Let $X(t) = A\cos(2\pi f_0 t+\theta)$ be a random process, where amplitude $A$ and phase $\theta$ are independent of each other, and are uniformly distributed in the intervals $[-2,2]$ and $[0, 2\pi]$, respectively. $X(t)$ is fed to an 8-bit uniform mid-rise type quantizer.

Given that the autocorrelation of $X(t)$ is $R_X(\tau) = \frac{2}{3} \cos(2\pi f_0 \tau)$, the signal to quantization noise ratio (in dB, rounded off to two decimal places) at the output of the quantizer is _________.

A random variable X, distributed normally as N(0, 1), undergoes the transformation Y = h(X), given in the figure. The form of the probability density function of Y is

(In the options given below, a, b, c are non-zero constants and g(y) is piece-wise continuous function)

Let X(t) be a white Gaussian noise with power spectral density $$\frac{1}{2}$$W/Hz. If X(t) is input to an LTI system with impulse response $$e^{-t}u(t)$$. The average power of the system output is ____________ W (rounded off to two decimal places).

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, f(x), as shown in the figure.

Consider a 1 bit quantizer that maps positive samples to value $$\alpha$$ and others to value $$\beta$$. If $$\alpha$$* and $$\beta$$* are the respective choices for $$\alpha$$ and $$\beta$$ that minimize the mean square quantization error, then ($$\alpha$$* $$-$$ $$\beta$$*) = ___________ (rounded off to two decimal places).