Design of modified second order frequency transformations based variable digital filters with large cutoff frequency range and improved transition band characteristics

CONTACT: PRAVEEN KUMAR. L (,+91 – 9791938249)
MAIL ID: sunsid1989@gmail.com, praveen@nexgenproject.com
Web: www.nexgenproject.com, www.finalyear-ieeeprojects.com
Design of Modified Second-Order Frequency
Transformations Based Variable Digital Filters with
Large Cutoff Frequency Range and Improved Transition
Band Characteristics
Abstract:
The frequency transformation based filters (FT filters) provide an absolute control over the
cutoff frequency. However, the cutoff frequency range (Ωc_range) of the FT filters is limited. The
second-order frequency transformations combined with coefficient decimation technique based
filter (FTCDM filter) has wider Ωc_range compared with the FT filter; however, the ratio of
transition bandwidth of the transformed filter to that of the prototype filter, tbwFT/tbwmod, is large
over a significant portion of Ωc_range. In this paper, we propose a novel idea of relaxing the one-
to-one mapping condition between the frequency variables, to overcome the issue of limited
Ωc_range for tbwFT ≤ tbwmod. In the proposed modified second-order frequency transformation
based filter (MSFT filter), we relax the one-to-one mapping condition between the frequency
variables and use low-pass to high-pass transformation on the prototype filter to achieve wider
Ωc_range with tbwFT ≤ tbwmod. Design example shows that the MSFT filter provides 3 and 1.22
times wider Ωc_range compared to FT and FTCDM filters, respectively. The proposed architecture
of this paper analysis the logic size, area and power consumption using Xilinx 14.2.
Enhancement of the project:
Existing System:
Frequency transformations are one of the first methods proposed to design the variable digital
filters. The frequency transformation based filter (FT filter) provides absolute control over ωc.
(Cutoff frequency resolution of 0.00001π can be obtained using the frequency transformations
based filter.) However, the cutoff frequency range of the transformed filter, Ωc_range, is very
limited and transition bandwidth of the transformed filter (i.e., tbwFT) can be significantly wider
than the prototype filter’s transition bandwidth (i.e., tbwmod). A variable digital filter based on the
second-order frequency transformations and CDM-I technique was proposed to increase Ωc_range.

(This variable filter is referred to as frequency transformations combined with coefficient
decimation technique based filter i.e. FTCDM filter in the rest of the paper.) However, the ratio
tbwFT/tbwmod is large over a significant portion of Ωc_range of the FTCDM filter. Hence, its
prototype filter needs to be overdesigned, i.e., the order of prototype filter needs to be increased
to account for this increase in the transition bandwidth of the transformed filter. In this paper, we
present a modified second-order frequency transformation based filter (MSFT filter), which
provides wider Ωc_range compared to FT and FTCDM filters and has advantage of maintaining
tbwFT ≤ tbwmod for the entire Ωc_range.
Disadvantages:
 Fixed high-pass, bandpass, as well as bandstop filter responses
 Complex circuit
Proposed System:
Use of Low-pass to High-Pass Transformation
The proposed MSFT filter can provide transformed lowpass filter responses that can be obtained
from the prototype low-pass filter of cutoff frequency ωc as well as the prototype low-pass filter
of cutoff frequency π − ωc. Consider a high-pass filter of order 2N with symmetric coefficients,
implemented in the Taylor form (same as in the case of low-pass filter).
The hardware realization of low-pass to high-pass transformation, i.e., reversing the sign of
every alternate “a” coefficient of the low-pass filter can be accomplished by changing every
alternate “add/sub” block of the add-delay chain of the filter structure to function as a subtractor.
The function addition or subtraction of the add/sub block can be selected using only one select
line. If the high-pass filter response is to be obtained simultaneously along with the low-pass
response, a separate add-delay chain can be implemented.
Relaxing the One-to-One Mapping Condition
For a particular value of ωc, Ωc_range of the second-order frequency transformations based filter is
determined by the range of parameters A0, A1, and A2. From (5), it is clear that
c can be controlled using only two parameters, say A1 and A0 (or equivalently A1 and A2) [15].
Even though A0 can vary from −1 to 1, the useful range of A0 (and hence Ωc_range) for any
particular value of A1 is limited by the constraints, such as the condition of one-to-one mapping

between ω and Ω, and the condition that cosΩc should be real. For instance, when A1 = 1, the
useful range of A0 is −0.5 to 0.5. For A1 = 0, there is not a single value of A0 that can satisfy all
the constraints.
Fig. 1. (a) Proposed MSFT filter. (b) Block D(Z)
Realization of the MSFT Filter
The proposed MSFT filter combines the two techniques presented above to provide four types of
low-pass filter responses:
1) Low-pass 1: the responses HL;
2) Low-pass 2: the responses HC H;
3) Low-pass 3: the responses obtained as first band of HL0;
4) Low-pass 4: the responses obtained as HC H0 + first band of the corresponding responses
HL0.
Hence, the proposed MSFT filter has wider Ωc_range compared to FT filter and is capable of
providing various high-pass, bandpass, and bandstop filter responses. In the MSFT filter, the
number of variables required to implement is reduced from three (i.e., A0, A1, and A2) to two
(A1 and A2). In the MSFT filter as A1 = {1, 0}, the multiplication with A1 can be implemented

simply using a multiplexer with one input as the other multiplicand and the other input as zero.
Low-pass to high-pass transformation (i.e., reversing the sign of alternate “a” coefficients) can be
implemented in any of the two ways, either by reconfiguring the alternate add/sub blocks or by
implementing a parallel add-delay chain. Unlike the FT and FTCDM filters, our MSFT filter
requires a low complexity masking filter to obtain two low-pass responses (Low-pass 3 and
Low-pass 4) from the response HL0.
Implementation of the MSFT filter is shown in Fig. 1. Block D(Z) implements the frequency
transformation. The responses HL, HC L, HH , and HC H (obtained when A1 = 1), and the
responses HL0, HC L0, HH0, and HC H0 (obtained when A1 = 0) are shown in Fig. 3(a). The
response HC L is the complementary response of the response HL. The response HH0 is
obtained at the output of high-pass add-delay chain when A1 = 0 and the response HC H0 is its
complementary response.
Advantages:
 variable high-pass, bandpass, as well as bandstop filter responses
 circuit complexity is reduced
Software implementation:
 Modelsim
 Xilinx ISE

Design of modified second order frequency transformations based variable digital filters with large cutoff frequency range and improved transition band characteristics

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Design of modified second order frequency transformations based variable digital filters with large cutoff frequency range and improved transition band characteristics