Let g(t) = p(t) * p(t), where * denotes convolution and p(t) = u(t) - (t-1) with u(t) being the unit step function. The impulse response of filter matched to the singal s(t) = g(t) - $$[\delta (t - 2)*g(t)]$$ is given as

The output y(t) of a linear time invariant system is related to its input x(t) by the following equation: y(t) = 0.5 x $$(t - {t_d} + T) + \,x\,(t - {t_d}) + 0.5\,x(t - {t_d} - T)$$. The filter transfer function $$H(\omega )$$ of such a system is given by

A causal system having the transfer function H(s) = $${1 \over {s + 2}}$$, is excited with 10 u(t). The time at which the output reaches 99% of its steady state value is

A rectangular pulse train s(t) as shown in Fig.1 is convolved with the signal $${\cos ^2}$$ ($$4\pi \,{10^{3\,}}$$t). The convolved signal will be a