Signal representation with triangular basis functions

The set of harmonically-related 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 convergence-completeness properties. The weights of this series, and the weights of the finite series having minimum mean-square 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 time-shared digital filter sections are also easily diverted to real-time 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, continuous-time and discrete-time triangular transforms for aperiodic and sampled signals can be enunciated. Several laboratory and computer-generated examples are given.