Wavelet techniques: A suitable tool to characterise and optimize encoders’ based systems J.M. Dias Pereira a,b, * , O. Postolache a,b , P. Gira ˜o b a LabIM, Escola Superior de Tecnologia de Setu ´ bal, Departamento de Sistemas e Informa ´ tica, Instituto Polite ´cnico de Setu ´ bal, Rua do Vale de Chaves, Estefanilha, 2910-761 Setu ´ bal, Portugal b Instituto de Telecomunicac ¸o ˜es, DEEC, IST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Received 30 December 2005; received in revised form 5 June 2006; accepted 5 June 2006 Available online 30 June 2006 Abstract Calibration data together with the approximation and details of wavelet decomposition functions can be used to obtain important parameters of digital transducers based systems. Besides offset and gain errors that are usually easy to evaluate and compensate, digital transducers exhibit differential and integral nonlinearity errors whose estimation, whenever possible, can be used to increase measurement’s system accu- racy. An error compensated system can easily be implemented by adding to the digital transducer, a microcontroller and a memory with a code conversion table. The proposed technique, based on wavelet data processing, can also be applied to estimated analogue-to-digital converters parameters, namely differential and integral nonlinearity errors. In the present paper, simulation and experimental results, obtained from a low-cost angular encoder, are used to val- idate the proposed characterisation method for different wavelet decomposition levels and wavelet mother function types. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Encoders; Wavelet transforms and error compensation 1. Introduction Digital encoders are used for linear and angular displacement and position measurements. For high resolution measurements the mechanical accuracy of the encoder must be very high in order to mini- mize errors. For example a 12-bit full-range encoder (span = 360°) corresponds to approximately 316 angular seconds resolution. As the resolution becomes lower the dimensional requests increase and the price of the transducer becomes higher (hun- dreds of dollars). It is also important to refer that even if incremental encoders are cheaper than abso- lute encoders, they are more sensitive to disturbing pulses (noise) and more affected by power failures. Even if in the literature several techniques are mentioned to improve digital sensor’s performance, namely dithering [1] and interleaving [2], their appli- cation to mechanical devices is always questionable 0263-2241/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2006.06.001 * Corresponding author. Address: LabIM, Escola Superior de Tecnologia de Setu ´bal, Departamento de Sistemas e Informa ´tica, Instituto Polite ´cnico de Setu ´ bal, Rua do Vale de Chaves, Estefanilha, 2910-761 Setu ´ bal, Portugal. Tel.: +351 265 790000; fax: +351 265 721869. E-mail address: joseper@est.ips.pt (J.M. Dias Pereira). Measurement 40 (2007) 264–271 www.elsevier.com/locate/measurement