The Self-Duality of Discrete Short-Time Fourier
Transform and Its Applications
Tzu-Hsien Sang, Member, IEEE
Abstract—The self-duality of short-time Fourier trans-form (STFT) is an elegant property and is useful in shedding light on the construction of STFT and its resolution capability. In this paper, the discrete version of self-duality is studied, and the property is interpreted in the context of resolution capabilities of time frequency distributions. In addition, two applications are provided as showcases of these insights obtained from the inter-pretation. In the first application, the problem of STFT synthesis is considered, and self-duality serves as an important indication of whether the synthesis problem at hands is properly formulated. In the second application, a new kind of high-resolution time-fre-quency distribution is constructed based on the understandings obtained by contrasting two of the most popular time-frequency analysis tools, namely, the STFT and the Wigner distribution.
Index Terms—Discrete short-time Fourier transform (STFT),
self-duality, signal synthesis, time-frequency analysis.
I. INTRODUCTION
T
HE self-duality of short-time Fourier transform (STFT) has been known for some time, for example, in the con-text of radar signal processing [1]. It is an elegant result, yet the usefulness of it has been far from fully demonstrated. To de-scribe the self-duality of STFT of continuous signals, a modified definition of STFT was used:(1) where is the signal-to-be-analyzed, is the window function, and denote the STFT. Notice the extra phase term which guarantees the symplectic covari-ance [2]–[4] and does not exist in the usual definition. With this little change, the following equation is easily shown to hold:
(2) where is the time-reversal of . In particular, if is time-symmetric, then the two-dimensional inverse Fourier transform of the STFT is proportional to the scaled version of the STFT
Manuscript received October 21, 2008; accepted August 20, 2009. First pub-lished September 09, 2009; current version pubpub-lished January 13, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Maria Hansson-Sandsten. This work was supported by NSC of Taiwan, R.O.C., under Grant NSC 98-2220-E-009-043.
The author is with the Department of Electronics Engineering, National Chiao-Tung University, Hsin-Chu 300, Taiwan, R.O.C. (e-mail: tzuh-sien54120@faculty.nctu.edu.tw).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2009.2032038
itself; hence we call the relation self-duality. The short-hand notation is used to denote such a relation. Notice that the commonly known Fourier duality between any two orthogonal directions on the time-frequency plane refers to the conjugation of pairs of operations along each direction. A quick review can be found in [5]. Here the self-duality refers to, however, the functional resemblance between the Fourier pairs in the and domain. One typical Fourier dual pair is the scaling operation, namely, expanding a function in one domain leads to the contraction of its Fourier pair in the other domain and vice versa. Applying the scaling duality to the STFT and its Fourier pair, one would get a duality which is, with a figu-rative short-hand notation, . Apparently this relation is quite different from the self-duality which reads . Nevertheless, they are not conflicting each other, as matters will be explained with more details in Section II.
In this paper, the self-duality is established for STFT based on discrete-time Fourier transform (DTFT). In Section II suf-ficient conditions for this version self-duality to hold will be identified and the implications will be discussed. A pure discrete version of self-duality of STFT based on discrete Fourier trans-form (DFT) is also trans-formulated for the implementation purpose. It can be viewed as an approximation to the exact self-duality, as long as the sampling and truncation of the signal and window function are done with precautions. Next, an interesting inter-pretation of self-duality, focusing on the supposed poor resolu-tion capability of STFT/spectrogram via contrasting it with the Wigner distribution (WD), raises the possibility of linking self-duality and the resolution issue. Furthermore, in Sections III and IV two algorithms based on the aforementioned understanding of self-duality were developed for applications in STFT syn-thesis and the construction of high-resolution time-frequency distribution (TFD). The algorithms powerfully demonstrate the beauty and usefulness of this elegant property.
II. SELF-DUALITY OFDISCRETESTFT
For a discrete signal , the STFT based on DTFT is de-fined as
(3) in which the signal and the window function are as-sumed to be band-limited within frequency bands and respectively, and both are properly sampled. The slice of along the frequency axis is a continuous and peri-odic function, while along the time axis is a discrete sequence.
