9:30, SPTM-P3.1
INSTANTANEOUS FREQUENCY ESTIMATION BASED ON THE ROBUST SPECTROGRAM
I. DJUROVIC, V. KATKOVNIK, L. STANKOVIC
Robust M-periodogram is defined for the analysis of signals with heavy-tailed distribution noise. In the form of a robust spectrogram (RSPEC) it can be used for the analysis of nonstationary signals. In this paper a RSPEC based instantaneous frequency (IF) estimator, with a time-varying window length, is presented. The optimal choice of the window length can resolve the bias-variance trade-off in the RSPEC based IF estimation. However, it depends on the unknown nonlinearity of the IF. The algorithm used in this paper is able to provide the accuracy close to the one that could be achieved if the IF, to be estimated, were known in advance. Simulations show good accuracy ability of the adaptive algorithm and good robustness property with respect to rare high magnitude noise values.

9:30, SPTM-P3.2
INSTANTANEOUS FREQUENCY ESTIMATION BY USING TIME-FREQUENCY DISTRIBUTIONS
V. IVANOVIC, M. DAKOVIC, I. DJUROVIC, L. STANKOVIC
Estimation of the instantaneous frequency by using quadratic distributions from the genaral Cohen class is analyzed. Frequency modulated signals corrupted with a white stationary noise are considered. Expression for the variance is derived. It is shown that the variance is closely related to the nonnoisy distribution of a predefined signal.

9:30, SPTM-P3.3
INSTANTANEOUS FREQUENCY ESTIMATION USING DISCRETE EVOLUTIONARY TRANSFORM FOR JAMMER EXCISION
L. CHAPARRO, R. SULEESATHIRA, A. AKAN, B. UNSAL
In this paper, we propose a method --based on the discrete evolutionary transform (DET)-- to estimate the instantaneous frequency of a signal embedded in noise or noise-like signals. The DET provides a representation for non-stationary signals and a time-frequency kernel that permit us to obtain the time-dependent spectrum of the signal. We will show the instantaneous phase and the corresponding instantaneous frequency (IF) can also be computed from the evolutionary kernel. Estimation of instantaneous frequency is of general interest in time-frequency analysis, and of special interest in the excision of jammers in direct sequence spread spectrum. Implementation of the IF estimation is done by masking and a recursive non-linear correction procedure. The proposed estimation is valid for monocomponent as well as multicomponent signals in the noiseless and noisy situations. Its application to jammer excision in direct sequence spread spectrum communication is considered as an important application. The estimation procedure is illustrated with several examples.

9:30, SPTM-P3.4
A FRACTIONAL GABOR TRANSFORM
A. AKAN, V. SHAKHMUROV, Y. CEKIC
We present a fractional Gabor expansion on a general, non-rectangular time-frequency lattice. The traditional Gabor expansion represents a signal in terms of time and frequency shifted basis functions, called Gabor logons. This constant-bandwidth analysis results in a fixed, rectangular time-frequency plane tiling. Many of the practical signals require a more flexible, non-rectangular time-frequency lattice for a compact representation. The proposed fractional Gabor expansion uses a set of basis functions that are related to the fractional Fourier basis and generate a non-rectangular tiling. The completeness and bi-orthogonality conditions of the new Gabor basis are discussed.

9:30, SPTM-P3.5
A FOUR-PARAMETER QUADRATIC DISTRIBUTION
D. ROBERTS, D. JONES
Quadratic distributions such as time-frequency distributions and ambiguity functions have many useful applications. In some cases it is desirable to have a quadratic distribution of more than two variables. Using the technique of applying operators to variables, general quadratic distributions of more than two variables can be developed. We use this technique to develop a four-parameter quadratic distribution that includes variables of time, frequency, lag, and doppler. A general distribution is first developed and some of the mathematical properties are discussed. The distribution is then applied to the improvement of an adaptive time-frequency distribution. An example signal is shown to evaluate the performance of the technique.

9:30, SPTM-P3.6
HARMONIC TRANSFORM
G. BI, F. ZHANG, Y. ZENG, Y. CHEN
The harmonic transform is designed for harmonic signals, which are composed of a base tone and some harmonics (e.g. voiced speech). As a generalization of the Fourier transform, harmonic transform represents signals by the sum of a base tone and some harmonics, which may gives more concise results for harmonic signals than Fourier transform. Some experiments of speech signals are used to demonstrate the advantages of harmonic transform on harmonic signal processing.

