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 _______________.

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________________

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.

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

The number of quantitization levels required to reduce the quantization noise by a factor of 4 would be