Abstract
In this paper, we analyze how to calculate the matrix transposition in continuous flow by using a memory or group of memories. The proposed approach studies this problem for specific conditions such as square and non-square matrices, use of limited access memories and use of several memories in parallel. Contrary to previous approaches, which are based on specific cases or examples, the proposed approach derives the fundamental theory involved in the problem of matrix transposition in a continuous flow. This allows for obtaining the exact equations for the read and write addresses of the memories and other control signals in the circuits. Furthermore, the cases that involve non-square matrices, which have not been studied in detail in the literature, are analyzed in depth in this paper. Experimental results show that the proposed approach is capable of transposing matrices of $8192 \times 8192$ 32-bit data received in series at a rate of 200 mega samples per second, which doubles the throughput of previous approaches.
Highlights
M ATRIX transposition is an essential operation in a wide range of signal processing applications
This is due to the fact that it is used for the calculation of multidimensional transforms. This makes it a key component for the 2D fast Fourier transform (FFT) in image processing and machine vision [1], multiple-input multipleoutput (MIMO) [2], [3], automotive [4] and synthetic aperture radars [5]–[7]. It is required for the 3D FFT in molecular dynamics [8], motion detection [9]; for the 2D discrete cosine transform (DCT) in image compression [10], [11]; for the 2D fast Hartley transform (FHT) in image processing and circular convolution [12], [13]; and for the 3D fast Wavelet transform (FWT) in video encoding [14]
Matrix transposition is considered in convolutional neural networks (CNN) [15], [16] for artificial intelligence
Summary
M ATRIX transposition is an essential operation in a wide range of signal processing applications. We provide a detailed analysis of the matrix transposition in a continuous flow using memories under any combination of specific conditions: Square and non-square matrices, use of limited access memories and use of several memories in parallel. For all these cases, efficient solutions that require a total memory size of order O(N) are presented. Whereas previous works study the problem based on examples, the proposed approach deepens in the mathematical and logical fundamentals of the problem This allows for obtaining the exact equations for the read and write addresses of the memories and other control signals.
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