Notch Filter Extends Spectrum-Analyzer Range

You don''t necessarily need specialized, expensive equipment to measure total harmonic distortion of today''s high-performance audio devices.

Bruce Tibbetts, Credence Systems -- Test & Measurement World, 1/1/2000

High-performance audio devices exhibit less than –100-dB total harmonic distortion (THD). You might expect that general-purpose spectrum analyzers—with dynamic ranges of about 90 dB—would be unable to measure this harmonic distortion. Further, you’d expect an ATE system’s 16-bit digitizer—with a 96-dB dynamic range—also to be deficient for measuring harmonic distortion below –100 dB. Yet, by adding a notch filter, you can turn a general-purpose spectrum analyzer into a valuable piece of audio test equipment, or you can convert a 16-bit digitizer into a high-performance component of your audio-device ATE systems.

Dynamic range is usually expressed as the difference, in dB, between the highest (|S|_MAX) and lowest (|S|_MIN) amplitude portions of a signal, or between the highest amplitude signal to which a device can respond linearly and the noise level of the device.¹ You can calculate dynamic range as follows:

Dynamic Range (dB) = 20 log₁₀ (|S|_MAX/|S|_MIN)

The total harmonic distortion (THD) of a complex waveform is the ratio of the root-mean-square (rms) value of the sum of the squared individual harmonic amplitudes (|SSQ|_RMS) to the rms value of the fundamental frequency (|f₁| _RMS):²

If you have fundamental and harmonics values in decibels instead of rms values, use Equation 1.

To calculate the THD using a spectrum analyzer, measure the amplitudes of the fundamental signal and the relevant harmonics; then apply the appropriate formula.

The Test Setup
To demonstrate what could happen if you attempted to measure a high-quality sine wave with a general-purpose spectrum analyzer, I set up a test using a signal source and a spectrum analyzer. I used the Audio Precision System Two, which has a residual distortion specification of less than –130 dBc for 1-kHz output signals. I set the System Two’s output to 1 Vrms (0 dBv) at 1 kHz. For the spectrum analyzer, I employed a Hewlett Packard HP 3585A, which has a typical dynamic range of 80 dB. I connected the source output directly to the spectrum analyzer’s high-impedance (1-MW) input and set the spectrum analyzer to center on the 1-kHz signal and to span it by 5 kHz. This span allows viewing of the second and third harmonics of the 1-kHz sine wave. For this test, I measured only the second and third harmonics, because the source has very low distortion and using only two harmonics simplifies the calculations. In a real application, you might need to measure 10 harmonics.

Figure 1. With an audio source connected directly to a spectrum analyzer’s input, I measured a –85.1-dBv second harmonic, which is nearly all attributable to spectrum-analyzer performance.

Figure 2. Employing a notch filter between the audio source and the spectrum analyzer permits a gain increase that enhances accuracy.

Figure 3. To make meaningful measurements with a notch filter and spectrum analyzer combination, you must know the filter’s frequency response.

The result of the sweep ( Fig. 1) shows a second harmonic with a level of –85.1 dBv and a third harmonic at –87.5 dBv. If you apply these values to Equation 1, you find the result for the THD ( Equation 2).

This result is far from correct, however; as noted above, the System Two’s total distortion is less than –130 dB. Consequently, the –83-dB measured harmonic distortion is virtually all due to the harmonic distortion of the spectrum analyzer itself.

Measurement Using the Notch Filter
As the first step in shielding the analyzer’s THD from the measurement of the source THD, I attenuated the 1-kHz signal using a passive notch filter from Analog Precision (see “Notch Filters Suppliers”). The filter I chose has a notch depth of about 70 dB. Figure 2 shows the spectrum-analyzer sweep using the notch filter.

