(a) In this instance, you will learn to use matlab's FFT facilities1, a very useful function. The FFT, or fast Fourier Transform is an O(Nlog(N)), algorithm for the calculation of the discrete Fourier transform (DFT) (or a finite discrete Fourier series). The DFT is defined as the pair i2T(k-1 n-1 and i2T(k-1)(-n For a fun demo that uses the fft library, type sunspots 2 where the second expression is the inverse DFT. Here, both x and X may be complex. The alert reader may notice that the indices do not seem to correspond to the traditional DFT definition (examine the inverse transform expression). There are actually more differ- ences between the DFT and the FFT, differences which are beyond not seem to correspond to the traditional DFT definition (examine the inverse transform expression). There are actually more differ- ences between the DFT and the FFT, differences which are beyond the scope of this course. Here we will learn of a way to make the correct correspondence between the FFT's and the DFT's spectral components, and how to scale any arbitrary domain to the [0, 2T space domain used by the code. What you will find is that the spec- tral components -N/2 + 1,-N/2 + 2,...,-1 are actually stored in locations N/2+1, N/2 +2...,N. Descrambling this is crucial The other thing that you will find is that there is a simple linear map between the domain [0, L], say, and [0,2. In the following exercise you'll learn a little trick that will help you determine how matlab's fft package, or any other fft package, is used correctly. The idea is the following: conjure up an arbitrary function f(t) over 0