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A novel approach for coefficient quantization of low-pass finite impulse response filter using differential evolution algorithm

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Abstract

Evolutionary computational techniques have been employed judiciously in various signal processing applications of late. In this paper, such an attempt has been made to design a low-pass linear-phase multiplier-less finite duration impulse response (FIR) filter using differential evolution (DE) algorithm. This particular evolutionary optimization technique has been explored to search the impulse response coefficients of the FIR filter in the form of sum of power of two (SPT) in order to avoid the multipliers during design process. The performance of the designed low-pass filter has been studied thoroughly in terms of its frequency characteristics and primitive requirement of fundamental hardware blocks. The superiority of our design has been ascertained over a number of existing techniques by various means. Finally, the proposed filter of different lengths has been implemented on a field programmable gate array (FPGA) chip for evaluating the competency of this work. The percentage improvement in hardware complexity produced by our design has also been computed and clearly listed in this paper for convenience.

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Abbreviations

CR::

Recombination or cross-over probability

D::

Dimension of parameter vector in DE

\({D_N^N}:\) :

N × N Square matrix of twiddle factors

e::

Input signal to FIR filter

E::

Input signal matrix

F,F 1 ,F 2::

Weighting factor(s)

\({\Im, \psi}:\) :

Cost function associated with DE

G::

Generation number in DE

h k ::

kth tap coefficient of FIR filter

H::

Impulse response vector of FIR filter

\({\mathcal{H}}:\) :

Frequency sample vector of FIR filter

L::

Length of the impulse response of FIR filter

M::

Total number of output samples from FIR filter under consideration

\({\mathbb{M}}:\) :

Binary mask set

\({\mathfrak{M}}:\) :

Symbolic mutation operation

N::

Total number of frequency samples

\({\mathbb{N}}\) :

Number of elements in a set

P::

Size of population in DE

\({\Re}:\) :

Symbolic recombination or cross-over operation

\({\mathbb{R}}:\) :

Set of real numbers

s::

Element of binary mask

\({\mathbb{S}}:\) :

Summation of all the elements from a set

\({\mathfrak{S}}:\) :

Symbolic selection operation

U::

Trial vector in DE

V::

Donor vector in DE

W::

Twiddle factor

ω p ::

Pass-band edge frequency

ω s ::

Stop-band edge frequency

X::

Parameter vector in DE

y::

Output signal from FIR filter

Y::

Output signal vector

\({\Upsilon}:\) :

Parameter/trial vector in DE

\({\lambda}:\) :

Control parameter of DE

Θ::

Set of the population members

\({\zeta}:\) :

Value of fitness function

Δ::

Word length of FIR filter coefficients

\({\emptyset}:\) :

Mathematical function

δ p ::

Maximum pass-band ripple in dB

δ s ::

Minimum stop-band attenuation in dB

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Authors and Affiliations

  1. Department of Electronics & Telecommunication Engineering, Bengal Engineering and Science University, Shibpur, Howrah, 711 103, India

    Abhijit Chandra

  2. Department of Electronics & Telecommunication Engineering, Jadavpur University, Kolkata, 700 032, India

    Sudipta Chattopadhyay

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  1. Abhijit Chandra
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  2. Sudipta Chattopadhyay
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Chandra, A., Chattopadhyay, S. A novel approach for coefficient quantization of low-pass finite impulse response filter using differential evolution algorithm. SIViP 8, 1307–1321 (2014). https://doi.org/10.1007/s11760-012-0359-4

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