A Low Noise, Low Distortion Design for Antialiasing and Anti-Imaging Filters

Home , Cutoff frequency

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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 L L analog low-pass filters that appear in many of our PCM 1 3 1 audio data sheets. They are used for antialiasing in front of 0.9852H 0.3350H ADCs or for smoothing on the output of DACs. The follow- C2 R4 ing bulletin is an excellent primer on the subject. —Ed. 0.8746F 1Ω 2 In any digitizing system, antialiasing and anti-imaging filters are used to prevent the signal frequencies from “folding 3 back” around the sample frequency and causing false (or alias) signals from appearing in the signal we are attempting FIGURE 1. Passive Third Order, Linear-Phase, Low-Pass to digitize. Very often, these filters must be very complex, Filter Prototype. high order analog filters in order to do their job effectively. 4 As sampling rates of converter systems have increased, the group delay is constant. These filters maintain phase however, oversampling may be used to reduce the filters’ information for sensitive DSP applications such as correla- stopband attenuation requirements(1)(2). In digital audio sys- tion, and preserve transient response. These characteristics tems, 4x oversampling may be used, and it can be shown(3) are critical in audio applications as well, because they affect that for an antialiasing filter (which precedes the ADC), a sound quality greatly. simple sixth order filter may be used. For the output side, Thus, we begin the design process by selecting a passive, after the DAC, a simple third order filter may be used. third order linear-phase filter design that will be realized Realizing these filters in a way that maintains extremely low using this active approach. The passive design shown in noise and low distortion then becomes a challenge. Figure 1 is neither a Butterworth nor a Bessel response; it is 6 Compact disk player manufacturers began using a filter something in between. The component values for this par- topology that was described many years ago—the General- ticular response, optimized for phase linearity and stopband ized Immittance Converter (GIC)(4). This topology allows attenuation, were found through exhaustive computer simu- one to easily realize active filters beginning from a passive lations and empirical analysis. Component values for stan- 7 filter design. In addition, the GIC filter provides extremely dard Butterworth and Bessel responses may be found in low distortion and noise, at a reasonable cost. Compared standard filter tables, such as those available in Huelsman with more familiar feedback filter techniques, such as Sallen and Allen(7). This circuit is then transformed to an active 8 & Key filter topologies, the GIC filter can be shown to have circuit by multiplying all circuit values by 1/s, which changes superior noise gain characteristics, making it particularly all inductors to resistors, all resistors to capacitors, and all suitable for audio and DSP type applications(5). capacitors to Frequency Dependent Negative Resistors 9 We use this type of filter on our demonstration fixtures for (FDNRs). These FDNRs have the characteristic impedance the PCM1750 and PCM1700, dual 18-bit ADC and DAC, of respectively. When sending out schematics of these demon- 1 10 stration fixtures, very often the first question is, “What are s2C those filters anyway?” Well, they’re GIC filters, and here’s and may be realized using the GIC circuit. Thus, L becomes how you design them and how they perform. Stepping 1 11 R , C becomes 1/s2C , L becomes R , and the terminating through this design process will allow you to modify these 1 2 2 3 3 designs for a different cutoff frequency for your particular

application. A more detailed treatment of the theory behind R1 R3 12 these filters may be found in Huelsman and Allen(6). 0.9852Ω 1 0.3350Ω As stated above, for oversampling digital audio applications, 2 C4 s C2 13 third and sixth order filters are adequate. Thus, we may 0.8746Fs 1F design our first GIC filter by designing a third order filter. The filter characteristic most desirable for sensitive DSP 14 type applications is linear-phase. The linear-phase filter is FDNR: units are farad-seconds (Fs) sometimes called a Bessel (or Thomson) filter. The linear- 15 phase filter has constant group delay. This means that the FIGURE 2. Filter of Figure 1 Transformed by Multiplying phase of the filter changes linearly with frequency, or that All Component Values by 1/s. 16

SBAA001 resistor R4 becomes C4, as shown in Figure 2. D = (R12 • R14 • C13 • C15)/R11

The FDNR is then realized by the GIC circuit shown in Thus by setting R11 = R12 = 1 and C13 = C15 = 1, D is entirely

Figure 3. The value of the FDNR is determined by determined by the value of R14. For the FDNR of Figure 2, Ω R14 = 0.8746 . The entire third order filter circuit is shown in Figure 4. This circuit now must be scaled in frequency to give the desired V V IN OUT cutoff frequency, and then must be scaled in impedance to R 11 allow for the use of reasonable sized component values. The filter circuits found in filter tables, such as that in Figure R12 A2 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 A1 C 13 make the filter have the cutoff frequency we desire, we must scale it in frequency by the scaling factor

