The average bit error rate at the input of a $(7,4,1)$ Hamming decoder is 0.10 . The probability that the decoder will fail to decode a received word correctly is $\_\_\_\_$ . (rounded off to two decimal places)

The information bit sequence {1 1 1 0 1 0 1 0 1} is to be transmitted by encoding with Cyclic Redundancy Check 4 (CRC-4) code, for which the generator polynomial is $C(x) = x^4 + x + 1$. The encoded sequence of bits is ____.

Consider a channel over which either symbol x_A or symbol x_B is transmitted. Let the output of the channel Y be the input to a maximum likelihood (ML) detector at the receiver. The conditional probability density functions for y given x_A and x_B are :

$${f_{\left. Y \right|{x_A}}}(y) = {e^{ - (y + 1)}}u(y + 1)$$,

$${f_{\left. Y \right|{x_B}}}(y) = {e^{(y - 1)}}(1 - u(y - 1))$$,

where, u( . ) is the standard unit step function. The probability of symbol error for this system is _________ (rounded off to two decimal places).

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 $\_\_\_\_$ .