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Instrumentation and Measurement Technology Conference

IMTC®/2001 - ABSTRACT


Numerical distortion in single-tone DDS

by Zs. Pápay

Keywords: DDS, NCO, amplitude quantization, phase truncation, numerical distortion, spectral purity, simulation

Introduction

Direct Digital Synthesis (DDS) is a technique using digital and mixed/analog processing blocks as a means to generate real-life waveforms and, specifically, a precise, fast frequency and phase tunable output. DDS solutions can be implemented in LSI (large-scale integration), and they play an ever-increasing role in digital waveform generating and modulation.

Fig. 1 shows the basic schematic, demonstrating the application of DDS as a single-tone (sine wave) generator. NCO-based DDS is a point(memory location)-skipping technique and runs at a constant update(clock)-rate. There are several applications for DDS that do not convert the digitized samples directly into an analog signal.

Fig. 1 The standard DDS structure

The generic DDS can be viewed as a simple assembly containing only three parts:
(1) fixed rate system clock - the time reference part
(2) NCO - the digital  part consists of a phase ACCumulator (register length: r - bit) and a look-up table (LUT, memory address length: m - bit, data length: n - bit). Numerical distortions (algorithmic nonlinearities) due to finite-wordlength are inherent in the system and contribute a major factor to the system complexity.
(3) (Re)construction of analog (sine)wave - the mixed/analog  part consist of a digital to analog converter (DAC, bit length: n - bit) and an anti-imaging filter (AIF).
There is a tradeoff between spectral spurs caused by numerical distortions and the ease of implementation. The two contributions to spectral impurity are amplitude quantization (AQ), being present permanently, and phase truncation (PT). Both errors are periodic (i.e. deterministic) in time, therefore appear as line spectra (spurs) in frequency domain. Knowing the level and location of spurs is a good initial point for the design, selection or length-customizing (FPGA) of an NCO for any specific applications .

Preliminaries

The spurious components mentioned above are predictable. The spurs depend strongly on the NCO parameters ( r, m, n ) and the numerical frequency ( f/fc = D/2r ). The analytical results (see References, in particular AQ) are rather complicated, not easy to be interpreted and not very useful for practical design. (E.g. spurs vary rather irregularly.)

The problem can be solved by interactive computer simulations, by calculating and interpreting data for two distortion mechanisms (AQ and PT) separately, as well as together, depending on the parameters.

Results

Analizing the effect of numerical distortion from another viewpoint, one can regard AQ as overlapped harmonic structure and PT as overlapped modulation structure in frequency domain. In this way, one can easily explain, visualize and interactively examine (with math tools) the parameter dependence of spurs.

On the basis of the previous method, one can answer some challenging questions easily, such as: Where is the quantization noise? [I.e. only loss (no distortion) at f/fc = 1/4, and coincident spurs (massive overlap), close-in spurs (fine offset) or a "sea" of fine spaced spurs are present in the spectrum depending on the value of numerical frequency.] Which effect (AQ or PT) is dominating? What is the right frequency tuning word if you can select the output frequency within a narrow band? What is the relation between periodicity's and spurs? How results depend on the initial phase? ... etc.

An example: AQ and PT interaction
        zero-phase sine, 8K FFT, BH7 (7-term Blackman-Harris) window, with and without AQ

Another example: multisine, and memory length effect: 8K (m=13) vs. 1K (m=10)
        cos components with -5dB level change (and 8K FFT, BH7 window)

Shortcomings of the method: limited length of FFT (or processing time of Megapoint FFT), extra large number of generated frequency. (But one can "zoom" on the interested area.)

Novelties

There is an interaction between AQ and PT.
The method can be used for multitone signals. Note: simple superposition is invalid (because of strong nonlinearities).
The method can be extended to dithered DDS.
Simulations of DDS in Mathcad8 woksheets [Online], Available: http://www.hit.bme.hu/people/papay/sci/DDS/simul.htm
(Note: Mathcad8 has a freely downloadable interactive viewer.)

References

The milestone paper
J. Tierney, C. M. Rader, B. Gold, "A digital frequency synthesizer", IEEE Trans. on Audio Electroacoust., vol. AU-19, pp. 48-57, Mar. 1971

A patent
J. A. Webb, "Digital signal generator synthesizer," U.S. patent  No. 3,654,450 - Apr. 1972  [Online], Available: http://eepatents.com/collection.html#015

Tutorials
Sciteq: "Frequency synthesis & RF subsystems," 1994
Qualcomm: "Synthesizer products data book," 1996
Stanford Telecom: "The DDS handbook," 7th ed., 1999
Analog Devices: "A technical tutorial on digital synthesis," 1999
IEE Colloquium on "DDFS", ref. No: 1991/172
V. F. Kroupa (Ed.), "Direct digital frequency synthesizers," IEEE Reprint Press, 1998
B. Goldberg, "Digital frequency synthesis demystified,"LLH Tech. Pub., 1999

