research-article

Adapting Recursive Sinusoidal Software Oscillators for Low-power Fixed-point Processors

Authors:

Matteo Ceriotti,

Pedro José MarrónAuthors Info & Claims

ACM Transactions on Embedded Computing Systems (TECS), Volume 19, Issue 3

Article No.: 17, Pages 1 - 26

https://doi.org/10.1145/3378559

Published: 18 May 2020 Publication History

Abstract

The growing field of the Internet of Things relies at the bottom on components with very scarce computing resources that currently do not allow complex processing of sensed data. Any computation involving Fast Fourier Transforms (FFT), Wavelet Transforms (WT), or simple sines and cosines is considered impractical on low-end devices due to the lack of floating point and math libraries. This article presents new techniques that make it possible to use these functions also on severely constrained target platforms.

Current literature abounds with schemes to compute sine and cosine functions, with focus on speed, hardware footprint, software size, target type, or precision. Even so, there is no practical exploration of the design space available for embedded devices with limited resources, in particular when only integer operations are possible. We select an efficient set of recursive sine and cosine generators and measure the frequency, amplitude, and phase error over a wide parameter range. We show that their simplicity allows them to be implemented on the most bare targets with good precision, reducing power consumption and size while being the fastest on integer-only processors. We also introduce specially tailored FFT and WT algorithms and show that they are usable in practice while having an extremely small code footprint, good precision, and high speed.

References

[1]

A. L. Abu-El-Haija and M. Al-Ibrahim. 1986. Improving performance of digital sinusoidal oscillators by means of error feedback circuits. IEEE Trans. Circ. Syst. 33, 4 (1986), 373--380.

[2]

M. M. Al-Ibrahim. 2001. A simple recursive digital sinusoidal oscillator with uniform frequency spacing. In Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS’01), Vol. 2. IEEE, 689--692.

[3]

M. M. Al-Ibrahim and A. M. Al-Khateeb. 1997. Digital sinusoidal oscillator with low and uniform frequency spacing. IEE Proc. Circ. Devices Syst. 144, 3 (1997), 185--189.

[4]

David L. Baughman. 2012. An early history of logarithms. Ph.D. Dissertation.

[5]

Sourav Bhattacharya, Henrik Blunck, Mikkel Baun Kjærgaard, and Petteri Nurmi. 2015. Robust and energy-efficient trajectory tracking for mobile devices. IEEE Trans. Mobile Comput. 14, 2 (2015), 430--443.

[6]

Amelie Bonde, Shijia Pan, Zhenhua Jia, Yanyong Zhang, Hae Young Noh, and Pei Zhang. 2018. VVRRM: Vehicular vibration-based heart RR-interval monitoring system. In Proceedings of the 19th International Workshop on Mobile Computing Systems (HotMobile’18). ACM, New York, NY, 37--42.

Digital Library

[7]

ChaN. 2005. Fixed-point FFT Routines for megaAVRs. Retrieved from http://elm-chan.org/docs/avrlib/fftavr.zip.

[8]

Glenn Chang, Ahmadreza Rofougaran, Mong-Kai Ku, A. A. Abidi, and Henry Samueli. 1994. A low-power CMOS digitally synthesized 0-13 MHz agile sinewave generator. In Proceedings of the 41st IEEE International Solid-State Circuits Conference (ISSCC’94). IEEE, 32--33.

[9]

Jen-Chuan Chih, Jun-Yei Chou, and Sau-Gee Chen. 2001. An efficient direct digital frequency synthesizer based on two-level table lookup. In Proceedings of the IEEE International Frequency Control Symposium and PDA Exhibition. IEEE, 824--827.

[10]

Jan L. Cieśliński, Leonid V. Moroz, and Cezary J. Walczyk. 2015. Fast exact digital differential analyzer for circle generation. Appl. Math. Comput. 271 (2015), 68--79.

Digital Library

[11]

W. T. Cochran, J. W. Cooley, D. L. Favin, H. D. Helms, R. A. Kaenel, W. W. Lang, G. C. Maling, D. E. Nelson, C. M. Rader, and P. D. Welch. 1967. What is the fast Fourier transform? Proc. IEEE 55, 10 (Oct. 1967), 1664--1674.

