A deterministic signal x(t) = $$\cos (2\pi t)$$ is passed through a differentiator as shown in Figure.
(a) Determine the autocorrelation R_xx ($$\tau $$) and the power spectral density S_xx(f).
(b) Find the output power spectral density S_yy( f ).
(c) Evaluate R_xy(0) and R_xy(1/4).

If the impulse response of a discrete-time system is $$h\left[ n \right]\, = \, - {5^n}\,\,u\left[ { - n\, - 1} \right],$$ then the system function $$H\left( z \right)\,\,\,$$ is equal to

Consider a sampled signal $$y\left( t \right) = 5 \times {10^{ - 6}}\,x\left( t \right)\,\,\sum\limits_{n = - \infty }^{ + \infty } {\delta \left( {t - n{T_s}} \right)} $$

where $$x\left( t \right) = 10\,\,\cos \,\left( {8\pi \times {{10}^3}} \right)\,\,t$$ and
$${T_s} = 100\,\,\mu \sec .$$ When $$y\left( t \right)$$ is passed through an ideal low-pass filter with a cutoff frequency of 5 KHz, the output of the filter is

A linear phase channel with phase delay $${\tau _p}$$ and group delay $${\tau _g}$$ must have