原创 【音频测试基础知识】加权滤波器

A-weighting in detail

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A, B, C and D weighting


The A-weighting curve is one of a set of four, defined in various standards relating to sound level measurement as A, B, C, and D. Curves A, B and C are for low, medium and high loudness sounds. D is specifically for measuring very loud aircraft noise.

The A-weighing curve is based on the 40-phon equal-loudness contour for typical human hearing, being a rough approximation of that curve after inversion to indicate gain rather than level. Although our absolute threshold of hearing is around 0 Phons, this is a very quiet level indeed, and only realised in specially isolated conditions. 40 phon is the sort of level likely to exist in a normal quiet environment.




2 Equal-loudness contours

In 1933 Fletcher and Munson (1933) investigated the variation in apparent loudness of sounds with frequency and found that they needed to plot a set of 'equal loudness' contours, because the ear’s frequency response varied with loudness level. They presented volunteer subjects with pure tones at various frequencies, and asked them to adjust the level until they judged each tone to sound equal in loudness to a 1kHz reference tone. Then they repeated the experiment with different levels of reference tone. This revealed that our ears are much less sensitive to low frequency sounds, especially at low levels. Robinson and Dadson repeated the experiment in 1956, and obtained somewhat different results, which were considered to be more accurate, and became the basis for the ISO standard, ISO226 until modern experiments began to cast doubt on their validity, resulting in a survey by the International Standards Organisation. This led to the release, in 2003, of a new standard: ISO 226:2003, which was derived from the reported results of various experimenters. The graphs shown here compare first the Fletcher-Munson and then the Robinson-Dadson results with the modern curves, and it can be seen that while the former appear to be badly out at high levels in the low-frequency region they are in quite good agreement at low levels. The Robinson-Dadson curves, which until 2003 formed the basis for ISO226, differ significantly, showing around 12dB greater sensitivity in the region around 150Hz. The recent ISO survey report commented that it was fortunate that the 40-phon contour of Fletcher and Munson, on which the A-weighting filter was based, does agree quite well with the new ISO226 standard.

3 Possible errors

The reasons for these discrepancies appear not to have been investigated, but several factors make the experiment very difficult. Firstly, loudness is only a subjective quality, and it is quite hard to say when two tones at different frequencies sound equally loud. This is especially true at frequencies below 100Hz where we sense the sound less as a tone and more as a feeling.

Then there is the difficulty of providing an undistorted sound source of known level. Even today, most loudspeakers have at least one percent of harmonic distortion at most frequencies, and very few indeed would be able to achieve even this at 20Hz and 100dB SPL. Fletcher and Munson used headphones, but in 1933 most headphones had poor low-frequency response, and it seems likely that what their subjects were hearing below 100Hz consisted mostly of higher harmonics, which would explain their apparently superior sensitivity at high levels. The steep slope of the curves at low frequencies means of course that we are very sensitive to harmonics, and it is now well known that our brains even have the special ability to ‘postulate’ a missing fundamental, so that we think we hear the fundamental even when it is not there, a phenomenon that makes small loudspeakers sound more acceptable than they otherwise would!

Robinson and Dadson used loudspeakers for their experiment, and one can only wonder at how they managed to generate a flat distortion-free level of 125dB SPL at 20Hz in 1956 (if they did)! Their results show an almost level response from 1kHz down to 200Hz compared to a 10dB loss of sensitivity on the modern ISO 226 curve, in a region that poses no special difficulties for loudspeaders. It is possible that a room resonance or reinforcing reflection off the floor caused the apparent increase in sensitivity around 150Hz, because today a large room surrounded by absorbent wedges on all sides, together with a false floor supporting the listener, would be necessary to avoid such effects. However, one would expect them to have been fully aware of these problems, which would also be expected to cause anomalies at lower frequencies.

To confuse matters, my own tests using headphones, carefully calibrated using an in-ear microphone, on my own ears, show no fall in sensitivity at 200Hz! Were Robinson and Dadson right after all, and if so why the discrepancy in the new ISO 226 curve? One explanation I can suggest is that the ear’s low frequency response depends on the tautness of the eardrum, which is governed by the stapedius and tensor timpani muscles, which tense under loud conditions, and remain tensed for hours and even days. Several modern studies were by Japanese researchers. Could it be that their subjects were adapted to noisy city life, and not kept in quiet conditions for several days prior to and during testing?

