Lowpass, 30kHz Bessel filter offers high performance for audio applications
( 01 May 2006 )
Troy Murphy, Analog Devices, San Jose, CA
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 multi-frequency 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, 30kHz, 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.
In this application, the filter processes audio signals that swing above and below ground, and its amplifiers draw power from positive and negative ±2.5V 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.
Figure 2 shows the filter’s measured magnitude response for a 1V-rms input signal. The filter’s passband gain of 0dB is flat within 1.2dB for frequencies as high as 20kHz. With its -3dB point at 30kHz, an eighth-order Bessel presents a theoretical attenuation of -110dB at 300kHz, decreasing at -160dB/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 300kHz and higher.
Figure 3 illustrates the filter’s phase shift and its group delay, which remains relatively constant at roughly 17µsec, even for frequencies as high as 40kHz. 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:
Group delay= ―δф/фf.
At dc, resistor R1 sets the filter’s input resistance at 383Ω. 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 ±15V power supplies, replace the AD8656 with a higher voltage amplifier, such as Analog Devices’ AD8672 low distortion, low-noise (3.8nV/√Hz), dual operational amplifier.
