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A Step Toward Real-Time Time–Frequency Analyses with Varying Time–Frequency Resolutions: Hardware Implementation of an Adaptive S-transform

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Abstract

This paper addresses the rarely considered issue of hardware implementation of the S-transform, a very important time–frequency representation with many practical applications. Various improved, adaptive, and signal-driven versions of the S-transform have been developed over the years, but only its basic (non-adaptive) form has been implemented in hardware. Here, a novel hardware implementation of the adaptive S-transform is proposed extending the previously developed design. To minimize hardware demands, the proposed approach is based on an appropriate approximation of the frequency window function considered in the S-transform. The adaptivity of the transform to the signal is achieved by an optimal choice of a window parameter from the set of predefined values, meaning that for each window parameter the S-transform is calculated. To additionally save hardware resources, the proposed design does not require storing all calculated values, but only two in each iteration. The proposed multiple-clock-cycle architecture is developed on the field-programmable gate array device, and its performance is compared with other possible implementation approaches such as the hybrid and single-clock-cycle ones. It is demonstrated that the developed design minimizes hardware complexity and clock cycle time compared to alternative approaches and is significantly more efficient than the software realization. Both noiseless and noisy multicomponent highly nonstationary signals were considered. An excellent match between the results of the hardware and the “exact” adaptive S-transform evaluation obtained through the MATLAB implementation is demonstrated. Lastly, the execution time that can be estimated in advance is also an important practical feature of the developed design.

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Notes

  1. There are several concentration measures for time–frequency representations. The commonly used measures are based on the Renyi entropy. However, all of them have drawbacks analyzed in [35]. Nevertheless, the measure used in this paper avoids some of the drawbacks outlined in the previous publication. In general, by keeping constraint (4), this measure minimizes the region of the time–frequency plane covered by signal components that correspond to high resolution. It is already used in various research papers including consideration of the adaptive S-transform in [8, 18, 20, 21, 33].

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Acknowledgements

The authors wish to thank Curtis Condon, Ken White, and Al Feng of the Beckman Institute of the University of Illinois for the bat data and for permission to use it in this paper.

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Authors and Affiliations

  1. Department of Electrical Engineering, University of Montenegro, Cetinjski Put bb, 81000, Podgorica, Montenegro

    Nevena Radović, Veselin N. Ivanović & Igor Djurović

  2. Faculty for Information Systems and Technologies, University of Donja Gorica, Oktoih 1, 81000, Podgorica, Montenegro

    Marko Simeunović

  3. The Edward S. Rogers Department of Electrical and Computer Engineering, Faculty of Applied Science and Engineering, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Ervin Sejdić

  4. North York General Hospital, Toronto, ON, M2K 1E1, Canada

    Ervin Sejdić

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  1. Nevena Radović
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Correspondence to Nevena Radović.

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Radović, N., Ivanović, V.N., Djurović, I. et al. A Step Toward Real-Time Time–Frequency Analyses with Varying Time–Frequency Resolutions: Hardware Implementation of an Adaptive S-transform. Circuits Syst Signal Process 42, 853–874 (2023). https://doi.org/10.1007/s00034-022-02203-3

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