Another problem that arises when trying to generate very high levels at low frequencies for these experiments is the need for an extremely low level of amplifier noise, because any hiss is not masked by the tone, and so must be kept 125dB or more below the tone if it is to be inaudible. This is just about impossible today, and inconceivable in the days of valves (vacuum tubes), though a possible way round the problem is to use an acoustic filter during the low-frequency tests (such as a quilt hung in front of the loudspeaker!).

4 Frontal versus side presentation

A complicating factor in the derivation of equal-loudness contours is the fact that our ears have a different frequency response in every direction. This becomes increasingly true above about 2kHz, and does not affect low frequencies, being caused by head-masking and the complicated shape of the outer ear, or pinna.

Very low frequencies pass around the head and are sensed equally by each ear purely as pressure variation, but as the wavelength of a sound becomes comparable with the dimensions of our head (1kHz equates to 320mm or roughly 1 foot) this starts to get in the way so that each ear becomes progressively less sensitive to sound from its opposite side as frequency increases. At still higher frequencies the outer ear begins to focus the sound, forming a cavity with various resonances that are excited to varying degrees by sounds from different directions, in both the horizontal and vertical planes. These two mechanisms, both involving difference in loudness between the ears are used by the brain to enable us to locate the direction of sounds in both the horizontal and vertical planes, along with another mechanism which uses the different time-of arrival at each ear, or phase difference, to determine direction. The latter mechanism takes over in the lower 100Hz to 2kHz region, and is made use of in stereo reproduction, though it cannot give us information about sounds originating above or behind us where the pinna comes into effect.

Head masking and pinna effects have been quantified in terms of another set of contours referred to as ‘Head-related transfer functions’, which also vary considerably between individuals at high frequencies. They are relevant to the determination of equal-loudness contours because we are considerably more sensitive to high frequency sound coming directly from each side, as in headphone listening, than to sound from a single loudspeaker at centre-front, with stereo presentation from two speakers at 60 degrees separation coming somewhere in between. Modern equal-loudness contours are for frontal presentation, but although Fletcher and Munson used headphones, the ISO survey lists their experiment as using ‘compensated headphones’. Most modern headphones intended for listening to stereo are in fact designed with a severe dip in their response around 3-10kHz, without which they would sound too bright; a fact that is not commonly recognised.

5 Towards a better noise-weighting standard

A modern version of A-weighting might be expected to take the form of the inverse 40-phon ISO 226 curve, as shown here compared to both the traditional A-weighting curve and the later ITU-R 468 weighting. This is not necessarily the best shape for it though, because above 2kHz each ISO curve is the result of averaging results from many individuals, some of whom have dips where others have peaks. A more sensible approach would therefore be to plot the results for many individuals and then draw a smooth curve through the maxima, since for noise measurement purposes we are more interested in what someone might hear than in what one person does not hear. We might then choose to err on the high side (by perhaps 6dB) around 6kHz, because it is now known that we are most sensitive to random noise in this region, as reflected in the ITU-R 468 weighting curve, and in practice environmental noise tends to be part tonal (music and some machinery) and part random (wind noise, traffic, and machinery).

Sound level meters currently use rms detection, which would be reasonable if our perception of loudness was related to power, but it in not. A series of clicks or tone-bursts sound almost as loud as the continuous sound, provided that the gaps between them are not too long, showing that our perception of loudness has more to do with peak level than average or rms. As the duration of a tone-burst is reduced below about 50ms, however, is starts to sound progressively quieter, showing that our ears take time to respond in each frequency band. To be perceptually valid therefore, sound level and noise level measurements must use some form of ‘quasi-peak’ detector with carefully devised integration times, as is the case with the ITU-R 468 standard. The traditional sound-level meter with traditional A-weighting is therefore starting to look rather irrelevant for all practical purposes.

6 Computed noise-measurement

Better still would be a system that aimed to model our hearing characteristics using a bank of overlapping narrow-band filters, with varying bandwidths, spaced across the audible spectrum. Their outputs would be combined using an algorithm that also incorporated quasi-peak detection and non-linearity to produce a result that remained reasonably valid for all types of sound at all levels. This need not be too difficult using modern digital processing, and would be relevant for all applications, from environmental and aircraft noise measurement to measurements on audio systems and telephone networks.

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