Ya Jun Yu

Design of Linear Phase FIR Filters With High Probability of Achieving Minimum Number of Adders

In this paper, an algorithm is proposed for the design of low complexity linear phase finite impulse response (FIR) filters with optimum discrete coefficients. The proposed algorithm, based on mixed integer linear programming (MILP), efficiently traverses the discrete coefficient solutions and searches for the optimum one that results in an implementation using minimum number of adders. During the searching process, discrete coefficients are dynamically synthesized based on a continuously updated subexpression space and, most essentially, a monitoring mechanism is introduced to enable the algorithms awareness of optimality. Benchmark examples have shown that the proposed algorithm can, in most cases, produce the optimum designs using minimum number of adders for the given specifications. The proposed algorithm can be simply extended for the optimum design with the maximum adder depth constraint.

Design of Linear Phase FIR Filters in Subexpression Space Using Mixed Integer Linear Programming

In this paper, a novel optimization technique is proposed to optimize filter coefficients of linear phase finite-impulse response (FIR) filter to share common subexpressions within and among coefficients. Existing approaches of common subexpression elimination optimize digital filters in two stages: first, an FIR filter is designed in a discrete space such as finite wordlength space or signed power-of-two (SPT) space to meet a given specification; in the second stage, an optimization algorithm is applied on the discrete coefficients to find and eliminate the common subexpressions. Such a two-stage optimization technique suffers from the problem that the search space in the second stage is limited by the finite wordlength or SPT coefficients obtained in the first stage optimization. The new proposed algorithm overcomes this problem by optimizing the filter coefficients directly in subexpression space for a given specification. Numerical examples of benchmark filters show that the required number of adders obtained using the proposed algorithm is much less than those obtained using two-stage optimization approaches.

Generalization of Orthogonal Frequency Division Multiplexing With Index Modulation

Recently, orthogonal frequency division multiplexing (OFDM) with index modulation (OFDM-IM) was proposed. By selecting a fixed number of subcarriers as active subcarriers to carry constellation symbols, the indices of these active subcarriers may carry additional bits of information. In this paper, we propose two generalization schemes of OFDM-IM, named OFDM with generalized index modulation 1 (OFDM-GIM1) and OFDM-GIM2, respectively. In OFDM-GIM1, the number of active subcarriers in an OFDM subblock is no longer fixed. Dependent on the input binary string, different numbers of active subcarriers are assigned to carry constellation symbols. In OFDM-GIM2, independent index modulation is performed on the in-phase and quadrature component per subcarrier. Through such ways, a higher spectral efficiency than that of OFDM-IM may be achieved. Since both generalization schemes proposed suffer from BER performance loss in low SNR region, an interleaving technique is proposed to tackle this problem. Finally, noting that the two generalization schemes are compatible with each other, the combination of these two schemes, named OFDM-GIM3, has also been investigated. Computer simulation results clearly show our proposed schemes superiority in both spectral efficiency and BER performance compared to existing works.

Optimum masking levels and coefficient sparseness for Hilbert transformers and half-band filters designed using the frequency-response masking technique

Hilbert transformers and half-band filters are two very important special classes of finite-impulse response filters often used in signal processing applications. Furthermore, there exists a very close relationship between these two special classes of filters in such a way that a half-band filter can be derived from a Hilbert transformer in a straightforward manner and vice versa. It has been shown that these two classes of filters may be synthesized using the frequency-response masking (FRM) technique resulting in very efficient implementation when the filters are very sharp. While filters synthesized using the FRM technique has been characterized for the general low-pass case, Hilbert transformers and half-band filters synthesized using the FRM technique have not been characterized. The characterization of the two classes of filter is a focus of this paper. In this paper, we re-develop the FRM structure for the synthesis of Hilbert transformer from a new perspective. This new approach uses a frequency response correction term produced by masking the frequency response of a sparse coefficient filter, whose frequency response is periodic, to sharpen the bandedge of a low-order Hilbert transformer. Optimum masking levels and coefficient sparseness for the Hilbert transformers are derived; corresponding quantities for the half-band filters are obtained via the close relationship between these two classes of filters.

FRM-Based FIR Filters With Optimum Finite Word-Length Performance

It is well known that filters designed using the frequency response masking (FRM) technique have very sparse coefficients. The number of nontrivial coefficients of a digital filter designed using the FRM technique is only a very small fraction of that of a minimax optimum design meeting the same set of specifications. A digital filter designed using FRM technique is a network of several subfilters. Several methods have been developed for optimizing the subfilters. The earliest method optimizes the subfilters separately and produces a network of subfilters with excellent finite word-length performance. Subsequent techniques optimize the subfilters jointly and produce filters with significantly smaller numbers of nontrivial coefficients. Unfortunately, these joint optimization techniques, that optimize only the overall frequency response characteristics, may produce filters with undesirable finite word-length properties. The design of FRM-based filters that simultaneously optimizes the frequency response and finite word-length properties had not been reported in the literatures. In this paper, we develop several new optimization approaches that include the finite word-length properties of the overall filter into the optimization process. These new approaches produce filters with excellent finite word-length performance with almost no degradation in frequency response performance

Single-Stage and Cascade Design of High Order Multiplierless Linear Phase FIR Filters Using Genetic Algorithm