[12]

James W. Cooley and John W. Tukey. 1965. An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 90 (1965), 297--301.

[13]

D. De Caro, E. Napoli, and A. G. M. Strollo. 2004. Direct digital frequency synthesizers with polynomial hyperfolding technique. IEEE Trans. Circ. Syst. II: Express Briefs 51, 7 (2004), 337--344.

[14]

Jack Dongarra and Francis Sullivan. 2000. Guest editors introduction: The top 10 algorithms. Comput. Sci. Eng. 2, 1 (2000), 22--23.

Digital Library

[15]

Ayman A. El-Badawy and Mohamed A. Bakr. 2016. Quadcopter aggressive maneuvers along singular configurations: An energy-quaternion based approach. J. Control Sci. Eng. 2016 (2016), 4.

Digital Library

[16]

Atis Elsts, Ryan McConville, Xenofon Fafoutis, Niall Twomey, Robert Piechocki, Raul Santos-Rodriguez, and Ian Craddock. 2018. On-board feature extraction from acceleration data for activity recognition. In the proceedings of EWSN 2018. 163--168.

[17]

L. Fanucci, R. Roncella, and R. Saletti. 2001. A sine wave digital synthesizer based on a quadratic approximation. In Proceedings of the IEEE International Frequency Control Symposium and PDA Exhibition. IEEE, 806--810.

[18]

Franz Franchetti, Stefan Kral, Juergen Lorenz, and Christoph W. Ueberhuber. 2005. Efficient utilization of SIMD extensions. Proc. IEEE 93, 2 (2005), 409--425.

[19]

Matteo Frigo and Steven G. Johnson. 2012. FFTW: Fastest Fourier Transform in the West. Astrophysics Source Code Library.

[20]

Gartner. 2016. Market Share Analysis: Microcontroller Revenue, Worldwide, 2015. Retrieved from https://www.gartner.com/doc/3293617/market-share-analysis-microcontroller-revenue.

[21]

Nils Gura, Arun Patel, Arvinderpal Wander, Hans Eberle, and Sheueling Chang Shantz. 2004. Comparing elliptic curve cryptography and RSA on 8-bit CPUs. In Proceedings of the International Workshop on Cryptographic Hardware and Embedded Systems. Springer, 119--132.

[22]

Idda Hervé. 2005. Cryptage du son et traitement numérique. Bup 879 99 (2005). Retrieved from http://agregation.capes.free.fr/bup/sommaires-2005bup.htm.

[23]

Barbara Burke Hubbard. 1998. The World According to Wavelets the Story of a Mathematical Technique in the Making. Universities Press.

[24]

Steven G. Johnson and Matteo Frigo. 2008. Implementing FFTs in practice. Fast Fourier Transforms, C. S. Burrus (Ed.).

[25]

Madhusudan Joshi, Chandrashakher, and Kehar Singh. 2007. Color image encryption and decryption using fractional Fourier transform. Optics Commun. 279, 1 (2007), 35--42.

[26]

George Lederman, Siheng Chen, James Garrett, Jelena Kovacevic, Hae Young Noh, and Jacobo Bielak. 2017. Track-monitoring from the dynamic response of an operational train. Mech. Syst. Signal Process. 87 (2017), 1--16.

[27]

Todd R. Littell, Robert D. Skeel, and Meiqing Zhang. 1997. Error analysis of symplectic multiple time stepping. SIAM J. Numer. Anal. 34, 5 (1997), 1792--1807.

Digital Library

[28]

Sanjoy Mahajan and Dennis Freeman. 2009. Discrete-time Signals and Systems. Retrieved from https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-003-signals-and-systems-fall-2011/readings/MIT6_003F11_front.pdf.

[29]

Aleksandar Milenkovic, Chris Otto, and Emil Jovanov. 2006. Wireless sensor networks for personal health monitoring: Issues and an implementation. Comput. Commun. 29, 13--14 (2006), 2521--2533. Wirelsess Sensor Networks and Wired/Wireless Internet Communications.

