Real-valued Discrete Gabor Transform Based On DCT And Fast Algorithms

In 1946, Dennis Gabor, a Hungarian-born British physicist, suggested expanding a signal into a set of elementary functions that consist of a time-and frequency-shift function, and then, used the expansion coefficients as the description of the signal's local property. The resulting representation is now known as the Gabor expansion (or Gabor transform). Although the Gabor expansion has been recognized as very useful for signal processing, its applications were limited due to the difficulties associated with computing the coefficients.As we review the research history of the Gabor expansion and transform, we find that there is very little research on real-valued Gabor theory. Although Gabor introduced the real-valued Gabor expansion based on continuous cosine transform in 1946; and in 1995, Stewart D F discreted it, but the above research is far too enough, for example, the completeness condition is not resolved. The related research paper is also very little.Based on the analysis above, this paper concentrates on the real-valued Gabor expansion and transform, the main contents as follows: first, we define the theory by introducing a series of time-frequency-shiftable basis functions based on discrete consine transform, and then we discuss the theory's completeness, finally, we study the theory's serial fast algorithm and its applications.The paper's characteristics and innovations just as follows:First, the proposed real-valued discrete Gabor transform can be applied to both the critical sampling condition and the over-sampling condition.Second, because the transform kernel of our theory contains DCT's, some features of the DCT are retained, eg. efficient data compress.Third, in allusion to real-valued signals, real-valued discrete Gabor transform is much simpler than complex discrete Gabor transform in calculation and implementation. By the way, our algorithm can be accelerated with the help of DCT and IDCT fast algorithms.