Design of an Unweighted Audio Filter with Flat Frequency Response

July 23, 2023 by John Woodgate

This article presents the design of an unweighted (Z-weighted) audio filter with a flat frequency response from 20 Hz to 20 kHz for the measurement of audio noise.

There are multiple methods for taking audio noise measurements that are used by sound level meters. These are generally characterized by the frequency response curves of the filters. While some audio filters are designed to mimic the frequency response of the human ear at various sound levels, this article will focus on the detailed design of a Z-weighted (unweighted) audio filter with a flat frequency response from 20 Hz to 20 kHz. The filter can be used in conjunction with the wideband voltage and current meter described in one of my previous articles.


The Unweighted (Z-Weighting) Frequency Response

The unweighted, flat frequency response curve is illustrated in Figure 1. The band limiting filter frequency response bounds are specified as a ‘mask’ in the ITU Recommendation ITU-R BS.468-4 for the Measurement of Audio-frequency Noise Voltage Level in Sound Broadcasting. International Standard IEC 61672 defines a similar, flat response over audio frequencies as “Z-weighted” or zero weighted.


Frequency response bounds for the ITU unweighted filter

Figure 1. Frequency response bounds for the unweighted filter. Image used courtesy of ITU


The measured response of the filter we design has to fit inside the mask, which requires a flat response over almost all of the audio frequency range. This technique gives results, measured as RMS values, that can be used to design for lower noise levels.


Design of the Unweighted Audio Filter

The unweighted filter is made up of two filters that will be connected in series:

  1. A 12 dB/octave high-pass filter with the -3dB cutoff frequency close to 20 Hz
  2. An 18 dB/octave lowpass filter with a -3 dB cutoff frequency close to 20 kHz.

It is possible to meet the mask requirements with Butterworth filters (which do not have a peak in the frequency response curve) tweaked to have 0.5 dB peaks. The frequency response, expressed as A in decibels of a Butterworth filter of order n is given by:

$$A = 10\log_{10}(1 + \Omega^{2n})$$

Ω depends on the type of filter:

  • Low-pass filter: \( \Omega = \omega \text{/} \omega_c \)
  • High-pass filter: \( \Omega = \omega_c \text{/} \omega \)


ω is the signal frequency

ωc is the -3 dB cut-off frequency


The mask requires a second-order high-pass filter to provide the 12 dB/octave rising response from well below 1 Hz to 22.4 Hz, and a third-order low-pass filter to provide the falling response from 22.4 kHz upwards.

Figure 2 shows the schematic of the tweaked Butterworth solution.


Schematic of the quasi-Butterworth unweighted audio filter

Figure 2. Schematic of the quasi-Butterworth unweighted audio filter (click to enlarge)


Adjusting the Filter Response

The filters use the equal component value versions of the Sallen and Key configuration. The tweaking is fairly easily done by increasing the values of the gain-setting resistors R7 and R12 until the response peaks by 0.5 dB compared with the response at 1 kHz at each end.

R2 is labeled “Adjust on test.” What that means is that with the filter connected to the Wideband Voltmeter, you put 1 Vrms at 1 kHz into the voltmeter (on the 1 V range, of course) and adjust R2 until the output is also 1 V.


Output Resistance Compensation

You may well wonder about amplifier U1B near the center top of Figure 2. It's there because the output resistance of U2A is in series with C3, and at frequencies much higher than 20 kHz, the resistance is not negligible compared to the reactance of C3, so the 18 dB/octave roll-off isn’t achieved. It is not negligible because the open-loop output resistance (not given in the data sheet) is reduced by negative feedback, but the open-loop gain is quite low at high frequencies, as is usual with opamps.

For a TL072 general-purpose op amp, the open-loop gain is only about 30 at 100 kHz, so feedback can't reduce the output resistance much. The LM4562 audio op amp is hardly any better in this respect. U1B provides a lower output resistance, allowing the response requirement to be met.


Designing for System Flexibility and Re-use

You may also notice J2. This makes frequent use of this and other external filters easier. A 5-pin DIN socket is added to block 4 of the Wideband Voltmeter, so that not only are the GO and RETURN signal connections established, but also the external filter gets its DC supplies from the Voltmeter.