To demonstrate the self duality, first take a mixed two-dimen-sional inverse Fourier transform of :
(4) It is now shown that if certain condi-tions are met. Continue with (4) [see (5), shown at the bottom of the page]. The inner-most integration produces a sinc function; therefore,
(6) The inner summation can be viewed as the modulated se-quence being convolved with a low-pass sinc filter. At the first glance, it seems necessary that shall be within the pass-band of the sinc filter, i.e., , to further proceed the derivation of self-duality. With a closer scrutiny, however, it becomes clear that the condition is unnec-essarily strict. The reason is that, as long as , the convolution of with the filtered version of will have the same result as in the case where is within the passing band of the sinc filter.
With the assumption that the afore-mentioned condition holds, we can proceed with the case where is band-limited within the passing band of the sinc filter and the output is
(7) If the window function is symmetric, then
(8) That is, the self-duality based on discrete-time Fourier transform is established with the sufficient conditions that both and are limited and properly sampled, and their band-widths satisfy .
To facilitate algorithm implementation, a version of approxi-mate self-duality based on discrete Fourier transform is needed. Next, such a version will be presented without elaborations on
derivation. The details pretty much follow the derivation of (8). The result can be viewed as a sampled version of
, and the self-duality holds as long as the “edge ef-fect” caused by substituting DTFT with DFT can be ignored.
Let be a length- discrete signal and its discrete STFT is defined as
(9)
for , and .
No-tice that the range of indexes is chosen such that the indexing can be simplified. The discrete inverse Fourier transform of
is
(10) The self-duality states that
(11) if the following conditions on and are met1:
1) the time supports should satisfy ; 2) the frequency supports should satisfy .
These sufficient conditions agree with the intuition derived from the argument of avoiding “edge effects.” The discrete ver-sion of self-duality is now established. In the following, an inter-pretation of self-duality, mainly focusing on the resolution issue on the time-frequency plane, is provided. Furthermore, the ob-tained insights will be utilized to develop two example applica-tions that powerfully demonstrate the beauty and usefulness of self-duality.
A. Interpretation of Self-Duality and the Resolution Capability
An interesting interpretation of the self-duality is presented to provide useful insights on the resolution capability of STFT, and with analogy, of general time-frequency distributions. To facilitate discussions, a single Gabor logon centered at the origin of the time-frequency plane will be used as an example, and the continuous instead of discrete version will be discussed for the simpler notations. A single Gabor logon is
(12)
1The window function and the signal should be essentially time and
band-limited, even though in strict sense this is not possible. Let their effective time supports be[0a; a] and [0b; b], and the effective frequency supports [0B; B] and[0W; W ] respectively.
and its STFT is
(13) satisfies the self-duality
(14) The corresponding spectrogram, i.e., the squared magnitude of the STFT, is
(15) In addition, the WD
(16) will be generated to demonstrate its superior resolution, which is regarded as its trademark [6]–[10], and its different yet some-what similar self-duality relation. Through the exercise, it is hoped to demonstrate the correspondence between resolution capabilities and different self-duality relations.
The WD of the same Gabor logon is
(17) Commonly, it is treated as an energy distribution function, and its corresponding “amplitude” distribution, which is to be com-pared with the STFT, is then
(18) Notice that it satisfies a different self-dual relation:
(19) or, in the short-hand notation, .
The contrast is clearly represented in Fig. 1. The different de-grees of concentration of the “amplitude” distribution functions on the time-frequency plane can be easily observed. Different dual relations govern the two “amplitude” distribution func-tions: for STFT, it is , i.e., the self-dual function is more concentrated on the plane; for , it is , i.e., the self-dual function remains the same shape on the plane. Notice the observation that the STFT is more concentrated on the plane corresponds to its “excessive” spreads on the time-frequency plane,
espe-Fig. 1. Scaling relation between the time-frequency domain and the- do-main. (a) shows an STFT and (b) is its two-dimensional inverse Fourier trans-form. (c) is the “amplitude” function of the Wigner distribution of the same signal, and (d) is its two-dimensional IFT.
