This example shows how to obtain designs for the most common weighting filters - A-weighting, C-weighting, C-message, ITU-T 0.41, and ITU-R 468-4 - using the audio weighting filter designer, fdesign.audioweighting, in the DSP System Toolbox™.

In many applications involving acoustic measurements, the final sensor is the human ear. For this reason, acoustic measurements usually attempt to describe the subjective perception of a sound by this organ. Instrumentation devices are built to provide a linear response, but the ear is a nonlinear sensor. Therefore, special filters, known as weighting filters, are used to account for the nonlinearities.

**A and C Weighting (ANSI® S1.42 standard)**

You can design A and C weighting filters that follow the ANSI S1.42 [1] and IEC 61672-1 [2] standards. An A-weighting filter is a band pass filter designed to simulate the loudness of low-level tones. An A-weighting filter progressively de-emphasizes frequencies below 500 Hz. A C-weighting filter removes sounds outside the audio range of 20 Hz to 20 kHz and simulates the loudness perception of high-level tones.

The ANSI S1.42 standard requires that the filter magnitudes fall within a specified tolerance mask. The standard defines two masks, one with stricter tolerance values than the other. A filter that meets the tolerance specifications of the stricter mask is referred to as a Class 1 filter. A filter that meets the specifications of the less strict mask is referred to as a Class 2 filter. You define the type of class you want in your design by setting the Class property to 1 or 2. The choice of the Class value will not affect the filter design itself but it will be used to render the correct tolerance mask in FVTOOL.

A and C-weighting filter designs are based on direct implementation of the filter's transfer function based on poles and zeros specified in the ANSI S1.42 standard. The filters only have one design method referred to as 'ansis142'.

The following code obtains an IIR Class 1 filter design for A-weighting with a sampling rate of 48 kHz.

h = fdesign.audioweighting('WT,Class','A',1,48e3)

h = Response: 'Audio Weighting' Specification: 'WT,Class' Description: {'Weighting type, Class'} NormalizedFrequency: false Fs: 48000 WeightingType: 'A' Class: 1

Ha = design(h,'ansis142','SystemObject',true); hfvt = fvtool(Ha); legend(hfvt, 'A-weighting')

The A and C weighting standards specify tolerance magnitude values for up to 20 kHz. In the following example we use a sampling frequency of 28 kHz and design a C-weighting filter. Even though the Nyquist interval for this sampling frequency is below the maximum specified 20 kHz frequency, the design still meets the Class 2 tolerances. The design, however, does not meet Class 1 tolerances due to the small sampling frequency value.

h = fdesign.audioweighting('WT,Class','C',2,28e3); Hcclass2 = design(h,'SystemObject',true); setfilter(hfvt,Hcclass2); legend(hfvt, 'C-weighting, Class 2')

**ITU-R 468-4 Weighting Filter**

ITU-R 468-4 recommendation [3] was developed to better reflect the subjective loudness of all types of noise, as opposed to tones. ITU-R 468-4 weighting was designed to maximize its response to the types of impulsive noise often coupled into audio cables as they pass through telephone switching facilities. ITU-R 468-4 weighting correlates well with noise perception, since perception studies have shown that frequencies between 1 kHz and 9 kHz are more "annoying" than indicated by A-weighting.

You design a weighting filter based on the ITU-R 468-4 standard for a sampling frequency of 80 kHz using the following code. You can choose from frequency sampling or equiripple FIR approximations, or from a least P-norm IIR approximation. In all cases, the filters are designed with the minimum order that meets the standard specifications (mask) for the sampling frequency at hand.

h = fdesign.audioweighting('WT','ITUR4684',80e3)

h = Response: 'Audio Weighting' Specification: 'WT' Description: {'Weighting type'} NormalizedFrequency: false Fs: 80000 WeightingType: 'ITUR4684'

Hitur1 = design(h,'allfir','SystemObject',true); setfilter(hfvt,Hitur1{1}); addfilter(hfvt,Hitur1{2}); legend(hfvt,'ITU-R 468-4 FIR equiripple approximation', ... 'ITU-R 468-4 FIR frequency sampling approximation')

Hitur2 = design(h,'iirlpnorm','SystemObject',true); setfilter(hfvt,Hitur2); legend(hfvt,'ITU-R 468-4 IIR least P-norm approximation')

While IIR designs yield smaller filter orders, FIR designs have the advantage of having a linear phase. In the FIR designs, the equiripple design method will usually yield lower filter orders when compared to the frequency sampling method but might have more convergence issues at large sampling frequencies.

