Orthogonal-Like Gabor Expansion

Overview

As shown earlier, given h[k] and the sampling steps, the solution of g[k] in general is not unique. Then, the question is how to select the g[k] that best meets our goal. Recall that the Gabor coefficients C_m,n are the sampled short-time Fourier transform with the window function g[k]. This means that the window function g[k] has to be localized in the joint time-frequency domain. Otherwise, the Gabor coefficients C_m,n, inner product of s[k] and g[k], would not characterize the signal's local behavior.



Fig. 3-13 Parts (b) and (c) are dual functions of h[k] in (a). They both yield the perfect reconstruction. g_opt[k] in (b) is optimally close to Gaussian-type function h[k] in (a) .

Note: The algorithm of computing different dual functions for a same given function h[k] is introduced in Appendix B.



Fig. 3-14 (a) is computed by g_opt[k], which well presents the linear chirp signal, (b) is computed by g[k] plotted in Fig. 3—13c, which does not provide desired information of the linear chirp signal in the joint time-frequency domain. However, both Gabor coefficients will lead to the perfect reconstruction by using the same synthesis function h[k].

Second, the behaviors of g[k] and h[k], such as time/frequency centers and time/frequency resolution, have to be close. Suppose that h[k] and its Fourier transform are centered in k = 0 and q = 0. If g[k] and h[k] are significantly different, for instance, they have completely different time or frequency centers, then C_m,n, will not reflect the signal behaviors in the vicinity of (mT,nW).

Fig. 3-13 depicts two different dual functions g[k] that both correspond to the same Gabor elementary function h[k] in (a). The shape of g[k] in (b) is optimally close to that of h[k], whereas the shape of g[k] in (c) significantly differs from h[k]. Although both g[k] will lead to a perfect reconstruction with the same h[k], the resulting Gabor coefficients have substantial differences.

Fig. 3—14 illustrates the magnitude of the Gabor coefficients computed by two different g[k] for the linear chirp signal, (a) is computed by the optimal g[k]. Because h[k] in Fig. 3-13a is concentrated in the joint time and frequency domain, the optimal g[k] is also concentrated. Consequently, the resulting Gabor coefficients well present the monotone linear chirp signal, (b) is computed by g[k] plotted in Fig. 3-13c, whose shape significantly differs from the Gabor elementary function h[k] in Fig. 3-13a. Due to the bad shape of the analysis function, the resulting Gabor coefficients plotted in Fig. 3-14b do not properly describe the time-varying nature of the linear chirp.

Because the Gabor elementary function h[k] is the Gaussian function that is optimally concentrated in the joint time-frequency domain, the natural selection of g[k] is that whose shape is closest to h[k], in the sense of the least square error, i.e.,




where



and h[k] is a normalized function, that is, ||h[k]||² = 1. For small G, g[k] » ah[k] where a is a real-valued constant. In this case, the discrete Gabor expansion can be written as




and



Note: Because (3.31) and (3.46) have the same form, discussions in this section apply for both periodic and non-periodic cases.


Obviously, as long as h[k] is localized in the joint time-frequency domain, the Gabor coefficients C_m,n, will well depict the signal's local time-frequency properties. Although the set of {h_m,n[k]} is not orthogonal and redundant, the coefficients C_m,n, are still good approximations of signal orthogonal projection on {h_m,n[k]}. Therefore, (3.59) is called an orthogonal-like Gabor expansion. The remaining problem is how to solve (3.56). Expanding Eq. (3.56) obtains [118]




Because (see (3.49))




(3.60) reduces to




which implies that the solution is the minimum energy of g[k]. According to the matrix analysis theory, g[k] exists as long as vector m. is in the range of matrix H. If matrix H is of full-row rank, the minimum energy of g[k] is equal to the pseudo inverse of H, i.e.,




When the rank of H is less than the number of rows, we can employ the singular value decomposition (SVD) to alleviate the rank deficiency problem.

Assume that the rank of H is p and the number of rows is q, where q ³ p. Applying SVD to matrix H, (3.46) becomes





where matrices U Î C^q,q and V Î C^L,L are unitary. S Î R^q,L . Multiplying (U^T to both sides of (3.63) yields




where S⁰ Î R^p,p is a diagonal matrix with non-negative main diagonal entries. The solution of (3.64) exists when m is orthogonal to U¹ (consistency condition). When the consistency condition is satisfied, we rewrite (3.64) as




where H = S⁰V⁰ Î C^P,L has full row rank p and m = (U⁰)^Tm. Then, the minimum energy solution of g[k] is






Fig. 3-15 h[k] (dotted line) and g_opt[k] (solid line).

Fig. 3—15 depicts the Gaussian function and corresponding dual functions at different sampling schemes. In order to utilize the FFT (fast Fourier transform), the number of frequency bins (or channels) N is usually chosen as two's power. For instance, N = 2^k, k = 1, 2 ,3, ... The oversampling rate is determined by a = N/DM, where DM denotes the time-sampling step. The length of h[k] and g_opt[k], L, has to be divided by both N and DM. That is, L/N and L/DM have to be integers.

In general, the difference between h[k] and g_opt[k] decreases as the over-sampling rate increases. To faithfully characterize a signal's local properties in the joint time-frequency domain, it is critical to make g_opt[k] and h[k] as close as possible. The minimum difference between g_opt[k] and h[k], G in (3.56), is related to the selection of h[k] and sampling steps. If the Gaussian function is used, for instance,



the minimum difference between g_opt[k] and h[k] is observed when




The discrete Gabor expansion and the concept of orthogonal-like Gabor expansion were first proposed by Qian et al. (see [148] and [149]) In addition to seeking g[k] that is optimally close to h[k], the algorithm introduced in this section could be easily modified to have g[k] optimally close to an arbitrary function, i.e.,




where d[k] is a normalized target function. Such generalized optimization is very useful, in particular, for those applications where analysis and synthesis functions are desired to have different properties. For instance, we could use the Gabor expansion for filter bank design. In many filter bank applications, analysis and synthesis functions usually have different requirements. In this case, while one filter can be predetermined by h[k], the other could be solved via (3.69). The general solution of (3.69) is discussed in Appendix B at the end of this chapter.
Related Links:
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
Fast Algorithm of Computing Dual Functions

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