2nd International Conference on Advances in Electrical and Electronics Engineering (ICAEE'2013) March 17-18, 2013 Dubai (UAE)
Cascading Sharpened CIC and Polyphase FIR
Filter for Decimation Filter
V.Jayaprakasan and M.Madheswaran
The essential function of decimation is to decrease the
sampling rate and to keep the passband aliasing within
prescribed bounds. In this paper CIC [1] filter introduced that
requires no multipliers and use limited storage hereby leading
to more economical hardware implementations. They are
designated cascaded integrator-comb (CIC) filters because
their structure consists of an integrator section operating at the
high sampling rate and a comb section operating at the low
sampling rate. Using CIC filters, the amount of passband
aliasing or imaging error can be brought within prescribed
bounds. Several schemes have been proposed to design comb
filters with improved magnitude response [2] - [4]. The
methods outlined in [5] and [6] use the filter sharpening
technique which was originally introduced to improve both
the passband and stopband of a symmetric FIR filter by using
multiple copies of the same filter. The main drawback of the
structure in [5], which uses sharpened comb filter with a
building block consisting of a K-stage comb filter, is that the
integrator section works at the high input rate resulting in
higher power consumption.
In 1973, McClellan, J. H & Parks [7] developed a
computer program to design a optimum FIR linear phase
digital filters. The program has options for designing standard
filters as low-pass, high-pass, band-pass, band-stop filters.
The program is also used to design filters which arbitrary
frequency specifications which are provided by users. This
program is written in FORTRAN. Bellanger, M.G., Bonnerot,
G. [8] designed a polyphase network structure and analyzed. It
permits the use of recursive devices for efficient sample-rate
alteration. The comparison with conventional filters shows
that, with the same active memory, a reduction of computation
rate approaching a factor of 2 can be achieved when the
alteration factor increases.
A.G. Dempster [9] examined the use of efficient shift and
adds multiplier structures and multiplier blocks to reduce
computational complexity in filter banks. He also examined
the farrow structure used in interpolator. Gustafsson, O.
Johansson, K [10] implemented a polyphase decomposed FIR
filters using MCM techniques i.e. realizing a number of
constant multiplications using minimum number of adders and
subtracters. For interpolations, direct form subfilters lead to
fewer registers as the shared among the subfilters but using
transposed direct form subfilters, the registers cannot be
shared. For decimation filters the opposite holds for direct
form and transposed direct form subfilters. Finally
implementation results for area, speed, and power for different
realizations are compared. Eghbali, A [11] this paper presents
the multiplierless FIR filter structures realization. They
derived the total number of adders required for the transposed
Abstract—This paper is concerned with design of cascading CIC
filter and FIR filter to improve the passband droop and stopband
attenuation for decimation filter. Decimation filter has wide
application in both analog and digital systems for data rate
conversion as well as filtering. We have three class of filters FIR, IIR
and CIC filters. IIR filters are simpler in structure, but not satisfy
