Multiwindow discrete Gabor transform using parallel lattice structures

IET Signal ProcessingVolume 14, Issue 7 p. 420-426 Research ArticleFree Access Multiwindow discrete Gabor transform using parallel lattice structures Rui Li, Corresponding Author lir@ahstu.edu.cn orcid.org/0000-0003-4372-1791 College of Information and Network Engineering, Anhui Science and Technology University, 1501 Huangshan Road, Bengbu, Anhui, 233100 People's Republic of ChinaSearch for more papers by this authorLiang Tao, School of Computer Science and Technology, Anhui University, 111 Jiulong Road, Hefei, Anhui, 233100 People's Republic of ChinaSearch for more papers by this authorHon Keung Kwan, Department of Electrical and Computer Engineering, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4 CanadaSearch for more papers by this author Rui Li, Corresponding Author lir@ahstu.edu.cn orcid.org/0000-0003-4372-1791 College of Information and Network Engineering, Anhui Science and Technology University, 1501 Huangshan Road, Bengbu, Anhui, 233100 People's Republic of ChinaSearch for more papers by this authorLiang Tao, School of Computer Science and Technology, Anhui University, 111 Jiulong Road, Hefei, Anhui, 233100 People's Republic of ChinaSearch for more papers by this authorHon Keung Kwan, Department of Electrical and Computer Engineering, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, N9B 3P4 CanadaSearch for more papers by this author First published: 01 September 2020 https://doi.org/10.1049/iet-spr.2019.0180Citations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract The multiwindow discrete Gabor transform (M-DGT) is an effective time-frequency analysis tool to analyse time-varying signals containing components with multiple frequencies. In this study, fast block time-recursive methods for computing the M-DGT coefficients of a signal and the reconstruction of the signal from the transform coefficients are presented with steps as listed, respectively, in Algorithms 1 and 2, and their implementations using unified parallel lattice structures are also given. The proposed algorithms consisting of Algorithms 1 and 2 for respective forward and inverse transforms are compared to (i) those of the existing serial algorithms in terms of computational complexity and time; and (ii) those of the existing parallel algorithms in terms of hardware complexity. The results indicate that the proposed algorithm is fast in computing M-DGT coefficients of a signal and reconstructing the signal with a reduced hardware complexity. 1 Introduction The Gabor transform (GT) [1] is an effective time-frequency analysis tool to represent a signal in time-frequency domain. The discrete Gabor transform (DGT) [2-4], a discrete version of GT, has been applied in many fields including speech processing, image processing, communication, and texture extracting. The advantage of the DGT is its structured system inherited from the uniform time-frequency lattice and time-frequency atom, which can utilise the fast Fourier transform (FFT)-based efficient algorithm to compute the Gabor coefficients of analysed signal and reconstruct the original signal from its transform coefficients [5, 6]. DGT can be formulated using complex transform coefficients as complex discrete Gabor transform or using real-value transform coefficients as real-valued discrete Gabor transform. In [7-11], real-valued DGT and its fast algorithm are introduced based on discrete Hartley transform (DHT) and discrete cosine transform. 2D DHT-based real-valued DGT and its fast algorithm are proposed in [12, 13]. According to the method of computation, the Gabor system can be classified into two categories: bi-orthogonal analysis method [4, 14], and frame theory including standard GT [15, 16], and non-stationary GT [17-21]. Each of the former methods for computing discrete Gabor coefficients and its inverse transform uses the complete set while each of the latter methods uses an overcomplete set to decompose a signal and therefore its reconstruction is no longer unique. Under the frame theory approach, the most popular way of reconstructing an original signal is based on the canonical dual Gabor frame, in which the canonical dual Gabor frame can be derived from a set of linear equations based on duality conditions and support constraints associated with the given Gabor frame. Due to the uncertainty principle, the traditional DGT with a single window has a fixed joint time-frequency resolution, which is not sufficient to analyse and process the dynamic time-frequency of a signal that contains a