R14 Ω π N = 2 fc This scaling factor is applied to all frequency-determining C 15 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 roll- off very slowly, causing 1-2dB attenuation before the cutoff FIGURE 3. Frequency Dependent Negative Resistor frequency; generally audio systems prefer to have their (FDNR) Realized Using Generalized Immit- frequency response out to 20kHz to be within 0.1dB. The tance Converter (GIC). 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

R1 R3 VIN VOUT Ω R Ω 0.9852 11 0.3350 C4 1Ω 1F

R12 1Ω A2

A1 C13 1F

R14 0.8746Ω

C15 1F

FIGURE 4. Third Order, Linear-Phase Realization of Circuit Shown in Figure 2.

R1 R3 VIN VOUT Ω Ω 0.9852 R11 0.3350 1Ω C4 3.98µF R12 1Ω A2

C A1 13 3.98µF

R14 0.8746Ω

C15 3.98µF

FIGURE 5. Circuit of Figure 4 Scaled to a 40kHz Cutoff Frequency.

2 2 6 R R A 1 3 2 VOUT VIN 3 Ω Ω OPA627 3.92k R11 1.33k Ω 3.92k C4 3 1000pF R12 1 3.92kΩ A1B 6 2

7 A C 1 1A 13 2 OPA2604 5 1000pF

1 2 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).

µ C13 = C15 = C4 = 3.98 F. with extremely high GBW would be required. An example The filter (Figure 5) could now be built, but the large of a sixth order, 40kHz Butterworth filter realized in this capacitor values and low resistance values could pose prac- fashion is shown in Figure 8, but its frequency response tical problems. To alleviate this, the impedances of the (Figure 9) is less than hoped for due to the GBW limitations circuit are scaled by an impedance scale factor: described above. A simpler solution is to cascade two of the third order Present C value Zn = Desired C value sections designed above. This cascaded design (Figure 10) works equally well for most applications. 3 By choosing the desired C value as 1000pF, Zn = 3.97x10 . This impedance scaling factor then is multiplied by all Figure 11 (a-d) shows the performance of this cascaded filter resistor values to find the new resistor values, and divides all design. Note that the phase linearity and THD + N are still the capacitor values, taking them from the present values to excellent using this approach. the desired capacitance. The final filter design is shown in Figure 6. Since the output REFERENCES impedance of this filter is relatively high, it’s a good idea to (1) R. Downs, “DSP Oversampling to Quiet Noise,” EE buffer the output using an op amp voltage follower. Ampli- Times, pg. 68, 8 August 1988. tude and phase response of this filter is shown in Figure 7a. Figure 7b is a closer look at the amplitude response in the (2) R. Downs, “High Speed A/D Converter Lets Users Reap Benefits of Oversampling,” Burr-Brown Update, Vol. XIV, passband—the frequency response is flat well within 0.1dB No. 2, pg. 3, May 1988. out to 20kHz. Figure 7c is a plot of the frequency response of the filter (3) R. Downs, “Unique Topology Makes Simple, Low- Distortion Antialiasing Filters,” to be published. (solid line) and the filter’s deviation from linear phase (dotted line). Note the phase scale; the phase response is (4) S.K. Mitra, Analysis and Synthesis of Linear Active well within 0.1° of linear phase in the 1kHz-20kHz region, Networks, John Wiley & Sons, Inc., New York, pg. 494, where the ear is most sensitive to phase distortion. 1969. Figure 7d is a plot of the total harmonic distortion plus noise (5) R. Downs, “Unique Topology Makes Simple, Low- (THD + N) of this filter versus frequency. At about –108dB, Distortion Antialiasing Filters,” to be published. this would be suitable for digital systems with true 18-bit (6) L.P. Huelsman, P.E. Allen, Introduction to the Theory converter performance! and Design of Active Filters, McGraw-Hill, New York, To make a sixth order filter, you can repeat the design 1980. process above from a passive realization and directly imple- (7) Ibid. ment 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

3 AMPLITUDE AND PHASE RESPONSE PASSBAND RESPONSE DETAIL OF FILTER CIRCUIT IN FIGURE 6 OF FILTER CIRCUIT IN FIGURE 6 40 1 30 0.8 20 0.6 10 Amplitude 0.4 0 0.2 –10 Amplitude 0 90 –20 90 –0.2 0 –30 0 Phase Response (dBu) Response (dBu) Phase Phase (degrees) –90 Phase (degrees) –40 –90 –0.4 –50 –180 –0.6 –180 –60 –270 –0.8 –270 –70 –360 –1 –360 20 100 1k 10k 100k 200k 20 100 1k 10k 100k 200k Frequency (Hz) Frequency (Hz)

