1 − z−1
4 1 − z−2
Hm (z) = Hk11 (z) .Hk22 zM1 ...HkNN zM1 ...MN-1 . (20)
−32 We use the cosine prefilters introduced in [16] to improve H 3 z8 = 41 1−z
.
(26)
1−z−8
In the next, we cascade filters H1 (z), H2 (z2 ) and
where ki is the number of the cascaded filters Hi.
H3 (z8 ) as shown in the following equation
We use the cosine prefilters introduced in [16] to
(27)
Hm (z) = H41 (z) H22 z2 H23 z8 .
improve the frequency characteristic of the modified
CIC filter (20).
HCCOS (z) =
K
Y
KCOS z
Ni
.
(21)
i=1
The stopband attenuation of this filter is further improved introducing cascade of the expanded cosine
filters
HCCOS (z) = H2COS z8 H4COS z4 H4COS z2 . (28)
where [16].
2
HCOS zN = 0.125 1 + z-2N 1 + z-N .
the frequency characteristic of the modified CIC filter
(20).
In general we have
[10]
K
H CCOS ( z ) = ∏ H COS ( z
i =1
Ni =
Ni
),
M
.
2i+1
(22)
(21)
(23)
( )
( )
(
)(
( )
2
4
4
H CCOS (z ) = H COS
z 8 H COS
z 4 H COS
z2 ,
2
N
-2N
1 + z-N .
whereHCOS z = 0.125 1 + z
where
( )
H COS z N = 0.125 1 + z −2 N 1 + z
).
(28)
(29)
−N 2
(29)
From (2) using R=8 the passband and stopband
where [16].
From
(2) using
R=8 are
the passband and stopband
frequencies
of interest
N
−2 N
−N 2
H COSThe
( z transfer
) = 0.125
(1 + z of)(the
1 + zcascaded
) . modified
(22) CIC
function
frequencies of interest are
and cosine prefilters is
0.00390625
ω c / πωc=/π0.=
00390625
.
(30)
In general we have [10]
K
= 0.05859375.
(30)
ω A / πωA=/π
0.05859375
Y
M= Hm (z) Hni zNi .
HmCCOS
(z)
(24)
COS
N i = i +1 .
(23)
Stopband attenuation in the frequency of interest is
Stopband attenuation in the frequency of interi=1
2
AAl=-116.7993dB, i.e., it is satisfied. However the
est is AAl =-116.7993dB, i.e., it is satisfied. However
The transfer function of the cascaded modified CIC and passband
droop isdroop
-0.3162dB.
In order
where ni is the number of cascaded cosine prefilters, the passband
is -0.3162dB.
In ordertoto improve
improve
cosine prefilters
is
in
passband,
we design
compensation
filterfilter
as described
and NK = M1 .
passband,
we design
compensation
as described
K
ni
Section
2.
Ni
For more details see [10].
in Section 2.
H mCCOS ( z ) = H m ( z )∏ H COS ( z ) ,
(24)
Solving
(9) we
Solving
(9)get
wethe
getcoefficients
the coefficients
i =1
DECIMATION
FILTER
where ni is 4.
the PROPOSED
number of cascaded
cosine prefilters,
and
⎡a ⎤ ⎡- 0.2435389959475⎤
NK=M1.
The proposed filter is the cascade of the modified
⎢b ⎥ = ⎢ 1.487077991895 ⎥ . (31)
For more CICcosine
details seefilter
[10].HmCCOS (z) [10] and the compensator
⎣ ⎦ ⎣
⎦
filter introduced in Section 2.
.
.
4. PROPOSED DECIMATION FILTER
G (z) = HmCCOS (z) HCOMP-CSD zM .
(25) Table II shows the corresponding quantized values(31)
aq
The proposed filter is the cascade of the modified CIC- and bq.
cosine filter HmCCOS(z) [10] and the compensator filter
Table II: Quantized Factors and Coefficients.
introduced in Section 2.
( ).
G ( z ) = H mCCOS (z )H COMP _ CSD z M
(25)
k
aq
bq
coefficients can control the desired passband droop of
the overall decimation filter. The overall filter exhibits
a low passband droop, and high stopband attenuation at
the frequencies of interest. Additionally, using the
- 0.24353899
59475
⎡ECTI
TRANSACTIONS
ON⎤ COMPUTERpolyphase
AND INFORMATION
TECHNOLOGY
2010
decomposition,
the filtersVOL.4,
at theNO.1
firstMAY
stage
⎢ 1.487077991895 ⎥ . (31) can be moved at the lower rate. Therefore, there is no
⎣
⎦
filtering at high input rate.
Section 2.
Solving (9) we get the coefficients
(24)
ters, and
⎡a ⎤
⎢b ⎥ =
⎣ ⎦
12
.
TableIIIIshows
showsthe
thecorresponding
correspondingquantized
quantized
values
Table
values
aq
a
and
b
.