wide range of spatial or frequency components. To improve the time-frequency resolution of the traditional DGT, the M-DGT based on biorthogonal analysis approach has been studied in [22-25] with applications in evolutionary spectral analysis. The multiwindow discrete Gabor expansion (M-DGE) in [26-29] using the frame theory has also been applied to signal processing [30], image processing [31], macromolecular sequence processing [32], and so on. Furthermore, the multiwindow real-valued discrete Gabor transform and its fast algorithms have been proposed in [33, 34] by using biorthogonal analysis approach. The M-DGT and M-DGE can extract local frequencies of a analysed signal in an effective manner, in which wide windows can be used to process slow-changing components and narrow windows can be aimed at detecting transient and fast-changing components. Among the existing algorithms using the biorthogonal analysis approach, the parallel lattice structures for implementing the block time-recursive methods has emerged as the main algorithms for calculating real or complex discrete Gabor coefficients and reconstructing a signal from its real or complex discrete Gabor coefficients, due to its low computational requirements and ease of hardware implementation [9, 35-37]. In this paper, the implementation of block time-recursive using parallel lattice structures for the fast computation of the transform coefficients of a signal and the fast reconstruction of the original signal from its transform coefficients in M-DGT is proposed, which offers reduced computational and hardware complexities. The remainder of the paper is organised as follows. In Section 2, the traditional M-DGT will be briefly introduced. In Sections 3 and 4, the block time-recursive methods for computing the discrete Gabor coefficients of a signal and reconstructing the signal in M-DGT are presented, and unified parallel lattice structures are used for implementing the proposed algorithms. In Section 5, comparisons of computational and hardware complexities between the proposed algorithms and existing algorithms are given. Finally, a conclusion is given in Section 6. 2 Review of M-DGT Let represent a periodic and discrete-time signal of a periodic length L, the discrete Gabor expansion with multiple windows (M-DGE) [27, 33] is given by (1)and the transform coefficients can be computed by (2)where stands for imaginary unit and . Equation (2) defines multiwindow discrete Gabor transform (M-DGT), and P is number of the windows. M and N are the numbers of Gabor sampling points in time domain and frequency domain, respectively. and are the frequency modulation size and time shift step, respectively. In (1) and (2), the original signal can be reconstructed according to its coefficients when . The Gabor oversampling case occurs when , with oversampling rate . Critical sampling occurs when and the original signal cannot be completely reconstructed in an undersampling conditions (). The completeness condition of M-DGT and M-DGE can be expressed by the bi-orthogonality between and (3)where , , and is the discrete version delta function. 3 Fast algorithm for computation of M-DGT coefficients In the oversampling case and the critical sampling case, the respective condition of and should be satisfied by properly selecting the parameters M and N to ensure that the oversampling rate must be a positive integer, in which and correspond to the critical sampling case and the oversampling case, respectively. Supposing (4) (5) (6) (7)Let , (2) can be rewritten as (8)and (8) can be expressed as matrix form as follows: (9)where is a block right-circulant matrix (10)with each block being an diagonal matrix (11)and is block diagonal matrix with each block being an N -point discrete Fourier transform (DFT) matrix (12)Equation (9) can be rewritten in the following form: (13)The block time signal is to be placed in a block delay line in arrays ranging to as shown in Fig. 1. The block time interval is equal to the time interval of block () inputs. The M-DGT at block time can be used to derive a block time-recursive relation between the state of updated M-DGT coefficients and the previous state of computed DGT coefficients, which is equivalent to the new signal block taking in and the old one taking out from the line. So (14)then, at block time , the transform coefficients can be expressed in the following form: (15)and when , compute (16)To summarise, a procedure for the above algorithm is described in Algorithm 1 (see Fig. 2), which can be implemented by a unified parallel lattice