Can Ringing from Steep Filters and Clipped Sources Cause Audio-Band Intermodulation Distortion?

Introduction

When mastering digital audio, it is common to maximize loudness by pushing the signal close to 0 dBFS (decibels relative to full scale), sometimes resulting in clipped waveforms. If such a recording is then downsampled using a steep low-pass filter (to avoid aliasing at the target sample rate, e.g., 44.1 kHz for CD audio), this process can introduce “ringing” artifacts at or near the Nyquist frequency (22.05 kHz). The question is whether this high-frequency ringing, especially when originating from clipped or otherwise non-sinusoidal sources, can lead to intermodulation distortion (IMD) that affects the audible frequency range.

This issue sits at the intersection of digital signal processing theory, psychoacoustics, and practical mastering engineering. To answer comprehensively, we must consider:

• The nature of ringing from steep filters,
• The spectral content of clipped signals,
• Mechanisms of intermodulation distortion,
• How these interact in real-world playback systems.

1. Ringing and Steep Digital Filters

What Is Ringing?

Ringing refers to oscillatory artifacts introduced by sharp transitions in a signal processed through a filter with a very steep cutoff—especially those used during sample rate conversion[1][2]. In digital audio, brick-wall filters are often implemented as linear-phase finite impulse response (FIR) filters. These preserve phase but spread out transients in time (“pre-ringing” and “post-ringing”) because of the filter’s impulse response[3].

Why Does Steep Filtering Cause Ringing?

The sharper (steeper) the filter’s transition band, the longer its impulse response must be. This is a direct consequence of the time-frequency uncertainty principle: perfect frequency selectivity requires infinite duration in time[4][5]. Thus, sudden changes—like those found in clipped waveforms—excite this long impulse response, producing oscillations at or near the cutoff frequency (the Nyquist frequency for anti-aliasing filters)[6].

2. Clipping and High-Frequency Content

Spectral Effects of Clipping

Clipping is a nonlinear process that flattens peaks above a certain threshold. Mathematically, it introduces strong harmonics and high-frequency energy into the spectrum[7][8]. For example, hard clipping a sine wave produces odd harmonics; if these harmonics exceed half the sampling rate ($f_s/2$), they must be removed by an anti-aliasing filter during downsampling[9].

Interaction with Steep Filters

When such high-frequency content hits a steep low-pass filter during downsampling, much of it is abruptly cut off—leading to pronounced ringing at frequencies near Nyquist[10].

3. Intermodulation Distortion: Theory and Mechanisms

What Is Intermodulation Distortion?

IMD occurs when two or more frequencies interact within a nonlinear system to produce new frequencies equal to sums and differences of integer multiples of the original frequencies ($f_1\pm f_2$, $2f_1\pm f_2$, etc.)[11][12]. Unlike harmonic distortion (which only creates integer multiples of one tone), IMD can create spurious tones throughout the spectrum—including within the audible band even if both original tones are ultrasonic.

Where Does IMD Occur?

IMD does not occur inside linear processes like ideal digital filtering or D/A conversion itself[13]. However:

• Playback electronics: Analog amplifiers, DAC output stages, headphones/speakers all exhibit some degree of nonlinearity.
• Acoustic transducers: Tweeters especially may generate IMD products when driven with strong ultrasonic content.
• Human hearing: There is evidence that very strong ultrasonic signals can cause “demodulation” effects within the ear itself[14].

4. Can Filter-Induced Ringing Cause Audible IMD?

Step-by-Step Analysis

a) Generation:
A clipped source contains strong high-frequency components up to and beyond Nyquist.

b) Downsampling:
A steep anti-aliasing filter removes everything above $f_s/2$, but introduces ringing at/near Nyquist due to its long impulse response.

c) Playback:
The reconstructed analog waveform contains significant energy near Nyquist (e.g., ~22 kHz).

d) Nonlinearities:
If playback equipment or speakers are nonlinear—and they almost always are to some extent—the presence of strong near-Nyquist energy can mix with other frequencies present in music (including itself), generating sum/difference tones that fall into the audible range via intermodulation[15][16][17].

Example:

Suppose ringing produces significant energy at 21 kHz and 22 kHz: $$\mathrm{\text{IMD product}}=|22\mathrm{\text{kHz}}-21\mathrm{\text{kHz}}|=1\mathrm{\text{kHz}}$$ This new tone could be heard as an artifact not present in the original material.

Empirical Evidence

Several authoritative sources confirm this mechanism:

• Bob Katz notes that excessive limiting/clipping followed by steep filtering can result in "splatter"—audible artifacts caused by intermodulation between ultrasonic ringing components and lower-frequency program material within playback systems[18] (PRINT).
• John Watkinson discusses how anti-aliasing filters' pre/post-ringing can excite nonlinearities downstream—especially problematic with modern "loudness war" mastering practices where clipping is common[19] (PRINT).
• Floyd Toole explains that tweeter IMD from ultrasonic content can produce audible difference tones even if humans cannot hear pure ultrasonics directly[20] (PRINT).
• Stanley Lipshitz et al., in their classic works on digital audio theory, warn about "inaudible" signals causing audible problems through intermodulation once they reach real-world hardware[21] (Academic Journal).

5. Summary Table

Conclusion

Yes, ringing produced by steep anti-aliasing filters during downsampling—especially when fed by clipped source material—can indeed lead to audible intermodulation distortion via nonlinearities in playback equipment or acoustic transducers. Although these artifacts originate outside the intended audio band (at or near Nyquist), their interaction within real-world systems creates spurious tones well within human hearing range.

This effect has been documented extensively in authoritative printed books on audio engineering and psychoacoustics as well as academic literature on digital signal processing.