AQ: Amplitude Quantization of sine wave (lookup-table wordlength effect, deterministic approach)
A. G. Clavier, P. F. Panter, D. D. Grieg, "Distortion in a pulse count modulation systems," Trans. AIEE, vol. 66, pp. 989-1005, 1947. Essential substance: "PCM distortion analysis," Electrical Eng., pp. 1110-1122, Nov. 1947
A. Fujii, K. Azegami, "Quantizing noise for a sine wave and a set of sine waves," Rev. of the Electrical Commun. Lab., vol. 15, pp. 145-152, Mar./Apr. 1967
M. J. Hawksford, "Unified theory of digital modulation," Proc. IEE, vol. 121, pp. 109-115, Feb. 1974
D. L. Duttweiler, D. G. Messerschmitt, "Analysis of digitally generated sinusoids with application to A/D and D/A converter testing," IEEE Trans. on Commun., vol. COM-26, pp. 669-675, May 1978
D. R. Morgan, A. Aridgides, "Discrete-time distortion analysis of quantized sinusoids," IEEE Trans. on Acoust., Speech, Signal Proc., vol. ASSP-33, pp. 323-326, Feb. 1985
J. H. Blythe, "The spectrum of the quantized sinusoid," GEC J. of Research, vol. 3, no. 4, pp. 229-242, 1985
N. M. Blachman, "The intermodulation and distortion due to quantization of sinusoids," IEEE Trans. on Acoust., Speech, Signal Proc., vol. ASSP-33, pp. 1417-1426, Dec. 1985
R. M. Gray, "Quantization noise spectra," IEEE Trans. on Inform. Theory, vol. 36, pp. 1220-1244, Nov. 1990
R. C. Maher, "On the nature of granulation noise in uniform quantization systems," J. Audio Eng. Soc., vol. 40, pp.12-19, Jan/Feb. 1992
P. E. K. Chow, "Performance in waveform quantization," IEEE Trans. on Commun., vol. COM-40, pp. 1737-1745, Nov. 1992
E. W. Multanen, Y. C. Jenq, "Harmonic quantization noise in oversampled analog to digital converters," IMTC'93, pp. 151-153, 1993
D. Bellan, A. Brandolini, A. Gandelli, "Quantizing theory in electrical and electronic measurements," IEEE Instrum. and Meas. Tech. Conf. (IMTC'95), pp. 494-499, 1995. Extended version: "Quantizing theory - a deterministic approach," IEEE Trans. on Instrum. and Meas., vol. 48, pp. 18-25, Feb. 1999
Zs. Pápay, "Comments on 'The Modulo Time Plot: A Useful Data Acquisition Diagnostic Tool' ", IEEE Trans. on Instrum. and Meas., Vol. IM-45, No.6, p. 959, 1996  and  (error correction)  Vol. IM-46, No. 3, p. 739, 1997

PT: Phase Truncation (lookup-table address-length effect)
S. Mehrgardt, "Noise spectra of digital sine-generators using the table-lookup method," IEEE Trans. on Acoust., Speech, Signal Proc., vol. ASSP-31, pp. 1037-1039, Aug. 1983
J. J. Olsen, P.M. Fishman, "Truncation effect in direct digital frequency synthesis," Proc. of the 20th Asilomar Conf. on Signal, Systems and Computers, pp. 186-190, Nov. 1986
Y. C. Jenq, "Digital spectra of non-uniformly sampled signals with application to digitally synthesized sinusoids," Proc. Int. Conf. on Acoust., Speech, Signal Proc., vol.2, pp. 689-692, Apr. 1987. Extended version: "Digital spectra of nonuniformly sampled signals - digital look-up table sinusoidal oscillators," IEEE Trans. on Instrum. and Meas., vol. 37, pp. 358-362, Sept. 1988
H. T. Nicholas, H. Samueli, "An analysis of the output spectrum of direct digital frequency synthesizers in the presence of phase-accumulator truncation," Proc. 41st Annual Frequency Control Symp., pp. 495-502, May 1987
H. T. Nicholas, H. Samueli, B. Kim, "The optimization of direct digital frequency synthesizer performance in the presence of finite word length effect," Proc. 42nd Annual Frequency Control Symp., pp. 357-363, May 1988
J. F. Garvey, D. Babitch, "An exact spectral analysis of a number controlled oscillator based synthesizer," Proc. 44th Annual Frequency Control Symp., pp. 511-521, May 1990
F. Cercas, M. Tomlinson, A. A. Albuquerque, "Designing with direct digital frequency synthesizers," RF EXPO EAST '90, pp.625-633, Nov. 1990
H. P. Benn, "Spurious frequency generation in direct digital synthesizers," IEE Colloquium on DDFS (Digest No: 1991/172), pp. 2/1-2/6, Nov. 1991
Zs. Pápay, "DDS signal generator" (1995), published in Zs. Pápay, "Waveform measurement and synthesis", Muegyetemi kiadó, Budapest, 1996 (in hungarian)
J. Vankka, "Spur reduction techniques in sine output direct digital synthesis," Proc. 50st Annual Frequency Control Symp., pp. 951-959,  1996
J. Vollmer, "Analysis and design of numerically controlled oscillators based on linear time-variant systems," Proc. of the IEEE-SP Int. Symp. on Time-Frequency and Time- Scale Analysis, pp. 453-456, 1998

Miscellaneous
"DDS" (Internet links), [Online], Available: http://www.hit.bme.hu/people/papay/sci/DDS/products.htm

Date:  29/9/2000

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