structures to compute M-DGT as shown in Fig. 3. Fig. 1Open in figure viewerPowerPoint Block delay line for Fig. 3 Fig. 2Open in figure viewerPowerPoint Algorithm 1: Block time-recursive method for computing M-DGT coefficients Fig. 3Open in figure viewerPowerPoint Implementation of block time-recursive using unified parallel lattice structure for M-DGT forward transform () 4 Fast algorithm for reconstruction of original signal Equation (1) can be rewritten in the following form by using , , and (17)then (17) can be simplified and expressed as (18)where is also a block diagonal matrix composed of diagonal block with an N -point inverse DFT matrix. is a block right-circulant matrix constructed by (19)which the block is an diagonal matrix as follows (20)where and . Equation (18) can be written in the following form: (21)By using a similar method adopted for computing M-DGT coefficients, the M-DGE at each block time t can be depicted in order to derive a block time-recursive relationship between the state of the renewed recovered signal and the previous state of computed reconstructed signal, which refers to the current block of the M-DGT coefficients taking in and the previous one taking out as shown in Fig. 4 for a block size N. Fig. 4Open in figure viewerPowerPoint Block delay line for Fig. 6 So (22)the reconstructed signal can be obtained at block time (23)The procedure for the above algorithm can be summarised as Algorithm 2 (see Fig. 5) and a unified parallel lattice structure for reconstructing the original signal is shown in Fig. 6. Fig. 5Open in figure viewerPowerPoint Algorithm 2: Block time-recursive method for reconstructing original signal Fig. 6Open in figure viewerPowerPoint Implementation of block time-recursive using unified parallel lattice structure for M-DGE inverse transform () 5 Results and discussion As the computational complexity of the proposed algorithms is proportional to the number of independent lattices, the computation time of the proposed algorithms is dependent on the computational complexity of each lattice. A comparison of the computational complexity of the proposed algorithms and the serial algorithms of [27] is shown in Table 1. We first assume that the FFT algorithm is of the order of [38]. In Fig. 3, each unified parallel lattice performs N -point additions and N -point multiplications during the each block time, and an N -point FFT at the last block time, so the proposed parallel algorithm for M-DGT (computing Gabor coefficients of a signal) is of the order of . In Fig. 6, each parallel coefficient array performs N -point inverse FFT and each parallel lattice performs N -point multiplications and -point additions, so the computational complexity of the parallel algorithm for M-DGE (reconstructing the original signal) is of the order of . In Tables 1–4, Coef and Reco denote, respectively, the computation of the M-DGT coefficients of a signal and the reconstruction of the signal, respectively. Table 1. Comparison of computational complexity of proposed algorithm and existing serial algorithm Algorithms Computational complexity Applicability [27] Coef Reco proposed algorithm Coef Reco Table 2. Comparison of computational time of proposed algorithm and existing serial algorithm M N Computational time, s [27] Proposed algorithm Applicability 232 232 21 0.069803 0.002181 Coef 0.089809 0.002806 Reco 264 264 24 0.096928 0.001512 Coef 0.142898 0.002234 Reco 2128 2128 216 0.201822 0.001577 Coef 0.265492 0.002074 Reco 2256 2256 264 0.389504 0.001523 Coef 0.462517 0.001807 Reco Table 3. Comparison of hardware complexity of proposed algorithm and existing parallel algorithm Algorithms Hardware complexity Applicability [34] PMN Coef PMN Reco proposed algorithm PM Coef PM Reco Table 4. Numerical comparison of hardware complexity of proposed algorithm and existing parallel algorithm P M N Hardware complexity [34] Proposed algorithm Applicability 1 32 32 1 1024 32 Coef, Reco 64 64 4 4096 64 Coef, Reco 2 128 128 16 32 768 256 Coef, Reco 256 256 64 13 1072 512 Coef, Reco 4 512 256 128 52 4288 2048 Coef, Reco 1024 256 256 1 04 8576 4096 Coef, Reco Computer simulations were conducted in Matlab on a personal computer (Intel(R) Core(TM) i7-8550U @2.0 GHz, memory 16 GB). Computational time results are given in Table 2, where the signal is of , and different Gabor sampling patterns were used in simulation. Since the each parallel lattice was computed in series using MATLAB simulation, computational time of the whole block of parallel lattices was obtained by taking the average time of all parallel lattices. As