量化噪声功率谱深度解析

本文开启量化噪声深度系列，概述指导探索的分析框架。

本系列延续两篇前作。第一篇探讨正交 (I/Q) 组合分离应模拟还是数字实现。评估 I/Q 调制解调器、ADC/DAC 性能，讨论高质量通信链路关键准则。

ADC 和 DAC（统称数据转换器）未针对现代通信波形完全表征。因此作者着手调研其性能并提供现实模型。第二篇探讨用 ENOB 和互调多项式建模 ADC/DAC，引入含低通滤波器的更全面模型。

系列目标

许多数据转换器性能分析常见场景：

图 1. 数据转换器使用简化框图

数据转换器在奈奎斯特带宽 (B_N) 上输出总噪声功率 N。后续滤波器——带通或低通——带宽 B_o。通常滤波器后噪声功率近似为：

滤波器输出噪声功率 = N (B_o / B_N)

公式 1. 此关系对 ADC 后任何合理滤波器成立，无论中心频率，前提是滤波器不过窄。

公式 1 假设白噪声（频率平坦）。作者质疑此伪量化噪声假设有效性，探究其成立条件。大量参考文献（见第 4 节）和仿真结果支撑探究。

仅聚焦均匀量化，因其为高速转换器标准。Sigma-Delta 架构刻意排除。

ADC 应用中，RF 链增益常提高使前级组件噪声比量化噪声高 3-5 dB，使量化谱无关紧要。但此做法因需更高 RF 增益和 ADC 动态范围可能推高系统成本。

相反，DAC 设计者倾向 DAC 噪声占主导，避免仅为强制白噪声频谱而添加下游噪声。

峰值、平均值与均方根值

定义输入信号电平至关重要。图 2 描绘 5 位量化正弦波。其振幅常表为 0 dBFS，FS 为量化器满量程。RF 工程师通常用均方根值；正弦波均方根比峰值低 3 dB，故信号为 –3 dBrmsFS 或 0 dBpeakFS。

图 2. 5 位量化正弦波

本系列余下部分信号电平用 dBrmsFS 或 dBpeakFS 表示。注意功率与电压平方成正比；故该恒包络正弦波峰均功率比 (PAPR) 为 3 dB。同样所有带通相位或频率调制恒包络信号（如 MSK）PAPR 为 3 dB。许多人称此类信号 PAPR 为 0 dB——仅考虑包络——但实际电压 PAPR 确实高 3 dB。