linear phase filter requirements, which are required for time sensitive
features like video and speech. CIC filter don’t have this drawback,
they are coefficient less, hardware requirement is much reduced, but
as they don’t have well defined frequency response. So another
structure is proposed which takes advantage of good features of both
CIC and FIR filter structures. Hence in this paper concerned about
cascading sharpened CIC filter and polyphase structure of FIR for
efficient compensation in decimation is designed, which has better
passband and stopband performance.
Keywords— Multirate filtering, CIC, Compensation Technique,
Polyphase structure, FIR
I. INTRODUCTION
T
HE decimator is one of the basic building blocks of a
sampling rate conversion system. The decimation filter
performs two operations. Low pass filtering and down
sampling. The filter converts low resolution high bit rate data
to high resolution low frequency data. It has been widely used
in audio, speech processing, radar systems, and
communication systems. Considerable attention has been
focused in the last few years on the design of high efficiency
decimation filters.
Eugene Hogenauer [1] in 1981 designed a new class of
economical digital filters for decimation and interpolation
called a cascaded integrator comb (CIC) filter. This filter has
two sections namely integrator and comb section, as well as
it’s a multiplier less structure. The implementation of CIC
filters in hardware also efficient due to its symmetric
structure. This CIC filter has two sections, namely integrator
and comb sections, which perform the operation of digital low
pass filtering, decimation simultaneously.
V.Jayaprakasan is with Jawaharlal Nehru Technological University
Anantapur as a Research Scholar in Department of Electronics and
Communication Engineering, Anantapur - 515 002, Andhra Pradesh, India.
(Phone: +919842645707; e-mail: jprakasan@gmail.com)
M.Madheswaran was with PSNA Engineering College and now associated
with the Centre for Advanced Research, Department of Electronics and
Communication Engineering, Muthayammal Engineering College, Rasipuram
– 637 408, Tamilnadu, India. He was also a Research fellow at Banaras Hindu
University,
Varanasi.
(Phone:
+918870011837;
email:madheswaran.dr@gmail.com.
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2nd International Conference on Advances in Electrical and Electronics Engineering (ICAEE'2013) March 17-18, 2013 Dubai (UAE)
direct form, polyphase and reduced complexity polyphase FIR
filter structures and compared.
The main idea of this paper is to integrate the advantages
of the structures presented in [4-6], [10] and [13] in order to
obtain the structure that can provide the best passband droop
and stop attenuation is designed and analyzed using
MATLAB. The paper is organized as follows. In Section 2,
we first describe the main characteristics of cascaded-comb
filter (CIC). Section 3, describes the FIR filters structures and
polyphase implementation. Section 4, we outline the new
efficient proposed structure. Finally in Section 5, shows
results obtained by the proposed design.
By operating at differentiator at the lower sampling rate the
power consumption is achieved.
A poor magnitude characteristic of the comb filter is
improved by cascading [3] several identical comb filters. The
transfer function of multistage comb filter composed of
identical single stage comb filter is given by,
1
H ( z) =
N
1 − z − N 

