Signal representation with triangular basis functions
Abstract
The set of harmonicallyrelated nonorthogonal triangle waves is shown to form a basis spanning the same function space representable by Fourier (trigonometric) series. The triangle function set is, further, equivalent to the trigonometric series in important convergencecompleteness properties. The weights of this series, and the weights of the finite series having minimum meansquare error, are calculated directly without resort to optimisation or other iterative techniques. This basis function set is most attractive for digital signal representation because these functions can be conveniently generated in a digital context. Unused 'time slots' of timeshared digital filter sections are also easily diverted to realtime signal representation. Thus, depending on the application, triangle waves can provide ease of implementation while maintaining the convergence properties of trigonometric series. For coding applications, continuoustime and discretetime triangular transforms for aperiodic and sampled signals can be enunciated. Several laboratory and computergenerated examples are given.
 Publication:

Journal of Electronic Circuits and Systems
 Pub Date:
 March 1979
 Bibcode:
 1979JECS....3...58G
 Keywords:

 Fourier Series;
 Function Space;
 Sawtooth Waveforms;
 Set Theory;
 Signal Analysis;
 Convergence;
 Digital Filters;
 Trigonometric Functions;
 Electronics and Electrical Engineering