Veselin N. Ivanovic

Performance of quadratic time-frequency distributions as instantaneous frequency estimators

General performance analysis of the shift covariant class of quadratic time-frequency distributions (TFDs) as instantaneous frequency (IF) estimators, for an arbitrary frequency-modulated (FM) signal, is presented. Expressions for the estimation bias and variance are derived. This class of distributions behaves as an unbiased estimator in the case of monocomponent signals with a linear IF. However, when the IF is not a linear function of time, then the estimate is biased. Cases of white stationary and white nonstationary additive noises are considered. The well-known results for the Wigner distribution (WD) and linear FM signal, and the spectrogram of signals whose IF may be considered as a constant within the lag window, are presented as special cases. In addition, we have derived the variance expression for the spectrogram of a linear FM signal that is quite simple but highly signal dependent. This signal is considered in the cases of other commonly used distributions, such as the Born-Jordan and the Choi-Williams distributions. It has been shown that the reduced interference distributions outperform the WD but only in the case when the IF is constant or its variations are small. Analysis is extended to the IF estimation of signal components in the case of multicomponent signals. All theoretical results are statistically confirmed.

An architecture for the VLSI design of systems for time-frequency analysis and time-varying filtering

A flexible system for time-frequency signal analysis is presented. It is based on the S-method, which has a significant advantage in implementation since it can involve, as a key intermediate step, the Short-time Fourier transform or the Hartley transform, each widely studied and commonly used in practice. Signal invariant and signal dependent system forms are presented. Hardware design, for a fixed-point arithmetic, is well-structured and suitable for vlsi implementation. The same hardware, without additional time requirements, may be shared for the simultaneous realization of the fourth order L-Wigner distribution, as well as for the realization of the cross-terms free fourth order polynomial Wigner-Ville distribution. This possibility makes the designed hardware suitable for wide range of the applications. The proposed hardware is applied to the realization of time-varying filtering, as well. Finally, it has been implemented with fpga chips (Field Programmable Gate Array) in order to verify the results on real devices.RésuméCet article présente un système souple pour l’analyse d’un signal en temps et en fréquence. Ce système est fondé sur la méthode S, ce qui facilite la réalisation grâce àl’emploi de la transformation de Fourier àcourt terme et de la transformation de Hartley, bien connues et largement en usage. L’article considère deux variantes suivant que la fenêtre dépend ou non du signal. La conception du matériel, en arithmétique àvirgule fixe, convient bien àune réalisation par vlsi. Le même matériel peut être utilisé simultanément et sans délai supplémentaire pour la réalisation d’une distribution L-Wigner du quatrième ordre et celle d’une distribution Wigner-Ville polynomiale d’ordre quatre. Le matériel proposé est appliqué àla réalisation de filtrages variables dans le temps. Les résultats ont pu être vérifiés grâce àune réalisation àbase de puces fpga.

Further results on the minimum variance time-frequency distribution kernels

Results for the minimum variance kernel, presented by Hearon and Amin (see ibid., vol.43, p.1258, 1995), for the complex Gaussian white noise with independent real and imaginary parts remain valid for real noise and approximately valid for analytic noise. These results are extended to the real and analytic noisy signals cases.

Multiple-clock-cycle architecture for the VLSI design of a system for time-frequency analysis

Multiple-clock-cycle implementation (MCI) of a flexible system for time-frequency (TF) signal analysis is presented. Some very important and frequently used time-frequency distributions (TFDs) can be realized by using the proposed architecture: (i) the spectrogram (SPEC) and the pseudo-Wigner distribution (WD), as the oldest and the most important tools used in TF signal analysis; (ii) the S-method (SM) with various convolution window widths, as intensively used reduced interference TFD. This architecture is based on the short-time Fourier transformation (STFT) realization in the first clock cycle. It allows the mentioned TFDs to take different numbers of clock cycles and to share functional units within their execution. These abilities represent the major advantages of multicycle design and they help reduce both hardware complexity and cost. The designed hardware is suitable for a wide range of applications, because it allows sharing in simultaneous realizations of the higher-order TFDs. Also, it can be accommodated for the implementation of the SM with signal-dependent convolution window width. In order to verify the results on real devices, proposed architecture has been implemented with a field programmable gate array (FPGA) chips. Also, at the implementation (silicon) level, it has been compared with the single-cycle implementation (SCI) architecture.

