APPLICATION BULLETIN ® Mailing Address: PO Box 11400 • Tucson, AZ 85734 • Street Address: 6730 S. Tucson Blvd. • Tucson, AZ 85706 Tel: (602) 746-1111 • Twx: 910-952-111 • Telex: 066-6491 • FAX (602) 889-1510 • Immediate Product Info: (800) 548-6132 A LOW NOISE, LOW DISTORTION DESIGN FOR ANTIALIASING AND ANTI-IMAGING FILTERS By Rick Downs (602) 746-7327 Many customers have requested more information about the analog low-pass filters that appear in many of our PCM audio data sheets. They are used for antialiasing in front of ADCs or for smoothing on the output of DACs. The following bulletin is an excellent primer on the subject. —Ed. In any digitizing system, antialiasing and anti-imaging filters are used to prevent the signal frequencies from “folding back” around the sample frequency and causing false (or alias) signals from appearing in the signal we are attempting to digitize. Very often, these filters must be very complex, high order analog filters in order to do their job effectively. As sampling rates of converter systems have increased, however, oversampling may be used to reduce the filters’ stopband attenuation requirements(1)(2). In digital audio systems, 4x oversampling may be used, and it can be shown(3) that for an antialiasing filter (which precedes the ADC), a simple sixth order filter may be used. For the output side, after the DAC, a simple third order filter may be used. Realizing these filters in a way that maintains extremely low noise and low distortion then becomes a challenge. Compact disk player manufacturers began using a filter topology that was described many years ago—the Generalized Immittance Converter (GIC)(4). This topology allows one to easily realize active filters beginning from a passive filter design. In addition, the GIC filter provides extremely low distortion and noise, at a reasonable cost. Compared with more familiar feedback filter techniques, such as Sallen & Key filter topologies, the GIC filter can be shown to have superior noise gain characteristics, making it particularly suitable for audio and DSP type applications(5). We use this type of filter on our demonstration fixtures for the PCM1750 and PCM1700, dual 18-bit ADC and DAC, respectively. When sending out schematics of these demonstration fixtures, very often the first question is, “What are those filters anyway?” Well, they’re GIC filters, and here’s how you design them and how they perform. Stepping through this design process will allow you to modify these designs for a different cutoff frequency for your particular application. A more detailed treatment of the theory behind these filters may be found in Huelsman and Allen(6). As stated above, for oversampling digital audio applications, third and sixth order filters are adequate. Thus, we may design our first GIC filter by designing a third order filter. The filter characteristic most desirable for sensitive DSP type applications is linear-phase. The linear-phase filter is sometimes called a Bessel (or Thomson) filter. The linearphase filter has constant group delay. This means that the phase of the filter changes linearly with frequency, or that © 1991 Burr-Brown Corporation 0.9852H 1 L3 L1 0.3350H C2 0.8746F R4 1Ω 2 3 FIGURE 1. Passive Third Order, Linear-Phase, Low-Pass Filter Prototype. 4 the group delay is constant. These filters maintain phase information for sensitive DSP applications such as correlation, and preserve transient response. These characteristics are critical in audio applications as well, because they affect sound quality greatly. Thus, we begin the design process by selecting a passive, third order linear-phase filter design that will be realized using this active approach. The passive design shown in Figure 1 is neither a Butterworth nor a Bessel response; it is something in between. The component values for this particular response, optimized for phase linearity and stopband attenuation, were found through exhaustive computer simulations and empirical analysis. Component values for standard Butterworth and Bessel responses may be found in standard filter tables, such as those available in Huelsman and Allen(7). This circuit is then transformed to an active circuit by multiplying all circuit values by 1/s, which changes all inductors to resistors, all resistors to capacitors, and all capacitors to Frequency Dependent Negative Resistors (FDNRs). These FDNRs have the characteristic impedance of 1 s 2C and may be realized using the GIC circuit. Thus, L1 becomes R1, C2 becomes 1/s2C2, L3 becomes