## Bark and ERB Bilinear Transforms (1999)

Citations: | 86 - 4 self |

### BibTeX

@MISC{Smith99barkand,

author = {Julius O. Smith and III and Jonathan S. Abel},

title = {Bark and ERB Bilinear Transforms},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

Use of a bilinear conformal map to achieve a frequency warping nearly identical to that of the Bark frequency scale is described. Because the map takes the unit circle to itself, its form is that of the transfer function of a first-order allpass filter. Since it is a first-order map, it preserves the model order of rational systems, making it a valuable frequency warping technique for use in audio filter design. A closed-form weighted-equation-error method is derived which computes the optimal mapping coefficient as a function of sampling rate, and the solution is shown to be generally indistinguishable from the optimal least-squares solution. The optimal Chebyshev mapping is also found to be essentially identical to the optimal least-squares solution. The expression...

### Citations

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Citation Context ...The error is also nonlinear in the parameter and to find its norm minimizer, an equation error is introduced, as is common practice in developing solutions to nonlinear system identification problems =-=[31]-=-. Consider mapping the frequency via the allpass transformation (12) Now, multiply (12) by the denominator and substitute from (8), to get (13) Rearranging terms, we have where is an equation error de... |

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Citation Context ...ss accurate than for the Bark scale. The paper concludes with a summary of findings. II. BARK FREQUENCY SCALE The Bark scale ranges from to Barks, corresponding to the first critical bands of hearing =-=[7]-=-. The published Bark band edges are given in Hertz as [0, 100, 200, 300, 400, 510, 630, 770, 920, 1080, 1270, 1480, 1720, 2000, 2320, 2700, 3150, 3700, 4400, 5300, 6400, 7700, 9500, 12 000, 15 500]. T... |

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Citation Context ...ate at 41 kHz. F. Filter Design Example We conclude discussion of the Bark bilinear transform with the filter design example of Fig. 10. A twelftth-order pole-zero filter was fit using Prony’s method =-=[33]-=- to the equalization function plotted in the figure as a dashed line. Prony’s method was applied normally over a uniformly sampled linear frequency grid in the example of Fig. 10(a), and over an appro... |

196 |
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Citation Context ...rectangular bandwidth (ERB) scale. The ERB of the auditory filter is assumed to be closely related to the critical bandwidth, but it is measured using the notched-noise method [13], [29], [28], [35], =-=[36]-=- rather than on classical masking experiments involving a narrowband masker and probe tone [5]–[7]. As a result, the ERB is said not to be affected by the detection of beats or intermodulation product... |

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Citation Context ...requency-scale warping is closely related to the topic of auditory filterbanks, which are nonuniform bandwidth bandpass filterbanks designed to imitate the frequency resolution of human hearing [10], =-=[11]-=-. Classical auditory filterbanks include constant- filterbanks such as the widely used thirdoctave filterbank. More recently, constant-Q filterbanks for audio have been devised based on the wavelet tr... |

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Citation Context ...or solution (21) can be multiplied by any desired application-dependent weighting. As another variation, an auditory frequency scale could be defined based on the cochlear frequency-to-place function =-=[38]-=-. In this case, a close relationship still exists between equal-place increments along the basilar membrane and equal bandwidth increments in the defined audio filterbank. Preliminary comparisons [38,... |

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Citation Context ...tory frequency-scale warping is closely related to the topic of auditory filterbanks, which are nonuniform bandwidth bandpass filterbanks designed to imitate the frequency resolution of human hearing =-=[10]-=-, [11]. Classical auditory filterbanks include constant- filterbanks such as the widely used thirdoctave filterbank. More recently, constant-Q filterbanks for audio have been devised based on the wave... |

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Citation Context ...plitting of the first formant as a result of four poles being used in the LPC fit instead of two. In 1983, the Bark bilinear transformation was developed independently for audio digital filter design =-=[19]-=-. In that work, the frequency response fit was carried out over an approximate Bark scale provided by the allpass transformation. The allpass coefficient was optimized as a function of sampling rate u... |

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Citation Context ...ing an FFT filterbank with nonuniformly spaced bins by taking the FFT of the outputs of a cascade chain of first-order allpass filters [17]. In 1980, “warped linear prediction” was proposed by Strube =-=[18]-=- for obtaining better formant models of speech: The frequency axis “seen” by LPC is made to approximate a Bark scale using the first-order allpass transformation. It was noted in [18] that setting the... |

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Citation Context ... bank [12]. Auditory filterbanks have also been based more directly on psychoacoustic measurements, leading to approximations of the auditory filter frequency response in terms of a Gaussian function =-=[13]-=-, a “rounded exponential” [14], and more recently the gammatone (or “Patterson–Holdsworth”) filterbank [10], [11]. The gammachirp filterbank further adds a level-dependent asymmetric correction to the... |

