Troy Murphy
EDN
Thanks to its property of applying an equal amount of delay to all frequencies below its cutoff frequency, the Bessel linear-phase filter sees service in audio applications in which it's necessary to remove out-of-band noise without degrading the phase relationships of a multifrequency in-band signal. In addition, the Bessel filter's fast step response and freedom from overshoot or ringing make it an excellent choice as a smoothing filter for an audio DAC's output or as an antialiasing filter for an audio ADC's input. Bessel filters are also useful for analyzing the outputs of Class D amplifiers and for eliminating switching noise in other applications to improve accuracy of distortion and oscilloscope-waveform measurements.
Although the Bessel filter provides flat magnitude and linear-phase – that is, uniform group-delay – responses within its passband, it has worse selectivity than Butterworth or Chebyshev filters of the same order, or number of poles. Thus, to achieve a given level of stopband attenuation, you need to design a higher order Bessel filter, which, in turn, requires careful selection of amplifiers and components to achieve the lowest levels of noise and distortion.
Figure 1 shows a schematic for a high-performance, eighth-order, 30-kHz, lowpass Bessel filter. This design uses standard values for 1%-tolerance resistors and 5%-tolerance ceramic capacitors. As an alternative, you can use 10%-tolerance capacitors at the expense of increased group-delay variance within the passband. For best results, use temperature-stable capacitors.
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| Figure 1. |
Two dual op amps and a handful of passive parts implement a high-performance, |
In this application, the filter processes audio signals that swing above and below ground, and its amplifiers draw power from positive and negative ±2.5 V supplies. Rail-to-rail output capability helps achieve maximum output-voltage swing at these low supply voltages. To achieve a high SNR in high-quality audio service, the amplifiers must exhibit unity-gain stability and low inherent noise. For example, Analog Devices' AD8656 low-noise, precision-CMOS dual op amp meets all of these requirements.
Connecting the amplifiers as inverting-gain stages maintains constant input-common-mode voltage and helps minimize distortion. Using less-than-1-kΩ resistors throughout the circuit reduces the resistors' thermal-noise contributions. Each AD8656 amplifier contributes less than 3 nV/√Hz of noise across a 30-kHz bandwidth, and the total noise over a 30-kHz bandwidth measures less than 3.5 µV rms. For a 1 V-rms input signal, the circuit yields an SNR of better than 109 dB, and, for a 1-kHz, 1 V-rms input signal, the circuit yields a THD+N (total-harmonic-distortion-plus-noise) factor of better than 0.0006%.
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| Figure 2. | The measured amplitude-versus-frequency response of the circuit in Figure 1 shows a change of scale on the right vertical axis. |
Figure 2 shows the filter's measured magnitude response for a 1 V-rms input signal. The filter's passband gain of 0 dB is flat within 1.2 dB for frequencies as high as 20 kHz. With its –3-dB point at 30 kHz, an eighth-order Bessel presents a theoretical attenuation of –110 dB at 300 kHz, decreasing at –160 dB/decade at higher frequencies. This characteristic provides sufficient attenuation of repetitive noise that switched-mode power supplies and other sources induce, which typically occurs at frequencies of 300 kHz and higher.
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| Figure 3. | Measured within the passband of DC to 30 kHz, the Bessel filter’s phase-shift and group-delay characteristics display excellent uniformity and linearity. |
Figure 3 illustrates the filter's phase shift and its group delay, which remains relatively constant at roughly 17 µsec, even for frequencies as high as 40 kHz. Note the linear scale on Figure 3's frequency axis, which clearly illustrates the filter's linear-phase behavior within the passband. The following equation defines group delay as the negative partial derivative of phase shift with respect to frequency:
At DC, resistor R1 sets the filter's input resistance at 383 Ω. If your application requires higher input impedance, you can insert a unity-gain buffer ahead of the filter at the expense of increased distortion and noise. For applications that require operation from ±15 V power supplies, replace the AD8656 with a higher voltage amplifier, such as Analog Devices' AD8672 low-distortion, low-noise (3.8-nV/√Hz), dual operational amplifier.
