Two 4-ray signal constellations are shown. It is given that $${\phi _1}$$ and $${\phi _2}$$ constitute an orthonormal basis for the two constellations. Assume that the four symbols in both the constellations are equiprobable. Let $${{{N_0}} \over 2}$$ denote the power spectral density of white Gaussian noise.

If these constellations are used for digital communications over an AWGN channel, then which of the following statements is true?

Let $$g\left( t \right){\mkern 1mu} {\mkern 1mu} \,\,\,\,\,{\mkern 1mu} = {\mkern 1mu} {\mkern 1mu} p\left( t \right){}^ * p\left( t \right)$$ where $$ * $$ denotes convolution and $$p(t) = u(t) - u(t-1)$$ with $$u(t)$$ being the unit step function

The impulse response of filter matched to the signal $$s(t) = g(t)$$ $$ - \delta {\left( {t - 2} \right)^ * }\,\,g\left( t \right)$$ is given as:

In the following figure the minimum value of the constant “C”, which is to be added to y₁(t) such that y₁(t) and y₂(t) are different, is

A symmetric three-level midtread quantizer is to be designed assuming equiprobable occurrence of all quantization levels.

If the input probability density function is divided into three regions as shown in figure, the value of 'a' in the figure is