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Application of matrix in signal processing
- 1 G social Radley College, Abingdon, OX14 2HR，United Kingdom
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Abstract
Signal processing, a foundational discipline in modern technology, encompasses a diverse array of applications, ranging from audio and image processing to communication systems and medical imaging. This review investigates how matrix-based techniques are widely used to advance signal processing methodologies. In order to discretize continuous-time signals for digital processing, which occurs in the first section of the paper, matrices play a crucial role in signal sampling. A key principle, the Nyquist-Shannon Sampling Theorem, directs appropriate sampling rates to prevent aliasing, with matrices permitting effective signal representation. The effectiveness of matrix-based filtering methods for frequency modulation and noise reduction, such as convolution and correlation, is then investigated. By utilising matrix operations, these methods enable real-time signal processing. The Fourier Transform and Wavelet Transform are also featured in matrix-driven signal transformation, providing insights into frequency analysis and non-stationary signal characterization. By reducing noise components, matrix-based approaches, particularly Singular Value Decomposition (SVD) denoising, are essential for improving signal quality. Additionally, image compression employs SVD. Matrix-based compressive sensing revolutionises signal recovery from sparse data and results in data-efficient reconstruction. Signal processing has been transformed by matrix-based approaches, which have enabled previously unheard-of levels of efficiency, accuracy, and adaptability. The review highlights their significant influence on several signal processing fields.
Keywords
signal processing, matrix-based techniques, signal sampling, signal transformation
[1]. Berggren J L 2007 Mathematics in medieval Islam. History of Indian Mathematics: A Source Book. Springer.
[2]. Cayley A 1858 A memoir on the theory of matrices. Philosophical Transactions of the Royal Society of London, 148, 17-37.
[3]. Gauss C F 1801 Theoria combinationis observationum erroribus minimis obnoxiae (Theory of the combination of observations least subject to error). Commentationes societatis regiae scientiarum Gottingensis recentiores.
[4]. Sylvester J J 1852 On a theory of the syzygetic relations of two rational integral functions, comprising an application to the theory of Sturm's functions; with numerous examples and applications to the investigation of quantities algebraical and transcendental. McGraw-Hill.
[5]. Chen M, Zheng C, Zhang Z 2020 Matlab function and instance quick check manual. People's Post and Telecommunications Publishing House.
[6]. Wiener N 1949 The theory of prediction. In Modern mathematics for the engineer: Second series. McGraw-Hill.
[7]. Padgett W T, et al. 2010 Discrete-Time Signal Processing. Pearson.
[8]. Cai Y 2000 Research on magnetic resonance image reconstruction based on hardware acceleration. University of the Chinese Academy of Sciences.
[9]. Boyd S, Vandenberghe L 2004 Convex optimization. Cambridge University Press.
[10]. Sun X B, Bao Z 1993 Intermediate frequency signal sampling and orthogonal coherent detection. Systems Engineering and Electronic Technology (5), 1-9.
[11]. Shannon C E 1949 Communication in the presence of noise. Proceedings of the IRE, 37(1), 10-21.
[12]. Proakis J G, Manolakis D G 2007 Digital Signal Processing: Principles, Algorithms, and Applications. Pearson.
[13]. Wang C B 2021 Research on the filtering and stability of two-dimensional systems with stochastic parameter matrices. Pearson.
[14]. Bahri M et al. 2013 Convolution Theorems for Quaternion Fourier Transform: Properties and Applications. Abstract and Applied Analysis, 1–10.
[15]. Good I J 2021 The Relationship Between Two Fast Fourier Transforms. in IEEE Transactions on Computers, 310-317.
[16]. Han X M 2021 Matrix singular value decomposition algorithm and application research. Working paper.
[17]. Qian Z W, et al. 2011 Signal noise reduction method using singular value decomposition. Working paper.
[18]. Zhang J C 2014 Research on sparse multi-band signal compression sampling method [D]. Harbin Institute of Technology.
[19]. Nyquist H 1928 Certain Topics in Telegraph Transmission Theory. in Transactions of the American Institute of Electrical Engineers, 617-644.
[20]. Donoho D L 2006 Compressed sensing. IEEE Transactions on Information Theory, 52(4), 1289-1306.
[21]. Candès E J, Tao T 2006 Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Transactions on Information Theory, 52(12), 5406-5425.
[22]. Li J F 2013 Research and design of centralized digital audio matrix. University of Electronic Science and Technology.
[23]. Luo Y P 2013 Design of image display system based on subpixel sampling. University of Electronic Science and Technology.
[24]. Rhodes W T 1981 Acousto-optic signal processing: Convolution and correlation. in Proceedings of the IEEE, 65-79.
[25]. Challis J H 2021 Signal Processing. In: Experimental Methods in Biomechanics. Springer, Cham.
[26]. Li Y J, et al. 2012 Advantages of Fourier transform in signal processing. Mathematics Learning and Research (13), 120-121.
[27]. Peng N, Kingdom A 2017 Research on wavelet transform parallel algorithm based on MPSoC platform. Computer Engineering and Applications, 33-38.
[28]. Abdi H 2007 Singular value decomposition (SVD) and generalized singular value decomposition. Encyclopedia of measurement and statistics, 907, 912.
[29]. Institute of Reconnaissance Information Equipment, Air Force Equipment Research Institute of the People's Liberation Army (2016-07-13). CN105763202A.
[30]. Qi Y, et al. 2022 Sparse signal recovery methods and devices. Beijing.
Cite this article
Du,H. (2024). Application of matrix in signal processing. Applied and Computational Engineering,35,41-49.
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Volume title: Proceedings of the 2023 International Conference on Machine Learning and Automation
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