Third-Octave and Bark Graphic-Equalizer Design with Symmetric Band Filters

Abstract

This work proposes graphic equalizer designs with third-octave and Bark frequency divisions using symmetric band filters with a prescribed Nyquist gain to reduce approximation errors. Both designs utilize an iterative weighted least-squares method to optimize the filter gains, accounting for the interaction between the different band filters, to ensure excellent accuracy. A third-octave graphic equalizer with a maximum magnitude-response error of 0.81 dB is obtained, which outperforms the previous state-of-the-art design. The corresponding error for the Bark equalizer, which is the first of its kind, is 1.26 dB. This paper also applies a recently proposed neural gain control in which the filter gains are predicted with a multilayer perceptron having two hidden layers. After the training, the resulting network quickly and accurately calculates the filter gains for third-order and Bark graphic equalizers with maximum errors of 0.86 dB and 1.32 dB, respectively, which are not much more than those of the corresponding weighted least-squares designs. Computing the filter gains is about 100 times faster with the neural network than with the original optimization method. The proposed designs are easy to apply and may thus lead to widespread use of accurate auditory graphic equalizers.Appl. Sci. 2020, 10, 1222 2 of 22 using a neural network, which handled the optimization calculation [19]. The second paper extended the neurally-controlled equalizer (EQ) method to the widely used third-octave GEQ design [20], where the 31 EQ bands also loosely approximated the bandwidths of human auditory filters [21].Compared to our previous conference paper [20], this article presents an improved design method using second-order band filters with a symmetric shape on the logarithmic frequency scale, which was recently proposed for an octave GEQ [22]. The proposed design allows the parametric EQ used as band filters in the GEQ to have a controllable gain at the Nyquist limit, in contrast to the earlier designs that forced the Nyquist gain to one (i.e., 0 dB) [3]. Furthermore, this paper extends both the weighted least-squares (WLS) GEQ design method and the neurally-controlled EQ method to operate in Bark bands, which are a more accurate approximation of the human auditory resolution than the third-octave bands [23].Currently, two important metrics in GEQ design are accuracy and computational complexity. Both aspects have an impact when a GEQ is controlled by a computer in automated tasks without a human listener fine-tuning the gains, e.g., when equalizing music to be above a masking threshold in time varying noise [24,25]. Previous GEQs that are not very accurate (see the measurements, e.g., in [10]) have typically required a human operator to listen to the effect of the command gains as he/she controls them, since the accuracy of different command gain settings, as well as the visual feedback of the command gain sliders can be highly inaccurate. One of the goals of this work was to get the proposed neurally controlled GEQs to provide ...

Special Issue on Digital Audio Effects

“…Third-Octave and Bark Graphic-Equalizer Design with Symmetric Band Filters by Jussi Rämö, Juho Liski, and Vesa Välimäki [8] uses an iterative weighted least-squares method to optimize the filter gains in a graphic equalizer, outperforming previous state-of-the-art, and also applies a recently proposed neural gain control relying on a two-level perceptron ensuring much faster optimization and only a slightly worse accuracy than the least squares method.…”

Sparse Graphic Equalizer Design

“…G RAPHIC equalizers (GEQs) that are commonly used in audio technology consist of parametric filters boosting or attenuating the signal gain in predefined frequency bands [1], [2], [3]. State-of-the-art GEQs [4], [5] require one secondorder filter per band to achieve an accuracy of ±1 dB from the target gain values, which is sufficient for human hearing [6], [7]. This letter presents a sparse GEQ design method that reduces the number of band filters when some frequency band gains are unused, requiring neither boost nor cut.…”

“…In 2017, Välimäki and Liski [4] proposed an accurate method for designing a cascade octave-band GEQ with 1-dB accuracy using the least squares (LS) method. Their design has subsequently been extended to one-third-octave design [5] and to a parallel structure [14]. Recently, Li et al proposed a modification to the parallel GEQ, where the optimized band-filter gains are compared to the neighboring ones.…”

“…An accurate cascade GEQ design with one-third-octave resolution [5] achieves a high accuracy with a single secondorder IIR filter per band by allowing the filter gains to differ from the target gains, thus accounting for the leakage of the filters to their neighboring bands. This is accomplished with the LS design based on an interaction matrix B:…”

“…where g are the optimized filter gains, vector t 1 contains the user-set gains in dB in odd rows and their linearly interpolated values in even rows, and T is the transpose operation [5].…”