Proportional Nonuniform Multi-Gabor Expansions

A nonuniform multi-Gabor expansion (MGE) scheme is studied under proportional time and fi'equency (TF) shifts among dif ferent window indices m. In particular, TF parameters for each m are different, but proportional and relevant to windows' TF patterns. The generation of synthesis waveforms for nonuniform MGE is generally difficult. 'Ne show constructively that there is a set of basic synthesis MGE waveforms at each window index under proportional parameter settings. Nonuniform MGE adapts to signal frequency dynamics effectively,and eliminates unnecessary overlapping redundancies of a uniform MGE. Examples of the evaluation of synthesis waveforms are provided. Efficiency comparison of TF analysis using nonuniform and uniform MGEs is also discussed.

If {XIl} is a frame, the standard dualJrame representation stands tor, lYl-l Nm-l Km-1 S = L L L (s,y(m,j,k)Jgj~Jm,kTm' (1) :11"1-=0 j=O k~O (3) 'ilx E Je. MGEs are frame representations.A frame in a Hilbert space Jf is a basis-like sequence {xll} for which there are constants 0 < A co; B < 00 such that where, for the window index m, Nm is the number of frequency bins (lINm is the frequency shift), Tm is the timetranslation parameter, Km =' LIT m, g(ln) is the mth window, and {y(m,j,k) : m,j,k} is a set of synthesis sequences to be constructed.We have also used the conventional notation {g)~~""kT",} for the given multi-Gabor family, namely, Typical Gabor expansions use one fixed window and its translates and complex modulates as basic elements in an attempt to analyze the time-frequency (TF) information of a signal.Studies on Gabor expansions are intense; for example, see [.1,2,3,4,5,6) 7, 8, 9, 10, 11, 12, 13, 14, 15, There is also a book on Gabor analysis and algorithm by Feichtinger and Strohmer [17].However, frequency-varying features of a signal or the multitrequency components in a signal require windows of different size (variance) for a refined TF resolution.lv1ulti-Gabor expansions (MGEs) were developed specifically to meet such requirements; for example, see [1, , ( ). MGEs are TF expansions using a set of multiple windows and their translates and modulates in a frame (overcomplete "basis") system.The set of windows are custom-tuned.
Typically, they range from a narrower window to a wider window to meet the requirements of TF representations of signals of timevarying frequency components.However, standard uniform MGEs apply the same TF shifts among all analysis windows.Such uniform MGEs do not take into consideration the distinct TF patterns of different windows, which gives rise to unnecessary redundancy.Nonuniform MGE schemes adapted to each window's TF characteristics are more natural.
Let 0 co; m CO; M -1 be the number of windows in an MGE system.Let j and k be the modulation and translation parameters, and let Jf L be an L-dimensional signal space.Then a discrete nonuniform multi-Gabor representation (of non un i-where S-1 is the inverse of the standard frame operator S defined by, The sequence {S 1 X,,} is the standard dual frame to the frame {xn}.In the case of nonuniform MGEs, {gj7.~m,kTm} is to form a frame in JfL, and [y(m,j,k)  : m,j,kJ would be a dual frame to {g;;~m,kT",}' A dual frame needs not be the standard dual.One can have infinitely many constructible dual frames for a frame that is redundant ].We will only be focusing on standard duals here.Gabor duals in a parametric form can be found in [12].The same dual formula can be transformed to nonuniform multi-Gabor scenarios discussed here.

Gabor expansions from a filtering point of view
A typical uni-Gabor expansion (with one window) is equivalent to shifted bandpass filtering in signal processing.The bandwidth of the filter is determined by the window size.Wider window in time has typically narrower frequency bandwidth and vice versa.Evidently, a fixed bandwidth system is insufficient to analyze signals of dynamic frequencies-a phenomenon known as poor resolution.This was the very reason why the multi-Gabor representations, for example, see [II, was introduced.With multi-Gabor systems, there are now multiple bandpass filters of varying bandwidth.They can be designed to adapt to signal frequency dynamics, and thereby provide flexible and adaptive TF analysis of a signal.
(5) nonuniform MGEs.We show that the standard dual muIti-Gabor frame is formed by translations and modulations of a set of basic windows at each window index m.This result in turn allows us to derive a fast algorithm for the evaluation of these basic dual windows.Numerical examples of the construction of such MGEs as well as the efficiencydiscussion of nonuniform MGEs in TF analysis are also presented.