36 |
III, “Body modeling techniques for string instrument synthesis
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Citation Context ...red less than half the predictor model order for comparable performance. Very recently, the first-order allpass transformation was used to implement audio-warped filters directly in the warped domain =-=[24]-=-, [25]. In this application, a digital filter is designed over the warped frequency axis, and in its implementation, each delay element is replaced by a first-order allpass filter, which implements th... |

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Citation Context ...nk further adds a level-dependent asymmetric correction to the basic gammatone channel frequency response, thereby providing a yet more accurate approximation to the auditory frequency response [15], =-=[16]-=-. All auditory filterbanks can be seen as defining some linear to warped frequency mapping, since the filterbank output signals are nonuniformly distributed versus frequency. While this paper is conce... |

32 |
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Citation Context ...ebyshev design internally as an intermediate step. Further filter-design examples, including more on the Hankel-norm case, may be found in [19]. V. EQUIVALENT RECTANGULAR BANDWIDTH Moore and Glasberg =-=[34]-=- have revised Zwicker’s loudness model to better explain 1) how equal-loudness contours change as a function of level, 2) why loudness remains constant as the bandwidth of a fixed-intensity sound incr... |

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Citation Context ...sing emphasis on psychoacousticbased spectral measures [1]–[4]. One of the classic approaches is to analyze and process signal spectra over the Bark frequency scale (also called “critical band rate”) =-=[5]-=-–[9]. Based on the results of many psychoacoustic experiments, the Bark scale is defined so that the critical bands of human hearing have a width of one Bark. By representing spectral energy (in dB) o... |

19 |
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Citation Context ..., preserving optimality in the Chebyshev sense [20], [21]. Higher order allpass transformations have been used to convert lowpass or highpass prototype filters into multiple bandpass/bandstop filters =-=[22]-=-. Allpass transformations of order greater than one appear not to have been used in frequency warping applications, since allpass transformations of order map the unit circle to traversals of the unit... |

19 | The Bark Bilinear Transform - Smith, Abel |

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Citation Context ...ks have also been based more directly on psychoacoustic measurements, leading to approximations of the auditory filter frequency response in terms of a Gaussian function [13], a “rounded exponential” =-=[14]-=-, and more recently the gammatone (or “Patterson–Holdsworth”) filterbank [10], [11]. The gammachirp filterbank further adds a level-dependent asymmetric correction to the basic gammatone channel frequ... |

18 |
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Citation Context ... equivalent rectangular bandwidth (ERB) scale. The ERB of the auditory filter is assumed to be closely related to the critical bandwidth, but it is measured using the notched-noise method [13], [29], =-=[28]-=-, [35], [36] rather than on classical masking experiments involving a narrowband masker and probe tone [5]–[7]. As a result, the ERB is said not to be affected by the detection of beats or intermodula... |

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Citation Context ... allpass transformation has been used traditionally in digital filter design to scale the cut-off frequency of digital lowpass and highpass filters, preserving optimality in the Chebyshev sense [20], =-=[21]-=-. Higher order allpass transformations have been used to convert lowpass or highpass prototype filters into multiple bandpass/bandstop filters [22]. Allpass transformations of order greater than one a... |

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Citation Context ...ilterbanks such as the widely used thirdoctave filterbank. More recently, constant-Q filterbanks for audio have been devised based on the wavelet transform, including the auditory wavelet filter bank =-=[12]-=-. Auditory filterbanks have also been based more directly on psychoacoustic measurements, leading to approximations of the auditory filter frequency response in terms of a Gaussian function [13], a “r... |

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Citation Context ...lterbank further adds a level-dependent asymmetric correction to the basic gammatone channel frequency response, thereby providing a yet more accurate approximation to the auditory frequency response =-=[15]-=-, [16]. All auditory filterbanks can be seen as defining some linear to warped frequency mapping, since the filterbank output signals are nonuniformly distributed versus frequency. While this paper is... |

10 |
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Citation Context ...g applications, since allpass transformations of order map the unit circle to traversals of the unit circle, and a one-to-one mapping of the unit circle to itself is desired. 2 More recently, in 1994 =-=[23]-=-, an allpass coefficient of 0.62 was used to generate a frequency warping closely approximating the Bark scale for a sampling rate of 22 kHz. Experiments comparing the performance of warped LPC and “n... |

9 |
A new auditory model for the evaluation of sound quality of audio system
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Citation Context ...ION WITH the increasing use of frequency-domain techniques in audio signal processing applications such as audio compression, there is increasing emphasis on psychoacousticbased spectral measures [1]–=-=[4]-=-. One of the classic approaches is to analyze and process signal spectra over the Bark frequency scale (also called “critical band rate”) [5]–[9]. Based on the results of many psychoacoustic experimen... |

