You have a 24-bit converter, a sample rate and a signal you care about. Between them sits an analogue filter that decides whether the converter's headline resolution means anything at all. Get it wrong and noise from a switching supply or an adjacent channel folds back into your passband, where no amount of digital post-processing can remove it. Once aliased, it is indistinguishable from your signal.
The reassuring part is that most of the work is arithmetic, and the arithmetic usually points to one sensible answer.
Start With the Numbers You Already Have
Write these down before you open a simulator:
- The converter's actual output sample rate, fs. For a delta-sigma part, that is the rate after decimation, not the modulator clock.
- The frequency band you must preserve intact, including the tolerance at the edges.
- The amplitude flatness you need across that band — ±0.1 dB, ±0.5 dB, whatever the application genuinely requires.
- The dynamic range you need, in dB, inside the band.
A worked case: a 24-bit delta-sigma running at 48 kHz output rate, signal content from DC to 20 kHz, target dynamic range 110 dB. Nyquist is 24 kHz. Everything above it folds, which is the whole problem.
It also helps to know the converter's real noise floor. Twenty-four bits suggests 144 dB, but a good audio-grade 24-bit converter delivers somewhere around 110 to 120 dB. Budget against the measured figure, not the marketing one.
Choosing the Cut-off Frequency
Put the corner where the maths says, not at Nyquist
The tempting answer is a −3 dB corner at 24 kHz. That gives you 3 dB of attenuation exactly where aliasing begins. Content at 24.1 kHz folds down to 23.9 kHz, and a filter that is only just rolling off will let most of it through. The corner must sit at the top of the passband — 20 kHz here — with full stopband attenuation reached by Nyquist.
That leaves a 4 kHz transition band, a ratio of 24/20 = 1.2 between the stopband edge and the passband edge. Ratio, not absolute frequency, is what sets the required filter order.
How much attenuation do you actually need?
Enough that the worst-case alias lands below the converter's noise floor. For a converter with a noise floor near 100 dB below full scale, that argues for something in the region of 100 dB of stopband attenuation.
Now run the numbers for a Butterworth response, whose attenuation is 10 log₁₀(1 + (f/fc)2n). With fc = 20 kHz and f = 24 kHz, reaching 100 dB means (1.2)2n must equal 1010. That works out to an order of about 63.
Sixty-three poles. Nobody builds that, and the reason they do not is the subject of the next section.
Oversampling Rewrites the Requirement
Suppose the modulator samples at 3.072 MHz and a digital decimation filter drops the rate to 48 kHz. The analogue stage no longer has to protect a 48 kHz sampler. It has to protect a 3.072 MHz one, and the first region that can fold into your 20 kHz band sits near 3.072 MHz − 20 kHz, around 3.052 MHz.
The ratio is now 3.052 MHz / 20 kHz ≈ 153. For 80 dB of attenuation, 2n log₁₀(153) = 8, so 2n ≈ 3.66 and a two-pole filter is sufficient.
Same converter, same passband. The analogue filter went from sixty-three poles to two.
This is the practical lesson: the analogue filter's job is to keep the modulator out of trouble, while the digital decimation filter does the sharp work. Check the converter datasheet for its own digital filter's stopband specification — whatever the part already rejects, your analogue stage does not have to. In many delta-sigma designs the analogue corner sits well above the audio band, at 100 kHz or more, purely to stop high-frequency energy from overloading the modulator input.
Topology and Component Choices
For the modest orders that oversampling makes possible, a Sallen-Key or multiple-feedback stage around a decent op-amp is usually enough. What matters for a 24-bit design:
- Topology. Keep noise gain low and avoid configurations that amplify the op-amp's own input noise at the corner.
- Op-amp. Gain bandwidth of at least 100 times the corner frequency, low voltage noise, and enough output current for the load. It should be driven from a low-impedance source.
- Resistors. Metal film, 0.1 % where the budget allows, and matched pairs from the same batch for differential paths.
- Capacitors. C0G/NP0 in the signal path. X7R and similar dielectrics change value with applied voltage and can be microphonic, which is a poor combination in front of a 24-bit converter.
- Differential signal path. A fully differential amplifier feeding a differential ADC input cancels even-order distortion and gives the converter a clean common-mode reference.
- A small RC at the ADC pins. Typically 22–100 Ω with 1 nF, to absorb charge-injection kickback from a switched-capacitor input.
Tolerances, Noise and the 24-bit Reality
Cut-off frequency is set by products of R and C. In a Sallen-Key stage, fc is proportional to 1/√(R₁R₂C₁C₂), so component spread translates fairly gently into frequency shift. With 1 % resistors and 5 % capacitors, worst-case stacking gives roughly a 6 % shift in corner frequency; with all 1 % parts, about 2 %.
If your stopband target is tight, design the nominal corner low enough that the worst-case build still passes, or spend the money on tighter capacitors — they usually dominate the error.
Then there is thermal noise. A 10 kΩ resistor
Photo: Kranich17 / Pixabay
