 Review Article
 Open Access
TimeFrequency Analysis and Its Application in Digital Watermarking
 Srdjan Stanković^{1}Email author
https://doi.org/10.1155/2010/579295
© Srdjan Stanković. 2010
 Received: 14 February 2010
 Accepted: 7 June 2010
 Published: 1 July 2010
Abstract
A review of timefrequency analysis and some aspects of its applications in digital watermarking are presented. The main advantages and drawbacks of various timefrequency distributions are first discussed. The aim of this theoretical overview is to facilitate an appropriate distribution selection in a specific application. Different aspects of the timefrequency analysis when applied to digital watermarking are then presented. In particular, the method that maps timefrequency characteristics of a host signal to the pseudo noise watermark sequence is thoroughly discussed. This approach is presented in the multidimensional form and then applied to digital audio, digital image, and digital video watermarking. Finally, the theoretical considerations are illustrated by various numerical and reallife examples.
Keywords
 Speech Signal
 Digital Watermark
 Wigner Distribution
 Watermark Embedding
 Chirp Signal
1. Introduction
Theoretical aspects of timefrequency analysis have been intensively studied over the last two decades [1–27]. In parallel, their various applications have been exploited as well. Namely, for an efficient analysis of nonstationary signals, such are radar, sonar, biomedical, seismic, and multimedia signals, timefrequency representations are required. Timefrequency distributions are most commonly used for this purpose. Many of the researchers have made significant efforts in defining a distribution that is optimal for a wide class of frequencymodulated signals [8–11]. As a result, a number of timefrequency distributions have been proposed. However, the efficiency of each of them is more or less limited to a specific class of signals and, consequently, to a specific application. One of the goals of this paper is to highlight the most important features of some popular timefrequency distributions and to give an idea of how to choose the most appropriate distribution depending on the signal form. The linear, quadratic, higherorder, and multiwindow timefrequency distributions are considered. The shorttime Fourier transform, as the most commonly used linear transform, is firstly discussed. Next, the Wigner distribution, as the best known quadratic distribution, is presented. Also, the Cohen class and some specific distributions belonging to this class are considered [1, 5–7]. It is shown that the quadratic distributions are optimal for a linear frequencymodulated signal. However, if the instantaneous frequency variation within the analysis window is faster, multiwindow, or higher order distributions should be used, [14–16, 19–27]. The Hermite functionsbased multiwindow approach is also discussed. Finally, highly concentrated distributions with complexlag argument are presented. To facilitate in better understanding of the presented theoretical considerations, numerous illustrative examples have been provided.
The second part of the paper considers timefrequencybased watermarking techniques. The watermarking of digital audio, digital image, and digital video is discussed [28–37]. A short overview of some existing approaches is first given and they are related to digital audio and digital image [28–50]. Watermarking using timefrequency techniques is usually employed in either of the following two ways. The first one uses the time/space domain of the host signal to embed the watermark with specific timefrequency characteristics. The timefrequency analysis is then used for detection. The second way uses timefrequency distributions to create or embed watermark in the timefrequency domain. A flexible procedure that can be used for different kinds of signals is discussed more extensively. Therein, the watermark is shaped according to the timefrequency characteristics of the host signal. The detection is performed in the timefrequency domain. This particular approach is presented in the multidimensional form, and it is applied to digital audio, digital image, and digital video. It provides a high degree of robustness and imperceptibility. Hence, even when the watermark is very weak, a reliably detection can still be achieved. Also, the watermark gets completely hidden by the timefrequency characteristics of the host signal.
2. TimeFrequency Analysis
The Fourier transform provides spectral content of a signal. It has been a valuable tool in various applications. However, for nonstationary signals the Fourier transform cannot give satisfactory results since the information about frequency components variations in time is required.
A question that naturally arises at this point is whether there exist a single representation that would be ideal for any signal at hand. The answer is no, hence a number of timefrequency distributions have been introduced.
2.1. The ShortTime Fourier Transform
Note that the spectral content is calculated for each windowed part of the signal. The central point of the sliding window is the time instant for which the spectrum is calculated. The influence of the window size is critical, as it will be discussed later.
are the measures of duration in time and frequency, respectively. The signal should satisfy .
An important property of the STFT is its linearity. Namely, the STFT of a multicomponent signal is . Consequently, if the signal components do not intersect in the timefrequency plane the spectrogram will be equal to the sum of spectrograms of each of the signal components. This is evident from Figure 3.
2.2. Quadratic TimeFrequency Distributions
Quadratic timefrequency distributions have been introduced in order to improve the timefrequency resolution. Namely, they remove the spread factors for linear frequencymodulated signals. Among them, the Wigner distribution is the most commonly used. It has also been used as a base to define several interesting timefrequency distributions.
The spread factor in this case is .
In this case, the auto components are well concentrated. However, the crossterms presence is significant. Namely, the timefrequency representation contains frequency components that do not exist within the signal itself. It could lead to a wrong analysis result.
Some distributions from the Cohen class.
Distribution  Kernel 