As shown, the amplitude of the second harmonic is much lower than the –85.1 dBv that the first measurement indicated. Because the notch filter removes most of the fundamental signal, I could set the spectrum analyzer gain to a much higher level. The increased gain improves the measurement accuracy of the low-level harmonics. Moreover, the harmonic distortion of the spectrum analyzer does not affect the measurement, since the ratio of the residual, or filtered, fundamental amplitude to the harmonic amplitudes is well within the spectrum analyzer’s dynamic range and THD specifications.

Notch-Filter Frequency Response
Because the notch filter filters out some of the harmonics, you must compensate for the notch filter’s frequency response (Fig. 3) before you can use the results of the second measurement. Figure 3 shows that the notch filter has an attenuation value of –9.6 dB at the second harmonic and –5.8 dB at the third harmonic. The corresponding levels for the second and third harmonics from Figure 2 are –140.8 and –139.9 dBv, respectively. Correcting for the notch-filter attenuation results in these values:

Second Harmonic = –140.8 dBv – (–9.6 dB)   = –131.2 dBv

Third Harmonic     = –139.9 dBv – (–5.8 dB)  = –134.1 dBv

You can substitute these values into Equation 1 to get the result shown in Equation 3.

The difference between the first measurement and the second measurement, corrected for the notch, is approximately 46 dB, which represents an improvement by a factor of 200. This result is a fairly accurate measurement of the THD of the test signal (within a few decibels) and is much closer to the actual specification of the System Two. For the measurement to have even greater accuracy, either the amplitude of the fundamental signal could be increased beyond 1 Vrms or the spectrum analyzer could be replaced with one that has an input voltage range lower than –38 dBv (12.5 mV).

Using the Notch Filter with a Digitizer
In an ATE environment, you would likely use a digitizer and digital-signal processing to evaluate an analog signal. The theoretical dynamic range, in decibels, for a digitizer’s N-bit ADC is 20log10 (2^N), where N is the ADC’s number of bits. For an ideal 16-bit ADC, the theoretical dynamic range is approximately 96.33 dB. From this result, you can see that the ADC’s dynamic range will limit the measurement capability, as did the spectrum analyzer’s. Figure 4 shows a test setup that uses a notch filter to extend a digitizer’s dynamic range. This application includes five functional blocks:

Figure 4. Notch-filter, selectable-gain, and programmable-low-pass-filter blocks combine with a digitizer to make a THD measurement system. With the notch filter switched out of the system during step 1 of a THD measurement, the post-notch gain is switched out as well.

• The input gain stage, which adjusts the amplitude of the input signal to fit the ADC’s full-scale range (FSR). In Figure 4, the FSR is 5V –(–5V), or 10 V.

• A programmable notch filter. Unlike the passive notch filter used with the spectrum-analyzer, the programmable filter uses a high-Q active filter that has negligible attenuation of the harmonics.

• The post-notch gain stage, which amplifies the signal output from the notch filter and has three gain settings: 0 dB for when the notch filter is switched out, and 20 dB or 40 dB to match the depth of the notch filter when active.

• The programmable low-pass filter, which limits signal bandwidth to prevent aliases from occurring when the signal is digitized.

• The ADC, which digitizes the input signal.

As in the test using the spectrum analyzer, the measurement involves two steps. The first requires you to measure the amplitude of the fundamental signal. Since the ADC is a 16-bit converter with a 10-V FSR, its LSB is:

FSR/(2^N) = 10 V/(2¹⁶) = 152.59 mV

Note that defined in terms of the LSB, dynamic range in decibels is 20 log₁₀(FSR/LSB). The dynamic range, when expressed this way, matches the dynamic range calculated earlier:

20 log (10 V / 152.59 mV) = 96.33 dB

This formula is used again in the next step of this process, which requires you to make a second measurement with the notch filter selected and the post-notch gain stage set as shown in Figure 5.

Figure 5. The THD measurement illustrated in Figure 4 continues with notch-filter and post-notch-gain stages switched in. A 40-dB post-notch gain compensates for 40-dB of notch-filter attenuation.