**ITU-T 0.41 and C-message Weighting Filters**

ITU-T 0.41 and C-message weighting filters are band pass filters used to measure audio-frequency noise on telephone circuits. The ITU-T 0.41 filter is used for international telephone circuits. The C-message filter is typically used for North American telephone circuits. The frequency response of the ITU-T 0.41 and C-message weighting filters is specified in the ITU-T O.41 standard [4] and Bell System Technical Reference 41009 [5], respectively.

You design an ITU-T 0.41 weighting filter for a sampling frequency of 24 kHz using the following code. You can choose from FIR frequency sampling or equiripple approximations. The filters are designed with the minimum order that meets the standard specifications (mask) for the sampling frequency at hand.

h = fdesign.audioweighting('WT','ITUT041',24e3); Hitut = design(h,'allfir','SystemObject',true); setfilter(hfvt,Hitut{1}); addfilter(hfvt,Hitut{2}); legend(hfvt,'ITU-T 0.41 FIR equiripple approximation', ... 'ITU-T 0.41 FIR frequency sampling approximation')

You design a C-message weighting filter for a sampling frequency of 51.2 kHz using the following code. You can choose from FIR frequency sampling or equiripple approximations or from an exact IIR implementation of poles and zeros based on the poles and zeros specified in [6]. You obtain the IIR design by selecting the 'bell41009' design method. The FIR filter approximations are designed with the minimum order that meets the standard specifications (mask) for the sampling frequency at hand.

h = fdesign.audioweighting('WT','Cmessage',51.2e3); Hcmessage1 = design(h,'allfir','SystemObject',true); setfilter(hfvt,Hcmessage1{1}); addfilter(hfvt,Hcmessage1{2}); legend(hfvt,'C-message FIR equiripple approximation', ... 'C-message FIR frequency sampling approximation')

Hcmessage2 = design(h,'bell41009','SystemObject',true); setfilter(hfvt,Hcmessage2); legend(hfvt,'C-message weighting (IIR)')

**Conclusions**

We have presented the design of A, C, C-message, ITU-T 0.41, and ITU-R 468-4 weighting filters. Some of the audio weighting standards do not specify exact pole/zero values, instead, they specify a list of frequency values, magnitudes and tolerances. If the exact poles and zeros are not specified in the standard, filters are designed using frequency sampling, equiripple, and/or IIR least P-norm arbitrary magnitude approximations based on the aforementioned list of frequency values, attenuations, and tolerances. The filter order of the arbitrary magnitude designs is chosen as the minimum order for which the resulting filter response is within the tolerance mask limits. Designs target the specification mask tolerances only within the Nyquist interval. If Fs/2 is smaller than the largest mask frequency value specified by the standard, the design algorithm will try to meet the specifications up to Fs/2.

In the FIR designs, the equiripple design method will usually yield lower filter orders when compared to the frequency sampling method but might have more convergence issues at large sampling frequencies.

**References**

[1] 'Design Response of Weighting Networks for Acoustical Measurements', American National Standard, ANSI S1.42-2001.

[2] 'Electroacoustics Sound Level Meters Part 1: Specifications', IEC 61672-1, First Edition 2002-05.

[3] 'Measurement of Audio-Frequency Noise Voltage Level in Sound Broadcasting', Recommendation ITU-R BS.468-4 (1970-1974-1978-1982-1986).

[4] 'Specifications for Measuring Equipment for the Measurement of Analogue Parameters, Psophometer for Use on Telephone-Type Circuits', ITU-T Recommendation 0.41.

[5] 'Transmission Parameters Affecting Voiceband Data Transmission-Measuring Techniques', Bell System Technical Reference, PUB 41009, 1972.

[6] 'IEEE® Standard Equipment Requirements and Measurement Techniques for Analog Transmission Parameters for Telecommunications', IEEE Std 743-1995 Volume , Issue , 25, September 1996.

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