ChoiWilliams 

BornJordan 

ZaoAtlasMarks 

Sinc distribution 

Wigner distribution 

Note that there is a tradeoff between the crossterms reduction and autoterm concentration (it depends on parameters of the kernel function). Obviously, distributions belonging to the Cohen class lie in between the two extreme cases: the spectrogram that eliminates the crossterms with a low autoterm concentration and the Wigner distribution that provides high resolution, but with emphatic crossterms. It is possible to obtain an ideal timefrequency concentration only if signal dependent kernels are used [8, 10].
The frequency domain finite window is denoted by . Observe that for the spectrogram and the Wigner distribution are obtained, respectively.
The calculation of the Smethod is illustrated for the central point of an autoterm as well as for the point located in between the two auto components (Figures 7(a) and 7(b)). It is important to observe that the summation has to be performed only over the autoterms. If the other terms are included, the concentration will not be improved. In addition, the noise could be also picked up. The window has to be narrower than the minimal distance between the autoterms. If this is not the case, the interactions between the autoterms will produce the crossterms. Namely, as illustrated in Figure 7(b), the crossterms appear if the window includes the summation terms marked by red color.
The adaptive Smethod with a variable window adjusted to the autoterms is introduced in [18]. However, in many applications the fixed window size of has been shown to provide very good results, since the convergence within the window is fast, and it is mostly achieved after a few summation terms.
Note that all signal components (with constant and linear frequency modulations) are well concentrated even for . By increasing , for the crossterms start to appear (the minimal distance between the autoterms becomes less than ).
The speech signal timefrequency resolution is improved by using the Smethod.
Observe that the spread factor in the quadratic distributions will be present if the instantaneous frequency contains third and higher order phase derivatives. Hence, further concentration improvement can be obtained by using the multiwindow approach or by using higher order timefrequency distributions, as discussed below.
2.3. Multiwindow TimeFrequency Distributions
The weighting coefficients for .










 


 


 



 




 





 






 







 








 









 










 