Unified approach to noise analysis in the Wigner distribution and spectrogram

An analysis of time-frequency representations of noisy signals is performed. Using the method for time-frequency signal analysis which was recently defined by Stankovic (the S-method), the influence of noise on the two most important distributions (spectrogram and Wigner distribution) is analyzed in unified manner. It is also shown that, for signals whose instantaneous frequency is not constant, an improvement over the spectrogram and the Wigner distribution performances in a noisy environment may be achieved using the S-method. The expressions for mean and variance are derived. Results are given for several illustrative and numerical examples.RésuméCet article présente une analyse de la représentation en temps-fréquence de signaux en présence de bruit. L’influence du bruit sur les deux distributions les plus importantes, spectrogramme et distribution de Wigner, est analysée de manière unifiée à l’aide de la méthode S définie par L. Stanković. Il est montré que pour des signaux à fréquence instantanée variable, cette méthode donne de meilleurs résultats que ceux obte-nus avec le spectrogramme ou la distribution de Wigner, en présence de bruit. Des expressions simples pour la valeur moyenne et la variance pour la méthode S sont donnés et deux exemples de traitement analytique et numérique illustrent la méthode.

Signal Adaptive Pipelined Hardware Design of Time-Varying Optimal Filter for Highly Nonstationary FM Signal Estimation

In this paper we develop a multiple-clock-cycle signal adaptive hardware design of an optimal nonstationary (time-varying) filtering system. The proposed design is based on the real-time results of time-frequency (TF) analysis and the estimation of instantaneous frequency (IF). It permits multiple detection of the local filter’s region of support (FRS) in the observed increment of time, resulting in the efficient filtering of multicomponent frequency modulated (FM) signals. The proposed design takes a variable number of clock (CLK) cycles–the only necessary ones regarding the highest quality of IF estimation–in different TF points within the execution. In this way it allows the implemented system to optimize the computational cost, as well as the time required for execution. Further, the proposed serial design optimizes critical design performances, related to the hardware complexity, making it a suitable system for real-time implementation on an integrated chip. Also, by applying the pipelining technique, it allows overlapping between different TF points within the execution, additionally improving the time required for time-varying filtering. The design has been verified by a field-programmable gate array (FPGA) circuit design, capable of performing filtering of nonstationary FM signals in real-time.

On the S-method based instantaneous frequency estimation

The performance analysis of the S-method (SM) as an instantaneous frequency (IF) estimator, for the case of multicomponent signals, is derived. It has been shown that, when the components are separated in time-frequency (TF) plane, the results obtained for each of them separately can be the same as in the case when all components exist. Also, these results can outperform the ones obtained by the spectrogram (and, consequently, by the reduced interference distributions (RID)) and pseudo Wigner distribution (WD) in the IF estimation of multicomponent signals. Theoretical results are statistically confirmed.

Signal Adaptive System for Space/Spatial-Frequency Analysis

This paper outlines the development of a multiple-clock-cycle implementation (MCI) of a signal adaptive two-dimensional (2D) system for space/spatial-frequency (S/SF) signal analysis. The design is based on a method for improved S/SF representation of the analyzed 2D signals, also proposed here. The proposed MCI design optimizes critical design performances related to hardware complexity, making it a suitable system for real time implementation on an integrated chip. Additionally, the design allows the implemented system to take a variable number of clock cycles (CLKs) (the only necessary ones regarding desirable—2D Wigner distribution-presentation of autoterms) in different frequency-frequency points during the execution. This ability represents a major advantage of the proposed design which helps to optimize the time required for execution and produce an improved, cross-terms-free S/SF signal representation. The design has been verified by a field-programmable gate array (FPGA) circuit design, capable of performing S/SF analysis of 2D signals in real time.

An Efficient Hardware Design of the Flexible 2-D System for Space/Spatial-Frequency Signal Analysis

An efficient multiple clock cycle implementation (MCI) of a flexible system for space/spatial-frequency signal analysis is proposed. It is designed by extending the 1-D MCI architecture to the 2-D case. The proposed system can implement various (almost all commonly used) 2-D space/spatial-frequency distributions (S/SFDs) based on the 2-D short-time Fourier transform (STFT) elements. The designed system is very flexible, since it allows the implemented S/SFDs to take different numbers of clock cycles and to share functional kernel, known as the STFT-to-SM gateway, , within their execution. These are major advantages of the MCI approach, that enable one to optimize critical design performances such as hardware complexity, energy consumption, and cost. The proposed approach is verified by a real-time design of the field-programmable gate array chip that is capable of performing 2-D S/SFDs in real time

Finite word-length effects in implementation of distributions for time-frequency signal analysis

This correspondence presents an analysis of the finite register length influence on the accuracy of results obtained by the time-frequency distributions (TFDs). In order to measure quality of the obtained results, the variance of the proposed model is found, signal-to-quantization noise ratio (SNR) is defined, and appropriate expressions are derived. Floating- and fixed-point arithmetic are considered, with the analysis of discrete random and discrete deterministic signals. It is shown that commonly used reduced interference distributions (RIDs) exhibit similar performance with respect to the SNR. We have also derived the expressions establishing the relationship between the number of bits and the required quality of representation (which is defined by the SNR), which may be used for register-length design in hardware implementation of time-frequency algorithms.