(Solved): (d) Use the Fourier transform synthesis equation to determine the inverse Fou ...

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(d) Use the Fourier transform synthesis equation to determine the inverse Fourier transform of (i) \( X_{1}(j \omega)=2 \pi \delta(\omega)+\pi \delta(\omega-4 \pi)+\pi \delta(\omega+4 \pi) \). (5 marks) (ii) \( X_{2}(j \omega)=\left|X_{2}(j \omega)\right| e^{j \angle X_{2}(j \omega)} \), where \( \left|X_{2}(j \omega)\right|=2\{u(\omega+3)-u(\omega-3)\}, \angle X_{2}(j \omega)=-\frac{3}{2} \omega+\pi \) (10 marks) (e) Find the Fourier transform of \( x_{3}(t)=\left(\frac{\sin t}{\pi t}\right)^{2} \) (10 marks) (f) Given that \( x(t) \) has the Fourier transform \( X(j \omega) \), express the Fourier transforms of the following signals in terms of \( X(j \omega) \). \[ x_{5}(t)=x(1-t)+x(3 t+6) \] (5 marks)