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.

The input to a 1-bit quantizer is a random variable X with pdf $${f_x}(x) = 2{e^{ - 2x}}\,\,for\,\,x \ge 0$$ and $${f_x}(x) = 0\,\,for\,\,x\, < \,0$$. For outputs to be of equal probability, the quantizer threshold should be _______________.

Four messages band limited to W, W, 2 W and 3 W respectively are to be multiplexed using Time Division Multiplexing (TDM). The minimum bandwidth required for transmission of this TDM signal is

A speech signal, band limited to 4 kHz and peak voltage varying between + 5V and - 5V is sampled at the Nyquist rate. Each sample is quantized and represented by 8 bits.

Assuming the signal to be uniformly distributed between its peak values, the signal to noise ratio at the quantizer output is