The
condition
(11)
is
satisfied
only
for
q
q
and b .
ied CICtor filter
6. ACKN
ne filter
stopband
δc in the
modified
d solving
b of the
d b are
aq
0
-0.125
-0.1875
-0.21875
-0.234375
-0.2421875
bq
1.25
1.375
1.4375
1.46875
1.484375
1.484375
-100
Gain(dB)
k
2
3
4
5
6
7
CIC-cosine
Compensated CIC-cosine, k=7
-50
Table II: Quantized Factors and Coefficients.
(25)
This
No.49640
-150
7. REFER
-200
-250
-300
0
0.2
a.
0.4
0.6
Normalized Frequency (ω /π)
0.8
1
Overall gain responses.
0.01
The condition (11) is satisfied only for k=7 resulting
r M=32,
erest less
requency
C-cosine
( ) (
)
)(
)
H COMP _ CSD z 32 = − 2 −2 + 2 −7 ⋅ 1 + z −64
Using
multirate identities we have efficient struc+ 21 − 2 −1 − 2 − 6 z − 32
.
(32)
ture shown in Fig. 5.
(
0
-0.01
Gain(dB)
in thepassband
corresponding
droop
ink=7
the resulting
corresponding
droop passband
of -0.001723dB,
of
-0.001723dB,
which
satisfies
the
desired
specificawhich satisfies the desired specification. Consequently,
Consequently,
we present for
quantized
coefficients
wetion.
present
quantized coefficients
k=7 in CSD
terms
for k=7 in CSD terms
-0.02
CIC-cosine
Compensated CIC-cosine, k=7
-0.03
-0.04
Overall magnitude response is shown in Fig. 6.a
Using multirate identities we have efficient structure
along with that of the CIC-cosine filter.
-0.05
0
0.5
1
1.5
2
2.5
3
3.5
-3
x 10
Gain(dB)
Normalized Frequency (ω/π)
. (26) shown in Fig. 5.
Corresponding
passband
is shown
Overall
magnitude
responsezoom
is shown
in Fig.in6.aFig.
b. Passband zooms.
Fig. 5: Efficient structure for Example 3.
6.b. with that of the CIC-cosine filter.
along
Corresponding
passband
zoom is shown
Fig. 6.b.
Fig. 6.c illustrates
attenuation
in theinneighborequations, and
expressed
in CSD
terms which
can be
Fig.
6.c
illustrates
attenuation
in
the
neighborhood
of
hood of the first zero.
implemented
the firstusing
zero.only adders and shifts. As a result
(z2) and
CIC-cosine
-120
the proposed filter is a multiplier free filter.
Compensated CIC-cosine, k=7
There is
a trade-off between the desired
-140
5. CONCLUSIONS
5. CONCLUSIONS
compensation
of the passband droop, and the
-160
(27)
complexity We
of proposed
the CSD here
representation.
Number
of
the multistage
decimation
filter
s further
adders based
mayWe
beonproposed
further
reduced
using
sub-expression
here the
multistage
decimation
the CIC-cosine
decimation
filter and fil-the
-180
d cosine
elimination
for
CSD
represented
coefficients
ter (CSE)
basedsecond
onthe
theorder
CIC-cosine
decimation
filter andof
symmetric
compensator.
The coefficients
[15]. In
contrast
to methods
[5-6],compensator.
thesolving
proposed
the
symmetric
second
The coef-200
the
compensator
are order
obtained
two
simple
compensation
design
is simpler,
the filter
ficientsfilter
of the
compensator
are and
obtained
solving two
-220
coefficients
can equations,
control theand
desired
passband
droop
of which
simple
expressed
in CSD
terms
the overall
filter. The
overall
filter exhibits
candecimation
be implemented
using
only adders
and shifts. As
-240
a low passband
and high stopband
at free fila resultdroop,
the proposed
filter is attenuation
a multiplier
0.059 0.06 0.061 0.062 0.063 0.064 0.065
the frequencies
of interest.
Additionally,
the comNormalized Frequency (ω/π)
ter. There
is a trade-off
between using
the desired
polyphase
decomposition,
the filtersdroop,
at theand
firstthe
stage
pensation
of the passband
complexity
c. Stopbands.
can be moved
the lower
rate. Therefore,
thereofisadders
no
of the atCSD
representation.
Number
may
Fig.6:
Example3.3.
filtering at
high
input
rate.
be further reduced using sub-expression elimination
Fig. 6: Example
Gain(dB)
0.066
(CSE) for the CSD represented coefficients [15]. In
contrast to methodsCIC-cosine
[5-6], the proposed compensaCompensated CIC-cosine, k=7
5. 1 ACKNOWLEDGEMENT
-50tion filter design is simpler, and the filter coefficients
can control the desired passband droop of the over- 6. ACKNOWLEDGEMENT
This work is part of the CONACYT project
-100all decimation filter. The overall filter exhibits a low
This work is part of the CONACYT project
No.49640.
passband droop, and high stopband attenuation at
No.49640.