cially when compared to the “amplitude” function of the Wigner distribution.2
On the alleged superior resolution of WD, there is an instruc-tive equation that shows the construction of the Wigner distri-bution through auto-convolution of the STFT on the time-fre-quency plane: [see equation (21) at the bottom of the page], in which and is the STFT. This equa-tion is a 2-dimensional extension of [13, eq. (3)] where the WD is related to the STFT via an auto-convolution in the frequency domain. There are two insights in (21) worth noting. The first is that, by the analogy to the claim that matching windows to lo-calized signals results in high-resolution adaptive spectrograms [14], the auto-convolution in (21) also contributes to the supe-rior resolution of WD. The second is that the oscillating term
2Please be noted that the self-duality relation presented here for the
“ampli-tude” function of the WD is only for the single Gabor logon and cannot be ex-tended to general signals. It should be viewed as merely an instructive example. For general signals, there is an interesting relation similar to what is found in [11], [12], regarding the “phased” WD, i.e.,e W (t; !). It goes like this:
IFfe W (t; !)g =1 2e x 2 X 2 (20)
in whichIF denotes the two-dimensional inverse Fourier transform. Notice the “scaling” in the -domain signal and the -domain spectrum on the right-hand side. This equation conforms to the reasoning that expansion in the 0 domain means tighter concentration of signal components and therefore a superior resolution in the time-frequency domain.
gives indications on how the distinctive fast-oscillating “cross terms” arise in the midpoint of “auto terms.” The self-dual relation does not conflict with the well-known tradeoff of the resolution capability of Fourier analysis on the time-frequency plane. In the tradeoff, when the resolu-tion capability along the frequency axis is enhanced through using a long window, the resolution along time axis is in-evitably compromised in a similar way as shown by the Fourier transform pairs of Gabor logon
and which demonstrate the nego-tiation of spreads between two domains. This tradeoff can be clearly stated not only by the scaling duality but also the development of the uncertainty principle for time-frequency distributions [15]–[18] and may be figuratively understood through the short-hand notation . The scaling dual relation and the self-duality
provide two different aspects to restrict the expression of STFT on the time-frequency plane.
The following sections present two applications of self-du-ality to general time-frequency analysis/synthesis. Notice that the main character of these applications is in recognizing the self-duality as an indication of the resolution limitation as well as a regulation on the expression of TFDs on the time-frequency plane.
III. APPLICATION I: SIGNALSYNTHESISFROMGIVEN
TIME-FREQUENCYTEMPLATES
The synthesis of STFT is to construct a valid STFT, that is, a 2-dimensional function which is mathematically related to a signal and a window function through (1), to match closely to a given time-frequency template. Some works have been done on related problems; for example, in [19], a finite-support dis-crete signal is reconstructed by using only the magnitude part of its STFT. TFDs other than STFT have also been studied in signal synthesis applications [20]–[22]. Here, the STFT syn-thesis problem is stated as follows.
1) The STFT Synthesis Problem: Given a power distribution
template on the time-frequency plane, find a legitimate STFT such that the spectrogram is closest to
.
The self-dual (2) is a necessary condition for a function to be an STFT with symmetric window. Therefore, if there exists an STFT whose magnitude is close to . , there must exist self-dual functions whose magnitude is also close to the given template. The STFT synthesis problem may be solved by first find a self-dual function such that its magnitude, i.e., , is close to . The function can be found with similar approaches adopted in the classical problem of two-dimensional signal retrieval using only the magnitude information of Fourier transform [23]–[29].
After the self-dual function is found, the next step is to construct a legitimate STFT such that is minimized. Assume a given symmetric window function as a design parameter, a convex norm-min-imization problem can be formulated and can be effectively solved through readily-available convex optimization software packages [30].
Fig. 2. STFT synthesizing algorithm is used to reconstruct a signal. (a) shows the time-frequency template function. (b) shows the real part of the self-dual functionC(n; k) generated by Step 1. The real part of the original signal (the dashed line) and the synthesized signal (the solid line) are shown in (c), and (d) shows the magnitude of the STFT generated by the reconstructed signal and the same window function.