Digital Library

[30]

Emiliano Miluzzo, Nicholas D. Lane, Kristóf Fodor, Ronald Peterson, Hong Lu, Mirco Musolesi, Shane B. Eisenman, Xiao Zheng, and Andrew T. Campbell. 2008. Sensing meets mobile social networks: The design, implementation and evaluation of the cenceme application. In Proceedings of the 6th ACM Conference on Embedded Network Sensor Systems. ACM, 337--350.

[31]

Masachika Miyata. 1988. Highly accurate digital sinusoidal oscillators using controlled rounding. In Proceedings of the IEEE International Symposium on Circuits and Systems. IEEE, 2213--2216.

[32]

Matteo Morelli and Marco Di Natale. 2014. Control and scheduling co-design for a simulated quadcopter robot: A model-driven approach. In Proceedings of the International Conference on Simulation, Modeling, and Programming for Autonomous Robots. Springer, 49--61.

Digital Library

[33]

Eva Murphy and Colm Slattery. 2004. Ask the application engineer (#33). All about direct digital synthesis. Analog Devices 38 (2004), 1--5.

[34]

Jouko Niiranen. 1999. Fast and accurate symmetric Euler algorithm for electromechanical simulations. In Proceedings of the 6th International Conference of the IMACS TC1 Committee (ElectrIMACS’99), Vol. 1. 71--78.

[35]

I. P. Omelyan, I. M. Mryglod, and Reinhard Folk. 2002. Optimized forest--ruth-and suzuki-like algorithms for integration of motion in many-body systems. Comput. Phys. Commun. 146, 2 (2002), 188--202.

[36]

Antonino Orsino, Giuseppe Araniti, Leonardo Militano, Jesus Alonso-Zarate, Antonella Molinaro, and Antonio Iera. 2016. Energy efficient IoT data collection in smart cities exploiting D2D communications. Sensors 16, 6 (2016), 836.

[37]

Kalle I. Palomaki and Jarkko Niittylahti. 2000. Direct digital frequency synthesizer architecture based on Chebyshev approximation. In Proceedings of the Conference Record of the 34th Asilomar Conference on Signals, Systems and Computers, Vol. 2. IEEE, 1639--1643.

[38]

Shijia Pan, Ningning Wang, Yuqiu Qian, Irem Velibeyoglu, Hae Young Noh, and Pei Zhang. 2015. Indoor person identification through footstep induced structural vibration. In Proceedings of the 16th International Workshop on Mobile Computing Systems and Applications (HotMobile’15). ACM, New York, NY, 81--86.

Digital Library

[39]

Laura Lo Presti and Giuseppe Cardamone. 1994. A direct digital frequency synthesizer using an IIR filter implemented with a DSP microprocessor. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’94), Vol. 3. IEEE, III--201.

[40]

Charles M. Rader. 1968. Discrete Fourier transforms when the number of data samples is prime. Proc. IEEE 56, 6 (1968), 1107--1108.

[41]

George U. Ramos. 1971. Roundoff error analysis of the fast Fourier transform. Math. Comput. 25, 116 (1971), 757--768.

[42]

Victor H. Rodriguez, Carlos Medrano, and Inmaculada Plaza. 2018. Embedded system based on an ARM microcontroller to analyze heart rate variability in real time using wavelets. Wireless Commun. Mobile Comput. 2018 (2018).

[43]

Cristian Rotariu, Vasile Ion Manta, and Hariton Costin. 2010. Patient monitoring using a low power wireless personal area network of sensors. Buletinul Institutului Politehnic din Iasi 56 (2010), 73--87.

[44]

John Spitzer. 2003. Implementing a GPU-efficient FFT. In Proceedings of the SIGGRAPH Course on Interactive Geometric and Scientific Computations with Graphics Hardware.

[45]

David A. Sunderland, Roger A. Strauch, Steven S. Wharfield, Henry T. Peterson, and Christopher R. Cole. 1984. CMOS/SOS frequency synthesizer LSI circuit for spread spectrum communications. IEEE J. Solid-State Circ. 19, 4 (1984), 497--506.

