Digital Signal Processing 20 (2010) 328–336 Contents lists available at ScienceDirect Digital Signal Processing www.elsevier.com/locate/dsp Flat magnitude response FIR halfband low/high pass digital filters with narrow transition bands Ishtiaq Rasool Khan ∗ Institute for Infocomm Research, A*STAR, Singapore article info abstract Article history: Available online 9 July 2009 Keywords: FIR digital filters Maximally flat MAXFLAT Linear phase Halfband Lowpass Highpass Filter transformation MAXFLAT FIR low/high pass digital filters are traditionally designed to satisfy the constraints of maximal flatness at the ends of the frequency band. In this paper, we show that by moving the points of flatness to the inner band, halfband filters with narrow transition bands can be realized. This, however, affects the smoothness of their magnitude responses at the ends of the frequency band. We propose a new design of halfband filters having their points of flatness at the middle of the pass and stop bands. The resulting filters have significantly narrow transition bands as compared to the existing MAXFLAT designs, and yet their magnitude responses are quite smooth in the entire frequency band.  2009 Elsevier Inc. All rights reserved. 1. Introduction Maximally flat (MAXFLAT) finite impulse response (FIR) digital filters (DFs), introduced by Hermann [1] in 1971, are known for their design simplicity, accuracy and high stopband attenuations. The basic idea behind MAXFLAT designs is to force the magnitude response and its derivatives to be as close to the ideal as possible, at one or more fixed points in the frequency band, to ensure the highest possible smoothness of the response at the desired points. The resultant filters have ripple-free magnitude responses and are very accurate in narrow regions centered at the points where the maximal flatness (MF) constraints are applied. Classical MAXFLAT designs involve approximation of the desired frequency response by some suitable polynomial like a Hermite [1], Krawtchouk [2] or Bernstein polynomial [3], etc. Equivalent forms of MAXFLAT designs were presented by Miller [4], and Fahmy [5]. The impulse response (IR) coefficients of the filters designed with these algorithms can be calculated using the inverse discrete Fourier transform. Halfband low/high pass FIR DFs have cutoff frequency at the middle of the frequency band ω = π /2. They are widely popular due to the fact that almost half of their IR coefficients are zeros, which leads to computationally efficient imple- mentations. Halfband filters find several applications in filterbanks, wavelets based compression, and multirate techniques. Several designs and implementation tricks have been proposed, both for MAXFLAT and other halfband filters [6–19]. Gu- macos [6] presented simplified expressions for the IR coefficients of MAXFLAT halfband designs in 1978. A class of filters proposed by Bahr [7] includes both linear and non-linear MAXFLAT halfband filters, besides several other FIR designs. Se- lesnick and Burrus [8] generalized these designs, and Samadi et al. [9] further simplified their transfer functions. An efficient implementation of these filters was given by Pei et al. [10]. Frequency response of a halfband DF can be obtained from that of a fullband digital differentiator (DD) by applying certain transformations [20]. We used this technique to derive simple explicit formulas for the coefficients of halfband MAXFLAT DFs [19] from an existing design of maximally linear (MAXLIN) DDs [21]. * Address for correspondence: 1 Fusionopolis Way, #21-01 Connexis (South Tower), Singapore 138632. E-mail address: irkhan@i2r.a-star.edu.sg. 1051-2004/$ – see front matter  2009 Elsevier Inc. All rights reserved. doi:10.1016/j.dsp.2009.07.001