It sometimes occurs that signals are captured with A-weighting applied to the data by the acquisition device. This can be a problem if, for example, you wish to use the data in a hearing test or to use it for a structural vibration analysis. Now, A-weighting allegedly mimics what the ear does to a signal. If we play back an A weighted sound then we perceive a double A-weighted signal which is clearly not intended. When doing structural work it is usually the lower frequencies, say 2kHz or less, that is generally required. A-weighting seriously attenuates the low frequencies and also applies gain above 1kHz.
Another scenario is that sometimes A-weighting is used as a convenient way of compressing the data.
The graphs below illustrate the ability to successfully unweight signals within Prosig’s DATS software. The signal used in this example was measured at the left ear of the subject without any of weighting (signal name L Ear, shown in green) so that we had a reference. The signal was then A-weighted (signal name L Ear_A, shown in red), and then un-A-weighted (signal name L Ear_A_N, shown in blue).
The data was sampled at 45450 samples/second and covered a 100 dB dynamic range. The first graph (Figure 1) below gives an overview of the entire frequency span from dc to sample rate/2. The original data (green) and the recovered data (blue) are practically indistinguishable. The low and high frequency roll-offs of the A-weighted spectrum (red) and the slight gain in the 2kHz to 6kHz region are evident.
Figure 2 illustrates the low frequency ends of the spectra. There is very good agreement down to 40 Hz and within 3dB at 25Hz. Signal recovery 40dB down is achieved.
The high frequency end (Figure 3) shows a similar excellent data recovery, with data recovery from 60 dB down, being within 0.1 dB at 22kHz and within 3dB at 22.4kHz.
The set of third octave spectra of the signals shown in Figure 4 further illustrates the recovery of the original signals.
We must emphasize that the un-weighting is carried out entirely in the time domain. No Fourier transforms, convolutions or similar analyzes are used. The method uses bilinear z transform filtering. A typical fragment of the time history signals is shown in Figure 5. Note the compressed range of the A weighted signal.
The recovered signal is not identical but is very, very similar to the original and is generally with the same phase. The accuracy of the recovery is further confirmed by viewing the transfer functions as shown below.
Wide range transfer function
Expansion around 0dB
Note that the phase of the combined transfer function is generally at 360o which is equivalent to a zero phase shift. Also observe that the transfer functions are all unity at 1kHz as required for A-weighting. This ensures correct scaling.
Finally we compare the set of statistical parameters in the table below. There is excellent agreement with the general result that the recovered signal is very slightly smaller by about 0.5 dB overall. The greatest differences are in the “odd” order statistics (mean, skew, M5) all of which tend to zero, and in consequence are of little significance. It is particularly interesting that the advance statistical parameters (Average Frequency, No. of peaks/sec, Irregularity Factor and Spectral Bandwidth) are in good agreement.
No. of peaks/sec
For the technically minded the A-weighting filter has a set of zeros which in a direct implementation of the inverse compensation filter become poles leading to an unstable infinite response. These need to be eliminated in a sympathetic and balanced manner. Equivalent unweighting is also available for B and C weighted signals.
Chief Signal Processing Analyst (Retired) at Prosig
Dr Colin Mercer was formerly at the Institute of Sound and Vibration Research (ISVR), University of Southampton where he founded the Data Analysis Centre. He then went on to found Prosig in 1977. Colin retired as Chief Signal Processing Analyst at Prosig in December 2016. He is a Chartered Engineer and a Fellow of the British Computer Society.