[46]

Microchip Technology. 2017. Cooley-Tukey FFT for 16-bit Integer Numbers. Retrieved from http://www.embeddedcodesource.com/codesnippet/cooley-tukey-fft-for-16-bit-integer-numbers.

[47]

Clay S. Turner. 2003. Recursive discrete-time sinusoidal oscillators. IEEE Signal Process. Mag. 20, 3 (2003), 103--111.

[48]

Chua-Chin Wang, Hsien-Chih She, and Ron Hu. 2002. A ROM-less direct digital frequency synthesizer by using trigonometric quadruple angle formula. In Proceedings of the 9th International Conference on Electronics, Circuits and Systems, Vol. 1. IEEE, 65--68.

[49]

Hae Young Noh, K. Krishnan Nair, Dimitrios G. Lignos, and Anne S. Kiremidjian. 2011. Use of wavelet-based damage-sensitive features for structural damage diagnosis using strong motion data. J. Struct. Eng. 137, 10 (2011), 1215--1228.

[50]

Nikos Zervas. 2014. 8051 Interrupt Latency: Designing with Modern 8-bit MCUs. Retrieved from http://www.cast-inc.com/blog/8051-interrupt-latency-designing-with-modern-8-bit-mcus.

[51]

Bin Zhou, Yingning Peng, and David Hwang. 2009. Pipeline FFT architectures optimized for FPGAs. Int. J. Reconfig. Comput. 2009 (2009), 1.

Digital Library

Cited By

Arts Lvan den Broek E(2022)The fast continuous wavelet transformation (fCWT) for real-time, high-quality, noise-resistant time–frequency analysisNature Computational Science10.1038/s43588-021-00183-z2:1(47-58)Online publication date: 27-Jan-2022
https://doi.org/10.1038/s43588-021-00183-z

Index Terms

Adapting Recursive Sinusoidal Software Oscillators for Low-power Fixed-point Processors

Recommendations

SC- and SS-wavelet transforms

Grigoryan [Fourier transform representation by frequency-time wavelets, IEEE Trans. Signal Process. 53 (7) (2005) 2489-2497] has proposed an alternative representation of the Fourier transform, called A-wavelet transform. In that paper, the Cosine and ...
Fourier transform representation by frequency-time wavelets

A new concept of the A-wavelet transform is introduced, and the representation of the Fourier transform by the A-wavelet transform is described. Such a wavelet transform uses a fully scalable modulated window but not all possible shifts. A geometrical ...
An algorithm for parallel calculation of trigonometric functions
CF '13: Proceedings of the ACM International Conference on Computing Frontiers

We propose a new way of calculating the sine and cosine functions. The method is based on recursive applications of a modified complex power algorithm. On a machine with multiple complex multipliers the method can be used to calculate sines and cosines ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Embedded Computing Systems

ACM Transactions on Embedded Computing Systems Volume 19, Issue 3

May 2020

156 pages

ISSN:1539-9087

EISSN:1558-3465

DOI:10.1145/3400880

Editor:
Sandeep K. Shukla
Indian Institute of Technology, India

Issue’s Table of Contents

Copyright © 2020 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Journal Family

ACM Journals for the Design of Smart and Connected Systems

Publication History

Published: 18 May 2020

Online AM: 07 May 2020

Accepted: 01 January 2020

Revised: 01 June 2019

Received: 01 July 2018

Published in TECS Volume 19, Issue 3

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

1
Total Citations
View Citations
147
Total Downloads

Downloads (Last 12 months)10
Downloads (Last 6 weeks)0

Reflects downloads up to 10 Aug 2024

Other Metrics

View Author Metrics

Citations

Cited By

Arts Lvan den Broek E(2022)The fast continuous wavelet transformation (fCWT) for real-time, high-quality, noise-resistant time–frequency analysisNature Computational Science10.1038/s43588-021-00183-z2:1(47-58)Online publication date: 27-Jan-2022
https://doi.org/10.1038/s43588-021-00183-z

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Media

Figures

Other

Tables

View Issue’s Table of Contents