The output of a system is given in difference equation form as $$y\left( k \right) = \,a\,\,y\left( {k - 1} \right) + x\left( k \right),$$ where $$x\left( k \right)$$ is the input. If $$x\left( k \right)$$ $$\, = \,\,0$$ for $$k\, \ne \,0,\,\,x\left( 0 \right)\, = \,1,$$ and $$y\left( 0 \right)\, = \,0,$$ find $$y\left( k \right)$$ for all $$k.$$

Determine the range of $$'a'$$ for which $$y\left( k \right)\,$$ is bounded.

A signal x(t) = $$\exp ( - 2\pi Bt)\,u(t)$$ is the input to an ideal low pass filter with bandwidth B Hz. The output is denoted by y(t). Evaluate $$\int\limits_{ - \infty }^\infty {{{[y(t) - x(t)]}^{2\,}}dt} $$.

The Laplace transform of a function f(t)u(t), where f(t) is periodic with period T, is A(s) times the Laplace transform of its first period. Then