The self-dual function serves as an intermediate vari-able in the synthesis process. Notice that the phase-retrieval problem is nevertheless a non-trivial problem. In practice, most popular procedures are based on Iterative Fourier transform al-gorithm (ITF) [23], [26]. Here a variation of ITF is adopted as the core of the synthesis procedure. Start with an initial function where is a random phase function, we proceed to the following algorithm.
2) The STFT Synthesis Algorithm:
Step 1) Generate , where de-notes the 2-dimensional inverse Fourier transform. Set . Go to Step 2. Step 2) Update with
where denotes the 2-dimen-sional Fourier transform. Set . Go back to Step 1 or go to Step 3 if the updating term is small.
Step 3) Given a window function as a synthesis pa-rameter, get via solving a norm-minimization problem. The details are in Appendix I.
Two examples are provided to illustrate the STFT synthesis algorithm. The results are shown in Figs. 2 and 3. In the first ex-ample (shown in Fig. 2), the spectrogram of a frequency-mod-ulated signal is used as the template. The STFT synthesis algo-rithm successfully reconstructs the signal up to a phase term, with only the magnitude of STFT is used. The next example demonstrates the role of the window function as a parameter in the synthesis process. Fig. 3(a) shows a template in the shape of a square with the length of 3.13 seconds and the width of 3.13 radians. Fig. 3(b) shows the real part of the self-dual function
Fig. 3. (a) shows the time-frequency template function. (b) shows the real part of the self-dual functionC(n; k) generated by Step 1. (c) and (d) show the optimal signal and the spectrogram generated with a short window; (e) and (f) show the results with a longer window.
found in Step 1. Fig. 3(c) and (e) shows, respectively, two norm-minimizing solutions obtained with a short and a long Gaussian window, and shown in Fig. 3(d) and (f) are the re-sulting spectrograms.
Since there are two duality relations (one is about self resem-blance and the other scaling duality), i.e.,
and , that regulate the expression of STFT on the time-frequency plane, a given template can be judged whether is reasonable in the sense whether it is possible to exit a legitimate STFT having magnitude reasonably close to the given template. Indeed, since the self-duality is the necessary condi-tion of an STFT with a symmetric window, Steps 1 and 2 can be used as a reality check for a given time-frequency template. If the magnitude of the generated self-dual function differs greatly from the given template, it means that the template function is ill chosen and there exists no signal that can possibly result in a spectrogram that is close to the given template.
Moreover, the scaling duality also regulates the concentration of signal energy, e.g., a signal cannot be concentrated along time axis without spreading on the frequency axis. With these two dual relations, sometimes it is straightforward to judge whether a given template can possibly be a suitable choice. For example,
Fig. 4. Effect of ill-chosen templates. (a) shows a square template with the length of 6.26 s and the width of 6.26 rad, and the magnitude of the self-dual function generated in Step 1 is shown in (b).
consider a square plateau with the length and width of . By self-duality, its two-dimensional IFT would have a magnitude function in the shape of a square plateau with the length and width of . At the same time, by the scaling duality, the length and width of the significant part of the IFT should be around . Hence, for choosing a reasonable template, one should set , i.e., should be around ; otherwise, the square plateau is not proper as a magnitude template for STFTs. Fig. 4 illustrates the effect of an ill-chosen template that has a plateau too large for a reasonable template. Fig. 4(a) shows such a template function, and the magnitude of the self-dual function generated in Step 1 is shown in Fig. 4(b). Since the template has a plateau much larger than what can be generated by a legitimate STFT, there is no self-dual function that can come close to the template and inevitably “hole” and “cracks” will develop in the magnitude of the self-dual function.
IV. APPLICATIONII: HIGH-RESOLUTIONTIME-FREQUENCY
DISTRIBUTIONS
The contrast between the self-dual relations regarding the spectrogram and the WD [(14) and (19)] and the fact that the WD holds a remarkable resolution capability despite the an-noying cross terms [31] leads to an interesting question: Could the self-duality be worked into the for-mulation of designing high-resolution TFDs with suppressed cross terms whose super-fine structure does not seem to con-form to self-duality? In the following, such a plan is devised to find a 2-dimensional function, call it , for a given signal such that satisfies the high-resolution version of self-duality and also optimizes a certain cri-terion regarding the relevance between the signal and , and will be shown to be worthwhile considering as a high-resolution TFD.