From Figure 10 it is obvious that the multiwindow versions of distributions outperform their standard counterparts. The multiwindow approach reduces the noise influence as well [27].
where is the order Laguerre function, and it is the Wigner distribution of the order Hermite function.
According to the previous consideration, the spread factor can be gradually reduced by increasing the number of Hermite functions, that is, by increasing the number of spectrograms in (16) and (22). Similar resolution improvements can be obtained by using polynomial timefrequency distributions, where each additional order of the distribution results in a removal of one more term within the spread factor [3].
2.4. Distributions with ComplexLag Argument
For a multicomponent signal, it is calculated for each component separately, where the STFT is used to separate them [20]. This procedure can be generalized for an arbitrary distribution order [24].
Note that the Wigner distribution produces poor results for both signals, since it cannot follow the instantaneous frequency variations.
3. Digital Watermarking
Digital watermarking has been used to protect multimedia data. Demands in this area increase proportionally with the number of internet applications. Namely, these applications are associated with a need for copyright protection of digital audio, digital image, and digital video. Note that the cryptographic methods could be used for this purpose. However, once the data are decoded they can be unlimitedly copied. This has been one of the primary reasons for developing the watermarking techniques. The watermarking, in general, consists of embedding a secret information that can be reliably detected within the host signal. Obviously, this information should be imperceptible within the host data. Depending on the application type, the watermarking can be robust, fragile, or semifragile. The robust watermark should be resistant to various nonmalicious or malicious attacks. Nonmalicious attacks are commonly used signal processing techniques such as compression algorithms, filtering, and so forth, while the malicious attacks are the signal processing techniques that are intentionally used to remove the watermark. The fragile watermark is used to prove data authenticity. Thus, if the content of a signal has been changed, the watermark should no longer exist. The semifragile watermark should be robust to a slight modification, such as for example a certain degree of compression.
Depending on the type of host signal (speech/audio signals, image, video, etc.) various watermarking approaches are developed. Also, different domains have been used: the time domain (or the space domain), the spectral domains such as DFT, DWT, and DCT domain, and a joint time/spacefrequency domain. The existing watermarking techniques are mainly based on either the time or frequency domain. However, in both cases, the timefrequency characteristics of the watermark do not correspond to the timefrequency characteristics of the host signal. It may result in the watermark being not imperceptible, because it is present in the timefrequency regions where the signal components do not exist.
3.1. An Overview of Some TimeFrequencyBased Watermarking Techniques
The timefrequency domain can be very efficient regarding the watermark imperceptibility and robustness. This section presents some key timefrequencybased watermarking procedures with the aim to inspire more contributions on this topic.
 (A)
Image Watermark with Specific TimeFrequency Characteristics
Observe that the Wigner distribution provides an ideal representation for this signal.
The watermark is embedded within the entire image: .
are compared with an assumed reference threshold. Also, multiple chirp watermarks with small and randomly chosen amplitude are used to increase flexibility of the proposed procedure. The parameters of the chirp signal as well as the random sequence that defines the amplitudes of chirp signals serve as the watermark key. Since the watermark is embedded within the entire image in the space domain, a proper masking that provides imperceptibility should be applied. An analysis of the performances giving an estimation of the detectable watermark amplitude level is provided in [38]. The robustness is tested on various attack, some being a median filter, geometrical transformations (translation, rotation and cropping simultaneously applied), a highpass filter, local notch filter, and Gaussian noise.
This provides a possibility to use the watermark that cannot be detected by considering a single block only. Thus, in such case it would be necessary to integrate all of them over the entire image. Note that it is also possible to generate different chirps for different blocks instead of using the same chirp for all blocks. It would make the detection even more difficult for unauthorized users.