−1 
 1− z 
K
(2)
II. CASCADED INTEGRATOR COMB [CIC] FILTER
Cascaded integrator comb [1] or Hogenauer filter, are
multirate filters used for realizing large sample rate changes in
digital systems. CIC filters are multiplierless structures,
consisting of only adders and delay elements which is a great
advantage when aiming at low power consumption. So the
CIC filters are frequently used in digital down converter and
digital up converters. The CIC filter is a class of hardware
efficient linear phase FIR digital filter consists of an equal
number of stages of ideal integrator and comb filter pairs. The
highly symmetric structure of this filter allows efficient
implementation in hardware. However the disadvantage of a
CIC filter is that is passband is not flat, which is undesirable
in many applications. This problem can be overcome through
the use of compensation filter. CIC filter achieve sampling
rate decrease (decimation) without using multiplication. The
CIC filter first performs the averaging operation then follows
it with the decimation.
Fig. 2 Structure of CIC filter with K stages
The magnitude response of this filter is given by
 1 sin(ω N / 2) 
H ( e jω ) = 

 N sin(ω / 2) 
K
(3)
The transfer function of the CIC filter in z-domain is given
as [1].
H ( z) =
1 1 − z − N 


N  1 − z −1 
(1)
Fig. 3 Magnitude response of multistage realization structure of CIC
filter with different K values
Where, N is the decimation factor
(
In equation (1) the numerator 1 − z − N
)
represents the
(
transfer function of comb and the denominator 1 / 1 − z −1
indicates the transfer function of integrator.
)
Fig. 4 Main performances of comb-based decimation filter
Fig. 1 Structure of first order CIC filter
Fig. 3 shows the multistage realization, which improves the
selectivity and the stopband attenuation of the overall filter. Fig. 4
Fig.1 shows the first order CIC filter; here the clock
divider circuit divides the oversampling clock signal by the
oversampling ratio, M after the integrator stage. The integrator
operates at the input sampling frequency, while the
.
differentiator operates at down sampled clock frequency
indicates the main performance of the comb based decimation
filter, here only the magnitude response is considered, since
the phase is linear.
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2nd International Conference on Advances in Electrical and Electronics Engineering (ICAEE'2013) March 17-18, 2013 Dubai (UAE)
The signal band at the decimator input occupies the
frequencies [0, F m ]. For the input signal sampling frequency
F x and the decimation factor N. The aliasing bands of the
bandwidths 2F m are located around the natural null
frequencies F x /N, 2 F x /N, 3 F x /N … J F x /N ; J = N/2 for
even and J = (N-1)/2 for odd. The main parameters that
characterize the comb filter performance are the passband
droop and the selectivity factor.
in fig. 4 indicates the
The passband droop denoted by
maximum attenuation at the edge of the useful signal
bandwidth compared to the ideal LPF. The selectivity factor
in figure (4) is defined as the ratio between the exact
values of the filter magnitude response achieved at the
passband edge frequency (F m ) and at the lower edge
frequency of the first aliasing band (Fx/N- F m ).
for k stage comb filter
Expression for passband droop
is given by,
G K (F ) sin (π N Fm / Fx )
d cK = c K m =
Gc (0) N sin (π N Fm / Fx )
conversion factor, since the low pass bandwidth is very small.
In multistage decimators with large conversion factor, the
comb filter is the best solution for first decimation stage,
whereas in interpolation, the comb filter is convenient for the
last stage.
B. CIC filter for decimation
The basic concept of CIC filter is given in fig. 6, which consists of
factor of N down sampler and K-stage CIC filter.
x[n]
Fx
d =
sin (π N f m / 2 )
(4)
K
sin (π (1 / N − Fm / Fx )
G K (F )
Φ = K c m
=
sin (π Fm / Fx )
Gc (Fx / N − Fm )
K
c
Fy = Fx/N
(5)
x[n]
The selectivity factor for the K stage comb filter is given
by,
N
Applying third identity, the factor of N down sampler is
moved and placed behind the integrator section and before the
comb section as shown in fig. 7. Finally the CIC decimator is
implemented as a cascade of K integrator, factor of N down
sampler and the cascade of K differentiator sections. The
integrator portion operates at the input date rate, whereas the
comb portion operates at N time’s lower sampling rate which
is shown in fig. 2.
K
N sin (π f m / 2 )
y[m]
K
Fig. 6 Cascade of CIC filter and down sampler
Normalization to the half of the sample frequency
f = F/( Fx/2), then
K
c
 1 1 − z −N 

−1 
 N 1− z 
Fx
1 1 
 N 1 − z −1 
K
N
y[m]
[1 − z ]
−1 K
Fy = Fx/N
K
(6)
Fig. 7 Cascade of integrator section, down-sampler
and comb section
C. Two stage sharpened comb filter
The structure consisting of a cascade [4] of a comb filter
based decimator and sharpened comb decimator [5, 6]. In this
realization allows the sharpened comb section to operate at a
lower rate that depends on the decimation factor of the first
section. Using the polyphase decomposition [3] the subfilters
of the first section can also be operates at this lower sampling
rate.
Magnitude response of two stage sharpened comb filter
2K
3K

 1 sin(ω N / 2)  

  1 sin(ω N / 2) 
3
2
−
 


 
sin(
/
2
)
sin(
/
2
)
N
N
N
N
ω
ω
  2
1
1

 2
 

H sh (eiω ) = 
L
 1 sin(ω N1 / 2) 