-150the frequencies of interest. Additionally, using the
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Dumonteix,
M. Engineering,
M. Loerat, and
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of Electrical
University of Sarajevo,
MSc degree from
H. Mehrezz, “Efficient
polyphaseandecomposition
University of Belgrade, and a PhD deof comb decimation
filters in Σ-Δ analog-to-digital
gree from the Faculty of Electrical Enconverters,” IEEE
Transactions
onofCircuits
gineering, University
Sarajevo.& She
professorand
at theDigital
Faculty ofSignal
Electrical
Systems – II:wasAnalog
University
of Sarajevo
Processing, vol. Engineering,
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October
2001. until 1993, and 1993-1995 she was with the
[4] A. Kwentus, Z.
Jiang,Mihailo
and A.
Willson,
Jr., In
Institute
Pupin,
Belgrade.
1995“Application
she joined Institute
INAOE,
Department
Electronics,
of filter
sharpening
to forcascaded
Puebla,
Mexico, wheredecimation
she works asfilters,”
a professor
and researcher.
integrator-comb
IEEE
Trans.
Gordana2001-2002
Jovanovic she
Dolecek
received
a BS of Electrical ComDuring
was at
Department
onEngineering,
Signal
Processing,
vol.45,
pp.457-467,
puter
University
SantaFebruary
Barbara, as
degree
from the
Department
ofof California,
Electrical
1997.
visiting
researcher
on of
a sabbatical
Engineering,
University
Sarajevo, anleave.
MSc She is the author of
Gain(dB)
equations, and expressed in CSD terms which can be
implemented
using only
adders
and shifts.
As pp.155-162,
a result
nal Processing,
vol.
ASSP-29,
No.2,
the proposed
filter is a multiplier free filter.
April 1981.
There
is aL. trade-off
betweenandtheH. desired
[2]
Y. Gao,
Jia, J. Isoaho
Tenhunen,
compensation
of
the
passband
the
A comparison design of droop,
comb and
decimators
complexity
of
the
CSD
representation.
Number
of
for sigmadelta analog-to-digital converters,
adders
may
be
further
reduced
using
sub-expression
Analog Integrated Circuits and Signal Processing,
elimination (CSE) for the CSD represented coefficients
vol. 22, pp. 51-60, 1999.
[15]. In contrast to methods [5-6], the proposed
[3] H. Aboushady, Y. Dumonteix, M. M. Locompensation filter design is simpler, and the filter
erat, can
andcontrol
H. Mehrezz,
Efficientdroop
polyphase
coefficients
the desired passband
of
decomposition
of
comb
decimation
the overall decimation
P filter. The overall filter exhibits filin droop,
−∆and analog-to-digital
converters,
a lowters
passband
high stopband attenuation
at
IEEE Transactions
on Additionally,
Circuits & Systems
the frequencies
of interest.
using theII:
Analog
and Digital Signal
Processing,
vol. 48,
polyphase
decomposition,
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898-903,
2001.
can be
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Application
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CIC-cosinedecimation
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vol.45,
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pp.457-467, February 1997.
[5] S.-100Kim, W. Lee, S. Ahn, and S. Choi, Design of
CIC roll-off compensation filter in a W-CDMA
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[6] K. S. Yeung and S. C. Chan, The design
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0.2 on Circuits
0.4
0.6 Systems-I:
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1
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No.12,Frequency
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[7] M. 0.01 Laddomada,
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multistage
(RS) dec0
0.5comb-modified
1
1.5
2 rotated
2.5
3sinc 3.5
-3
Normalized Frequency
(ω/π)
imator with sharpened
magnitude
response,
x 10
zooms.
b. Passband
IEICE Transactions
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new cascaded modified CIC-Cosine decimation fil-
three
editor ofof one
book,and
and
author of more than 200
degree books,
from University
Belgrade,
a PhD
papers.
Hertheresearch
digital signal processing
degree from
Faculty ofinterests
Electrical include
Engineering,
and digital communications. She is a Senior member of IEEE,
University
of
Sarajevo.
She
was
professor
at
the
the member of Mexican Academy of Science,
and the member
Faculty
of Electrical
Engineering,
of
of
National
Researcher
SystemUniversity
(SNI) Mexico.
Sarajevo until 1993, and 1993-1995 she was with
the Institute Mihailo Pupin, Belgrade. In 1995 she
joined Institute INAOE, Department for
Electronics, Puebla, Mexico, where she works as a
professor and researcher. During 2001-2002 she
was at Department of Electrical & Computer
14
ECTI TRANSACTIONS ON COMPUTER AND INFORMATION TECHNOLOGY VOL.4, NO.1 MAY 2010
Fernando Javier Torres received a
BS degree from the Department of Electrical Engineering, University of Las
Americas, Puebla, 2005, Mexico and
an MSc degree from Institute INAOE,
Puebla, Mexico, 2007. His research interest is digital signal processing and
communications.
Biography:
Fernando Javier Torres received a BS degree
from the Department of Electrical Engineering,
University of Las Americas, Puebla, 2005,
Mexico and an MSc degree from Institute
INAOE, Puebla, Mexico, 2007. His research
interest is digital signal processing and
communications.