where is the unmarked compressed block. The watermark is completely removed by compression if is obtained. The quality of the proposed technique is tested on the image Lena, and it is proven that, for this case, it outperforms the standard spread spectrum technique.
where the watermark is a Gaussian white noise with the variance . The watermark key consists of the watermark sequence and the angles ( _{1}, _{2}). Thus, the algorithm provides two more degrees of freedom, and it offers more possibility to generate watermarks. The watermarking procedure is tested on various images and attacks.
(A.4) Barkat and Sattar have proposed a fragile watermarking procedure for image authentication [43]. The watermark with a particular timefrequency signature is inserted in the image pixels. Although, in general, pixels (according to the image size) exist, a significantly lower number of them is used. The pixels location can be chosen arbitrarily. The authors have used diagonal pixels, modulated by a pseudonoise sequence as a secret key. Various frequencymodulated nonstationary signals can be a watermark, as well. However, the features that could be easily identified should be used. Consequently, different timefrequency distributions should be used for watermark detection. Barkat and Sattar have used a quadratic frequencymodulated signal. It is detected by using the Wigner distribution. The proposed scheme is tested on the following attacks: cropping, translation, JPEG compression, and scaling. Very week and imperceptible attacks were applied (e.g., JPEG with 99% quality is used). It is shown that the watermark cannot be identified after these attacks.
(B) Watermark Created in the TimeFrequency Domain
However, the previous equation holds only if the twodimensional function (45) is a valid Wigner distribution. Namely, it is well known that any twodimensional function cannot be the Wigner distribution. It introduces a very restrictive condition on the function . In the proposed method it is determined by using the timefrequency representation of the corresponding row and taking the middle frequency region.
Alkhassaweneh and Aviyente have suggested a nonblind detection procedure. Namely, the second part of the function in (46) that depends on the watermark is selected. The detection is performed by using the standard correlation detection. A threshold that provides a minimal probability of error is derived. The proposed method is tested on different images and under various attacks. The average probability of error was found to be 0.03.
(B.2) Foo et al. in [48] have defined a method for digital audio watermarking based on the timefrequency domain. Here the audio frames are changed, so that the logical value of 1 is assigned. If the original frame is lengthened or shortened, the logical value 1 is assigned, otherwise the "normal frames" correspond to the logical value 0. The watermark is a sequence obtained as a binary code of the alphabet letters, converted to the ASCII code (the example with the binary code 010001100101001101010111 for the letters FSW is used). The crucial part of this method is the selection of frames that will be lengthened or shortened (the frame size of 1024 samples is used). The frames with signal energy level above the masking threshold are selected (the psychoacoustic model is used to determine the masking threshold in each subband). The frames length is changed by adding or removing samples with amplitudes that do not exceed the masking threshold. Four samples are added or removed within the frame of 1024 samples. It ensures that a perceptual distortion will not appear. In order to preserve the total length of the watermarked audio signal, the same number of the lengthened and shortened frames is used. The pair of frames called Diamond frames is used to represent the binary 1, while the logical values 0 are assigned to the unaltered frames.
The detection procedure is nonblind, that is, the original signal is required. A significant difference between the watermarked and the original signals will appear only if a pair of changed frames exists. Thus, it is used for logical values detection. The proposed watermarking scheme has been tested on various musical signals, as well as on a speech signal, and a set of different attacks has been applied (filtering, resampling, noise, cropping, and MP3 compression). Although the results vary for different signals and attacks, in general they are good. The worst results are obtained for the rock and pop music signals with MP3 compression. However, in all cases the owner can be identified.