 N1 sin(ω / 2) 
Fig. 5 Gain response of CIC filter: Passband droop
A. CIC filter for sample rate conversion
The CIC filters are utilized in multirate systems for
constructing digital up converter and down converter. The
ability of comb filter to perform filtering without
multiplication is very attractive to be applied to high rate
signals; moreover CIC filters are convenient for large
(7)
where N is decimation factor and N = N 1 * N 2
K is number of stages
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2nd International Conference on Advances in Electrical and Electronics Engineering (ICAEE'2013) March 17-18, 2013 Dubai (UAE)
[H 1 (z )]
x (n )
[H 2 (z )]L
Fx
y[m]
[H 1 (z )]
2K
K
-2
N1
Fy = Fx/N
z − ( N 2 −1) K / 2
First stage
N2
Sharpened Second Stage
3
Fig. 8 Two-stage implementation of the modified sharpened comb decimator
B. Polyphase structure for FIR decimator
A higher order FIR filter can be realized in parallel structure
based on the polyphase [8] decomposition of the transfer
function. The FIR transfer function is decomposed into Mlower order transfer function called polyphase components,
III. FIR FILTER STRUCTURE
The non-recursive nature of FIR filter [7] offers the
opportunity to create implementation schemes that
significantly improve the overall efficiency of the decimator.
Let us consider the factor M-decimator of which uses a FIR
antialiasing filter with the impulse response h[n].The time
domain relation for the filter are expressed by the convolution
sum,
Which are added together to compose the original transfer
function.
N −1
H ( z ) = ∑ h[k ] z − k
N −1
v(n ) = ∑ h[k ] x[n − k ]
(8)
k =0
In expanded form,
Where N is the order of the filter
H ( z ) = h[0] + h[1] z −1 + h[2] z −2 + ⋅ ⋅ ⋅ + h[N − 1] z − ( N −1)
The decimated signal y[m] is obtained after applying down
sampling is given by,
y (m ) = ∑ h[k ] x[nM − k ]
(9)
H (z ) = h[0] + h[2] z −2 + h[4] z −4 + ⋅ ⋅ ⋅ + h[N − 1] z − ( N −1) +
k =0
x[n]
v[n]
h[1] z −1 + h[3] z − 3 + ⋅ ⋅ ⋅ + h[N − 4] z − ( N − 4 ) + h[N − 4] z − ( N − 4 )
(12)
and
y[m]
M
h[n]
Fx
Fy = Fx/M
H (z ) = h[0] + h[2] z −2 + h[4] z −4 + ⋅ ⋅ ⋅ + h[N − 1] z − ( N −1) +
Fig. 9 Factor of M-decimator with FIR filter
A.
(11)
Here we consider N is odd number and the equation (10)
can be expressed as a sum of two terms.
N −1
Fx
(10)
k =0
(
z −1 h[1] + h[3] z −2 + ⋅ ⋅ ⋅ + h[N − 4] z −( N −5 ) + h[N − 4] z −( N −3 )
Efficient implementation of M-factor decimator
)
(13)
In conventional FIR filter implementation of figure (9), the
number of multiplications per input sample in the decimator is
equal to FIR filter length N. The efficient implementation
structure of fig. 10 reduces the number of multiplications per
input sample to N/M. This property significantly improves the
efficiency of multirate FIR filter.
By using the notation, equations (11) and (12) can be
expressed as,
E0 ( z ) = h[0] + h[2] z −1 + h[4] z −2 + ⋅ ⋅ ⋅ + h[N − 1] z − ( N −1) / 2
(14)
E1 (z ) = h[1] + h[3] z −1 + h[5] z −2 + ⋅ ⋅ ⋅ + h[N − 2] z − ( N − 3 ) / 2
(15)
( )
( )
H ( z ) = E0 z 2 + z −1 E1 z 2
(16)
In generalized form polyphase transfer function is written as,
( )
M −1
H ( z ) = ∑ z − k Ek z M
k =0
where, Ek ( z ) =
Fig.10 Efficient implementation of M-factor decimator
151
[N
M]
∑ h[Mn + k ] z
n =0
(17)
−N
, 0 ≤ k ≤ M − 1 (18)
2nd International Conference on Advances in Electrical and Electronics Engineering (ICAEE'2013) March 17-18, 2013 Dubai (UAE)
IV. PROPOSED STRUCTURE
In the proposed structure, sharpened CIC filter is cascaded
with polyphase FIR filter to improve the passband droop and
stopband attenuation.
The overall conversion ratio is factored as, M = N x R.
The overall sampling rate conversion systems can be
implemented by cascading a factor N CIC decimator, which is
again factored into N 1 , N 2 . A factor of ‘R’ FIR decimator
which is shown in fig. (12 -14).
Fig. 11 Efficient polyphase decimator
x[n]
Fx
 1 1− z−N 

−1 
 N 1− z 
y[m]
τ (z )
K
N
R
Fy = Fx/M
Fig. 12 Cascade implementation of two-stage decimator composed of a CIC filter and an FIR filter
x[n]
Fx
K
y[m]
( )
 1 1− z−N  τ z N

−1 
 N 1− z 
N
Fy = Fx/M
Fig. 13 Single-stage equivalent of two-stage decimator composed of a CIC filter and an FIR filter
x (n )
Fx
[H 1 (z )]K
[H 2 (z )]L
-2
N1
z − ( N1 −1) K / 2
N2
Sharpened Second Stage
First stage
3
v0 (m )
v1 (m )
v[n]
[H 1 (z )]2 K
v2 (m )
vR −1 (m )
E0 (z )
E1 ( z )
E 2 (z )

