Engineering Homework Help

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Spectral Derivatives (20 pts)One of the key application areas of numerical analysis is the accurate numerical calculation of derivatives.This can be used within multiphysics simulations (such as AutoCAD, Solidworks, and COMSOL) in orderto accurately model a given physical scenario. These numerical models can be used to test hypotheses beforea physical object is built, as well as be used to directly help design something like an antenna.We will explore three different ways of calculating the derivative numerically. The first method is calledthefinite differencemethod, and basically involves directly implementing the standard limit formula for aderivative, but evaluating at anh6= 0 but instead some small number. The second method is called aspectralmethod. Here, we take the Fast Fourier Transform (FFT) of a signal in order to express it in terms of knownfunctions. We can then use derivative properties of the FFT to calculate the derivative, which is applied inthek-wavesoftware shown above to solve for the acoustic propagation in general media.1. For these problems, assume that the sampling rateFs= 50 samples per second, and the time is between[−1,1].(a)•Plot the signalx1(t) = sin(ω0t) whereω0= 2π·3•Plot the signalx′1(t) =ω0cos(ω0t)(b)•Plot the signalx2(t) = 6t4−2t2+ 5•Plot the signalx′2(t) = 24t3−4t(c)•Plot the signalx3(t) =J1(6t) (Bessel function, you may find thebesseljfunction helpful)•Plot the signalx′3(t) = 3(J0(6t)−J2(6t))2. A finite-difference derivative, as the name might imply, simply takes the standard limit definition of thederivative:x′(t) = lim∆t→0x(t+ ∆t)−x(t)∆tand uses a finite non-zero ∆t. In this problem, the ∆t=Tsthe sampling period. Then we can calculatethe finite difference derivative as a difference equation:d[n] =x[n+ 1]−x[n]∆tA problem arises dealing with the boundaries of the domain, wherex[n+ 1] doesn’t necessarily exist.We will simply avoid this problem by directly setting the value of the derivative to the known derivativefunctiond[N] =x′[N]. In a physical model, these boundary terms would be addressed by the boundaryconditions.(a) Find the finite difference derivatived1(t)≈x′1(t) of the sinusoidal signalx1(t). Plot the result, andcalculate the mean error, mean(|d1(t)−x′1(t)|).(b) Find the finite difference derivatived2(t)≈x′2(t) of the polynomial signalx2(t). Plot the result,and calculate the mean error, mean(|d2(t)−x′2(t)|).(c) Find the finite difference derivatived3(t)≈x′3(t) of the Bessel functionx3(t). Plot the result, andcalculate the mean error, mean(|d3(t)−x′3(t)|).Page 2

3. In order to perform the FFT spectral derivative, we first take the Fourier transform of the given signalX(Ω) =F{x(t)}. Then using the derivative property of the FFT, we can then write:F(Ω) =jΩX(Ω)The derivative is then found by simply performing the inverse Fourier transform (taking the real part,since we started with a real signal):f(t) =R{F−1{F(Ω)}}(a) Find the FFT spectral derivativef1(t)≈x′1(t) of the sinusoidal signalx1(t). Plot the result, andcalculate the mean error, mean(|f1(t)−x′1(t)|).(b) Find the FFT spectral derivativef2(t)≈x′2(t) of the polynomial signalx2(t). Plot the result, andcalculate the mean error, mean(|f2(t)−x′2(t)|).(c) Find the FFT spectral derivativef3(t)≈x′3(t) of the Bessel functionx3(t). Plot the result, andcalculate the mean error, mean(|f3(t)−x′3(t)|)