First the problem of formulating a proper criterion of rele-vance is addressed. The Wigner distribution, despite its problem of cross terms, has been widely regarded as a plausible fre-quency representation with high-resolution capability for time-varying signals. Therefore, a straightforward choice is to mini-mize the norm between and , i.e., to min-imize the term
Intuitively, the approach can be understood as to create the func-tion by harnessing two forces which compensate each other. The minimization of the norm makes the resulting assemble the Wigner distribution and hopefully keep the high resolution capability; while self duality eliminates spu-rious cross terms whose ultra-localized features on the time-fre-quency plane cannot meet the requirement. Later an example will be provided to illustrate this point. Take the formulation with continuous signals as a model, we now proceed to define the first version of the algorithm to construct high-resolution TFD for discrete signals.
1) The High-Resolution TFD Problem (I): Find a 2-D
func-tion which satisfies the high-resolution version of self-duality, i.e., , and whose squared magni-tude should be as close to the Wigner distribution as possible.
A procedure based on the ITF algorithm can also be used to solve this problem. First, start with a discrete function where is an arbitrary random phase function. Then it goes to the following iterative procedure.
2) The High-Resolution TFD Algorithm (I):
Step 1) Generate . Set . Go to Step 2. Step 2) Update with
. Set
. Go to Step 1 or exit if the updating term is small.
More often than not the TFD design problem involves time and frequency marginal functions, since they represent the in-formation available on the time and frequency axes to which most attention is often given. These kinds of considerations usu-ally are formulated as equality constraints in the TFD designing posed as a convex optimization problem [32].
Generalized marginal functions derived from the fractional Fourier transform [33]–[35] have also been proposed for the TFD design problem, for instance, in [36]. Here a novel ap-proach towards considering multiple marginal functions is pro-posed, and it is called the collective marginal error. In short, the goal is to find a TFD that does not aim at satisfying a specific set of marginal functions, but to minimize the collective error of all generalized marginal functions. This criterion is particularly suitable for exploratory signal analysis where no preference for any marginal function is set a priori.
In [37], the authors proposed to use the Radon transform [38], to facilitate the kernel design [39] procedure for TFDs to achieve high resolution with suppressed cross terms. The Radon trans-form is pertrans-formed on the modulus of the ambiguity function [6], [7] for observing of how the auto terms are distributed in the ambiguity domain (the domain) and use that infor-mation to design an adaptive kernel function. In this paper, the Radon transform will be used on the time-frequency domain to develop the idea of collective marginal error. In addition, the Radon transform only serves as a conceptual tool and is not cal-culated in the final algorithm.
Let denote the Radon transform on the axis obtained by counterclockwise rotating the time axis with the angle and be the signal’s generalized marginal on the rotated axis.
The collective marginal error of a power distribution is defined as
(23) Let , i.e., the difference between the target power distribution and the Wigner distribution, then the collective marginal error has an intuitive expression in the ambiguity domain as follows:
(24)
where is the 2-dimensional Fourier transform of . The mathematical derivation is presented in Appendix II.
Notice that the collective marginal error emphasizes the low-pass part on the ambiguity, i.e., - , domain. Usually it is the case that the so-called cross terms occupy regions away from the origin of the ambiguity domain. Accordingly, the collective marginal error criterion is expected to be compatible with de-signing TFDs aimed at suppressing cross terms. Furthermore, notice that is also a measure of how close is a TFD to the WD. Therefore, the resulting TFD is also expected to retain the resolution capability of the WD.
3) The High-Resolution TFD Problem (II): Find a 2-D
func-tion which satisfies the self-duality
and whose squared magnitude minimizes the discrete collective marginal error:
(25)
Notice that is made to be zero by forcing the total energy of equal to that of .
The minimization of only involves the magnitude of . Indeed is a convex function of , and it is straightforward to find the -minimizing . Once given a magnitude template, via ITF, a self-dual function whose magnitude close to the template can be found. The overall procedure contains iterations between two operations. The operations result in the following heuristic algorithm.
