TAILIEUCHUNG - Model-Based Design for Embedded Systems- P24

Model-Based Design for Embedded Systems- P24:The unparalleled flexibility of computation has been a key driver and feature bonanza in the development of a wide range of products across a broad and diverse spectrum of applications such as in the automotive aerospace, health care, consumer electronics, etc. | 666 Model-Based Design for Embedded Systems Fourier transform can be implemented by one of the numerous fast Fourier transform FFT techniques. The computational order of the FFT for a 2D input is O N2log2N obviously more efficient when compared to the direct integration method. We show this speed increase later through an example. In continuous theory the angular spectrum method is an exact solution of the Rayleigh-Sommerfeld formulation. However when solving the algorithm on a digital computer a discrete Fourier transform DFT must be used resulting in the accuracy of the angular spectrum method being dependent on the resolution or spacing of the aperture and observation plane meshing. We call the physical size of the aperture and observation planes the bounding box defining the size of the optical wave front being propagated. Since the complex wave function is only nonzero for a finite space in the bounding box the signal is not always bandwidth limited and the Nyquist sampling theory does not always apply. It can be shown however that the resolution of the aperture and observation meshing must be A 2 or smaller 39 . For many simulation systems without large degrees of tilt and hard diffractive apertures the resolution can be coarser. In systems with high tilts the resolution is most sensitive. With a mesh spacing of A 2 the angular spectrum decomposition will model plane waves propagating from the aperture to the observation plane in a complete half circle that is between -90 and 90 degrees. Other inaccuracies that can occur when using a DFT are aliasing and window truncation. Aliasing occurs when frequencies exist greater than the critical sampling frequency. In this case these high frequencies are folded over into the sampled frequency range 40 . The effect of this is seen in our simulations as optical power reflecting off of the walls of the bounding box. If significant optical power reflects off the wall interference between the propagating beam and these .

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