Bits is Bits? Page 7
Analogy to Phase-Intermodulation distortion in Audio Amplifiers: Amplitude errors caused by timing jitter at the ADC or DAC gateway can be examined in a wider perspective by comparing the jitter-error mechanism with artifacts found in analog electronics. Hawksford (footnote 18) has shown that jitter errors in DACs can be compared to slew-rate limiting in transimpedance amplifiers located at DAC outputs. The jitter-error mechanism can also be likened to phase-intermodulation distortion (PID) in analog amplifiers. Otala (footnote 19, footnote 20) has shown that PID occurs when open-loop amplitude non-linearity in a feedback amplifier is mapped to a closed-loop phase non-linearity. Cordell (footnote 21) refined the PID model by writing the timing error t(x) in the phase-distorted output voltage x from a feedback amplifier in terms of the normalized open-loop non-linearity e(x) and closed-loop cutoff frequency fco;
If we substitute typical values of e(xmax) = 1% (0.01) and fco = 1MHz for an IC operational amplifier, we find that the peak timing error is equal to 1.6ns. This is of the same order of magnitude as the jitter found in digital audio interfaces. It is also of interest that a typical analog transfer function non-linearity will result in PID where the timing error is highly correlated with the audio signal. This observation lends weight to the analogy made between PID and digital audio interface jitter, where we have shown that the jitter resulting from a band-limited digital interface can also strongly correlate with the transmitted audio signal. Of course, PID in analog amplifiers is intimately linked with closed-loop amplitude non-linearity, and is not in itself a cause of additional error, while jitter in digital audio interfaces is a source of error in A/D or D/A conversion.
Nevertheless, the analogy between PID timing error and interface jitter is useful if the overall timing error in a system is to be minimized; there is little point in minimizing digital audio interface jitter if the analog circuitry preceding or following conversion is of poor quality. In general, it is rewarding to make comparisons between analog and digital system artifacts, an exercise that has shown interesting results before (footnote 22, footnote 23). In truth, the boundaries between analog and digital systems are not as clear-cut as they may at first appear, and we encourage system appraisal in a global sense.
Audibility of jitter errors
How much jitter is tolerable in a reconstructed stereo PCM transmission? One of the earliest studies of the audible consequences of jitter is due to Manson (footnote 24), who carried out a series of tests using a monophonic analog signal input to a sampling device with a stable clock. The sampled (but unquantized) audio signal is then converted back to the analog domain using a sample-and-hold unit with a controlled amount of clock jitter, and auditioned. Manson suggests that 35ns RMS jitter represents the threshold of subjective impairment using critical source material. However, we believe that several factors make this an unrealistically high figure of "minimum audible jitter."
First, Manson's experiments were carried out using a monophonic test signal. Much evidence suggests that the audibility of low-level distortion (as would be caused by jitter) increases when music is reproduced in stereo, as acoustic objects are now perceived in two-dimensional space, and masking of errors by the primary signal is not so effective. Second, these tests were carried out some time ago, using tape recordings; the advances in reproduction equipment now available to the consumer should result in a lower threshold.
A better estimate of the audible jitter threshold can be obtained by examining the jitter error sequence, and assuming that it will be inaudible if below the level of quantization noise present in the system for any possible excitation frequency. Lidbetter (footnote 25) thus arrives at a value of 120 picoseconds for a 16-bit, 100% sample DAC, and an incredibly low 8ps for 20-bit system. Shelton (footnote 26), Fourré (footnote 27), Harris (footnote 28), van Willenswaard (footnote 29), and the recommendations embodied in AES11-1991 (footnote 30) all quote similar values.
Are these lower limits reasonable? How audible is jitter when its level is at the quantization noisefloor? An attractive approach to answering these questions is to model the hearing process itself; ie, find out whether a given jitter error is below the masked threshold due to the jittered audio signal—a method adopted by Julian Dunn (footnote 31). Correspondingly, a simple hearing model was developed in order to assess the audibility of jitter in a band-limited interface.
Our model assumes that the error due to jitter is inaudible if it is below the threshold of hearing (minimally audible field) at all frequencies. This approach will yield pessimistic results as far as error audibility is concerned, since the additional masking effect of signal tones is not considered, although masking of low-frequency noise by high-frequency tones is minimal. Recent work by Bob Stuart (footnote 32) suggests that the audibility of errors in isolation may well be of higher significance than has previously been thought.
We define the threshold of hearing in the frequency domain by passing a cubic spline through the threshold data of ISO226 (footnote 33) and scaling by the gain of a typical audio system under critical listening conditions such that "0dB" refers to a sound-pressure level of 112dB at 1m per speaker (footnote 34). The error signal is then integrated at each frequency bin across a bandwidth defined by the equivalent rectangular noise bandwidth (footnote 34) at that frequency, and compared to the threshold. Fig.33 indicates the audibility of 16-bit triangular probability-density (TPD) flat dither noise assessed in this manner, clearly showing the dither to be audible in the 2-6kHz frequency range: this is similar to the result achieved by Stuart in fig.2 of his AES paper (footnote 35).
Fig.33 Audibility of spectrally flat TPD 16-bit dither. Top curve is interpolated from the minimally audible field threshold data of ISO226, with 0dB equivalent to a sound-pressure level of 112dB at 1m per loudspeaker.
