humans are continuously exposed to noise as they go about their daily life , from household 
 appliances through to transportation systems such as automobiles or trains .
however , 
 exposure to noise in the workplace can be a source of discomfort leading to health related 
 problems such as temporary hearing loss , or even permanent hearing loss if the correct 
 protective measures are not in place .
limits on the amount of daily noise exposure to which 
 workers may be exposed were tightened across europe in 20031 and are being enforced by member states ( for example , in the uk , the 
 directive is being enforced through the control of noise at work regulations2 ) .
similar legislation has been introduced in 
 the usa and japan . in order to assess the risk of noise to an operative
, one course of action is to measure 
 the noise level present in the operator s working environment .
one problem for the equipment 
 designer that exists in noise measurement is correlating the measured sound pressure level 
 ( spl ) to the perceived loudness level of noise ( iso 2263 ) .
1.moduli of the a and c frequency weightings used in iec 61672 - 1 and ansi s1.43 .
are defined in iec 61672 - 14 ) and the equivalent ansi 
 s1.43 - 1997/2007 standard5 .
they are 
 standardised for use in sound level meters and are required for the assessment of noise 
 levels for legislative regulations such as the european directive1 .
the a - weighting is an approximation of the 40-phon curve in 
 iso 226 and the c - weighting is an approximation to the 100-phon curve .
the a - weighting is a 
 general purpose model of the human response to noise and used for environmental noise and 
 hearing damage risk assessment .
both the a - weighting and the c - weighting are used to help in 
 the specification of hearing protection through the use of the 
moduli of the a and c frequency weightings used in iec 61672 - 1 and ansi s1.43 .
this paper takes the definitions of the a- and c - weightings and presents them in a format 
 that can be used in a digital implementation without the need for specialist digital signal 
 processing knowledge .
the paper takes the same form as one published previously by the 
 authors where digital weighting filters were presented for assessment of workers exposed to 
 whole - body and hand - arm vibration7 , 8 .
there are three different frequency weighting filters described in iec 61672 - 1 : a , c and z. 
 as described above , the a- and c - weightings are approximations to the equi - loudness contours 
 given in iso 226 .
the z - weighting is specified as having a flat frequency magnitude response 
 of zero db , and is thus equivalent to not having a filter at all ; therefore it will not be 
 discussed any further in this paper .
ansi s1.43 describes three frequency weighting filters : 
 a , b and c ; however only a and c are discussed in this paper because they are ones specified 
 in the health legislation1 .
both iec 61672 - 1 and ansi s1.43 present equations to calculate the frequency magnitude 
 responses of the weighting filters .
ansi s1.429 gives the equations required to calculate the analogue ( s - domain ) 
 frequency response , impulse response and step response of both the a- and c - weightings . 

equation 1 and table 1table 1.filter coefficients for the a - weighting filtera64 + ( 16 2 . 1 . 

4 ) + ( 64 
 1 ) + ( 32 2 ) + ( 32 3 ) + ( 64 
 4 ) + ( 32 2 .
1 ) + ( 16 1 ) + ( 16 2 . 1 . 
 3 .
3 ) + ( 4 
 4 . 2 . 3)a-128 + ( 64 2 . 1 . 

1 ) + ( 32 
 1 ) + ( 96 2 . 1 . 3 . 

( 320 
 1 )  ( 160 2 )  ( 160 3 )  ( 320 
 4 )  ( 96 2 .
1 )  ( 48 1 ) + ( 208 2 . 1 . 
 3 .
3 ) + ( 52 
 4 . 2 . 3)a512  ( 128 2 .
3 ) + ( 32 4 . 
 2 . 3)a128  ( 224 2 .
4 ) + ( 640 
 1 ) + ( 320 2 ) + ( 320 3 ) + ( 640 
 4 ) + ( 64 2 .
4 ) + ( 192 4 ) + ( 768 1 . 
 4)a128 + ( 224 2 . 1 . 