(a) (b)

AMPLITUDE AND DEVIATION FROM LINEAR PHASE TOTAL HARMONIC DISTORTION + NOISE FOR FILTER IN FIGURE 6 OF FILTER CIRCUIT IN FIGURE 6 vs FREQUENCY 40 –40

20 –5 Amplitude –60 0 .040 –70 –20 .020 –80 –40 0 Phase –90 THD + N (dBr) Response (dBu) –60 –.020 Phase (degrees) –100

–80 –.040 –110

–100 –120 20 100 1k 10k 100k 200k 20 100 1k 10k 20k Frequency (Hz) Frequency (Hz)

Note phase scale—deviation from linear phase in critical 1kHz-20kHz region NOTE: Referred to 6Vp-p full-scale signal typical of most digital audio is well within 0.1°. converters.

FIGURE 7. Performance Details of Figure 6 Circuit.

4 OUT V 100pF 3B A Ω Ω Ω

39.2k 39.2k 100pF 10.2k 100pF Phase (degrees) Phase 3A A Ω 90 0 –90 –180 –270 –360 29.4k Amplitude 2B A Ω Ω Ω Phase 39.2k 39.2k 100pF 8.87k 100pF Frequency (Hz) 2A sponse of Filter Circuit in Figure 8. (Note flattening of stopband response near 150kHz due to inad- equate GBW of operational amplifiers used.) A Ω 20 100 1k 10k 100k 200k 60.4k 0

40 30 20 10

–10 –20 –30 –40 –50 –60 –70 Response (dBu) Response FIGURE 9. Amplitude (solid line) and Phase (dotted Re- 1B A Ω Ω Ω 39.2k 39.2k 100pF 68.1k 100pF 1A A Ω 60.4k IN V FIGURE 8. Sixth Order Butterworth Filter Realized by Method Outlined in Text (actual circuit would require output buffer amplifier to lower impedance). 5 OUT V 7 OPA2604

3B 1 A 6 5 1000pF 1 Ω OPA2604

2 2B

1 1.33k A 3 2 Ω Ω Ω 1000pF 1000pF 7.32k 7.32k 3.48k 6 5 2A A Ω 3.92k OPA2604 7

1 1 OPA2604

3A 1 A 2 3 1000pF 1 Ω OPA2604

2 1B

1 1.33k A 3 2 Ω Ω Ω 1000pF 1000pF 7.32k 7.32k 3.48k 6 5 1A A Ω 3.92k OPA2604 7

1 IN V FIGURE 10. Sixth Order Linear-Phase Filter Made by Cascading Two Third Filters. 6 AMPLITUDE AND PHASE RESPONSE PASSBAND RESPONSE DETAIL OF FILTER CIRCUIT IN FIGURE 10 OF FILTER CIRCUIT IN FIGURE 10 40 1 30 0.8 20 0.6 10 Amplitude 0.4 0 0.2 –10 Amplitude 0 90 –20 90 Phase –0.2 0 –30 0 Phase Response (dBu) Response (dBu) Phase (degrees) Phase (degrees) –90 –40 –90 –0.4 –50 –180 –0.6 –180 –60 –270 –0.8 –270 –70 –360 –1 –360 20 100 1k 10k 100k 200k 20 100 1k 10k 100k 200k Frequency (Hz) Frequency (Hz)

(a) (b)

AMPLTIUDE AND DEVIATION FROM LINEAR PHASE TOTAL HARMONIC DISTORTION + NOISE FOR FILTER IN FIGURE 10 FOR FILTER CIRCUIT IN FIGURE 10 vs FREQUENCY 40 –40

20 –5 Amplitude –60 0 .040 –70 –20 .020 –80 –40 0 –90

Phase THD + N (dBr) Response (dBu) –60 –.020 Phase (degrees) –100

–80 –.040 –110

–100 –.060 –120 20 100 1k 10k 100k 200k 20 100 1k 10k 20k Frequency (Hz) Frequency (Hz)

Note phase scale—deviation from linear phase in critical 1kHz-20kHz region NOTE: Referred to 6Vp-p full-scale signal typical of most digital audio is well within 0.1°. converters.

FIGURE 11. Performance Details of Figure 10 Circuit.

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