shown in Table 2, we can see that the computational time of the proposed algorithms is less than that of the existing serial algorithms. The hardware complexity of the proposed algorithms depends on the number of independent parallel lattices. From Figs. 6 and 3, we can see that the total number of parallel lattices is determined by frequency sampling interval , Gabor sampling rate , and the number of windows P, which is equivalent to . A comparison between the proposed algorithms and the parallel algorithms [34] is given in Table 3. By using the formulas in Table 3, the corresponding numerical results are given in Table 4 under different Gabor sampling patterns and the number of windows. According to Tables 3 and 4, we can see that the proposed algorithms require reduced hardware complexity than that of the parallel algorithms in [34]. To demonstrate the effectiveness of the proposed algorithms for signal analysis, two examples are shown below. Example 1.Fig. 7 shows a transient signal [39, 40] defined by (24) consisting of four () transient atoms of length points, sampled at , and its Fourier spectrum is shown in Fig. 8(24)where denote the amplitude, damping coefficient, arrive time, frequency, and phase, respectively, of the i th transient component, with values shown in Table 5.The transient signal contains four transients and each of them has a single frequency and an arrival time, the traditional Fourier spectrum in Fig. 8 cannot reveal how the frequencies change over time. However, the combined time-frequency spectrum of M-DGT of in Fig. 9 is displayed with high time and frequency resolutions, where , , , and is defined by the geometric average of of M-DGT.The reconstruction of the original signal is also simulated, which gives virtually error-free reconstruction (). Fig. 7Open in figure viewerPowerPoint Transient signal consisting of four transients with length Fig. 8Open in figure viewerPowerPoint Fourier spectrum corresponding to Fig. 9Open in figure viewerPowerPoint Combined time-frequency spectrum of with and Table 5. Parameters of transients of signal i , Hz 1 2.9 2.0 256 23.75 2 3.0 1.8 256 41.25 3 2.0 1.0 512 23.75 4 3.0 1.7 640 41.25 Example 2.In this example, the proposed algorithm is used to analyse a signal consisting of a transient pulse, a sine wave, and a cosine wave, in which the equation, waveform, and its Fourier spectrum are shown in (25), Figs. 10 and 11, respectively (25)where the parameters , , , and sampling frequency . The traditional Fourier spectrum in Fig. 11 can sufficiently analyse the frequency components of the sine and cosine waves but fail to detect the transient pulse. In Fig. 12, the combined time-frequency spectrum by the arithmetic average of of M-DGT of can effectively detect the components of the sine and cosine waves and the pulse wave, where , , and . Fig. 10Open in figure viewerPowerPoint Signal composed of a transient pulse, a sine wave, and a cosine wave Fig. 11Open in figure viewerPowerPoint Fourier spectrum corresponding to Fig. 12Open in figure viewerPowerPoint Combined time-frequency spectrum of with and 6 Conclusion In this paper, the M-DGT has been shown to be an effective and fast time-frequency tool to analyse signals with multiple frequency components and/or time-varying frequencies. Based on the FFT, the implementation of block time-recursive methods for computing the M-DGT coefficients of a signal and reconstructing the original signal from the its M-DGT coefficients have been developed in both critical sampling case and oversampling case, which can be implemented in unified parallel structures. Comparison results have shown that the proposed algorithms can provide a fast method to compute M-DGT coefficients of a signal of the order of and reconstruct the original signal of the order of in computational complexity, and require reduced hardware complexity to implement the parallel lattice structures. The corresponding numerical results are shown in Tables 2 and 4. 7 Acknowledgments This work was supported by Anhui Provincial Natural Science Foundation under Grant No. 2008085QF329, the Natural Science Foundation for the Higher Education Institutions of Anhui Province under Grant No. KJ2020A0061, and the Natural Science Foundation of Anhui Science and Technology University under Grant No. XWYJ201802. 8 References 1Gabor, D.: 'Theory of communication', J. Inst. Electr. Eng., 1946, 93, (3), pp. 429– 457Google Scholar 2Bastiaans, M.J.: 'Gabor's expansion of a signal into Gaussian elementary signals', Opt. 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