9:30, SPTM-P3.7
RECURSIVE ZAK TRANSFORMS AND WEYL-HEISENBERG EXPANSIONS
A. BRODZIK
We develop algorithms for computing block-recursive Zak transforms and Weyl-Heisenberg expansions, which achieve p/logL and (logM+p)/(logN+logL+1) multiplicative complexity reduction, respectively, over direct computations, where p'=pM, and N-p' is the number of overlapping samples in subsequent signal segments. For each transform we offer a choice of two algorithms that is based on two different implementations of the Zak transform of the time-evolving signal. These two algorithm classes exhibit typical trade-offs between computational complexity and memory requirements.

9:30, SPTM-P3.8
INFORMATION BOUNDS FOR RANDOM SIGNALS IN TIME-FREQUENCY PLANE
S. AVIYENTE, W. WILLIAMS
Renyi entropy has been proposed as one of the methods for measuring signal information content and complexity on the time-frequency plane by several authors. It provides a quantitative measure for the uncertainty of the signal. All of the previous work concerning Renyi entropy in the time-frequency plane has focused on determining the number of signal components in a given deterministic signal. In this paper, we are going to discuss the behaviour of Renyi entropy when the signal is random, more specifically white complex Gaussian noise. We are going to present the bounds on the expected value of Renyi entropy and discuss ways to minimize the uncertainty by deriving conditions on the time-frequency kernel. The performance of minimum entropy kernels in determining the number of signal elements will be demonstrated. Finally, some possible applications of Renyi entropy for signal detection will be discussed.

9:30, SPTM-P3.9
HIGH RESOLUTION TIME-FREQUENCY ANALYSIS BY FRACTIONAL DOMAIN WARPING
A. OZDEMIR, L. DURAK, O. ARIKAN
A new algorithm is proposed to obtain very high resolution time-frequency analysis of signal components with curved time-frequency supports. The proposed algorithm is based on fractional Fourier domain warping concept introduced in this work. By integrating this warping concept to the recently developed directionally smoothed Wigner distribution algorithm [1], the high performance of that algorithm on linear, chirp-like components is extended to signal components with curved time-frequency supports. The main advantage of the algorithm is its ability to suppress not only the cross-cross terms, but also the auto-cross terms in the Wigner distribution. For a signal with N samples duration, the computational complexity of the algorithm is O(N log N) flops for each computed slice of the new time-frequency distribution.

9:30, SPTM-P3.10
PARAMETER SELECTION FOR OPTIMISING TIME-FREQUENCY DISTRIBUTIONS AND MEASUREMENTS OF TIME-FREQUENCY CHARACTERISTICS OF NON-STATIONARY SIGNALS
V. SUCIC, B. BOASHASH
Selecting a time-frequency distribution (TFD) which represents a signal in an optimal way is commonly done by visually comparing plots of different TFDs. This paper presents a procedure that allows the analyst to perform the same task in an automatic way. Using the resolution performance measure for TFDs, the procedure optimises all distributions considered and selects the one which results into best concentration of signal components around their instantaneous frequency laws, as well as best suppression of the interference terms in the time-frequency plane. To do this requires to define a methodology to measure the time-frequency characteristics of a signal from its optimal TFD. An algorithm which implements this methodology is described and results are presented.

9:30, SPTM-P3.11
FRACTIONAL, CANONICAL, AND SIMPLIFIED FRACTIONAL COSINE TRANSFORMS
S. PEI, J. DING
Fourier transform can be generalized into the fractional Fourier transform (FRFT), linear canonical transform (LCT), and simpli-fied fractional Fourier transform (SFRFT). They extend the utilities of original Fourier transform, and can solve many prob-lems that can't be solved well by original Fourier transform. In this paper, we will generalize the cosine transform. We will derive fractional cosine transform (FRCT), canonical cosine transform (CCT), and simplified fractional cosine transform (SFRCT). We will show they are very similar to the FRFT, LCT, and SFRFT, but they are much more efficient to deal with the even, real even functions. For digital implementation, FRCT and CCT can save 1/2 of the real number multiplications, and SFRCT can save 3/4. We also discuss their applications, such as optical system analysis and space-variant pattern recognition.