(Solved): A raised-cosine pulse has a narrower amplitude spectrum than a square pulse, which makes it useful ...

A raised-cosine pulse has a narrower amplitude spectrum than a square pulse, which makes it useful in digital telecommunications. The raised-cosine pulse is defined as $s(t)=\frac{1+\cos \left(\frac{2\pi t}{T}\right)}{2}\mathrm{rect}\left(\frac{t}{T}\right).$ Below there are two questions. 1. Find the expression for the Fourier transform $S(f)$ of the pulse $s(t)$. Remember to consult the equation sheet for the terms of $s(t)$. (max. 15 points) 2. Find the energy spectrum $|S(f){|}^2$. (max. 5 points) Write your solutions into the Octave code below. See section below to get your parameters. Listing 29 Energy spectrum of a raised-cosine pulse $1\mathrm{f}=-10:0.01:10$; $2\%$ Write below your expression for the Fourier transform $S(f)$ $3S=2\ast 4/3\ast \cos (5\ast f)+6\ast \mathrm{sinc}(f/4);\%$ replace this line with your solution 4 $5\%$ Write below your expression for the energy spectrum $|S(f){|}^{\wedge }2$. You can use $6\%$ the Fourier transform $S$ which you defined above. 7 S_energyspectrum $=\sin (2\ast f)+0.5\ast S$; $\%$ replace this line with your solution 8 9 subplot $(2,1,1)$ $10\mathrm{plot}(f,S)$ 11 xlabel(' $f(Hz)$ '); legend('Fourier transform $S(f)$ '); 12 subplot $(2,1,2)$ 13 plot(f, 10*log10(S_energyspectrum)); 14 xlabel(' $f(Hz{)}^{\prime })$; ylabel('dB'); legend('Energy spectrum $|S(f)|\wedge 2$ ');