R3, and the terminating R1 0.9852Ω 7 8 9 10 11 12 R3 0.3350Ω 1 s2C2 0.8746Fs 6 C4 1F 13 14 FDNR: units are farad-seconds (Fs) FIGURE 2. Filter of Figure 1 Transformed by Multiplying All Component Values by 1/s. AB-026A Printed in U.S.A. March, 1991 15 16 resistor R4 becomes C4, as shown in Figure 2. D = (R12 • R14 • C13 • C15)/R11 Thus by setting R11 = R12 = 1 and C13 = C15 = 1, D is entirely determined by the value of R14. For the FDNR of Figure 2, R14 = 0.8746Ω. The FDNR is then realized by the GIC circuit shown in Figure 3. The value of the FDNR is determined by VIN The entire third order filter circuit is shown in Figure 4. This circuit now must be scaled in frequency to give the desired cutoff frequency, and then must be scaled in impedance to allow for the use of reasonable sized component values. The filter circuits found in filter tables, such as that in Figure 1 and the active realization of this passive circuit (Figures 2 and 4), are designed for a cutoff frequency of ω = 1 rad/s. To make the filter have the cutoff frequency we desire, we must scale it in frequency by the scaling factor ΩN = 2πfc VOUT R11 R12 A1 A2 C13 R14 This scaling factor is applied to all frequency-determining components—capacitors in this case. The example filter will be designed for audio, so we might consider a cutoff frequency of 20kHz. However, linear-phase filters tend to rolloff very slowly, causing 1-2dB attenuation before the cutoff frequency; generally audio systems prefer to have their frequency response out to 20kHz to be within 0.1dB. The example filter then will have a cutoff frequency of 40kHz, commonly used in many of today’s CD players. All capacitor values are divided by the frequency scaling factor, so C15 FIGURE 3. Frequency Dependent Negative Resistor (FDNR) Realized Using Generalized Immittance Converter (GIC). VIN R3 R1 VOUT R11 1Ω 0.9852Ω R12 1Ω 0.3350Ω C4 1F A2 C13 1F A1 R14 0.8746Ω C15 1F FIGURE 4. Third Order, Linear-Phase Realization of Circuit Shown in Figure 2. VIN R3 R1 VOUT R11 1Ω 0.9852Ω R12 1Ω C13 3.98µF A1 R14 0.8746Ω C15 3.98µF FIGURE 5. Circuit of Figure 4 Scaled to a 40kHz Cutoff Frequency. 2 0.3350Ω C4 3.98µF A2 2 R1 VIN R3 R11 3.92kΩ 3.92kΩ 3 6 7 1 2 A1A 5 OPA2604 6 VOUT OPA627 1.33kΩ C4 1000pF 3 R12 3.92kΩ A2 A1B 2 C13 1000pF 1 2 1 OPA2604 R14 3.48kΩ C15 1000pF FIGURE 6. Circuit of Figure 5 Scaled in Impedance (note use of buffer amplifier to reduce output impedance of the filter). C13 = C15 = C4 = 3.98µF. with extremely high GBW would be required. An example of a sixth order, 40kHz Butterworth filter realized in this fashion is shown in Figure 8, but its frequency response (Figure 9) is less than hoped for due to the GBW limitations described above. The filter (Figure 5) could now be built, but the large capacitor values and low resistance values could pose practical problems. To alleviate this, the impedances of the circuit are scaled by an impedance scale factor: Zn = A simpler solution is to cascade two of the third order sections designed above. This cascaded design (Figure 10) works equally well for most applications. Present C value Desired C value By choosing the desired C value as 1000pF, Zn = 3.97x103. This impedance scaling factor then is multiplied by all resistor values to find the new resistor values, and divides all the capacitor values, taking them from the present values to the desired capacitance. Figure 11 (a-d) shows the performance of this cascaded filter design. Note that the phase linearity and THD + N are still excellent using this approach. The final filter design is shown in Figure 6. Since the output impedance of this filter is relatively high, it’s a good idea to buffer the output using an op amp voltage follower. Amplitude and phase response of this filter is shown in Figure 7a. Figure 7b is a closer look at the amplitude response in the passband—the frequency response is flat well within 0.1dB out to 20kHz. REFERENCES (1) R. Downs, “DSP Oversampling to Quiet Noise,” EE Times, pg. 68, 8 August 1988. Figure 7c is a plot of the frequency response of the filter (solid line) and the filter’s deviation from linear phase (dotted line). Note the phase scale; the phase response is well within 0.1° of linear phase in the 1kHz-20kHz region, where the ear is most sensitive to phase distortion. (3) R. Downs, “Unique Topology Makes Simple, LowDistortion Antialiasing Filters,” to be published. Figure 7d is a plot of the total harmonic distortion plus noise (THD + N) of this filter versus frequency. At about –108dB, this would be