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.

Specify the filter type if its voltage transfer function H(s) is given by

H(s) = $${{K({s^2} + {\omega _0}^2)} \over {{s^2} + ({\omega _0}/Q)s + {\omega _0}^2}}$$

A signal containing only two frequency components (3 kHz and 6 kHz) is sampled at the rate of 8 kHz, and then passed through a low pass filter with a cutoff frequency of 8 kHz. The filter output

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 $$\int\limits_{ - \infty }^\infty {{{[y(t) - x(t)]}^{2\,}}dt} $$.