In this paper, a novel recursive realization method with word length reduced is proposed to synthesize linear-phase FIR filters for arbitrary bandwidth. By exploring the sparsity in the frequency domain, it constructs the desired impulse response with few complex exponential series (CES) and very sparse additional coefficients (AC) with equal length. The design problem is formulated as a combinatorial optimization problem by confining the frequencies of CES to a finite dictionary. To reduce the computational complexity at optimization, an iterative algorithm is presented, where appropriate frequencies of CES and zero locations of AC can be quickly determined. Thereafter, an efficient parallel structure is introduced to implement the proposed filter. To verify the performance of the proposed method, MATLAB simulation and Field-programmable gate array (FPGA) implementation are made. It shows that the proposed method requires 18 bits × 23 bits word length while the existing advanced recursive realization method —the piecewise-polynomial-sinusoidal recursive—needs 59 bits × 59 bits for the same filter specifications. Meanwhile, the method can achieve comparable multiplier reduction compared with the state-of-the-art low-complexity techniques but produce a low extra group delay. Moreover, it can realize variable bandedges with a fixed filter structure.

中文翻译：

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具有复指数脉冲响应的线性相位 FIR 滤波器的合成


本文提出了一种新颖的减少字长的递归实现方法来合成任意带宽的线性相位FIR滤波器。通过探索频域中的稀疏性，它用很少的复指数级数（CES）和非常稀疏的等长度附加系数（AC）构建了所需的脉冲响应。通过将 CES 的频率限制在有限字典中，设计问题被表述为组合优化问题。为了降低优化时的计算复杂度，提出了一种迭代算法，可以快速确定适当的 CES 频率和 AC 的零位置。此后，引入有效的并行结构来实现所提出的滤波器。为了验证该方法的性能，进行了 MATLAB 仿真和现场可编程门阵列（FPGA）实现。结果表明，对于相同的滤波器规格，所提出的方法需要 18 位 × 23 位字长，而现有的高级递归实现方法（分段多项式正弦递归）需要 59 位 × 59 位字长。同时，与最先进的低复杂度技术相比，该方法可以实现可比的乘数降低，但产生较低的额外群延迟。此外，它可以在固定滤波器结构的情况下实现可变带边。