Least squares linear phase FIR filter design and its VLSI implementation
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Least squares linear phase FIR filter design and its VLSI implementation Mansoor Khan1
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Shahrukh Agha1
Received: 15 September 2019 / Revised: 17 May 2020 / Accepted: 11 July 2020 Ó Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work we present least squares (LS) approach to design linear phase Finite Impulse Response (FIR) filter. Since the design of FIR digital filters is non-analytic, we aim at ideal zero-phase magnitude response and minimize the weighted error in passband and stopbands. The problem of least squares can then be solved non-iteratively by solving system of linear equations. Solution of which yields impulse response that is both real and symmetric. Frequency response of the proposed LS FIR filter shows a flat passband, and higher stop-band attenuation than traditional window based FIR design and comparable attenuation with Parks–McClellan method of the same order. In addition we have implemented LS FIR filter on FPGA based VLSI architectures. Performance evaluation of proposed LS FIR design on VLSI architecture shows comparable throughput, area and power consumption compared to classical filter design approaches. Keywords Linear phase FIR filter Parks–McClellan algorithm Window method VLSI architecture
1 Introduction Least squares approach to design linear phase finite impulse response filters has been considered by a number of authors before [1–4]. Most least squares methods of designing FIR filters are based on solution of system of linear equations. In sense of providing the minimum order that would satisfy the desired frequency constraints, Parks– McClellan is by far the most popular numerical technique for the design of equal ripple and linear phase FIR filter [5]. Unlike classical IIR filter design which is based on transformation of an analog filter design, FIR filter design is semi-analytic that is given the frequency bounds calculation of exact order that will satisy the desired specifications is not possible [6]. Hence the numerical design of linear phase FIR filter aims at minimizing the squared error between an ideal filter and actual magnitude response (Fig. 1a) with either symmetric or antisymmetric impulse sequence [7].
& Mansoor Khan [email protected] 1
Department of Electrical and Computer Engineering, COMSATS University Islamabad, Islamabad 44000, Pakistan
Since the development of Parks–McClellan algorithms [8], computing resources have grown and many new and flexible design methadologies have appeared in digital filter design literature. Most notable are peak constrained least squares approach (PCLS) [9]. The peak constrained least squares unlike Parks–McClellan algorithm which is based on minimax Remez exchange algorithm is useful when the peak errors are more important than total squared errors. LS on the other hand are used in applications where the total squared error is more important than the peak errors. In PCLS optimization problems, we constrain the peak error while minimizing the total squared error [10]. On
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