Fast Algorithm of Computing Dual Functions
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In Sections 3.3 and 3.4, we discuss the algorithms of computing the dual function g[k] for the periodic discrete Gabor expansion and infinite discrete Gabor expansion, respectively. In both cases, g[k] are the solutions of the linear complex systems. In this section, we shall show that those complex matrices in fact could be converted into real-valued matrices and thereby save a considerable amount of computations [118]. The following discussions are mainly based on the case of the infinite samples (3.49), but the results should be easily extended to the periodic cases (3.29) simply by replacing the auxiliary function h[k] by the periodic function h[k].Given the finite Gabor elementary function h[k], the dual function g[k] for the discrete Gabor expansion can be computed via Eq. (3.49)
where 0 £ p < DM and 0 £ q < 2(DN-1). The auxiliary periodic function h[k] is defined in (3.47). Therefore, computing g[k] is to solve the linear complex system with DM(2DN-1)-by-L complex matrix H.
Let's take the inverse DFT with respect to (3.49), i.e.,
Eq. (3.70) reduces to
where 0 £ k < DM and 0 £ q < 2DN-1. Eq. (3.72) suggests that g[k] can be computed by DM-separated real-valued linear systems, i.e.,
where Hk are (2DN-1)-by-M matrices with the entries:
and
While (3.49) is a DM(2DN-1)-by-L linear complex system, the solution of (3.72) is a DM independent (2DN-1)-by-M linear real system. Therefore, considerable computations are saved when (3.72) is used. Moreover, it is interesting to note that the minimum norm of (3.31) is equal to the minimum norms computed by each individual linear system in (3.73). We leave the proof for the reader to exercise.
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Short-Time Fourier Transform and Gabor Expansion (Introduction)
Short-Time Fourier Transform
Gabor Expansion - Inverse Sampled STFT
Gabor Expansion for Discrete Periodic Samples
Discrete Gabor Expansion
Orthogonal-Like Gabor Expansion
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