The probability of energy for each frequency (within a window used for the spectrogram calculation) is denoted by , while is the maximum frequency. A half of the maximum entropy is taken as a threshold between noiselike and tonelike characteristics. If the entropy is lower than it is considered as a tonelike, otherwise it is a noiselike characteristic.
Finally, after the watermark is modulated and shaped, it is embedded in the time domain as .
A simple watermark detection procedure is applied. First, demodulation is performed by using the timevarying carrier, and then the watermark is detected by using the standard correlation procedure with the pseudonoise sequence.
The proposed method has been tested on several music files. It has been shown that, under various attacks, the bit error rates are mostly between 0.02 and 0.08.
(B.4) An interesting audio watermarking approach based on linear chirps has been proposed in [47]. The watermark is created as a chirp signal, which is perceptually shaped according to the host signal samples. Different chirp rates, each representing a unique watermark message, produce different slopes in the timefrequency domain. The efficient timefrequency representation is obtained by using the Wigner distribution. The extracted chirps are postprocessed in the timefrequency plane by an optimal line detection method based on the HoughRadon transform. It can correctly estimate the slope of the watermark signal despite the broken lines caused by attacks. The simulation results show that the HoughRadon transform applied to a timefrequency distribution can detect the watermark message correctly at bit error rates up to 20%.
3.2. Watermaking Approach Based on the TimeFrequencyShaped Watermark
The approach that will be presented can be used either for audio signals or images [41, 42]. Thus, the embedding and detection procedures for both kinds of signals will be defined and discussed simultaneously, by using the multidimensional notation.
In order to ensure imperceptibility constraints, the watermark should be modeled according to the timefrequency characteristics of the signal components. The concept of nonstationary multidimensional filtering [52] is adapted and used to create a watermark with timefrequency characteristics that correspond to the characteristics of the host signal. The corresponding algorithm consists of the following steps:
()selection of the nonstationary parts of signal suitable for watermark embedding;
()watermark modeling according to the multidimensional timefrequency characteristics of the host signal;
()watermark embedding and watermark detection procedure within the multidimensional timefrequency domain.
Multidimensional timefrequency distributions are employed in order to determine the nonstationary regions. As it will be shown later, the Smethod can be efficiently used to analyze dynamics of the regions of speech signals and images. Although the crossterms are usually undesirable in the timefrequency analysis, they have found to be useful in watermarking. Namely, they may increase performances of a speech watermark detector, and also, increase the efficiency of dynamic regions selection within an image.
where is the shorttime Fourier transform of a multidimensional random sequence . The function contains the information about the components within the region . It is used to create the watermark that will be adjusted to these components. Thus, we may start with an arbitrary random multidimensional sequence and, by using , its multidimensional timefrequency characteristic is modeled.
A value of between 0 and 1 is taken.
where and are the shorttime Fourier transforms of the multidimensional watermarked data, the host data, and the watermark, respectively.
The multidimensional timefrequency domainbased detector provides a low probability of error, even when the number of watermarked samples in the signal domain is small.
3.2.1. Digital Audio Signal
The watermark is embedded in the time domain: . The timefrequency representation of the watermark is shown in Figure 12(c).
As expected, the timefrequency characteristics of the watermark follow those components of the speech signal. Consequently, the watermark is inaudible within the speech signal.
In this case, the fourthorder complexlag distribution is more appropriate for the region selection than the Smethod, because it better follows the frequency variations in the signal (Figures 13(a) and 13(b)). Thus, by using this distribution an inaudible watermark is created.
The second term in (60) is the result of crossterms.
Note that this form of detector can be used in other existing detector structures.
is used. The mean value and standard deviation of the detector response are denoted by and . Indices and indicate the right and the wrong keys, respectively.
