Instead of calculating the FFT digitally by computer using the Cooley–Tukey algorithm 8, there are some alternative methods to implement FFT physically. Similar to the 2-D FFT computation, the computational complexity would become high if the number N (the sampling points of the signal) is large. Additionally, the computational complexity of the 1-D FFT for signals is O( NlogN). In these scenarios where repetitive FFT computations are usually needed, excessive energy will be consumed and the efficiency could be also low if the resolution of the input signal is high (large 1-D input data). Usually, real-time frequency response is required for applications such as object detection, recognition, distance measurement, etc. Unlike 2-D image processing, signal processing 7 based on electromagnetic waves is usually based on 1-D temporal input data. Obviously, the computational complexity would be exponentially increased if the number N (the size of the image) becomes larger. The 2-D FFT for an image with \(N \times N\) pixels has a computational complexity of \(O(N^2 logN)\). For example, FFT acceleration using photonics for AI is an ongoing heated topic 4, and photonic integrated circuits can do the FFT physically rather than digitally 5, 6. In this case, FFT also can be used for convolutional calculation, given that the convolution of two images is equivalent to the multiplication of the FFT results of the two images. Repetitive CNN calculation is needed for training or running a trained model with various inputs. Image processing techniques have flourished in the recent years with the rapid development of deep learning methods, especially for those based on convolutional neural networks (CNN) 3. Thus, FFT could transform the image from spatial domain to the corresponding frequency counterpart 2. For example, many types of image processing are implemented in frequency/spectral domain such as de-noising, edge detection, etc. Repetitive FFT computations could lead to considerable power consumption and prevent real-time signal/image processing, especially when the dimension of the input data is extremely large.
The FFT results from FD and FEM agree well with the digitally computed FFT, with computational complexity reduced from \(O(N^2 logN)\) to O( N) for 2-D FFT, and from O( NlogN) to O( N) for 1-D FFT with a large number of signal sampling points N.įourier transform (FT) is commonly used in a wide variety of digital computations 1, including signal and image processing, solving differential equations, artificial intelligence (AI) models, etc.
SYSTEM OF EQUATIONS IN MATLAB FUNCTION HANDLE FULL
We validate our proposed 2-D full wave FEM simulation method by comparing our results with the theoretical FFT and Fresnel diffraction method.
Unlike the previously reported WFC system which catered only for 2-D images, our proposed method also can solve the 1-D FFT effectively. We also compared the analytical FFT solution with our Fourier transform (FT) results from 3-D and 2-D FD and novel 2-D full wave FEM simulations of a multi-level Fresnel lens with 1-D signals and 2-D images as inputs. To effectively simulate the WFC system for inputs of 1-D signals and 2-D images, we verified the design parameters and focal length of an ideal plano-concave lens using the ray-tracing method. We validate the design of the WFC system using ray-tracing, Fresnel diffraction (FD), and the full-wave finite element method (FEM).
In this paper, we present comprehensive studies on the modeling and simulation methods for ultrasonic WFC computation. The recent emerging alternative to classic numerical Fast Fourier transform (FFT) computation, based on GHz ultrasonic waves generated from and detected by piezoelectric transducers for wavefront computing (WFC), is more efficient and energy-saving.
0 kommentar(er)