suitable for digital systems with true 18-bit converter performance! (5) R. Downs, “Unique Topology Makes Simple, LowDistortion Antialiasing Filters,” to be published. (2) R. Downs, “High Speed A/D Converter Lets Users Reap Benefits of Oversampling,” Burr-Brown Update, Vol. XIV, No. 2, pg. 3, May 1988. (4) S.K. Mitra, Analysis and Synthesis of Linear Active Networks, John Wiley & Sons, Inc., New York, pg. 494, 1969. (6) L.P. Huelsman, P.E. Allen, Introduction to the Theory and Design of Active Filters, McGraw-Hill, New York, 1980. To make a sixth order filter, you can repeat the design process above from a passive realization and directly implement a filter. This implementation is very sensitive to the gain-bandwidth product (GBW) match of all of the op amps used, however; for a 40kHz cutoff frequency, an op amp (7) Ibid. 3 AMPLITUDE AND PHASE RESPONSE OF FILTER CIRCUIT IN FIGURE 6 PASSBAND RESPONSE DETAIL OF FILTER CIRCUIT IN FIGURE 6 40 1 30 0.8 0 –10 90 –20 –30 0 Phase –40 –90 Response (dBu) Amplitude 0.4 0.2 Amplitude 90 0 0 –0.2 Phase –0.4 –50 –0.6 –180 –60 –270 –0.8 –270 –70 –360 –1 20 100 1k 10k Frequency (Hz) 100k 200k –360 20 100 1k 10k Frequency (Hz) (a) 100k 200k (b) TOTAL HARMONIC DISTORTION + NOISE OF FILTER CIRCUIT IN FIGURE 6 vs FREQUENCY AMPLITUDE AND DEVIATION FROM LINEAR PHASE FOR FILTER IN FIGURE 6 –40 40 –5 20 Amplitude .040 –20 .020 –40 0 Phase –60 –.020 –80 –.040 THD + N (dBr) 0 –60 Phase (degrees) Response (dBu) –90 –180 Phase (degrees) 0.6 10 Phase (degrees) Response (dBu) 20 –70 –80 –90 –100 –110 –120 –100 20 100 1k 10k Frequency (Hz) 20 100k 200k 100 1k Frequency (Hz) 10k 20k Note phase scale—deviation from linear phase in critical 1kHz-20kHz region is well within 0.1°. NOTE: Referred to 6Vp-p full-scale signal typical of most digital audio converters. (c) (d) FIGURE 7. Performance Details of Figure 6 Circuit. 4 5 A1A 60.4kΩ 100pF 100pF 100pF 8.87kΩ A2A 68.1kΩ 100pF 39.2kΩ 39.2kΩ A1B 39.2kΩ 39.2kΩ 60.4kΩ A2B A3A 29.4kΩ 100pF 10.2kΩ 100pF 39.2kΩ 39.2kΩ 20 100 100k 200k –270 –360 –60 –70 1k 10k Frequency (Hz) –180 –50 –90 0 –40 90 –30 Phase Amplitude –20 –10 0 10 20 30 40 FIGURE 9. Amplitude (solid line) and Phase (dotted line) Response of Filter Circuit in Figure 8. (Note flattening of stopband response near 150kHz due to inadequate GBW of operational amplifiers used.) Response (dBu) A3B VOUT 100pF FIGURE 8. Sixth Order Butterworth Filter Realized by Method Outlined in Text (actual circuit would require output buffer amplifier to lower output impedance). VIN Phase (degrees) 6 1 2 A1A OPA2604 7 3.92kΩ 5 6 1000pF 3.48kΩ 1000pF 7.32kΩ 7.32kΩ 2 3 1 2 1 OPA2604 A1B 1.33kΩ 1000pF 3 1 2 A3A OPA2604 1 1 2 A2A OPA2604 7 3.92kΩ FIGURE 10. Sixth Order Linear-Phase Filter Made by Cascading Two Third Order Filters. VIN 2 5 6 1000pF 3.48kΩ 1000pF 7.32kΩ 7.32kΩ 2 3 1 2 1 OPA2604 A2B 1.33kΩ 1000pF 5 6 1 2 A3B OPA2604 7 VOUT PASSBAND RESPONSE DETAIL OF FILTER CIRCUIT IN FIGURE 10 AMPLITUDE AND PHASE RESPONSE OF FILTER CIRCUIT IN FIGURE 10 40 1 30 0.8 0 –10 –20 90 Phase –30 0 Response (dBu) Amplitude 0.4 0.2 Amplitude 90 0 –0.2 –90 –40 –90 –50 –180 –0.6 –180 –60 –270 –0.8 –270 –70 –360 –1 20 100 1k 10k Frequency (Hz) –360 20 100k 200k 100 1k 10k Frequency (Hz) (a) 100k 200k (b) AMPLTIUDE AND DEVIATION FROM LINEAR PHASE FOR FILTER IN FIGURE 10 TOTAL HARMONIC DISTORTION + NOISE FOR FILTER CIRCUIT IN FIGURE 10 vs FREQUENCY 40 –40 –5 20 Amplitude .040 –20 .020 –40 0 Phase –60 THD + N (dBr) 0 Phase (degrees) Response (dBu) 0 Phase –0.4 Phase (degrees) 0.6 10 Phase (degrees) Response (dBu) 20 –70 –80 –90 –60 –.020 –80 –.040 –110 –.060 –120 –100 20 100 1k 10k Frequency (Hz) –100 100k 200k 20 100 1k Frequency (Hz) 10k 20k Note phase scale—deviation from linear phase in critical 1kHz-20kHz region is well within 0.1°. NOTE: Referred to 6Vp-p full-scale signal typical of most digital audio converters. (c) (d) FIGURE 11. Performance Details of Figure 10 Circuit. The information provided herein is believed to be reliable; however, BURR-BROWN assumes no responsibility for inaccuracies or omissions. BURR-BROWN assumes no responsibility for the use of this information, and all use of such information shall be entirely at the user’s own risk. Prices and specifications are subject to change without notice. No patent rights or licenses to any of the circuits described herein are implied or granted to any third party. BURR-BROWN does not authorize or warrant any BURR-BROWN product for use in life support devices and/or systems. 7

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