The problem of reducing the number of arithmetic operations in digital filtering algorithms is highly relevant, as it directly impacts power consumption, processing speed, and hardware costs. Under strict power efficiency requirements for mobile and embedded systems, minimizing multiplication and addition operations becomes a critical design factor. This paper presents a method for implementing a recursive filter with a finite impulse response (FIR) based on a truncated sinc function smoothed by a window (weighting function), represented as a sum of quasi-harmonic functions. These quasi-harmonic functions with different frequencies are polynomials of degree r. The study adopts a second-degree polynomial as a baseline and proposes a numerical method for increasing the polynomial order to improve the accuracy of the approximation. Accuracy analysis demonstrates that using 4th- and 6th-order polynomials achieves an approximation error of less than 1%. The coefficients of the non-recursive part of the filter are computed via inverse finite differences of the original FIR impulse response. These coefficients are integers whose values depend on the number of samples (length) of the half-period of the quasi-sinusoidal function, simplifying the implementation of such a recursive FIR (RFIR) filter on a field-programmable gate array (FPGA). Numerical analysis of finite differences for each quasi-sinusoid revealed that quadratic approximation requires only 16 samples but results in relatively high side-lobe levels (–30 dB). Switching to 4th-order approximation increases the number of non-zero coefficients to 20 and significantly reduces (by 13 dB) the stopband magnitude of the frequency response, reaching –43 dB.
This overview article covers finite impulse response filters and filter banks. The use of these filters for hearing aids is considered. Ways to compensate for hearing loss and ways to increase loudness using broadband amplification are considered. A schematic diagram of a method for digital signal processing using a bank of filters, as well as a technique for synthesizing interpolation filters with low computational complexity, is presented. Also, the application of the MATLAB system for the synthesis of narrow-band non-recursive FIR filters, their design procedure, methodology and examples are considered. Finite Impulse Response (FIR) filters and filter banks have specific properties that guarantee stability. Therefore, they are popular in many applications such as communication systems, audio signal processing, biomedical instruments, and so on. Unfortunately, due to the longer wavelength, the cost of implementing an FIR filter is usually not higher than an infinite impulse response (IIR) filter that meets the same requirements. It is well known that the length of an FIR filter is inversely proportional to its transition bandwidth. Therefore, the disadvantage becomes acute when a given filter has a narrow transition band. The main goal is to consider computationally efficient methods for designing FIR filters and filter banks. The masking method (FRM) results in significant savings in the number of multipliers. Next, a 16-band, low group delay, non-equal-spacing digital FIR filter bank is considered. Overall latency is significantly reduced as a result of a new filter structure that reduces the interpolation factor for prototype filters. Masking filter may be an interpolated finite impulse response (IFIR) filter that helps reduce complexity.
There are various types of weighting functions, the so-called windows in digital signal processing, such as rectangular (Dirichlet window), triangular (Bartlett window), Vallee-Poussin window, Kaiser-Bessel window, Barsilon-Temesh window, Hann, Bohman, Blackman, Gauss (Weierstrass), Dolph - Chebyshev, Hamming windows and many others and ideal characteristics of standard filters such as low-pass, high-pass, bandpass filters. The purpose of this review article is to determine the most suitable weighting function for implementation on its basis of a controlled recursive low-pass filter with a finite impulse response. This article presents an analysis of only some of the above windows and their approximations, namely the Dolph - Chebyshev window, the Gauss (Weierstrass) window and the Hamming window. In addition to the analysis, the synthesis of recursive filters with a finite impulse response for weighting data based on the selected windows and their approximations was considered. The method of synthesis of Dolph-Chebyshev windows is considered. The implementation of the Gauss (Weierstrass) window is considered. Methods for approximating the Hamming window and methods and several algorithms for developing filters with FIR in the form of this window are considered. The estimation of parameters dependencies some quick window of the maximum level of the side lobes. Based on the data obtained, conclusions were drawn about the selection of the most suitable and demonstrating maximum performance windows, suitable for implementation on its basis of a controlled recursive low-pass filter with a finite impulse response.
