J. M. Morris and Y. Lu, "Discrete Gabor Expansion of DiscreteTime Signals in l 2 (Z) via Frame Theory," Signal Processing, Vol. 40, No. 3, Nov. 1994, pp. 155-191.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(henceforth, just called the Gabor expansion) generalizes all concepts mentioned above. There are several theories addressing the relationship between fgmng and ffl mn g, and suggesting computation methods for the analysis and synthesis filters. The most important concepts are frame theory [11], 12] and biorthogonal like sequence theory [12] 13] A. Frame Theory Let fgmn g be generated by Eq. 2, then it is a frame if there exist constants A and B called frame bounds with 1 B A 0 such that Akfk 2 X m X n j hf; gmn i j 2 Bkfk 2 ; 8f 2 l 2 (Z) 6) 1. If A B, the ....

....then the frame is said to be exact. If fgmng is a frame, then ffl mn g used in the Gabor expansion exists and is called the dual frame, and convergence and stability of the Gabor expansion is guaranteed. The computation of dual frames for discrete time signals (signal sequences) was presented in [11], 12] and summarized as follows. Define the frame operator such that S( Delta) X m X n h Delta; gmn i gmn ; 7) then it is shown that S is a positive definite operator if fgmn g is a frame. This implies that the inverse operator S Gamma1 exists. The frame theory proves that fl mn = S ....

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J. M. Morris and Y. Lu, "Discrete Gabor Expansion of DiscreteTime Signals in l 2 (Z) via Frame Theory," Signal Processing, Vol. 40, No. 3, Nov. 1994, pp. 155-191.

....from a pair of sequences (windows) called the analysis and synthesis sequences. Computing pairs of analysis and synthesis sequences, therefore, is essential in computing the DTGT. For the reconstruction to be stable, the collection defined by the analysis sequence should constitute a frame ( 1] [4]) The BiorthogonalLike Function (BLF) theory introduced in ( 3] 5] offers an easier approach for computing the analysis and synthesis sequences. In this paper we show an easy method to compute the analysis sequence given a synthesis sequence. This paper is organized as follows. The rest of ....

....by ONR Grant # N00014 89 J1210. generated by the corresponding synthesis sequence fl(k) via f(k) M Gamma1 X m=0 X n c mn fl mn (k) 2) Here fl mn is defined similarly to gmn above. Equations 1 and 2 represent a DTGT if fgmng and ffl mn g are dual frames of each other ( 1] [4], 6] 2. Biorthogonal Like Sequences It was shown ( 3] 5] that if g(k) and fl(k) satisfy the biorthogonal like condition (BLC) X n g(k Gamma nN )fl (k Gamma nN Gamma mM ) 1 M ffi (m) 3) where k = 0; N Gamma 1, then Eqs. 1 and 2 represent the DTGT. The sequences g ....

J. M. Morris and Y. Lu, "Discrete Gabor expansion of discrete-time signals in l 2 (\Delta) via frame theory," Signal Processing, Vol. 40, No. 2-3, Nov. 1994, pp. 155-181.

....a pseudo orthogonality condition (POC) derived in the next section. Two methods for designing pseudo orthogonal (PO) sequences are then derived. The conclusions and a brief discussion on further research appear at the end. 2. DISCRETE TIME GABOR TRANSFORM The discrete time Gabor transform (DTGT) [3, 8, 17] is an important tool for representing and analyzing a signal in the joint time frequency domain (JTFD) Using the DTGT a signal can be represented in terms of a collection fgmng as f(k) P M Gamma1 m=0 P n c mn gmn (k) where c mn are the Gabor coefficients that constitute the DTGT and P ....

....condition [9] X n g(k Gamma nN )fl (k Gamma nN Gamma mM ) 1 M ffi (m) 1) This condition is also stated in a different form as the WexlerRaz condition [17] Assume that the support of the window g(k) is [0; Q 1 Gamma 1] Then, for M = Q 1 N , a simple solution to Eq. 1 is given by [8] fl(k) g(k) M P n jg(k Gamma nN )j 2 ; 0 k Q 1 Gamma 1: 2) Though other solutions to Eq. 1 exist [7, 8, 9] the solution in Eq. 2 results in a simple pseudo orthogonal condition as discussed later. We will, hence, restrict ourselves to M = Q 1 N and the corresponding solution in Eq. ....

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J. M. Morris and Y. Lu, "Discrete Gabor expansion of discrete-time signals in l 2 (Z) via frame theory," Signal Processing, Vol. 40, No. 2-3, Nov. 1994, pp. 155181.

....we have to consider the N Q 1 case only. The condition in Eq. 14 is similar to the biorthogonal like condition in [7] and on p. 63 in [11] X n g(k Gamma nN)fl (k Gamma nN Gamma mM) 1 M ffi(m) 16) with M = 2N . If the length of g(k) Q 1 , is finite and M Q 1 , then it can be shown [7, 9] that fgmn g constitutes a frame, and that the dual frame (window) can be obtained as fl(k) g(k) MG(k) 17) where G(k) X n jg(k Gamma nN)j 2 : 18) Then g(k) and fl(k) are also a pair of biorthogonal like functions satisfying Eq. 16. Therefore, a solution fl 1 (k) to Eq. 16, as ....

....6 3.2. N Q 1 Gamma1 2 case We used an approximation of Eq. 22 fl(k) X X x= GammaX N Gamma1 X y=0 b xy g xy (k) 27) where X is an integer. It was seen in [4] that the solution of the BLC via the moment equation gives the dual frame, therefore, we used a faster algorithm of [9] for calculation of the dual frames. Figures 5 to 8 show the dual (analysis) windows for various synthesis windows for N = 30. Table 2 shows the MSEs in reconstruction. In the algorithm for the dual frames computation, the dual frame is approximated by a sequence of a larger length than that of g. ....

J. M. Morris and Y. Lu, "Discrete Gabor expansion of discrete-time signals in l 2 (Z) via frame theory," Signal Processing, Vol. 40, No. 2-3, Nov. 1994, pp. 155-181.

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