The external filter can have either a flying lead with a plug or another socket, so that a connecting cable can be used. It would be good to have the wires to pins 1 and 5 individually shielded so as to prevent stray capacitance coupling across the filter. Four-core individually shielded cables, quite small in diameter, can be obtained, and the shields, of course, provide the necessary fifth conductor.

Figure 3 shows the modified block 4 of the Wideband Voltmeter, incorporating the 5-pin DIN socket. It is worthwhile keeping the BNC connectors so that measuring instruments and experimental external filters can easily be connected.


Wideband voltmeter schematic with added 5-pin connector

Figure 3. Modified Block 4 of the wideband voltmeter to add the 5-pin connector (click to enlarge)


Measurement Results for the Unweighted Audio Filter Design

Figure 4 shows the measured frequency response of the Quasi-Butterworth filter. The decibel scale is large so as to show the small peaks at each end of the frequency range. The noise level is -50 dB referred to 1 V, and because of the way the gain switching of the Voltmeter is arranged, the signal applied to the filter can always be between 100 mV and 1 V, except on the most sensitive range.


Butterworth filter unweighted audio response

Figure 4. Frequency response of the quasi-Butterworth unweighted filter


Alternative Audio Filter Design Using Chebyshev Filters

It would be technically superior to use Chebyshev filters. These are filters whose frequency response has the sharpest possible fall-off in gain beyond a peak of a specified height. The math for designing such filters is somewhat complicated, so pre-calculated results are given in textbooks. One such source is the ‘Active Filter Cookbook’ written by Don Lancaster (ISBN 9780750629867).

Figure 5 shows the schematic of an unweighted audio filter using Chebyshev building blocks. Don't forget to adjust the value of R2, as explained above.


Schematic of the Chebyshev unweighted filter

Figure 5. Schematic of the Chebyshev unweighted filter (click to enlarge)


Figure 6 shows the measured frequency response of the Chebyshev filter. The curve has slightly steeper slopes over the first 4 dB of attenuation.


Chebyshev unweighted filter response

Figure 6. Frequency response of the Chebyshev filter


Figure 7 shows the overall frequency response of the wideband voltmeter with the Chebyshev filter inserted. The bandwidth of the voltmeter itself is so much wider than that of the filter that it has no effect on the filtered response.


Figure 7. Frequency response of the wideband voltmeter with Chebyshev filter


Weighting for the Response of the Human Ear

An alternative to the Z-weighted flat frequency response is the A-weighted response which corresponds to the sensitivity of the otologically normal human ear to sounds at a level of 40 phon. Th ‘A-weighting’ is specified in the sound-level meter standard IEC 61672-1. Figure 8 shows the frequency response curves for the A, C, and Z-weightings defined by the standard.


Comparison of the frequency response bounds for A, C, and Z-weighting

Figure 8. Comparison of the frequency response for A, C, and Z-weighting. Image used courtesy of Comsol


It is used with an RMS meter and has become traditionally used for measuring other things that it shouldn’t be, because it grossly underestimates the effect of low-frequency sounds at levels above 40 phons. For example, it underestimates a 100 phon 20 Hz signal by 40 dB.

Conversely, a sound system with a noise level of 40 phon would be a very noisy system; a good system would not exceed 30 phon. A-weighting overestimates the level of a 20 Hz signal at 30 phon by 5 dB. But the use of A weighting is so well established that no-one wants to make any changes.


Choosing the Filter Response When Measuring Audio Noise

It is important to know the frequency response of the audio noise measurement device you are designing or using. Depending upon your application, you may want to mimic the response of the human ear using an A-weighted filter. But, you need to always remember that this is heavily filtering the lower frequency signals.


All images provided by the author, unless otherwise noted.
  • J
    jlm1948 July 29, 2023

    The announced frequency response is achieved only with a load impedance of about 50-70 kohms. Otherwise there is a significant LF hump (2dB@34Hz).
    Simulation does not show any significant difference between this circuit, featuring the additional voltage-follower (U1b) in the FB loop and a simpler version using a direct connection between the output of U2a and C3.
    It’s not a surprise, since the output impedance of U2a and U1b are very similar.
    The transient behaviour may be different but should not matter for measurement purpose.

    Like. Reply
  • M
    MisterBill2 September 24, 2023

    Where would I find the balance of the circuit for this useful audio voltmeter? Or is it commercially available? It would be quite useful in evaluating microphones and microphone circuits.

    Like. Reply