( 640 
 1 )  ( 320 2 )  ( 320 3 )  ( 640 
 4 ) + ( 64 2 .
4 ) + ( 320 
 1 ) + ( 160 2 ) + ( 160 3 ) + ( 320 
 4 )  ( 96 2 .
1 )  ( 48 1 ) + ( 208 2 . 1 . 
 3 .
1 ) + ( 32 
 1 ) + ( 96 2 . 1 . 3 . 

3 ) + ( 24 4 . 
 2 . 3)a64  ( 16 2 . 1 . 

4 )  ( 64 
 1 )  ( 32 2 )  ( 32 3 )  ( 64 
 4 ) + ( 32 2 .
1 ) + ( 16 1 ) + ( 16 2 . 1 . 
 3 .
4b128 4b32 4b192 4b32 4b128 4b48 4b32 4b16 4ga10the normalization constant ga is used to provide 0 db gain at 1 khz . in ansi s1.42 can be re - formatted to give the following s - domain 
 equations : the normalization constant ga is used to provide 0 db gain at 1 khz .
where : =2f and 
 f1=20.598997  hz , f2=107.65265  hz , 
 f3=737.86223  hz , f4=12194.217  hz , 
 and ga and gc are the gains ( i.e. 
 normalization constants ) required to make the response equal to 0 db at 1 khz . the 
 normalization constants are defined in iec 61672 - 1 as : ga=2.0 db and gc=0.062 db . in iec 61672 - 1
the frequencies are given as : f1=20.6  hz , 
 f2=107.7  hz , f3=737.9  hz and 
 f4=12194.0  hz . equations 1 and 2 are presented as analogue 
 system equations and can not directly be used in a digital ( sampled ) system .
firstly , the 
 equations need to be converted from analogue ( s - domain ) to digital ( z - domain ) .
modern data acquisition systems used for recording sound pressure level for analysis of 
 human exposure are usually digital systems . assuming a high quality digital recording has 
 been obtained , the first stage of analysis using iec 61672 - 1 or ansi s1.43 is to apply the 
 weighting filter to the data . as the data are recorded in the time - domain , and subsequent 
 processing
is most straight forward if carried out in the time - domain , it is better to also 
 perform filtering in the time - domain .
some metrics often quoted ( e.g. peak pressure ; r.m.s . ) 
 require a time - domain solution .
an important parameter in the digital data acquisition 
 process is the selection of an appropriate sampling rate .
the nyquist - shannon sampling 
 theorem states that the sampling frequency should not be less than twice the maximum 
 frequency that is to be recorded10 . for 
 time - domain analysis
higher sample rates are desirable due to waveform distortion that 
 occurs as the frequency of the signal of interest approaches the nyquist frequency .
the 
 actual sample rates required to meet the standards are discussed later in the paper . with a digital filter it is possible to modify the frequency and phase response of a system 
 by applying a series of multiplications and additions to the time - domain data ( for a 
 detailed description of digital filtering see oppenheim and schafer11 or rabiner and gold12 ) .
there are essentially two methods of filtering a signal 
 digitally : finite impulse response ( fir ) and infinite impulse response ( iir ) .
fir filters 
 work by multiplying the current and previous data samples by a set of coefficients and 
 summing the results .
the value of the current output sample , 
 y[n ] , is given by : where bk is the k filter 
 coefficient , x[n-1 ] is the previous input sample and m is 
 the filter order .
an iir filter uses previous output values in addition to previous input 
 values to calculate the current output sample value , which may be given by : whilst fir filters have some advantages ( for example , they are always stable and they can be 
 designed to have linear phase ) , their main disadvantage is that they would need a large number 
 of coefficients ( possibly thousands ) to match the frequency magnitude response of the 
 weighting filters given in the standards . as will be seen later , the iir a - weighting and 
 c - weighting filters only require twenty two and fourteen coefficients respectively . as the 
 number of coefficients are small , it is convenient to publish generic equations that are a 
 function of sampling frequency for each coefficient
. it is possible that a poor iir filter 
 design could be unstable ( due to the recursive nature of the filter ) ; however , the filters 
 presented here are stable .
there are a number of methods for deriving a digital filter from an analogue s - domain 
 filter .
one possible method is to convert the analogue ( s - domain ) impulse response given in 
 ansi s1.42 to a digital ( z - domain ) impulse response using the impulse invariance method ; 
 however , this is not straight forward because the selection of sample rate needs to account 
 for aliasing in the frequency domain . in this paper
, the authors propose the bilinear 
 transform method with frequency warping as it maps the entire j-axis in 
 the s - plane to one revolution of the unit circle in the z - plane11 which eliminates the aliasing problem . in the bilinear 

