The impulse response function of four linear system S1, S2, S3, S4 are given respectively by

$${h_1}$$(t), = 1;

$${h_2}$$(t), = U(t);

$${h_3}(t)\, = \,{{U(t)} \over {t + 1}}$$;

$${h_4}(t)\, = {e^{ - 3t}}U(t)$$ ,

where U (t) is the unit step function. Which of these system is time invariant, causal, and stable?

The Fourier transform $$G(\omega )$$ of the signal g(t) in Fig.(1) is given as
$$G(\omega ) = {1 \over {{\omega ^2}}}({e^{j\omega }} - j\omega {e^{j\omega }} - 1)$$.

Using this information, and the time-shifting and time-scaling properties, determine and Fourier transform of signals in Fig (2), (3) and (4).

A band limited signal is sampled at the Nyquist rate. The signal can be recovered by passing the samples through

A base band signal g(t) band limited to 100 Hz modulates a carrier of frequency $${{f_0}}$$ Hz. The modulated signal g(t) $$\cos 2\,\pi \,{f_0}t$$ is transmitted over a channel whose input x and output y are related by $$y = 2x + {x^2}$$. The spectrum of g(t) is shown in Fig. Sketch the spectrum of the transmitted signal and the spectrum of the received signal.