CONSTRUCTING NONUNIFORM MGEs
To construct a nonuniform MGE system, one has to make sure that a set of multi-Gabor waveforms form a hame.
Intuitively, the requirement is to have combined window waveforms covering the entire TF plane.vVeshow in the following some direct conditions for the construction of nonuniform MGEs.This will provide users with an intuitive guideline.The proof of the following result can be found in A.I.

Pm-)
A -I Nm I a(m)(INm) 2 Ao > 0, (6) m~O 1~1 Reasons for using nonuniform TF shifts vVhileMGE marks one step forward in a refined TF analysis of Gabor expansions, uniform MGE schemes do not reflect the nature of TF patterns of different windows, causing excessiveand unnecessary TF-tiling overlaps.
For instance, a wider window in time has narrower effective frequency bandwidth and vice versa.Hence, in a more natural MGE scheme, a wider window in time should be assigned with larger time translates and smaller frequency shifts (i.e., larger Nm).Likewise, a narrower window should be given smaller time but larger frequency shifting parameters.
MGEs of nonuniform time translates but fixed frequency shifts among windows have been analyzed and constructed in detail in [18].
When both TF shifts are varying among window indices, the theory and construction of nonuniform MGEs are generally difficult.In this paper, we report the study on the construction of a nonuniform MGE with proportional TF shifts, namely, the time translates and frequency shifts are proportional among window indices and relevant to windows' TF patterns, folloV\ing the basic TF tiling principle described earlier.
We provide an analysis and construction of the structure of the dual multi-Gabor frames of such proportional

Assume that
we assume that the following proportional relationship hold: T.
A sketch of the proof can be found in A.2.Note that condition (2) of Theorem 2 may be removed in finitedimensional space since it always holds.Nevertherless, since the ratio of the frame bounds determines the numerical stability of the frame system, condition (2) can be used towards that end.
The improvement of conditions in 'rheorem 2 lies in the fact that pointwise comparison is done between the sampled autocorrelation and the sampled cross-correlation (shifted correlation) 1, the infimum of the autocorrelation must be greater than the supreme of the cross-correlation.vI/hen the redundancy of a nonuniform lvfGE system is close to the minimum, the conditions in Theorem 2 will become substantially superior.
The remaining fundamental task is to determine dual multi -Gabor waveforms {y(!Ii,j,k)} for a given multi -Gabor PIl !Ii~O k~O

NONUNIFORM MGEs WITH FIXED PROPORTIONS OF TF SHIFTS
Assume that the window "width" increases as index m increases (from 0 to IvI -1), and hence assume that Tm increases.Since a wider window corresponds to narrower frequency bandwidth, a smaller frequency shift is desired, which in turn corresponds to a larger number of frequency bin Nm as see in Therefore, both Tm and Nm are assumed increasmg as 111 goes up.In particular, if Nm divides T for all m (which is practically so since T is the largest translation parameter among all), the constant is then always L Examples will be presented later.
For instance, let 1\1 = 2 and a = f3 = 2.We have a nonuniform MGE of2-window system.Then for the window index In = 0, there are 2 basic dual windows for p = 0 and q = 0, 1.For In = 1, there are 2 basic dual windows as well for p = 0, 1 and q = O.Moreover, the 2 basic dual windows for m = 0 are both real-valued windows (p = 0), while the second basic dual windows for m = 1 are generally complex valued since p = L See (11).It turns out that a similar dual structure result holds.
The standard dual multi-Gabor sequences {y,~;;;:~:q) : p, q, u, v} at index m are formed by ( for all 0 :-<: u :-<: N -1, 0 :-<: v:-<: LIT -1, and for all p and q specified in Here T = TM-I and N = No are the largest time-shifting step and the smallest frequency bins, respectively Wnile MGEs affixed proportional TF shifts (presented in the last section) are natural enough for a majority of applications, it is conceivable that some applications may demand a nonuniform MGE scheme, where the TF shifts do not fol-Iowa fixed proportion.Under an assumption that each of the translation and modulation parameters Tm and Nm satisfies an integer-multiple relationship with respect to the reference parameter set (T, N), it is possible to analyze the dual structure similarly.
Again, let T = TMI, the largest time translate, and let N = No, the smallest frequency bin parameter.For integers am and 13mfor all m, we now assume that