s in the analogue s - domain 
 equation is replaced by the bilinear transform defined by equation 512 .
there is , however , a non - linear relationship between the analogue frequency and the digital 
 frequency . pre
- warping the frequencies used in the analogue s - domain equations ( with the 
 substitution shown in equation 6 ) can eliminate this problem . where n is the normalised filter design frequency ( i.e. 
 2c /s or 2 
 fc /fs , where 
 c and fc are the centre frequency 
 and s and fs are the sampling 
 frequency ) and n is the normalised warped frequency .
to design the equivalent digital iir filter , equation 5 is substituted into the analogue 
 s - domain equation , which is simplified until it is in the form of equation 7 ( shown here for a 
 second - order filter , where m=2 ) .
the values for 
 n should then be warped using equation 6 to give the final 
 filter coefficients .
the filter coefficients ( a0 to 
 am and b0 to 
 bm ) may then be read directly .
this section contains a worked example to show the process of designing an iir digital 
 filter from an analogue s - domain equation .
consider a generic second - order 
 ( m=2 ) low - pass filter : and let  = 1570.796 rad / s ( i.e. f = 250 hz ) , q = stage one of the process is to apply the bilinear transform .
substituting equation 5 into 
 equation 8 gives : the second stage of the process is to expand and simplify equation 9 , to place it in the form 
 of equation 7 : ( 10 ) the third stage is to perform the frequency warping procedure .
the normalised centre 
 frequency , n = 2 / s = 
 2f / fs = 2  250/1000 = 1.57  rad / s . from equation 6 ,
the 
 warped frequency is given by n.thus , the iir filter coefficients are as 
 follows ( where n = 2.0 and q 
 = ):
2n = 1.66 a0 = 4 q + n 
 q + 2n = 9.66 once these six quantities have been calculated , they can be used with equation 4 in order to 
 calculate the filtered signal .
by applying the bilinear transform to equations 1 or 2 , it is possible to design the iir 
 weighting filter with the corresponding frequency magnitude response .
the general form for 
 an iir filter section is given by equation 7 , where ak to 
 bj are the coefficients defined in tables 1 and 2table 2.filter coefficients for the c - weighting filtera(16 4 ) + ( 16 
 1 ) + ( 4 1 
 2 ) + ( 16 4 . 

1 ) 
 + ( 8 1 ) + ( 16 
 4 . 
 1)a16 + ( 20 4 . 

1 )  ( 48 
 4 )  ( 4 4 
 2 )  ( 4 1 ) + ( 15 
 4 . 

1 )  ( 4 
 4 ) + ( 48 
 1 )  ( 16 4 . 