9 |
Warped filters and their audio applications
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Citation Context ...ss than half the predictor model order for comparable performance. Very recently, the first-order allpass transformation was used to implement audio-warped filters directly in the warped domain [24], =-=[25]-=-. In this application, a digital filter is designed over the warped frequency axis, and in its implementation, each delay element is replaced by a first-order allpass filter, which implements the unwa... |

7 |
Constraints of filter banks used for perceptual measurements
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Citation Context ...DUCTION WITH the increasing use of frequency-domain techniques in audio signal processing applications such as audio compression, there is increasing emphasis on psychoacousticbased spectral measures =-=[1]-=-–[4]. One of the classic approaches is to analyze and process signal spectra over the Bark frequency scale (also called “critical band rate”) [5]–[9]. Based on the results of many psychoacoustic exper... |

7 | Analytical expressions for critical band rate and critical bandwidth as a function of frequency - Zwicker, Terhardt - 1980 |

4 | 1/SC 29/WG 11, \ISO/IEC 11172-3: Information technology - coding of moving pictures and associated audio for digital storage media at up to about 1.5 mbit/s - part 3: Audio - JTC - 1993 |

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3 |
Stimulus variability and auditory filter shape
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Citation Context ...alent rectangular bandwidth (ERB) scale. The ERB of the auditory filter is assumed to be closely related to the critical bandwidth, but it is measured using the notched-noise method [13], [29], [28], =-=[35]-=-, [36] rather than on classical masking experiments involving a narrowband masker and probe tone [5]–[7]. As a result, the ERB is said not to be affected by the detection of beats or intermodulation p... |

2 |
Auditory filter asymmetry at 2000 Hz in 80 normal-hearing ears
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(Show Context)
Citation Context ...byshev fit. In fact, the optimal warping is within 0.04 Bark of the optimal warping. Since the experimental uncertainty when measuring critical bands is on the order of a tenth of a Bark or more [27]–=-=[30]-=-, we consider the optimal Chebyshev and least-squares maps to be equivalent psychoacoustically. (2) (3) (4)s700 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 7, NO. 6, NOVEMBER 1999 A. Comput... |

2 | Spectral pre-processing for audio digital filter design - Smith - 1983 |

2 | Signal reconstruction from modi#ed auditory wavelet transform - Irino, Kawahara - 1993 |

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2 | Glasberg, "A revision of Zwicker's loudness model - Moore, R - 1996 |

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1 |
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Citation Context ...l Chebyshev fit. In fact, the optimal warping is within 0.04 Bark of the optimal warping. Since the experimental uncertainty when measuring critical bands is on the order of a tenth of a Bark or more =-=[27]-=-–[30], we consider the optimal Chebyshev and least-squares maps to be equivalent psychoacoustically. (2) (3) (4)s700 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL. 7, NO. 6, NOVEMBER 1999 A. C... |

1 |
Auditory filter shapes at 8 and 10
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- 1990
(Show Context)
Citation Context ...by the equivalent rectangular bandwidth (ERB) scale. The ERB of the auditory filter is assumed to be closely related to the critical bandwidth, but it is measured using the notched-noise method [13], =-=[29]-=-, [28], [35], [36] rather than on classical masking experiments involving a narrowband masker and probe tone [5]–[7]. As a result, the ERB is said not to be affected by the detection of beats or inter... |

1 |
Steiglitz-McBride algorithm revisited-convergence analysis and accuracy aspects
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Citation Context ...ed solution. This is analogous to the Steiglitz–McBride algorithm for converting an equation-error minimizer to the more desired “output-error” minimizer using an iteratively computed weight function =-=[32]-=-. (a) (b) Fig. 4. (a) Optimal allpass coefficients &3I Y&3 PY and &3 iY plotted as a function of sampling rate ��X Also shown is the arctangent approximation &3 a IXHTUR�‘@Pa%A —�™�—�@HXHTSVQ��A“ IaP ... |

1 |
Expressions relating frequency, critical-band rate, and critical bandwidth
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Citation Context ... is 100 Hz wide for center frequencies below 500 Hz, and 20% of the center frequency above 500 Hz. Also plotted is the emprically determined formula, CB bandwidth in Hz % WR C UI� QaP Y with � in kHz =-=[37]-=-. The ERB’s are computed from (28), and the Bark CB bandwidths were computed by differencing the band-edge frequencies listed in Section II, plotting each difference over its corresponding band center... |

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1 | Constantinides, "Spectral transformations for digital filters - G - 1970 |