THE EVALUATION OF THE DUAL WAVEFORMS
Remark 3. It seems that at m = 0, the f30 (since ao = 1) basic synthesis windows are simply translates of each other.A further notice that can be made is that these basic window functions do have a shifting property, but not exactly the shift to each other in general.This observation applies to both scenarios of fixed as well as varying proportional parameters as in Sections 3 and 4, respectively.
The proof of this result mimics that of Theorem 3 which can be found in B.
The evaluation of the dual multi-Gabor waveforms takes an algorithmic structure similar to that of uniform frequency shifting.In fact, the frame operator matrix is still a sparse and "banded" matrix corresponding to the smallest frequency bin N. The computation of each synthesis waveform at each index m is broken into a set of Nm x Nm systems oflinear equations.We refer to [18, Algorithm 4] for a similar implementation algorithm that breaks the system into a set of smaller systems of equations.Readers that are interested in the implementation of the construction algorithm may contact the author for the Matlab code.

x,gjk (•+qTo) gjk (•+qTo). m jk
We present the results of nonuniform multi-Gabor representations of proportional TF shifts among window indices.Theorems 3 and 4 make possible efficient implementations of the construction of nonuniform multi -Gabor dual-frame systems.It is shown that the standard dual system can be generated by a set of basic waveforms as well.8) The ratio of the coefficients (nonuniform versus uniform) is merely 12.5%.This is expected to substantially reduce the postprocessing complexity.
We consider the efficiency of the proportional nonuniform MGEs by calculating the number of coefficients and by comparing with that of the uniform MGEs, under the assumption that the corresponding TF resolution for each corresponding window stays equivalent.Take a 2-window system for instance.Assume that one narrower window (m = 0) has a support of 8 units, and the wider window (m = 1) has 64 units.This system is exactly the same as illustrated in Figures 1,2, and 3.
The parameter selection follows the principle that the TF resolution is equivalent in corresponding window indices beh\Teenthe proportional nonuniform MGE and the uniform MGE.Specifi.cally,atIn = 0, since the TF pattern is fundamentally stretched in the frequency direction and very narrow in time, the time resolution is the fine one.The uniform MGE parameter T is necessarily small and is set at T = 2. Meantime, at m = 1, the IF pattern is stretched in the time direction and very narrow in frequency.The frequency parameter N is set at N = 128 to attain a certain frequency resolution in the uniform MGE.
In the corresponding nonuniform MGE, To is set at To = T = 2. Hence the fine time resolution emphasized at m = 0 is kept the same between the uniform and nonuniform MGE systems, and No is set at No = 16 « N = 128.At In = 1, the fine frequency resolution is to be preserved.Hence, Ni is set as Ni = N = 128, while Ti is enlarged substantially: With the equivalent TF resolutions, the efficiency improvement of the representation using the nonuniform TF shifts is clearly substantial.The number of coefficients of the proportional nonuniform MGE is where L is the length of the signal.Meantime, the number of coefficients of the uniform MGE is A sketch of the proof can be found in C. Note that, except for index m = 0, the translation by Tm of the window g(m)(.+ qTo) for all m 1= 0 no longer covers the signal span in exactly the same way as the Tm-translates of gem).Consequently, S 1= SqTo in general.In fact, for each q, 5'1To is generally different.Nevertheless, since the translates of (. +qTo) still cover the signal span, [g;;'\.+qToHm,j,k remains a multi-Gabor frame.Because of the redundancy, the difference among SqTo for different q is generally unnoticeable.Numerical examples are provided.Proportional nonuniform MGEs eliminate unnecessary redundancies due to superfluous overlapping using identical TF shifts among all windows (in uniform MGEs).This substantially reduces the cost of post-decomposition processing, which is valuable in applications of TF signal analysis and processing.Applications of proportional nonuniform MGEs remain an open area.Practical benefits llsing sllch a more natural MGE system are expected.

