A binary random variable $X$ takes the value +2 or -2 . The probability $P(X=+2)=\alpha$. The value of $\alpha$ (rounded off to one decimal place), for which the entropy of $X$ is maximum, is $\_\_\_\_$ .

In a digital communication system, a symbol $S$ randomly chosen from the set $\left\{s_1, s_2, s_3, s_4\right\}$ is transmitted. It is given that $s_1=-3, s_2=-1, s_3=+1$ and $s_4=+2$. The received symbol is $Y=S+W . W$ is a zero mean unit - variance Gaussian random variable and is independent of $S . P_i$ is the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol $S=s_i$. The index $i$ for which the conditional symbol error probability $P_i$ is the highest is $\_\_\_\_$ .

$S_{P M}(t)$ and $S_{F M}(t)$ are defined below, are the phase modulated and the frequency modulated waveforms, respectively, corresponding to the message signal $m(t)$ shown in the figure.

$$ \begin{aligned} & S_{P M}(t)=\cos \left[1000 \pi t+k_p m(t)\right] \\ & S_{F M}(t)=\cos \left[1000 \pi t+k_f \int_{-\infty}^t m(\tau) d \tau\right] \end{aligned} $$

Where $k_p$ is the phase deviation constant in radians/volt and $k_f$ is the frequency deviation constant in radians/second/volt. If the highest instantaneous frequencies of $S_{P M}(t)$ and $S_{F M}(t)$ are same, then the value of the ratio $\frac{k_p}{k_f}$ is $\_\_\_\_$ seconds.

For the modulated signal $x(t)=m(t) \cos \left(2 \pi f_c t\right)$, the message signal $m(t)=4 \cos (1000 \pi t)$ and the carrier frequency $f_c$ is 1 MHz . The signal $x(t)$ is passed through a demodulator, as shown in figure below. The output $y(t)$ of the demodulator is