Efficiency of the proposed procedure is demonstrated on various examples. The results for speech signals with maximum frequencies of 4 kHz and 11,025 kHz are presented in [41]. This approach provides a reliable detection for a high SNR (SNR = 32 dB has been used) and under various attacks. The watermark sequence was created by using a pseudorandom Gaussian sequence of 1000 samples.
The probability of error was of order 10^{−7} for: MP3 (constant bit rate 8 kbps and variable bit rate 75–120 kbps are considered), delay monolight echo (180 ms, mixing 20%), echo 200 ms, deep flutter (deep 10, sweeping rate 5 kHz), amplitude (normalize 100%), and additive Gaussian noise (SNR = −35 dB). The worst case is obtained for pitch scaling and it is of order 10^{−5}. The results for other attacks (time stretch wow delay 20%, wow delay 10% and bright flutter, MP3 variable bit rate 40–50 kbps) are of order 10^{−6}.
3.2.2. Digital Image
By increasing the size of a twodimensional window, the crossterms start to appear. Thus, when compared to the spectrogram, the number of frequency components increases, hence making the region characterization easier. A pixel that belongs to the dynamic region can be selected by using the following procedure.
()The Smethod is calculated for a window (windows of size up to are used). The middle frequency range is used.
()The energy floor is obtained by using the experimentally determined .
where , while is the total number of the points within the region .
A twodimensional pseudorandom sequence is used.
This procedure is tested on several images (Lena, Peppers, Boat, F16, and Barbara), under various attacks (JPEG80JPEG40, Median , Median , Average , Impulse noise, Gaussian noise, Lightening, and Darkening). The PSNR was around 50 dB. The number of the selected pixels varied from 3304 for F16 to 7833 for Barbara. The probability of error was compared with the standard DCTbased procedure (with different detector forms), where 22050 coefficients are used. It was shown that the proposed procedure significantly outperforms the standard DCT procedures.
3.2.3. Digital Video
where , while is a constant. The stationarity of the selected pixels, along the time axis, is examined by using the onedimensional Smethod. The experiments show that the minimal number of pixels for reliable watermark detection is about 600. This can be easily achieved, even for a very short video sequence (note that more than 2500 stationary pixels are obtained for a signal of duration of 2 s in the example provided in [42]). This approach was tested under the presence of MPEG4 compression. The obtained probabilities of errors were found to be within the range .
4. Conclusion
An overview of most important timefrequency analysis techniques is presented. An appropriate distribution selection procedure for a specific type of signal is discussed. Timefrequencybased watermarking algorithms for digital audio, digital image, and video are reviewed, as well. The watermark is either a signal with specific timefrequency characteristics or a pseudonoise sequence shaped according to the timefrequency characteristics of the host signal. The main advantages of the timefrequency domain over the Fourier, DCT, and signal domain are emphasized. Finally, the presented theory could be used to generalize the existing watermarking approaches defined in either the Fourier or the DCT domain.
Declarations
Acknowledgments
The author is very thankful to Dr. Irena Orović and Professor Victor Sučić for their help and useful suggestions during the work on this paper.
Authors’ Affiliations
References
 Cohen L: Timefrequency distributions—a review. Proceedings of the IEEE 1989, 77(7):941981. 10.1109/5.30749View ArticleGoogle Scholar
 Hlawatsch F, BoudreauxBartels GF: Linear and quadratic timefrequency signal representations. IEEE Signal Processing Magazine 1992, 9(2):2167. 10.1109/79.127284View ArticleGoogle Scholar
 Boashash B, Ristić B: Polynomial timefrequency distributions and timevarying higher order spectra: application to the analysis of multicomponent FM signals and to the treatment of multiplicative noise. Signal Processing 1998, 67(1):123. 10.1016/S01651684(98)000188View ArticleMATHGoogle Scholar
 Boashash B: TimeFrequency Analysis and Processing. Elsevier, Amsterdam, The Netherlands; 2003.MATHGoogle Scholar
 Choi H, Williams WJ: Improved timefrequency representation of multicomponent signals using exponential kernels. IEEE Transactions on Acoustics, Speech, and Signal Processing 1989, 37(6):862871. 10.1109/ASSP.1989.28057View ArticleGoogle Scholar
 Zhao Y, Atlas LE, Marks RJ: The use of coneshaped kernels for generalized timefrequency representations of nonstationary signals. IEEE Transactions on Acoustics, Speech, and Signal Processing 1990, 38(7):10841091. 10.1109/29.57537View ArticleGoogle Scholar
