P. R. Pérez–Alcázar¹*, A. Santos²

¹ Departamento de Ingeniería Electrónica Facultad de Ingeniería, UNAM, Mexico. *pperezalcazar@fi–b.unam.mx

² Departamento de Ingeniería Electrónica Universidad Politécnica de Madrid, Spain.

ABSTRACT

This paper presents a brief revision of several studies presented in the literature about the behavior of quantization noise for sinusoidal signals and for uniform quantizers. From this revision, the conclusion is that quantization noise has been assumed to be additive and has a white spectrum, although some published studies, considering the problem either from a deterministic point of view or from a stochastic one, have shown a different noise behavior for some specific cases. Some of these cases are related with the parameters that characterize the sinusoidal signal and other with the conditions under which the process of conversion is realized. There are some cases that have not very been well considered in the previous literature and about which it is convenient to call attention. By this reason and using computer simulations with sinusoidal input signals, it is illustrated here that the quantization noise spectrum can show a discrete or complex structure depending on the relation between the sampling rate used and the frequency of the signal. Moreover, some points to consider in order to get a better description of the quantization noise are presented.

RESUMEN

Este artículo presenta una breve revisión de varios estudios presentados en la literatura sobre el comportamiento del ruido de cuantización señales sinusoidales y cuantificadores uniformes. A partir de esta revisión, la conclusión es que generalmente se ha asumido que el ruido de cuantización es aditivo y tiene un espectro blanco; aunque algunos estudios publicados, considerando el problema desde un punto de vista determinístico o desde un punto de vista estocástico, han mostrado un comportamiento diferente del ruido para algunos casos específicos. Algunos de estos casos están relacionados con los parámetros que caracterizan la señal sinusoidal y otros con las condiciones bajo las cuales se realiza el proceso de conversión. Hay algunos casos que no han sido muy bien considerados en la literatura previa y sobre los cuales conviene llamar la atención. Por esta razón, utilizando simulaciones mediante computadora, con señales de entrada sinusoidales, se muestra aquí que el espectro del ruido de cuantización puede mostrar una estructura discreta o compleja dependiendo de la relación entre la velocidad de muestreo utilizada y la frecuencia de la señal. Además, algunos puntos a considerar para obtener una mejor descripción del ruido de cuantización son presentados.

Keywords: Analog–digital conversion, quantization noise, noise spectrum.

DESCARGAR ARTÍCULO EN FORMATO PDF

References

[1] Pérez P., Adquisición de imágenes de resonancia magnética nuclear mediante técnicas de submuestreo, PhD Thesis, Universidad Politécnica de Madrid, Madrid, 1999, pp. 122–125.         [ Links ]

[2] Oliver B., Pierce J., & Shannon C., The philosophy of PCM, Proceedings of the IRE, Vol. 36, No. 11, November, 1948, pp. 1324–1331.         [ Links ]

[3] Bennett W., Spectra of quantized signals, Bell System Technical Journal, Vol. 27, No. 4, July, 1948, pp. 446–472.         [ Links ]

[4] Panter P. & Dite W., Quantization distortion in pulse–count modulation with nonuniform spacing of levels, Proceedings of the IRE, Vol. 39, No. 1, January, 1951, pp. 44–48.         [ Links ]

[5] Sripad A. & Snyder D., A necessary and sufficient condition for quantization errors to be uniform and white, IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 25, No. 5, October, 1977, pp. 442–448.         [ Links ]

[6] Widrow B., A study of rough amplitude quantization by means of Nyquist sampling theory, IRE Transactions of the Professional Group on Circuit Theory, Vol. 3, No. 4, December, 1956, pp. 266–276.         [ Links ]

[7] B. Widrow, Statistical analysis of amplitude quantized sampled data systems, Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry, Vol. 79, No. 52, January, 1961, pp. 555–568.         [ Links ]

[8] Clavier A., Panter P & Grieg D., Distortion in a pulse Count modulation system, Transactions of the American Institute of Electrical Engineers, Vol. 66, November, 1947, pp. 989–1005.         [ Links ]

[9] Claasen T. & Jongepier A., Model for the power spectral density of quantization noise, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 29, No. 4, August, 1981, pp. 914–917.         [ Links ]

[10] Oppenheim A. & Schafer R., Digital Signal Processing, Prentice Hall, 1975, pp. 413–418.         [ Links ]

[11] Oppenheim A. & Schafer R., Discrete–time Signal Processing, Prentice Hall, 1989, pp. 119–123.         [ Links ])

[12] Gray R., Quantization noise spectra, IEEE Transactions on Information Theory, Vol. 36, No. 6, November, 1990, pp. 1220–1244.         [ Links ]

[13] Gray R. & Neuhoff D., Quantization, IEEE Transactions on Information Theory, Vol. 44, No. 6, October, 1998, pp 1–63.         [ Links ]

[14] Oppenheim A. & Willsky A., Signals and systems, 2d Ed., Prentice Hall, 1996, pp. 25–28.         [ Links ]

[15] Bellan D., Brandolini A., & Gandelli A., Quantization theory—A deterministic approach, IEEE Transactions on Instrumentation and Measurement, vol. 48, No. 1, February, 1999, pp. 18–25.         [ Links ]

[16] Gray R. & Stockham T., Dithered quantizers, IEEE Transactions on Information Theory, Vol. 39, No. 3, May, 1993, pp 805–812.         [ Links ]

[17] Schuchman L., Dither signals and their effect on quantization noise, IEEE Transactions on Communication Technology, Vol. 12, No. 4, December, 1964, pp. 162–165.         [ Links ]

[18] Wannamaker R., Lipshitz S., Vanderkooy J., & Wright J., A theory of nonsubtractive dither, IEEE Transactions on Signal Processing, Vol. 48, No. 2, February, 2000, pp. 499–516.         [ Links ]

[19] Louzon P., Decipher high–sample–rate ADC specs, Electronic Design, Vol. 43, No. 5, March, 1995, pp. 91100.         [ Links ]

[20] Schweber B., Converters restructure communication architectures, EDN, Vol. 8, No. 8, August, 1995, pp. 51–64.         [ Links ]

[21] Wepman J., Analog–to–digital converters and their applications in radio receivers, IEEE Communications Magazine, Vol. 33, No. 5, May, 1995, pp. 39–45.         [ Links ]

[22] Proakis J. & Manolakis D., Digital signal processing: principles, algorithms, and applications, 3d Ed., Prentice Hall, 1996, pp. 33–35.         [ Links ]