( 16 4 ) 
 + ( 4 4 ) + ( 4 
 1)b4 4b8 4b4 4b16 4b4 4b8 4b4 4gc10the normalization constant , gc , is used to provide 0 db gain at 1 khz .. the 
 values of  from these tables are obtained by warping the normalised 
 values of  ( from equations 1 and 2 ) using equation 6.for 
 example , .
the normalization constant , gc , is used to provide 0 db gain at 1 khz . as stated previously
, it is necessary to specify a minimum sample rate in order for the 
 filters to meet the tolerances specified in the standards .
3.digital implementation of the c - weighting filter ( fs=48 khz ) show the frequency magnitude responses of the a- and c - weighting filters designed 
 using a sample - rate of 48 khz and with the coefficients calculated according to tables 1 and 
 2 , respectively .
the frequency magnitude responses are calculated as 
 follows : digital implementation of the a - weighting filter ( fs=48 khz ) digital implementation of the c - weighting filter ( fs=48 khz ) where fft { } is a real to complex fast fourier transform , 
 mag ( ) is the magnitude of a complex number and 
 h[n ] is the impulse response of the filter calculated 
 from equation 4 where : as the impulse response of the filter decays to zero , it is not necessary to use a 
 windowing function .
this method of calculating the frequency response of a digital filter is 
 similar to the freqz ( ) function in matlab .
2 and 3 are the weighting curves and the class 1 & class 2 tolerances 
 presented in iec 61672 - 1 . from figs . 2 and 3
it may be seen that the digital filter presented 
 here meets the standard of the class 1 filter .
although the specification of the a- and 
 c - weighting curves is the same in both iec 61672 - 1 and ansi s.1 - 43 , the tolerances are 
 different .
iec 61672 - 1 defines two sets of tolerances : class 1 and class 2 ; whereas , ansi 
 s.1 - 43 defines three sets of tolerances : type 0 , type 1 and type 
 2 .
table 3table 3.minimum sampling frequency required to meet the full tolerances specified in iec 
 61672 - 1 and ansi s.1 - 43a - weightingc - weightingiec 61672 - 1 class 135 khz35 khziec 61672 - 1 class 220 khz20 khzansi s.1 - 43 type 071 khz72 khzansi s.1 - 43 type 135 khz35 khzansi s.1 - 43 type 220 khz20 khz shows the minimum sample rate required to meet the full tolerances specified 
 in the standards .
note that this is the sample rate of the filter and not necessarily that 
 of the recorded data , as the recorded data could be re - sampled before being filtered .
excessive noise exposure 
 can cause health problems and , therefore , it is important that the worker s noise exposure 
 is measured and/or monitored by the equipment manufacturer or by the employer .
human 
 exposure to noise may be measured using microphones ; however , weighting filters are also 
 required to correlate the physical noise measurements to the perceived loudness of the 
 noise .
the weighting filters are presented in the relevant standards ( iec 61672 - 1 and ansi 
 s1.43 ) , but not in a format that can be implemented directly as a digital filter .
this paper 
 has presented a method of implementing the frequency weightings as digital filters , which 
 can be used with any numerical processing software ( such as a spreadsheet or programming 
 language running on a pc , mobile device or embedded system ) .
weighting filters were 
 generated using the method described in this paper and the frequency responses compared to 
 those given in the standards and were thus verified to be within the allowable tolerances . 
 using the information presented ,
it is possible to design an iir digital filter of arbitrary 
 sample rate to weight the recorded time - domain noise spl data , necessary for calculating the 
 a- or c - weighted exposure and thus the maximum health - related exposure times in accordance 
 with the international standards and the eu directive .
 many workers are exposed to noise in their industrial environment . 
 excessive noise 
 exposure can cause health problems and therefore it is important that the worker s noise 
 exposure is assessed . this may require measurement by an equipment manufacturer or the 
 employer . human exposure to noise may be measured using microphones ; however , weighting 
 filters are required to correlate the physical noise sound pressure level measurements to 
 the human s response to an auditory stimulus . 
 iec 61672 - 1 and ansi s1.43 describe suitable 
 weighting filters , but do not explain how to implement them for digitally recorded sound 
 pressure level data . by using the bilinear transform , 
 it is possible to transform the 
 analogue equations given in the standards into digital filters . 
 this paper describes the 
 implementation of the weighting filters as digital iir ( infinite impulse response ) filters 
 and provides all the necessary formulae to directly calculate the filter coefficients for 
 any sampling frequency . 
 thus , the filters in the standards can be implemented in any 
 numerical processing software ( such as a spreadsheet or programming language running on a 
 pc , mobile device or embedded system ) . 
Introduction
Standard Definition of the A- and C-Weightings
Design of Digital Weighting Filters Using the Bilinear Transform
Bilinear Transform
Worked example: Low-pass filter
IIR Digital Weighting Filter Coefficients
Conclusions