 Stanković L: Autoterm representation by the reduced interference distributions: a procedure for kernel design. IEEE Transactions on Signal Processing 1996, 44(6):15571563. 10.1109/78.506622View ArticleGoogle Scholar
 Baraniuk RG, Jones DL: A signaldependent timefrequency representation. Optimal kernel design. IEEE Transactions on Signal Processing 1993, 41(4):15891602. 10.1109/78.212733View ArticleMATHGoogle Scholar
 Hlawatsch F, Urbanke RL: Bilinear timefrequency representations of signals: the shiftscale invariant class. IEEE Transactions on Signal Processing 1994, 42(2):357366. 10.1109/78.275608View ArticleGoogle Scholar
 Baraniuk RG, Jones DL: Signaldependent timefrequency analysis using a radially Gaussian kernel. Signal Processing 1993, 32(3):263284. 10.1016/01651684(93)90001QView ArticleMATHGoogle Scholar
 Amin MG, Williams WJ: High spectral resolution timefrequency distribution kernels. IEEE Transactions on Signal Processing 1998, 46(10):27962804. 10.1109/78.720381View ArticleGoogle Scholar
 Bastiaans MJ, Alieva T, Stanković L: On rotated timefrequency kernels. IEEE Signal Processing Letters 2002, 9(11):378381. 10.1109/LSP.2002.805118View ArticleGoogle Scholar
 Boashash B: Estimating and interpreting the instantaneous frequency of a signal—part 1: fundamentals. Proceedings of the IEEE 1992, 80(4):520538. 10.1109/5.135376View ArticleGoogle Scholar
 Barkat B, Boashash B: Design of higher order polynomial WignerVille distributions. IEEE Transactions on Signal Processing 1999, 47(9):26082611. 10.1109/78.782225View ArticleMATHGoogle Scholar
 Viswanath G, Sreenivas TV: IF estimation using higher order TFRs. Signal Processing 2002, 82(2):127132. 10.1016/S01651684(01)001682View ArticleMATHGoogle Scholar
 Stanković L: A multitime definition of the Wigner higher order distribution: LWigner distribution. IEEE Signal Processing Letters 1994, 1(7):106109. 10.1109/97.311805View ArticleGoogle Scholar
 Stanković L: A method for timefrequency analysis. IEEE Transactions on Signal Processing 1994, 42(1):225229. 10.1109/78.258146View ArticleGoogle Scholar
 Stanković S, Stanković L: An architecture for the realization of a system for timefrequency signal analysis. IEEE Transactions on Circuits and Systems II 1997, 44(7):600604. 10.1109/82.598433View ArticleMATHGoogle Scholar
 Stanković S, Stanković L: Introducing timefrequency distribution with a "complextime" argument. Electronics Letters 1996, 32(14):12651267. 10.1049/el:19960849View ArticleGoogle Scholar
 Stanković L: Timefrequency distributions with complex argument. IEEE Transactions on Signal Processing 2002, 50(3):475486. 10.1109/78.984717View ArticleGoogle Scholar
 Cornu C, Stanković S, Ioana C, Quinquis A, Stanković L: Generalized representation of phase derivatives for regular signals. IEEE Transactions on Signal Processing 2007, 55(10):48314838.MathSciNetView ArticleGoogle Scholar
 Stanković S, Orovic I, Ioana C: Effects of Cauchy integral formula on the precision of the IF estimation. IEEE Signal Processing Letters 2009, 16(4):327330.View ArticleGoogle Scholar
 Morelande M, Senadji B, Boashash B: Complexlag polynomial WignerVille distribution. Proceedings of IEEE Speech and Image Technologies for Computing and Telecommunications (TENCON '97), December 1997, Brisbane, Australia 1: 4346.View ArticleGoogle Scholar
 Stanković S, Žarić N, Orović I, Ioana C: General form of timefrequency distribution with complexlag argument. Electronics Letters 2008, 44(11):699701. 10.1049/el:20080902View ArticleGoogle Scholar
 Frazer G, Boashash B: Multiple window spectrogram and timefrequency distributions. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '94), 1994 4: 193296.Google Scholar
 Çakrak F, Loughlin PJ: Multiwindow timevarying spectrum with instantaneous bandwidth and frequency constraints. IEEE Transactions on Signal Processing 2001, 49(8):16561666. 10.1109/78.934135View ArticleGoogle Scholar
 Bayram M, Baraniuk RG: Multiple window timefrequency analysis. Proceedings of the IEEESP International Symposium on TimeFrequency and TimeScale Analysis, June 1996 173176.Google Scholar
 Cox IJ, Miller ML, Bloom JA: Digital Watermarking. Academic Press, New York, NY, USA; 2002.Google Scholar
 Barni M, Bartolini F: Watermarking Systems Engineering. Marcel Dekker, New York, NY, USA; 2004.Google Scholar
 Special issue on "Identification and protection of multimedia information" Proceedings of the IEEE 1999., 87(7):Google Scholar
 Muharemagic E, Furht B: Survey of watermarking techniques and applications. In Multimedia Watermarking Techniques and Applications. Edited by: Furht B, Kirovski D. Auerbach Publication; 2006:91130.Google Scholar
