The concept of rotations in continuous-time, continuousfrequency is extended to discrete-time, discrete-frequency as it applies to the Wigner distribution. As in the continuous domain, discrete rotations are defined to be elements of the special orthogonal group over the appropriate (discrete) field. Use of this definition ensures that discrete rotations will share many of the same mathematical properties as continuous ones. A formula is given for the number of possible rotations of a prime-length signal, and an example is provided to illustrate what such rotations look like. In addition, by studying a 90 degree rotation, we formulate an algorithm to compute a prime-length discrete Fourier transform (DFT) based on convolutions and multiplications of discrete, periodic chirps. This algorithm provides a further connection between the DFT and the discrete Wigner distribution based on group theory.
A discrete-time, discrete-frequency Wigner distribution is derived using a g,roup-theoretic approach. It is based upon a study of the Heisenberg group generated by the integers mod N , which represents the group of discrete-time and discrete-frequency shifts. The resulting Wigner distribution satisfies several desired properties. An example demonstrates that it is a full-band time-frequency representation, and, a s such, does not require special sampling techniques to suppress aliasing. It also exhibits some interesting and unexpected interference properties. The new distribution is compared with other discrete-time, discretefrequency Wigner distributions proposed in the literature.
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