下期将详细考察 ADC 输出频谱。

本系列缩写

请参考下表贯穿系列的术语。

参考文献

本系列依据以下来源：

• 引言与动机
• [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
• [4] Lever, K. V.; Cattermol, K.W., "Quantising noise spectra," Electrical Engineers, Proceedings of the Institution of, vol.121, no.9, pp.945‑954, September 1974
• 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
• [6] Gersho, A, "Quantization," Communications Society Magazine, IEEE, vol.15, no.5, pp.16‑16, September 1977
• [7] Schuchman, L., "Dither Signals and Their Effect on Quantization Noise," Communication Technology, IEEE Transactions on, vol.12, no.4, pp.162‑165, December 1964
• [8] Walden, R.H., "Analog‑to‑Digital Converters and Associated IC Technologies," Compound Semiconductor Integrated Circuits Symposium, 2008. CSIC '08. IEEE, vol., no., pp.1‑2, 12‑15 Oct. 2008
• [9] Walden, R.H., "Performance trends for analog to digital converters," Communications Magazine, IEEE, vol.37, no.2, pp.96‑101, Feb 1999
• [10] Walden, R.H., "Analog‑to‑digital converter technology comparison," Gallium Arsenide Integrated Circuit (GaAs IC) Symposium, 1994, Technical Digest 1994., 16th Annual, vol., no., pp.217‑219, 16‑19 Oct. 1994
• [11] Walden, R.H., "Analog‑to‑digital converter survey and analysis," Selected Areas in Communications, IEEE Journal on, vol.17, no.4, pp.539‑550, Apr 1999
• [12] Morgan, D.R., "Finite limiting effects for a band‑limited Gaussian random process with applications to A/D conversion," Acoustics, Speech and Signal Processing, IEEE Transactions on, vol.36, no.7, pp.1011‑1016, Jul 1988
• [13] Chow, P.E.-K., "Performance in waveform quantization," Communications, IEEE Transactions on, vol.40, no.11, pp.1737‑1745, Nov 1992
• [14] Dardari, D., "Exact analysis of joint clipping and quantization effects in high speed WLAN receivers," Communications, 2003. ICC '03. IEEE International Conference on, vol.5, no., pp.3487‑3492, vol.5, 11‑15 May 2003
• [15] Gray, R.M., "Quantization noise spectra," Information Theory, IEEE Transactions on, vol.36, no.6, pp.1220‑1244, Nov 1990
• [16] Echard, J.; Watt, M.L., "The quantization noise spectrum of a sinusoid in colored noise," Signal Processing, IEEE Transactions on, vol.39, no.8, pp.1780‑1787, Aug 1991
• [17] He Jing; Li Gang; Xu Xibin; Yao Yan, "Estimation for the quantization noise spectrum of linear digital filter," Communication Technology Proceedings, 2000. WCC - ICCT 2000. International Conference on, vol.1, no., pp.184‑187, vol.1, 2000
• [18] Bennett, W.R., "Spectra of quantized signals," Bell System Technical Journal, The, vol.27, no.3, pp.446‑472, July 1948
• [19] Mohamed, E.M., "Low complexity channel estimation technique for MIMO-Constant Envelope Modulation," Wireless Technology and Applications (ISWTA), 2013 IEEE Symposium on, vol., no., pp.97‑102, 22‑25 Sept. 2013
• [20] Clavier, A G.; Panter, P. F.; Grieg, D.D., "Distortion in a Pulse Count Modulation System," American Institute of Electrical Engineers, Transactions of the, vol.66, no.1, pp.989‑1005, Jan. 1947
• 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
• [22] Mazo, J.E., "Asymptotic distortion spectrum of clipped, DC‑biased, Gaussian noise [optical communication]," Communications, IEEE Transactions on, vol.40, no.8, pp.1339‑1344, Aug 1992
• [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
• [24] Dardari, D.; Tralli, V.; Vaccari, A, "A theoretical characterization of nonlinear distortion effects in OFDM systems," Communications, IEEE Transactions on, vol.48, no.10, pp.1755‑1764, Oct 2000
• [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
• [26] Van Vleck, J.H.; Middleton, D., "The spectrum of clipped noise," Proceedings of the IEEE, vol.54, no.1, pp.2‑19, Jan. 1966
• Other relevant mathematical treatments
• [27] Ermolova, N.Y.; Haggman, S.-G., "An extension of Bussgang's theory to complex‑valued signals," Signal Processing Symposium, 2004, NORSIG 2004. Proceedings of the 6th Nordic, vol., no., pp.45‑48, 11‑11 June 2004
• [28] Requicha, Aristides A G, "Expected Values of Functions of Quantized Random Variables," Communications, IEEE Transactions on, vol.21, no.7, pp.850‑854, Jul 1973
• [29] Pirskanen, J.; Renfors, M., "Quantization and jitter requirements in multimode mobile terminals," Communications, 2001. ICC 2001, IEEE International Conference on, vol.4, no., pp.1182‑1186, vol.4, 2001
• [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
• [31] Widrow, B., "A Study of Rough Amplitude Quantization by Means of Nyquist Sampling Theory," Circuit Theory, IRE Transactions on, vol.3, no.4, pp.266‑276, Dec 1956
• [32] Rowe, H.E., "Memoryless nonlinearities with Gaussian inputs: Elementary results," Bell System Technical Journal, The, vol.61, no.7, pp.1519‑1525, Sept. 1982
• [33] VanTrees, Harry L; Detection, Estimation, and Modulation Theory, Part III, Radar/Sonar Signal Processing and Gaussian Signals in Noise; John Wiley and Sons; 1971. Appendix: "Complex Representation of Bandpass Signals, Systems, and Processes" AES‑1, Issue: 6, 1979, Page(s): 840‑848.