A typical graphic equalizer frequency resolution is one-third octave comprising 31 bands. A previous design based on a least-squares optimization of the band-filter gains with a single second-order section per band has an accuracy of 1 dB. However, the design always uses all the band filters even when a small number of gains is adjusted. This letter proposes a sparse design of a one-third-octave graphic equalizer, where the number of active bands is minimized using orthogonal matching pursuit and linear programming before the least-squares gain optimization. In addition, the bandwidths of the high-frequency band filters are gain dependent in order to optimize their shape relative to an analog prototype. The optimal bandwidth is obtained with linear interpolation during the filter design. The proposed design achieves approximately the same or better accuracy in comparison to the state-of-the-art non-sparse design and can be used to automatically reduce the computational load of equalization when only some band filters are active.

This paper discusses the audibility of group-delay variations. Previous research has found limits of audibility as a function of frequency for different test signals, but extracting the tolerance for group delay to help audio reproduction system designers is hard. This study considers four critical test signals, three synthetic and one recorded, modified with digital allpass filters. The signals are filtered to produce a positive or negative group-delay peak covering the most sensitive frequency range from 500 Hz to 4 kHz, without changing the delay at other frequencies. ABX listening tests using headphones reveal the audibility thresholds for each signal. The perception is highly dependent on the signal, and the unit impulse and pink impulse are the most critical test signals. Negative group-delay variations are more easily audible than positive ones. The smallest mean threshold for the negative group delay was $-$0.56 ms and 0.64 ms for the positive group delay, obtained with a pink impulse. The thresholds are smaller than those obtained in previous studies. A synthetic hi-hat sound decaying 60 dB in 80 ms hides a positive group-delay variation. The variation is more difficult to hear in a recorded castanet sound than in the most critical synthetic signals. This work demonstrates how the group-delay response of headphones and loudspeakers can be perceptually tested, and leads to a better understanding of how audio systems should be equalized to avoid audible group-delay distortion.

A graphic equalizer is a high-order filter controlling the gain of several frequency bands. For good accuracy, graphic equalizers consisting of cascaded IIR filters have been of very high order. A previously proposed parallel graphic equalizer entailing twice as many second-order filter sections as there are bands can have a maximum approximation error of less than 1 dB, but its design is complicated. This letter proposes a cascade graphic equalizer having an accuracy comparable to the best parallel graphic equalizer, although only one second-order section is assigned per command gain. A key idea is to use band filters whose interaction with the two neighboring filters at their center frequency is exactly controlled. The filter gains are obtained using the least-squares method with one iteration step, which involves linear interpolation of the target gain vector, inversion of a square matrix, and a few matrix multiplications. The proposed method is compared with previous designs and is shown to be the most accurate one. The new graphic equalizer is widely useful for audio and music processing.