APPENDiCES
A. PROOFS OF THEOREMS 1 AND :2 omitted in the presentation.Now for all x E :HI, consider Notice that for each 0 s r s L -1, The proof is in line with similar results for continuous cases; see, for example, .Again, we assume that Tm and Nm both divide L so that L = T",K", and L = NmP", for all window indices m.We further agree that the translate index k = km and the modulate index j = jm, although for simplicity, the subindex m for k and j or their substitutes are Note that it is easy to verify that "f NIl KIl I (x,gj~j)) where, at the last step, we have used the periodicity of sequences x and g, and rearranged the summation over I from o to Pm -1.Now, assume that condition (1) holds in Theorem 1, that lv[-l Ken'''-l I-I Pm-[q/Nmj-l where ora ~b) 0= I if a 0= b, orab) 0= 0 if a f= b, and [q/N] is the greatest integer less than or equal to q/N.So now, for given r, (A.9) Al-1 ') " "1/ (m))  So now it~for 0 "S P "S a111 -1 and 0 "S q "S j3M-m-l -1, We have assumed that N = No is the smallest number of frequency bins corresponding to the window index 0, and that T = hil is the largest time-translation parameter corresponding to the window index iYI -1.The frame operator S is defined by (12).We will also be using the notation Therefore, writing j = uam + p and k = vfyYI-m-1 + q, where u = 0,1, ... , N -1, v = 0,1, ... , LIT -I, P = 0,1, ... , am -I, and q = 0,1, ... ,t3M-m~1 -1, we see that 0 :<: j :<: Nm -1, and 0 s; k s; LlTm -1 for all m, and = e .r. re2JIipvT/Nms 1 ('gcm) (. _ qT .)e2rrip(,)/N,,,) ].where in the second step, we have used the periodicity assumption of the window functions g(m) for each m.
where (lNm)== max I Ky~(m)(q-kTm)g(m)(q+INm-kTm) ; m s; ]V[ -1, 0 s; j S; Pm -1, and 0 S; k S; Km -I) forms a multi-Gabor frame of JfI.Remark 1.(a) Notice that the indices j, k, and I should all have carried a subindex m as they all have different ranges for different m.Nevertheless, the subindex m is omitted for the simplicity of the presentation.(b)If the combined translates of all windows cover all the signal span (condition (I)), and if N711 is large enough (so that the relative shifts in the cross-correlation arm) (lNm) is amply extended) to sufficiently lower the value of a(m) (lNm) (condition (2)), then {gj~n)} forms a nonuniform multi-Gabor frame..It is also possible to improve the conditions in Theorem 1 following an observation by Casazza and Christensen Theorem 2. Let {gem)}tI-I E :If.I be a set of M windovv functions.For each m, let Tm and Nm be given in Tileorern I.
j,k: for all 0 0; U :5: N -1, 0 :5: 11 :5: LIT -], and for all p and q specified in (l Here T = TM-l and N = No are the largest time-shifting step and the smallest frequency bins, respectively, r '" '" the largest time translate, and let N = No, the smallest frequency bin (which corresponds to the largest frequency shift).Then, for integers a and 13, Remark 2. (a) Theorem 3 states that the construction of a nonuniform MGE with proportional TF shifts is reduced to the evaluation of a set of basic dual window waveforms at each index 111 by (11).This result greatly facilitates the evaluation of proportional nonuniform multi-Gabor dual frames.It also allows us to derive a fast algorithm for the construction of the dual multi-Gabor frames.(b) The standard dual multi-Gabor sequences are formed by translations of the basic windows the largest step size T and multiplied by a constant (for each parameter p and each translate v) and then modulated by the smallest frequency bins N. Notice that the constant e2ifipvT/Nm is mostly 1 if pvT/Nm is an integer.ElJRASIP Journal on Applied Signal Processing 0

Figures 2 and 3
Figures 2 and 3 are the sets of dual multi-Gabor windows at two window indices.When pictures are shown in two columns, the left column (a) represents the real parts of the basic dual window functions, while the right column (b) represents the imaginary parts.The parameters in this example are T = 16 (the largest time-translation parameter), N = 16 (the smallest frequency-shift parameter); and ao = 1, a) = 8, [30 = 8, and PI = 1.

Corollary 1 .
Let {g;;') }m,j,k be a given multi-Gabor frame for eJfJwhose parameters Tm and Nm are given as in Theorem 4, Let 5qTo be the frame operator associated with the multi-Gabor frame sequence .+ qTo)} ""j,b that is, ] = 128.(a) Real part of the basic dual windows.(b) Imaginary part of the basic dual windows.
Proportional Nonuniform Multi-Gabor Expansion:~2 731 Shidong Li received the B.S. degree in electrical engineering from the Hefei University ofTechnology, China, in 1982, and the M.S. degree in electrical engineering from the Graduate School of the Chinese Academy of Sciences in 1985.He received his second M.S. degree in applied mathematics in 1989, and the Ph.D. degree in applied mathematics in 1993 from the Graduate School, University of Maryland, Baltimore.Prom 1985 to 1987, he was a Lecturer at the Graduate School of the Chinese Academy of Sciences.He was a Visiting Assistant Professor at Dartmouth College from 1993 till 1994 and the University of Maryland, College Park, from 1994 till 1996.He has been with San Francisco State University since 1996.Dr. Li is currently an Associate Professor of mathematics.His major research interests include wavelet theories, frame and frame extensions and their applications in signal analysis and processing, time-frequency representations, Gabor expansions, signal processing in digital communication systems, and applied mathematics.