Understanding the Power Spectrum of Quantization Noise

This article opens our in‑depth series on quantization noise, outlining the analytical framework guiding our exploration.

This series continues work from two prior posts. The first examined whether in‑phase and quadrature (I/Q) combination and separation should be performed analogously or digitally. We assessed the performance of I/Q modulators and demodulators, as well as ADCs and DACs, and discussed the key criteria for a high‑quality communications link.

ADCs and DACs—collectively known as data converters—have not been fully characterized for modern communications waveforms. Consequently, I set out to investigate their performance and provide a realistic model. The second post explored ADC and DAC modeling using ENOB and an intermodulation polynomial, and introduced a more comprehensive model that incorporates a low‑pass filter.

Series Goal

In many analyses of data converter performance, the following scenario is common:

Figure 1. Simplified block diagram of a data converter's use

The total noise power emerging from the data converter over the Nyquist bandwidth (B_N) is N. A subsequent filter—either band‑pass or low‑pass—has a bandwidth B_o. Typically, the noise power after the filter is approximated by:

Noise power out of the filter = N (B_o / B_N)

Equation 1. This relationship holds for any reasonable filter following the ADC, regardless of center frequency, provided the filter is not excessively narrow.

Equation 1 assumes white (frequency‑flat) noise. I questioned the validity of this pseudo‑quantization‑noise assumption and investigated the conditions under which it holds. Extensive references (see Section 4) and simulation results support this inquiry.

Uniform quantization was the sole focus, as it is standard in high‑speed converters. Sigma‑delta architectures were deliberately excluded.

For ADC applications, RF chain gain is often increased so that preceding component noise is 3–5 dB above the quantization noise, rendering the quantization spectrum inconsequential. However, this practice can inflate system cost by demanding higher RF gain and ADC dynamic range.

In contrast, DAC designers prefer the DAC noise to dominate, avoiding the addition of downstream noise merely to enforce a white spectrum.

Peak, Average, and rms Values

Defining the input signal level is essential. Figure 2 depicts a 5‑bit quantized sine wave. Its amplitude is commonly expressed as 0 dBFS, where FS denotes full scale on the quantizer. RF engineers typically use rms values; for a sine wave, the rms level is 3 dB below the peak, so the signal is –3 dBrmsFS or 0 dBpeakFS.

Figure 2. 5‑bit quantized sine wave

For the remainder of the series, signal levels are expressed in dBrmsFS or dBpeakFS. Note that power is proportional to voltage squared; therefore, the peak‑to‑average power ratio (PAPR) of this constant‑envelope sine wave is 3 dB. Likewise, all band‑pass phase or frequency‑modulated constant‑envelope signals, such as MSK, exhibit a 3 dB PAPR. While many refer to the PAPR of such signals as 0 dB—considering only the envelope—the real‑voltage PAPR is indeed 3 dB higher.

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In the next installment, we will examine the spectrum of ADC outputs in detail.

Abbreviations Used

Refer to the following table for terminology throughout the series.

References

The following sources underpin this series:

• Introduction and Motivation
• [1] Digital or Analog? How Should I and Q Combining and Separation Be Done?
• Requirements for Good Communications Link Performance: IQ Modulation and Demodulation
• [2] How Should Data Converters Be Modeled for System Simulations?
• Modeling ADCs Using Effective Number of Bits (ENOB)
• Modeling ADCs Using Intermodulation Polynomial and Effective Number of Bits
• Adding a Low‑pass Filter to an ADC Model and DAC Modeling
• Quantization noise with or without clipping effects
• ADC & DAC
• [3] Maloberti, Franco; Data Converters; Springer Publishing; 2007
• ADC Specific, with and without Clipping Effects
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• Lever, K.V.; Cattermole, K.W., "Erratum: Quantising noise spectra," Electrical Engineers, Proceedings of the Institution of, vol.122, no.3, pp.272, March 1975
• [5] Gersho, A, "Principles of quantization," Circuits and Systems, IEEE Transactions on, vol.25, no.7, pp.427‑436, July 1978
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• DAC Specific, with and without Clipping Effects
• [21] Ling, W.A, "Shaping Quantization Noise and Clipping Distortion in Direct‑Detection Discrete Multitone," Lightwave Technology, Journal of, vol.32, no.9, pp.1750‑1758, May 1, 2014
• Clipping effects only; ADC Only
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• [23] Dakhli, M.C.; Zayani, R.; Bouallegue, R., "A theoretical characterization and compensation of nonlinear distortion effects and performance analysis using polynomial model in MIMO OFDM systems under Rayleigh fading channel," Computers and Communications (ISCC), 2013 IEEE Symposium on, vol., no., pp.000583‑000587, 7‑10 July 2013
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• [25] Giannetti, F.; Lottici, V.; Stupia, I, "Theoretical Characterization of Nonlinear Distortion Noise in MC‑CDMA Transmissions," Personal, Indoor and Mobile Radio Communications, 2006 IEEE 17th International Symposium on, vol., no., pp.1‑5, 11‑14 Sept. 2006
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• Other relevant mathematical treatments
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• [30] Irons, Fred H.; Riley, K.J.; Hummels, D.M.; Friel, G.A, "The noise power ratio-theory and ADC testing," Instrumentation and Measurement, IEEE Transactions on, vol.49, no.3, pp.659‑665, Jun 2000
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