We can use the error-audibility model to assess the validity of these claimed limits to jitter audibility. Consider the case where an audio tone is corrupted by spectrally white jitter. Fig.34a shows that for a 100% DAC reproducing a 0dB, 20kHz sinusoid, rectangular probability-density (RPD) jitter of peak amplitude 180ps should be on the threshold of audibility, although it should be noted that the error level reduces as the audio sinewave frequency is reduced. This can be compared to fig.34b for an impulsive DAC, where the error signal associated with 550ps peak jitter noise still lies below the audibility curve. It should be noted that the error curve in this case is constant with audio frequency (this diagram was obtained using 100Hz). The problem with making predictions about the audibility of jitter artifacts using noise-like jitter is that the error tends to be spread across the audio band; more stringent jitter specifications are required when the jitter is sinusoidal.
Fig.34 Simulated jitter errors for RPD white jitter noise: a) 20kHz at 0dBFS, 180ps peak jitter, 100% DAC (top); b) 100Hz at 0dBFS, 550ps peak jitter, impulsive DAC (bottom).
Fig.35a shows the worst-case 100% DAC jitter error resulting from a 22kHz audio signal and 18.5kHz jitter—only 20ps peak jitter is required for audibility. The 75ps limit for an impulsive DAC occurs when reproducing a low-frequency tone, and both jitter sidebands are coincident due to reflection about DC (fig.35b).
Fig.35 Simulated worst-case jitter errors for sinusoidal jitter: a) 22khz at 0dBFS, 20ps peak jitter at 18.5kHz (top); b) impulsive DAC, 100Hz at 0dBFS, 75ps peak jitter at 4kHz (bottom).
Are there any circumstances under which these critical combinations of audio and jitter signals could occur at the same time? In the "Jitter in the Digital Audio Interface" section we have shown that digital audio interface jitter can be highly correlated with the transmitted audio data—and, when it is remembered that digital filters with delays up to a few milliseconds often follow interface receivers before D/A conversion (causing the jitter to precede the associated audio signal at the DAC), such combinations may indeed occur.
Footnote 17: B. Adams, "Comments on 'Chaos, Oversampling and Noise-Shaping in Digital-to-Analog Conversion'," JAES, October 1990, Letter to the Editor, Vol.38, pp.766-768.
Footnote 18: M.O. Hawksford, "Digital-to-Analog Converter with Low Inter-Sample Transition Distortion and Low Sensitivity to Sample Jitter and Transresistance Amplifier Slew Rate," presented at the 93rd AES Convention, San Francisco, October 1992.
Footnote 19: M. Otala, "Feedback-Generated Phase Intermodulation in Audio Amplifiers," presented at the 65th AES Convention, February 1980, Preprint 1576.
Footnote 20: M. Otala, "Phase Modulation and Intermodulation in Feedback Audio Amplifiers," presented at the 68th AES Convention, March 1981, Preprint 1751.
Footnote 21: R.R. Cordell, "Phase Intermodulation Distortion—Instrumentation and Measurement Results," presented at the 70th AES Convention, October 1981, Preprint 1842.
Footnote 22: M.J. Hawksford, "Fuzzy Distortion in Analog Amplifiers: A Limit to Information Transmission?," JAES, October 1983, Vol.31, pp.745-754.
Footnote 23: M.J. Hawksford, "Nth.-Order Recursive Sigma-ADC Machinery at the Analogue-Digital Gateway," presented at the 78th AES Convention, May 1985, Preprint 2248.
Footnote 24: C. Manson, "Digital Sound Signals: Subjective Effect of Timing Jitter," BBC Research Department Engineering Report 1974/11 (March 1974).
Footnote 25: P.S. Lidbetter, "Basic Concepts and Problems of Synchronization of Digital Audio Systems," presented at the 84th AES Convention, March 1988, Preprint 2605.
Footnote 26: T. Shelton, "Synchronization of Digital Audio," Proceedings of the AES 7th International Conference: Audio in Digital Times (Toronto, Ontario, Canada, 1989).
Footnote 27: R.D. Fourré, "Testing 20-Bit Audio Digital-to-Analog Converters," Proceedings of the AES 7th International Conference: Audio in Digital Times, (Toronto, Ontario, Canada, 1989).
Footnote 28: Peter van Willenswaard, Stereophile, November 1988, Vol.11 No.11, pp.51-53.
Footnote 29: S. Harris, "The Effects of Sampling Clock Jitter on Nyquist Sampling Analog-to-Digital Converters, and on Oversampling Delta-Sigma ADCs," JAES, July/August 1990, Vol.38, pp.537-542.
Footnote 30: AES11-1991, "AES Recommended Practice for Digital Audio Engineering—Synchronization of Digital Audio Equipment in Studio Operations," JAES, March 1991, Vol.39, pp.156-162.
Footnote 31: J. Dunn, "Considerations for Interfacing Digital Audio Equipment to the Standards AES3, AES5 and AES11," Proceedings of the AES 10th International Conference: Images of Audio (London 1991).
Footnote 32: J.R. Stuart, Hi-Fi News & Record Review, January 1991, Letter to the Editor, p.15.
Footnote 33: ISO226:1987, "Acoustics—Normal Equal-Loudness Level Contours for Pure Tones under Free-Field Listening Conditions," 1987.
Footnote 34: J.R. Stuart, "Predicting the Audibility, Detectability, and Loudness of Errors in Audio Systems," presented at the 91st AES Convention, New York, October 1991, Preprint 3209.
Footnote 35: J.R. Stuart, "A Search for Efficient Dither for DSP Applications," presented at the 92nd AES Convention, Vienna, March 1992, Preprint 3334.