4) The High-Resolution TFD Algorithm (II):
Step 1) Find the -minimizing mask function via the steepest-gradient method.
Step 2) Generate . Set . Go to Step 3. Step 3) Update with
. Set
and normalize the energy. Go back to Step 2 or exit if the updating term is small.
Simulation results show that this procedure performs better than the first algorithm and it does generate superb results
Fig. 5. Several time-frequency distributions of a pair of crossing chirped Gabor logons are shown for comparison. (a) and (b) are spectrograms generated with a short and a long window respectively. The Wigner distribution is shown in (c) and (d) shows the RID. Finally, (e) and (f) show the results generated with the high-resolution TFD algorithm I and II, respectively.
for signals that are typically difficult to be handled by other well-known time-frequency analysis tools. Fig. 5 illustrates an example of two chirped Gabor logons crossing each other. Fig. 5(a) and (b) shows the spectrograms generated with a relatively short and a long Gaussian window respectively. Ei-ther spectrogram fails to represent these chirped Gabor logons in a visually appealing way for an intuitive interpretation. The WD is shown in Fig. 5(c), and its signature features (the so-called cross terms) are so obtrusive that they obscure the superb resolution of the WD in representing the crossing chirp signals. The Choi–Williams distribution [39], a member of the reduced interference distribution (RID) class, is shown in Fig. 5(d) for comparison. While generating the RID, careful adjustment of the kernel function is made via minimizing the Rényi information measure [40]–[42] in an attempt to achieve a visual balance between showing details of the signal and reducing interference terms. However, for chirped signals, the end result is not very satisfying. Fig. 5(e) and (f) shows the results of high-resolution TFD algorithm I and II respectively. The TFDs approach the high resolution typically associated
Fig. 6. WD is shown in (a) while the high-resolution TFD in (b). Notice that the spurious cross terms are substantially reduced in (b).
with the WD and substantially suppressing the annoying cross terms, especially in the case of Fig. 5(f).
Finally, an example of two well-separated Gabor logons is provided to show that fast-oscillating “cross terms” can be sub-stantially reduced by the ITF which enforces self-duality. No-tice that applying collective marginal error does reduce the cross terms; yet a heavy suppression is only achieved after the ITF.
V. CONCLUSION
The sufficient conditions for the approximate self-duality of discrete STFT to hold are presented. Self-duality is interpreted in the context of resolution capabilities of time-frequency dis-tributions. Furthermore, two signal processing applications , the first be the STFT synthesis and the other the construction of high-resolution TFD, are provided to demonstrate the power and beauty of the concept.
APPENDIXI
NORMMINIMIZATION INSTFT SYNTHESIS
Let be the self-dual function found in Step 1 of the STFT synthesis algorithm. We would like to find the discrete signal such that its STFT with the given window function minimized the norm
(26) The definition of STFT in (1) can be viewed as an inner product:
(27) where
(28) representing the combination of windowing operation and Fourier transform. The norm-minimizing thus comes from the complex-valued least-squares solution of the equation (29) in which is an matrix stacked up with the row vectors , and is an vector stacked up with corresponding .
(31)
(32)
Notice that not all row vectors of any and need to be selected for the equation system. A system with much re-duced size can be constructed with only corresponding to region of interests. According to the sampling theory of dis-crete STFT, as few as equations may be sufficient to deter-mine a legitimate STFT; as the number of selected equations goes small, however, solving the norm-minimization problem encounters the problem of numerical accuracy. The best selec-tion of and efficient numerical method to solve for op-timal is beyond the scope of this paper.
APPENDIXII
THECOLLECTIVEMARGINALERROR
First consider the case of continuous signals. The term in the collective marginal error can be written as
(30) where denotes the Fourier transform and the convolution in the variable . By invoking the projection theorem [38], the expression becomes [see (31), shown at the top of the page], and the collective marginal error becomes [see (32), shown at the top of the page]. Notice that the final expression em-phasizes the low-pass part of the error through the weighting factor ; the suppression of high-pass components is a common feature of the so-called reduced-interference distri-bution [39].