With a notch-filter depth of greater than 40 dB, the fundamental signal amplitude is reduced by a factor greater than 100. Because the amplitude of the signal is reduced, it no longer reaches the ADC’s full-scale range. To compensate, the 40-dB post-notch gain stage amplifies the notched signal back to the full-scale range of the ADC. As a result of the post-notch gain, the LSB value is reduced by 40 dB (a factor of 100). As with the passive notch-filter and spectrum-analyzer example, the programmable notch filter serves to extend the dynamic range of the 16-bit ADC, yielding a new LSB value:

LSB = 152.59 mV / 100 = 1.5259 mV

You can use the new value can to recalculate the effective dynamic range in decibels:

Dynamic range = 20 log (10 V/1.5259 mV)  = 136.33 dB

From this result, notice that the dynamic range was increased by the amount of post-notch gain that was applied (96.33 dB + 40 dB = 136.33 dB).

Notching out the fundamental and amplifying the remaining harmonics raised the harmonics above the noise floor and increased the resolution of the notched measurement. The dynamic range of the first measurement was the limiting factor in making an accurate measurement of the harmonics of the input signal. The result is that you can use a two-pass test using a notch filter and post-notch gain stage to accurately evaluate single-tone sine waves that have a THD greater than –100 dB.

a)

b)

Figure 6. (a) Using a 20-bit digitizer but not a notch filter yields a –102-dB second harmonic. (b) Adding the notch filter yields the more accurate –138.6-dB result.

Resolution of the converter alone is not enough to ensure the best measurement of low-level harmonics. For this example, the limiting specification is the distortion of the 20-bit converter, which is –100 dB relative to the analyzer’s FSR. Figure 6b shows the result when using the notch filter. The second harmonic is at approximately –138.6 dB while the third harmonic is at approximately –145.6 dB.

The “Q” of the notch filter will determine the level of attenuation seen when using the filter. For the test with the spectrum analyzer, the notch filter used was a passive twin-T design. Table 1 lists the harmonic attenuation for ideal notch filters with several different values for Q.

Another item to be aware of is the input impedance of the spectrum analyzer. If you use a passive notch filter, you’ll need a spectrum analyzer that has a high input impedance (for instance, 1 MV), such as the HP 3585A. Note too that any passive twin-T notch filter connected to the spectrum analyzer’s input will have a series capacitance that forms a capacitive divider with the spectrum analyzer’s input-to-ground capacitance (greater than 30 pF for the HP 3585A).

The effect of this capacitive divider is especially evident for higher frequency filters (greater than 100 kHz), where the notch filter’s capacitors are relatively small. The capacitive divider has the interesting effect that a swept sine wave never comes back up to the 0-dB level after it has passed through the bottom of the notch. Harmonic distortion measurements will not be affected, however, as long as you adjust the measured harmonics to compensate for the effect of the notch filter’s frequency response (which includes the capacitive divider). T&MW

FOOTNOTES
1. Metzler, Bob, Audio Measurement Handbook, Audio Precision, Beaverton, OR, p. 154. www.audioprecision.com/techsupport/audio_measurement_handbook.html.

2. The IEEE Standard Dictionary of Electrical and Electronics Terms, 6th ed., IEEE, New York, NY. 1997. ISBN 1-55937-833-6.

3. The Engineering Staff of Analog Devices, Analog-Digital Conversion Handbook, 3rd ed., Prentice Hall, Englewood Cliffs, NJ, 1986, ISBN 0-13032-848-0.

4. Williams, Arthur B., and Fred J. Taylor, Electronic Filter Design Handbook, 2nd ed., McGraw-Hill Publishing Co., 1988.

Bruce Tibbetts is a principal applications engineer for Credence Systems. His focus is on the development of mixed-signal test solutions and new test methodologies for consumer devices. He has more than 15 years experience in the ATE industry and holds a B.S.E.E. from Northeastern University in Boston. E-mail: bruce_tibbetts@credence.com.