High spectral purity digital direct synthesizer implementation by means of a fuzzy approximator

Abstract Fuzzy systems are used in many applications because of their implementation simplicity both in software and in hardware. Some papers in literature have shown the behavior of SISO fuzzy systems as function approximators. If the approximation function have certain analytical properties, it can be used to develop interesting solutions of classical problems. In particular, the authors have studied a new method suitable to implement systems whose behavior is described through a function having the derivative continuity. This paper shows how this approach can be used to implement a sinusoidal wave synthesizer that is the core of the digital direct synthesizers (DDS), widely used in modern communications and measurements devices. This work presents how tuning the fuzzy system in order to achieve a very clean sinusoidal wave in output. Both the theoretical behavior of the algorithm and the performance taking into account the errors due to the fixed point arithmetic have been inspected in detail. The results of the simulations and those obtained from the hardware implementation of the device show excellent features especially in the case that a very high spectral purity is required.

[1]  M. MendelJ.,et al.  Fuzzy basis functions , 1995 .

[2]  Dario Petri,et al.  Accuracy of effective bit estimation methods , 1999 .

[3]  Y. F. Kroupa.  Spectral properties of DDFS: computer simulations and experimental verifications , 1994, Proceedings of IEEE 48th Annual Symposium on Frequency Control.

[4]  Kari Halonen,et al.  Direct Digital Synthesizers , 2001 .

[5]  B. Daneshrad,et al.  Direct digital frequency synthesis using a modified CORDIC , 1998, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187).

[6]  Miriam Leeser,et al.  An area/performance comparison of subtractive and multiplicative divide/square root implementations , 1995, Proceedings of the 12th Symposium on Computer Arithmetic.

[7]  Michael J. Flanagan,et al.  Spur-reduced digital sinusoid synthesis , 1993, IEEE Trans. Commun..

[8]  Xiao-Jun Zeng,et al.  Approximation theory of fuzzy systems-SISO case , 1994, IEEE Trans. Fuzzy Syst..

[9]  Hussein Baher,et al.  Analog & digital signal processing , 1990 .

[10]  Xiao-Jun Zeng,et al.  Approximation theory of fuzzy systems-MIMO case , 1995, IEEE Trans. Fuzzy Syst..

[11]  C. V. Freiman,et al.  Statistical Analysis of Certain Binary Division Algorithms , 1961, Proceedings of the IRE.

[12]  Miriam Leeser,et al.  Division and square root: choosing the right implementation , 1997, IEEE Micro.

[13]  Kari Halonen,et al.  Preface , 2021 .

[14]  M. Ercegovac,et al.  Division and Square Root: Digit-Recurrence Algorithms and Implementations , 1994 .

[15]  James E. Robertson,et al.  A New Class of Digital Division Methods , 1958, IRE Trans. Electron. Comput..

[16]  Venceslav F. Kroupa.  Methods of Mapping from Phase to Sine Amplitude in Direct Digital Synthesis , 1999 .

[17]  H. Eisenson.  Frequency Synthesis Using DDS/NCO Technology , 1991, Electro International, 1991.

[18]  G. A. Zimmerman,et al.  Spur-reduced numerically-controlled oscillator for digital receivers , 1992, [1992] Conference Record of the Twenty-Sixth Asilomar Conference on Signals, Systems & Computers.

[19]  L X Wang,et al.  Fuzzy basis functions, universal approximation, and orthogonal least-squares learning , 1992, IEEE Trans. Neural Networks.

[20]  Amir M. Sodagar,et al.  Mapping from phase to sine-amplitude in direct digital frequency synthesizers using parabolic approximation , 2000 .

[21]  Diego Bellan,et al.  Effects of ADC nonlinearities in sine-wave amplitude measurement , 1998, 1998 IEEE International Conference on Electronics, Circuits and Systems. Surfing the Waves of Science and Technology (Cat. No.98EX196).

[22]  V. S. Reinhardt,et al.  Sine output DDSs. A survey of the state of the art , 1998, Proceedings of the 1998 IEEE International Frequency Control Symposium (Cat. No.98CH36165).

[23]  Michael J. Flynn.  On Division by Functional Iteration , 1970, IEEE Transactions on Computers.

[24]  Jouko Vankka.  1996 IEEE INTERNATIONAL FREQUENCY CONTROL SYMPOSIUM METHODS OF MAPPING FROM PHASE TO SINE AMPLITUDE IN DIRECT DIGITAL SYNTHESIS , 1996 .

[25]  Dario Petri,et al.  Accuracy of effective bits estimation methods , 1996 .

[26]  R. Rovatti,et al.  On the approximation capabilities of the homogeneous Takagi-Sugeno model , 1996, Proceedings of IEEE 5th International Fuzzy Systems.

[27]  Hao Ying,et al.  Sufficient conditions on general fuzzy systems as function approximators , 1994, Autom..

[28]  Diego Bellan,et al.  Quantization theory in electrical and electronic measurements , 1995, Proceedings of 1995 IEEE Instrumentation and Measurement Technology Conference - IMTC '95.

[29]  V.F. Kroupa.  Phase and amplitude disturbances in direct digital frequency synthesizers , 1999, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[30]  E. P. Nowicki,et al.  Optimal approximation of nonlinear functions by fuzzy systems , 1997, CCECE '97. Canadian Conference on Electrical and Computer Engineering. Engineering Innovation: Voyage of Discovery. Conference Proceedings.

[31]  Xiao-Jun Zeng,et al.  Approximation accuracy analysis of fuzzy systems as function approximators , 1996, IEEE Trans. Fuzzy Syst..

[32]  R. Rovatti,et al.  Sobolev approximation by means of fuzzy systems with overlapping Gaussian concepts , 1999, FUZZ-IEEE'99. 1999 IEEE International Fuzzy Systems. Conference Proceedings (Cat. No.99CH36315).

[33]  Arianna Mencattini,et al.  Function approximation using non-normalized SISO fuzzy systems , 2001, Int. J. Approx. Reason..

[34]  Diego Bellan,et al.  Quantization theory-a deterministic approach , 1999, IEEE Trans. Instrum. Meas..

[35]  Alan V. Oppenheim,et al.  Discrete-time Signal Processing. Vol.2 , 2001 .

[36]  Arianna Mencattini,et al.  Performance optimization of SISO fuzzy systems used as function approximators , 2002 .

[37]  M. J. Flanagan,et al.  Spur-reduced digital sinusoid generation using higher-order phase dithering , 1993, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers.

[38]  H. Samueli,et al.  The optimization of direct digital frequency synthesizer performance in the presence of finite word length effects , 1988, Proceedings of the 42nd Annual Frequency Control Symposium, 1988..

[39]  Bart Kosko,et al.  Fuzzy Systems as Universal Approximators , 1994, IEEE Trans. Computers.

[40]  K. D. Tocher.  TECHNIQUES OF MULTIPLICATION AND DIVISION FOR AUTOMATIC BINARY COMPUTERS , 1958 .

[41]  A. L. Bramble,et al.  Direct Digital Frequency Synthesis , 1981 .