As for discrete signals, the weighting term
with and being not simultaneously zero is used in evalu-ating the collective marginal error in double summation, and is made to be zero by forcing the total energy of equal to that of .
REFERENCES
[1] C. A. Stutt, “Some results on real-part/imaginary part and ’ magnitude-phase relations in ambiguity functions,” IEEE Trans. Inf. Theory, vol. 10, no. 4, pp. 321–327, 1964.
[2] G. B. Folland, Harmonic Analysis in Phase Space. Princeton, NJ: Princeton Univ. Press, 1989.
[3] K. Gröchenig, Foundations of Time-Frequency Analysis. Boston, MA: Birkhäuser, 2000.
[4] E. Chassande-Mottin, I. Daubechies, F. Auger, and P. Flandrin, “Differential reassignment,” IEEE Signal Process. Lett., vol. 4, pp. 293–294, 1997.
[5] H. M. Ozaktas and U. Sümbül, “Interpolating between periodicity and discreteness through the fractional Fourier transform,” IEEE Trans.
Signal Process., vol. 54, no. 11, pp. 4233–4243, Nov. 2006.
[6] J. Ville, “Theorie et applications de la notion de signal analytique,”
Cables et Transmission, vol. 2A, pp. 61–74, 1948.
[7] J. E. Moyal, “Quantum mechanics as a statistical theory,” in Proc.
Camb. Phil. Soc., 1949, vol. 45, pp. 99–124.
[8] T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner dis-tribution—A tool for time-frequency signal analysis, Part I: Contin-uous-time signals,” Philips J. Res., vol. 35, no. 3, pp. 217–250, 1980. [9] T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner
dis-tribution—A tool for time-frequency signal analysis, Part II: Discrete-time signals,” Philips J. Res., vol. 35, no. 4/5, pp. 276–300, 1980. [10] T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner
distri-bution—A tool for time-frequency signal analysis, Part III: Relations with other time-frequency signal transformations,” Philips J. Res., vol. 35, no. 6, pp. 372–389, 1980.
[11] J. G. Kirkwood, “Quantum statistics of almost classical ensembles,”
Phys. Rev., vol. 44, pp. 31–37, 1933.
[12] P. J. Loughlin, J. W. Pitton, and B. Hannaford, “Fast approximations to positive time-frequency distributions, with applications,” in Proc. IEEE
Int. Conf. Acoust., Speech, Signal Process. (ICASSP), 1995, vol. II, pp.
1009–1012.
[13] L. Stankovic, “A method for time-frequency analysis,” IEEE Trans.
Acoust., Speech, Signal Process., vol. 42, no. 1, pp. 225–229, 1994.
[14] G. Jones and B. Boashash, “Generalized instantaneous parameters and window matching in the time-frequency plane,” IEEE Trans. Signal
Process., vol. 45, no. 5, pp. 1264–1275, May 1997.
[15] W. Heisenberg, Physics and Philosophy; The Revolution in Modern
Science. New York: Harper & Row, 1958.
[16] P. Flandrin, Time-Frequency/Time-Scale Analysis. San Diego, CA: Academic, 1999.
[17] P. Korn, “Some uncertainty principles of time-frequency transforms of the Cohen class,” IEEE Trans. Signal Process., vol. 53, no. 2, pp. 523–527, Feb. 2005.
[18] P. J. Loughlin and L. Cohen, “The uncertainty principle: Global, local, or both?,” IEEE Trans. Signal Process., vol. 52, no. 5, pp. 1218–1227, May 2004.
[19] S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Signal Process., vol. 31, no. 4, pp. 986–998, Aug. 1983.
[20] A. Francos and M. Porat, “Analysis and synthesis of multicomponent signals using positive time-frequency distributions,” IEEE Trans.
[21] S. Peleg and B. Friedlander, “Multicomponent signal analysis using the polynomial-phase transform,” IEEE Trans. Aerosp. Electron. Syst., vol. 32, no. 1, pp. 378–387, 1999.
[22] L. Le and S. Krishnan, “Time-frequency signal synthesis and its appli-cation in multimedia watermark detection,” EURASIP J. Appl. Signal
Process., vol. 2006, pp. 1–14, 2006.
