The discrete Fourier transform (DFT) is the most widely used technique for determining the frequency spectra of digital signals. However, in the sliding transform scenario where the transform window is shifted one sample at a time and the transform process is repeated, the use of DFT becomes difficult due to its heavy computational burden. This paper proposes an optimal sliding DFT (oSDFT) algorithm that achieves both the lowest computational requirement and the highest computational accuracy among existing sliding DFT algorithms. The proposed oSDFT algorithm directly computes the DFT bins of the shifted window by simply adding (or subtracting) the bins of a previous window and an updating vector. We show that the updating vector can be efficiently computed with a low complexity in the sliding transform scenario. Our simulations demonstrate that the proposed algorithm outperforms the existing sliding DFT algorithms in terms of computational accuracy and processing time.

Guaranteed-Stable Sliding DFT Algorithm With Minimal Computational Requirements

The Hopping Discrete Fourier Transform [sp Tips

We deal with efficiently computing the far-field radiation/scattering by volumetric sources/objects. The approach consists of three steps: first, representation of “nonuniformly sampled” Fourier exponentials by a finite number of “uniformly sampled” ones, with such number controlling the accuracy; second, “pruned” fast Fourier transform; and third, interpolation, again with controlled accuracy, of the “uniformly sampled” exponentials onto the “nonuniformly sampled” ones corresponding to the relevant observation directions. The performance of the approach in terms of accuracy and complexity is evaluated.

Efficient Computing of Far-Field Radiation in Two Dimension

We present a new algorithm for the 2D Sliding Window Discrete Fourier Transform (SWDFT). Our algorithm avoids repeating calculations in overlapping windows by storing them in a tree data-structure based on the ideas of the Cooley- Tukey Fast Fourier Transform (FFT). For an $N_0 \times N_1$ array and $n_0 \times n_1$ windows, our algorithm takes $O(N_0 N_1 n_0 n_1)$ operations. We provide a C implementation of our algorithm for the Radix-2 case, compare ours with existing algorithms, and show how our algorithm easily extends to higher dimensions.

The 2D Tree Sliding Window Discrete Fourier Transform

A fast algorithm based on the gray code kernel (GCK) for computing the conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) in a sliding window is presented. The proposed algorithm computes the current projection value from the previously computed ones. In order to obtain the peculiar computation order of the projection values, we construct the CS-SCHT matrix tree and also introduce the $\alpha $ -related concept. The properties of the elements of the CS-SCHT matrix are also given for deriving the GCK sliding CS-SCHT algorithm. The proposed algorithm only needs N/2+log $_{2}N-2$ (or log $_{2}N-1$ ) multiplications with $j$ and $4N-2$ (or $2N-1$ ) real additions for complex (or real) input data, which is more efficient than the block-based CS-SCHT and other existing sliding complex transform algorithms, such as the radix-4 sliding CS-SCHT algorithm, sliding FFT algorithm, and sliding DFT algorithm. A comparison of the proposed algorithm with other sliding transforms in terms of computation time is also presented to validate the theoretical results.

Fast Gray Code Kernel Algorithm for the Sliding Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform

The two-dimensional (2D) discrete Fourier transform (DFT) in the sliding window scenario has been successfully used for numerous applications requiring consecutive spectrum analysis of input signals. However, the results of conventional sliding DFT algorithms are potentially unstable because of the accumulated numerical errors caused by recursive strategy. In this letter, a stable 2D sliding fast Fourier transform (FFT) algorithm based on the vector radix (VR) 2 × 2 FFT is presented. In the VR-2 × 2 FFT algorithm, each 2D DFT bin is hierarchically decomposed into four sub-DFT bins until the size of the sub-DFT bins is reduced to 2 × 2; the output DFT bins are calculated using the linear combination of the sub-DFT bins. Because the sub-DFT bins for the overlapped input signals between the previous and current window are the same, the proposed algorithm reduces the computational complexity of the VR-2 × 2 FFT algorithm by reusing previously calculated sub-DFT bins in the sliding window scenario. Moreover, because the resultant DFT bins are identical to those of the VR-2 × 2 FFT algorithm, numerical errors do not arise; therefore, unconditional stability is guaranteed. Theoretical analysis shows that the proposed algorithm has the lowest computational requirements among the existing stable sliding DFT algorithms.

/pdf/vector-radix-2-2-sliding-fast-fourier-transform-1c5l38j964.pdf

Chun Su Park

Papers

Vector Radix 2 × 2 Sliding Fast Fourier Transform