Consider the common emitter amplifier shown below with the following circuit parameters:

$$\beta = 100,\,{g_m} = 0.3861\,{\rm A}/V,\,{r_0} = \infty ,\,{r_\pi } = 259\,\Omega, $$
$${R_s} = 1\,K\Omega ,{R_B} = 93\,K\Omega ,\,{R_C} = 250\,\Omega, $$
$${R_L} = 1\,K\Omega ,\,{C_1} = \infty \,\,and\,\,{C_2} = 4.7\,\mu F.$$

The lower cut-off frequency due to C₂ is

The transfer characteristic for the precision rectifier circuit shown below is (assume ideal OP-AMP and practical diodes)

Assuming the OP-AMP to be ideal, the voltage gain of the amplifier shown below is

Consider a base band binary PAM receiver shown below. The additive channel noise $$n(t)$$ is white with power spectral density $${S_N}\left( f \right) = {N_0}/2 = {10^{ - 20}}$$ $$W/Hz$$. The low-pass filter is ideal with unity gain and cut -off frequency $$1MHz$$. Let $${Y_k}$$ represent the random variable $$y\left( {{t_k}} \right)$$.
$${Y_k} = {N_k}$$ if transmitted bit $${b_k} = 0$$
$${Y_k} = a + {N_k}$$ if transmitted bit $${b_k} = 1$$
Where $${b_k} = 0$$ represents the noise sample value. The noise sample has a probability density function, $${P_{{N_k}}}\left( n \right)\,\,\,\,\,\,\, = 0.5\alpha {e^{ - \alpha \left| n \right|}}$$ (This has mean zero and variance $$2/{\alpha ^2}$$). Assume transmitted bits to be equiprobable and threshold $$z$$ is set to $$a/2 = {10^{ - 6}}V$$.

The value of the parameter $$\alpha $$( in V^-1 ) is