Consider the following interconnection of the three LTI systems (Fig.1). $${h_1}(t)$$ , $${h_2}(t)$$ and $${h_3}(t)$$ are the impulse responses of these three LTI systems with $${H_1}(\omega )$$, $${H_2}(\omega )$$, and $${H_3}(\omega )$$ as their respective Fourier transforms. Given that $${h_1}\,(t)\, = \,{d \over {dt}}\left[ {{{\sin ({\omega _0}t)} \over {2\,\pi \,t}}} \right],{H_2}(\omega ) = \exp \left( {{{ - j2\pi \omega } \over {{\omega _0}}}} \right)$$
$${h_3}\,(t)\, = u(t)\,and\,x(t)\, = \,\sin \,2\,{\omega _0}t\, + \,\cos \,({\omega _0}t/2),$$ find the output y(t).

Sketch the waveform (with properly marked axes) at the output of a matched filter matched for a signal s(t), of duration T, given by $$s(t) = \left\{ {\matrix{ {A\,\,\,\,for} & {0 \le t < {2 \over 3}T} \cr {0\,\,\,\,\,\,for} & {{2 \over 3}T \le t < T} \cr } } \right.$$

Obtain an expression for the signal in figure, for the signal $${v_3}(t)$$ in Fig for $${v_1}(t) = 100\cos (2000\pi t) + 4\sin (200\pi t)$$. Assume that $${v_2}(t)$$=$${v_1}(t)$$+0.1 $$v_1^2(t)$$ and that the BPF is an ideal unity gain filter with pass band from 800 Hz to 1200 Hz.

A low pass signal m(t) band-limited to B Hz is sampled by a periodic rectangular pulse train, $${p_\tau }(t)$$ of period $${T_s}$$ = 1/(3B) sec. Assuming natural sampling and that the pulse amplitude and pulse width are A volts and 1/(30B) sec, respectively, obtain an expression for the frequency spectrum of the sampled signal $${m_s}$$(t)