In this work, a novel genetic algorithm (GA) is proposed for the design of multiplierless linear phase finite impulse response (FIR) filters. The filters under consideration are of high order and wide coefficient wordlength. Both the single-stage and cascade form are considered. In a practical filter design problem, when the filter specification is stringent, requiring high filter order and wide coefficient wordlength, GAs often fail to find feasible solutions, because the discrete search space thus constructed is huge and the majority of the solution candidates therein can not meet the specification. In the proposed GA, the discrete search space is partitioned into smaller ones. Each small space is constructed surrounding a base discrete coefficient set which is obtained by a proposed greedy algorithm. The partition of the search space increases the chances for the GA to find feasible solutions, but does not sacrifice the coverage of the search. The proposed GA applies to the design of single-stage filters. When a cascade form filter is designed, for each single-stage filter meeting the filter specification generated during the course of GA, an integer polynomial factorization is applied. Design examples show that the proposed GA significantly outperforms existing algorithms dealing with the similar problems in terms of design time, and the hardware cost is saved in most cases.

Design of Discrete-Valued Linear Phase FIR Filters in Cascade Form

Digital filters in cascade form enjoy many advantages over their equivalent single-stage realizations in that lower coefficient sensitivity, higher throughput, reduced computational and smaller implementation cost can be achieved. However, the numerical design and optimization of such structure are of much more difficulty than the single-stage case if the filter coefficients are restricted to be of discrete values. This is mainly due to the non-convexity of the constraints, which rules out the possibility of employing sophisticated convex optimization techniques as well as the guaranteed global optimality. In this work, a general-purpose algorithm is proposed for the design of linear phase finite impulse response (FIR) filters in cascade form with discrete coefficients. The proposed algorithm decomposes the overall filter into subfilters during the traverse of a tree search of the overall filter. Discrete-valued linear phase FIR filters are able to be searched and decomposed into both symmetric and non-symmetric subfilters. The optimization complexity is of the same order as the single-stage filter optimization. Design examples have shown that the proposed algorithm is capable of achieving notable reduction in both implementation cost and adder depth compared with their single-stage optimum designs.

Fixed-Point Analysis and Parameter Selections of MSR-CORDIC With Applications to FFT Designs

Mixed-scaling-rotation (MSR) coordinate rotation digital computer (CORDIC) is an attractive approach to synthesizing complex rotators. This paper presents the fixed-point error analysis and parameter selections of MSR-CORDIC with applications to the fast Fourier transform (FFT). First, the fixed-point mean squared error of the MSR-CORDIC is analyzed by considering both the angle approximation error and signal round-off error incurred in the finite precision arithmetic. The signal to quantization noise ratio (SQNR) of the output of the FFT synthesized using MSR-CORDIC is thereafter estimated. Based on these analyses, two different parameter selection algorithms of MSR-CORDIC are proposed for general and dedicated MSR-CORDIC structures. The proposed algorithms minimize the number of adders and word-length when the SQNR of the FFT output is constrained. Design examples show that the FFT designed by the proposed method exhibits a lower hardware complexity than existing methods.

Orthogonal frequency division multiplexing with generalized index modulation

Recently, an Orthogonal Frequency Division Multiplexing (OFDM) with index modulation (OFDM-IM) [1] was proposed. By selecting a fixed number of subcarriers as active subcarriers to carry constellation symbols, the indices of these active subcarriers may carry additional bits of information. In this paper, a generalization of OFDM-IM, named as OFDM with generalized index modulation (OFDM-GIM) is proposed. In this scheme, the number of active subcarriers in an OFDM subblock of the overall structure is no longer fixed. Dependent on the input binary string, different number of active subcarriers are assigned to carry constellation symbols. In such a way, a higher spectral efficiency than that of OFDM-IM may be achieved. To facilitate the implementation of our scheme, a generalized index modulation block for the active subcarrier selection and an upgraded Log-likelihood Ratio (LLR) detector for the active subcarriers and symbol constellation detection have also been proposed. Computer simulation results show that our proposed scheme does offer us a much higher spectral efficiency at the cost of a marginal error performance loss.

Bit-Level Multiplierless FIR Filter Optimization Incorporating Sparse Filter Technique

Multiplierless FIR filter optimization has been extensively studied in the past decades to minimize the number of adders. A more accurate measurement of the implementation complexity is the number of full adders counted at bit-level. However, the high computational complexity of the optimization at bit-level hinders the technique from practical applications. In this paper, the sparse filter technique is exploited and makes the search space at bit-level significantly reduced. Thus, the bit-level optimization of multiplierless FIR filters for the first time becomes possible. When the sparse filter technique is employed for the multiplierless filter design, the sparsity of the filter is properly controlled so that the feasibility of the bit-level optimization in discrete space is maintained. Thereafter, in the reduced search space, a tree search algorithm is formulated at bit-level, and techniques to estimate the bit level hardware cost and to accelerate the search are presented. Design examples show that the proposed bit-level optimization method generates designs with lower hardware cost and power consumption than that of the best word-level optimization methods, while the design time is still at an acceptable level. The average power savings to 3 recent published techniques are 13.6%, 8.0% and 26.1%, respectively.