Exploiting Coefficient Symmetry in Conventional Polyphase FIR Filters

Abstract

The conventional polyphase architecture for linear-phase finite impulse response (FIR) filter loses its coefficient symmetry property due to the inefficient arrangement of the filter coefficients among its subfilters. Although, existing polyphase structures can avail the benefits of coefficient symmetry property, at the cost of versatility and complex subfilters arrangement of the conventional polyphase structure. To address these issues, in this paper, we first present the mathematical expressions for inherent characteristics of the conventional polyphase structure. Thereafter, we use these expressions to develop a generalized mathematical framework which exploits coefficient symmetry by retaining the direct use of conventional FIR filter coefficients. Further, the transfer function expressions for the proposed Type-1/ transposed Type-1 polyphase structures using coefficient symmetry are derived. The proposed structures can reduce the requirement of multiplier units in polyphase FIR filters by half. We also demonstrate the decimator design using the proposed Type-1 polyphase structure and the interpolator design using the proposed transposed Type-1 polyphase structure. Moreover, the phase and magnitude characteristics of the proposed Type-1/transposed Type-1 polyphase structures are presented. It is revealed via numerical examples that all subfilters of the proposed symmetric polyphase structure possess linear-phase characteristics. INDEX TERMS Coefficient symmetry, frequency response, polyphase FIR structures, pre/post processing, sampling rate conversion.

Design of low complexity parallel polyphase finite impulse response filter using coefficient symmetry

“…subfilter index c ( k ) is still required to be incorporated in Equation () as per Theorem 1 of the ref. [19]. For the brevity, this correspondence demonstrates the identification of symmetric subfilters for s th polyphase structure only, otherwise, ref.…”

“…For the application of polyphase FIR symmetry [19] with M × M traditional parallel filter [2, 3], it is required to generate the output samples correspondingly for each s th (where 0 ≤ s ≤ M − 1) M order polyphase structures. Subsequently, the output of the system y ( n ) can be generated by the interpolation of these M outputs.…”

“…For the brevity, this correspondence demonstrates the identification of symmetric subfilters for s th polyphase structure only, otherwise, ref. [19] should be referred for implications of coefficient symmetry. Now, we are assuming c ′( c ( k )) and c ″( c ( k )) as the indices of symmetric subfilters for deriving the mathematical expression with coefficient symmetry.…”

“…Moreover, these structures could not be suitable for signal processing applications such as adaptive filtering and transceivers [17, 18] where the generality of the algorithm is one of the major concerns. A Type‐1 polyphase symmetry [19] may be used with the traditional structures [2, 3] but sharing of subfilter coefficients among the parallel units and position of associated delay elements (i.e. z − D in ref.…”

Design and Implementation of Two- Dimensional Pipelined& Parallel FIR Filter Architectures

“…The efficiency of the algorithm that is being utilized in any system or design requires it to be more efficient. In such instances, using the pipelining technique to measure the algorithm's efficiency is highly recommended [21,28,29]. The author examines pipelined and non-pipelined FIR and IIR filters utilizing a full comparative investigation of the designs using several accessible FPGAs.…”