In Fig. 1, a linear time invariant discrete system is shown. Blocks labeled D represent unit delay elements. For $$n\, < 0,$$ you may assume that $$x\left( n \right),$$ $${y_1}\left( n \right),\,\,{y_2}\left( n \right)$$ are all zero.

(a) Find the expression for $${y_1}\left( n \right)$$ and $${y_2}\left( n \right)$$ in terms of $$x\left( n \right).$$
(b) Find the transfer function $${y_2}\left( z \right)/X\left( z \right)$$ in the $$z$$-domain.
(c) If $$x\left( n \right) = 1$$ at $$n = 0$$ or $$x\left( n \right) = 0$$ otherwise

Find $${y_2}\left( n \right).$$

The power spectral density of a deterministic signal is given by $${\left[ {\sin (f)/f} \right]^2}$$, where 'f' is frequency. The autocorrelation function of this signal in the time domain is

Match each of the items 1, 2 on the left with the most appropriate item A, B, C or D on the right.

In the case of a linear time invariant system

List - 1
(1) Poles in the right half plane implies.
(2) Impulse response zero for $$t \le 0$$ implies.

List - 2
(A) Exponential decay of output
(B) System is causal
(C) No stored energy in the system
(D) System is unstable

The function f(t) has the Fourier Transform g($$\omega $$). The Fourier Transform of $$$g(t) = \left( {\int\limits_{ - \infty }^\infty {g(t){e^{ - j\omega t}}} } \right)\,is$$$