Abstract
This paper presents a forward-path, novel, two-dimensional (2-D) sliding discrete Fourier transform (SDFT) algorithm based on the column-row 2-D DFT concept and the shifted window property. After applying a descending dimension method (DDM), a mixed-radix and butterfly-based structure can be further employed to effectively implement the proposed algorithm. Conceptually, it has many advantages, including greater stability, more accuracy, and less computational complexity because there are no extra feedback loops in the calculation. The evaluation results are based on the following conditions: (1) the window size (N) to 16 × 16; (2) the test pattern is an SVC grayscale video, and the formats are CIF and 4CIF with 30fps; (3) one multiplication involves four real multiplications and two real additions. The proposed 2-D SDFT method clearly reduced the number of multiplications by 43.8% and only increased the number of additions by 33%, compared with the state-of-the-art Park's method. Additionally, for the first 100 frames of the CIF and 4CIF sequences, the proposed method saves 10.8% and 10.9% of the processing time, respectively, on average. Overall, the proposed DDM-based 2-D SDFT algorithm can be applied to calculate not only 1-D but also 2-D SDFT spectrum, and are especially appropriate for hybrid applications.
Highlights
The discrete Fourier transform (DFT) has been widely employed in many digital signal processing applications to analyze signal power spectral density (PSD)
Computational requirement significantly lower than that for the traditional DFT. Both the sliding DFT (SDFT) algorithm and the DFT computation have to perform all of the time indices, and the output data rate of the spectral bin is the same as that of the input data rate. This implies that SDFT would have better resolution in the time domain than the traditional DFT and fast Fourier transform (FFT)
COMPUTATIONAL COMPLEXITY For the sliding issues in various 2-D SDFT algorithms, the performance metrics in terms of number of multiplications and additions are evaluated in Table 2, where RM and real additions (RA) denote the number of real multiplications and additions, respectively
Summary
The discrete Fourier transform (DFT) has been widely employed in many digital signal processing applications to analyze signal power spectral density (PSD). The overall processes can be calculated as 2N times the length-of-N 1-D DFT computation, and it takes (2N × N 2) complex multiplications and (2N × N 2) complex additions for each complexinput sample This column-row-based 2-D algorithm has more computational complexity than the vector-radix 2-D FFT algorithm [25], but it can be simplified into a 1-D computation. In the column-row-based case, the 2-D sliding process is equivalent to perform DFT computation once along one new column followed by the N -point SDFT along the N rows, as shown in Fig 3 (b) We make another assumption: that the 1-D sliding window is shifted in the vertical direction for each column.
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