E M −1 ( z )
Fig. 14 Proposed Structure
152
y (m )
Fy = Fx / M
2nd International Conference on Advances in Electrical and Electronics Engineering (ICAEE'2013) March 17-18, 2013 Dubai (UAE)
V. RESULTS
Design the two stage decimator and compute the single
stage equivalent for the specification.
The decimation factor M=16; The overall decimation filter
H(z) is specified by, Passband edge P = 0.05 , and the
decimation of the passband magnitude response are bounded
to a p = 0.15dB. Stopband edge frequency s = /M = 0.1
with the minimal stopband attenuation a s = 50dB. Consider
the phase characteristics is linear.
Solution:
The overall factor 16 decimator is composed cascade of two
decimators.
Fig. 18 Magnitude response of CIC, FIR and Proposed cascaded
Structure
M=16; N=8, N 1 =4, N 2 =2, R=2
Fig. 19 Passband Zoom
VI. CONCLUSION
Fig. 15 Sharpened CIC filter Magnitude response with
N=8 and N 1 =4, N 2 =2
Decimation of a signal at high frequency using FIR filter
structure is very complex, since it needs a lot of
multiplications and hence the cost is increased. In CIC filter,
as the number of stages increases, the stopband attenuation
improves but passband droop increases. So this passband
droop is compensated with help of some compensation
techniques. CIC filter are very economic, computationally
efficient and simple to implement in comparison with FIR or
IIR for large rate change due to its multiplierless structure.
This paper analyzed the performance of CIC decimation filter
for efficient compensation by using sharpened CIC filter as a
first stage in decimation and FIR filter as second stage as in a
cascaded form. This will improve the passband droop of the
overall decimation filter. Results show this improvement of
passband and stopband attenuation.
Fig. 16 Polyphase FIR filter Magnitude response with R = 2
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Fig. 17 Magnitude response of Proposed Structure
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V.Jayaprakasan received his Bachelor’s Degree in Electronics and
Communication Engineering from Bharadhidasan University, Tiruchirappalli,
India in the year 1999 and Master’s Degree in Communication Systems from
Anna University, Chennai, India in the year 2006. He has started his teaching
profession in the year 2006 in Ganadipathy Tulsi’s Jain Engineering College,
Vellore. Earlier he has 11 years industrial experience in an electronics based
industry. At present he is a Professor in Electronics and Communication
Department. He has published 1 research papers in International Journal. He is
a part time research scalar in Jawaharlal Nehru Technological University
Anantapur. His areas of interest are Wireless communication, Networking and
Signal Processing. He is a life member of ISTE.
M.Madheswaran received the BE Degree from Madurai Kamaraj
University in 1990, ME Degree from Birla Institute of Technology, Mesra,
Ranchi, India in 1992, both in Electronics and Communication Engineering.
He obtained his PhD degree in Electronics Engineering from the Institute of
Technology, Banaras Hindu University, Varanasi, India, in 1999. At present
he is a Principal of Muthayammal Engineering College, Rasipuram, India. He
has authored over one hundred and twenty five publications in international
and national journals and conferences. Currently he is the chairman of IEEE
India Electron Devices Society Chapter and Vice Chair of IEEE India Circuits
and System Chapter. His areas of interest are theoretical modeling and
simulation of high-speed semiconductor devices for integrated optoelectronics
application, Bio-optics and Bio-signal Processing and Clinical decision
Support Systems development. He was awarded the Young Scientist
Fellowship (YSF) by the State Council for Science and Technology,
Tamilnadu, in 1994 and Senior Research Fellowship (SRF) by the Council of
Scientific and Industrial Research (CSIR), Government of India in 1996. Also
he has received YSF from SERC, Department of Science and Technology,
Govt. of India. He is named in Marquis Who’s Who in Science and
engineering in the year 2006. He is a fellow of IETE and IE (India), life
member of ISTE and also a senior member of IEEE. He is also the member of
International Association of Computer Science and Information Technology
(IACSIT) Singapore.
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