A binary random variable X takes the value of 1 with probability 1/3. X is input to a casade of 2 independent identical binary symmetric channels (BSCs) each with crossover probability 1/2. The output of BSCs are the random variables $${Y_1}$$ and $${Y_2}$$ as shown in the figure.

Let $$X(t)$$ be a wide sense stationary $$(WSS)$$ random procfess with power spectral density $${S_x}\left( f \right)$$. If $$Y(t)$$ is the process defined as $$Y(t) = X(2t - 1)$$, the power spectral density $${S_y}\left( f \right)$$ is .

A real band-limited random process $$X( t )$$ has two -sided power spectral density $$${S_x}\left( f \right) = \left\{ {\matrix{ {{{10}^{ - 6}}\left( {3000 - \left| f \right|} \right)Watts/Hz} & {for\left| f \right| \le 3kHz} \cr 0 & {otherwise} \cr } } \right.$$$

Where f is the frequency expressed in $$Hz$$. The signal $$X( t )$$ modulates a carrier cos $$16000$$ $$\pi t$$ and the resultant signal is passed through an ideal band-pass filter of unity gain with centre frequency of $$8kHz$$ and band-width of $$2kHz$$. The output power (in Watts) is ______.

In a PCA system, the signal $$m(t) = \{ \sin (100\,\pi \,t)\, + \cos (100\,\pi \,t\} $$ V is sampled at the Nyquist rate. The samples are processed by a uniform quantizer with step size 0.75 V. The minimum data rate of the PCM system in bits per second is________________