 Kirovski D, Malvar HS: Spreadspectrum watermarking of audio signals. IEEE Transactions on Signal Processing 2003, 51(4):10201033. 10.1109/TSP.2003.809384MathSciNetView ArticleGoogle Scholar
 Steinebach M, Dittmann J: Watermarkingbased digital audio data authentication. EURASIP Journal on Applied Signal Processing 2003, 2003(10):10011015. 10.1155/S1110865703304081View ArticleGoogle Scholar
 Nikolaidis A, Pitas I: Asymptotically optimal detection for additive watermarking in the DCT and DWT domains. IEEE Transactions on Image Processing 2003, 12(5):563571. 10.1109/TIP.2003.810586View ArticleGoogle Scholar
 Hernández JR, Amado M, PérezGonzález F: DCTdomain watermarking techniques for still images: detector performance analysis and a new structure. IEEE Transactions on Image Processing 2000, 9(1):5568. 10.1109/83.817598View ArticleGoogle Scholar
 Briassouli A, Strintzis MG: Locally optimum nonlinearities for DCT watermark detection. IEEE Transactions on Image Processing 2004, 13(12):16041617. 10.1109/TIP.2004.837516View ArticleGoogle Scholar
 Cheng Q, Huang TS: An additive approach to transformdomain information hiding and optimum detection structure. IEEE Transactions on Multimedia 2001, 3(3):273284. 10.1109/6046.944472View ArticleGoogle Scholar
 Stanković S, Djurović I, Pitas L: Watermarking in the space/spatialfrequency domain using twodimensional RadonWigner distribution. IEEE Transactions on Image Processing 2001, 10(4):650658. 10.1109/83.913599View ArticleMATHGoogle Scholar
 Djurović I, Stanković S, Pitas I: Digital watermarking in the fractional Fourier transformation domain. Journal of Network and Computer Applications 2001, 24(2):167173. 10.1006/jnca.2000.0128View ArticleMATHGoogle Scholar
 Wickens T: Elementary Signal Detection Theory. Oxford University Press, Oxford, UK; 2002.Google Scholar
 Stanković S, Orović I, Žarić N: Robust speech watermarking procedure in the timefrequency domain. EURASIP Journal on Advances in Signal Processing 2008, 2008:9.Google Scholar
 Stanković S, Orović I, Žarić N: An application of multidimensional timefrequency analysis as a base for the unified watermarking approach. IEEE Transactions on Image Processing 2010, 19(3):736745.MathSciNetView ArticleGoogle Scholar
 Barkat B, Sattar F: A new timefrequency based private fragile watermarking scheme for image authentication. Proceedings of the 7th International Symposium on Signal Processing and Its Applications, July 2003 2: 363366.Google Scholar
 Mobasseri BG, Zhang Y, Amin MG, Dogahe BM: Designing robust watermarks using polynomial phase exponentials. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '05), March 2005 833836.Google Scholar
 Kutter M, Winkler S: A visionbased masking model for spreadspectrum image watermarking. IEEE Transactions on Image Processing 2002, 11(1):1625. 10.1109/83.977879View ArticleGoogle Scholar
 Esmaili S, Krishnan S, Raahemifar K: Audio watermarking using timefrequency characteristics. Canadian Journal of Electrical and Computer Engineering 2003, 28(2):5761. 10.1109/CJECE.2003.1532509View ArticleGoogle Scholar
 Erküçük S, Krishnan S, Zeytinoǧlu M: A robust audio watermark representation based on linear chirps. IEEE Transactions on Multimedia 2006, 8(5):925926.View ArticleGoogle Scholar
 Foo SW, Ho SM, Ng LM: Audio watermarking using timefrequency compression expansion. Proceedings of IEEE International Symposium on Cirquits and Systems (ISCAS '04), May 2004 201204.Google Scholar
 Alkhassaweneh M, Aviyente S: A timefrequency based perceptual and robust watermarking scheme. Proceedings of 13th European Signal Processing Conference (EUSIPCO '05), September 2005, Antalya, TurkeyGoogle Scholar
 Erküçük S: Timefrequency analysis of spread spectrum based communication and audio watermarking systems, Ph.D. thesis. 2003.Google Scholar
 Orović I, Stanković S, Thayaparan T, Stanković L: Multiwindow Smethod for instantaneous frequency estimation and its application in radar signal analysis. IET Signal Processing 2010, 90(5):7.Google Scholar
 Stanković L, Stanković S, Djurović I: Space/spatialfrequency analysis based filtering. IEEE Transactions on Signal Processing 2000, 48(8):23432352. 10.1109/78.852015MathSciNetView ArticleMATHGoogle Scholar
 Stanković S: About timevariant filtering of speech signals with timefrequency distributions for handsfree telephone systems. Signal Processing 2000, 80(9):17771785. 10.1016/S01651684(00)000876View ArticleMATHGoogle Scholar
 Stanković S, Stanković L, Uskokovic Z: On the local frequency, group shift, and crossterms in some multidimensional timefrequency distributions: a method for multidimensional timefrequency analysis. IEEE Transactions on Signal Processing 1995, 43(7):17191724. 10.1109/78.398736View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.