[23] R. Gerchberg and W. O. Saxton, “A practical algorithm for the deter-mination of phase from image and diffraction plane pictures,” Optik, vol. 35, pp. 237–246, 1972.
[24] P. J. Napier and R. H. T. Bates, “Inferring phase information from mod-ulus information in two-dimensional aperture synthesis,” Astron.
Astro-phys. Suppl., vol. 15, pp. 427–430, 1974.
[25] M. H. Hayes, J. S. Lim, and A. V. Oppenheim, “Signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, and Signal
Process., vol. ASSP-28, pp. 672–680, Dec. 1980.
[26] J. R. Fienup, “Phase retrieval algorithms: A comparison,” Appl. Opt., vol. 21, pp. 2758–2769, 1982.
[27] M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase of magnitude of its Fourier transform,” IEEE Trans. Acoust.,
Speech, Signal Process., vol. ASSP-30, pp. 140–154, Apr. 1982.
[28] Z. Mou-yan and R. Unbehauen, “Methods of reconstruction of 2-d sequences from Fourier transform magnitude,” IEEE Trans. Image
Process., vol. 6, no. 2, pp. 222–233, Feb. 1997.
[29] A. E. Yagle and A. E. Bell, “One- and two-dimensional minimum and nonminimum phase retrieval by solving linear systems of equations,”
IEEE Trans. Signal Process., vol. 47, no. 11, pp. 2978–1989, Nov.
1999.
[30] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004.
[31] L. Cohen, Time-Frequency Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1995.
[32] R. M. Nickel, T. H. Sang, and W. J. Williams, “New signal adaptive approach to positive time-frequency distributions with suppressed in-terference terms,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal
Process. (ICASSP), 1998, vol. 3, pp. 1777–1780.
[33] V. Namias, “The fractional order Fourier transform and its application to quantum mechanism,” J. Inst. Math. Appl., vol. 25, pp. 241–265, 1980.
[34] L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process., vol. 42, no. 11, pp. 3084–3091, Nov. 1994.
[35] A. I. Zayed, “On the relationship between the Fourier and fractal Fourier transforms,” IEEE Signal Process. Lett., vol. 3, pp. 310–311, Dec. 1996.
[36] J. R. Fonollosa, “Positive time-frequency distributions based on joint marginal constraints,” IEEE Trans. Signal Process., vol. 44, no. 8, pp. 2086–2091, Aug. 1996.
[37] B. Ristic and B. Boashash, “Kernel design for time-frequency signal analysis using the radon transform,” IEEE Trans. Signal Process., vol. 41, no. 5, pp. 1996–2008, May 1993.
[38] J. S. Lim, Two-Dimensional Signal and Image Processing. Engle-wood Cliffs, NJ: Prentice-Hall, 1990.
[39] J. Jeong and W. J. Williams, “Kernel design for reduced interference distributions,” IEEE Trans. Signal Process., vol. 40, no. 2, pp. 402–412, Feb. 1992.
[40] A. Rényi, “On measures of entropy and information,” in Proc. 4th
Berkley Symp. Math., Stat., Probab., 1961, vol. 1, pp. 547–561.
[41] T. Sang and W. J. Williams, “Rényi information and signal-depen-dent kernel design,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal
Process. (ICASSP), 1995, vol. 2, pp. 509–512.
[42] S. Aviyente and W. J. Williams, “Minimum entropy time-frequency distributions,” IEEE Signal Process. Lett., vol. 12, no. 1, pp. 37–40, Jan. 2005.
Tzu-Hsien Sang (S’95–M’99) received the B.S.E.E.
degree from the National Taiwan University in 1990 and the Ph.D. degree from the University of Michigan at Ann Arbor in 1999.
He previously worked on physical layer design for broadband technologies at Excess Bandwidth, Sunnyvale, CA, a start-up company. He is currently with the Department of Electronics Engineering, National Chiao-Tung University (NCTU), Taiwan. His research interests include signal processing for communications, time-frequency analysis for biomedicl signals, and RF circuit noise modeling.
