Consider a communication scheme where the binary valued signal X satisfies P{X = + 1} = 0.75 and P {X = - 1} = 0.25. The received signal Y = X + Z, where Z is a Gaussian random variable with zero mean and variance $${\sigma ^2}$$. The received signal Y is fed to the threshold detector. The output of the threshold detector $${\hat X}$$ is: $$$\hat X:\left\{ {\matrix{ { + \,1,} & {Y\, > \tau } \cr { - \,1,} & {Y\, \le \,\,\tau .} \cr } } \right.$$$ To achieve a minimum probability of error $$P\{ \hat X\, \ne \,X\} $$, the threshold $$\tau $$ should be

Consider a discrete-time channel Y = X + Z, where the additive noise Z is signal- dependent. In particular, given the trasmitted symbol $$X\, \in \,\{ \, - \,a,\,\, + \,a\} $$ at any instant, the noise sample Z is chosen independently from a Gaussian distribution with mean $$\beta X$$ and unit variance. Assume a threshold detector with zero threshold at the receiver. When $$\beta $$ = 0 the BER was found to be $$Q\,(a) = 1\, \times \,{10^{ - 8}}$$. $$\left( {Q\,\,(v)\, = {1 \over {\sqrt {2\,\pi } }}\,\int\limits_v^\infty {{e^{ - {u^2}/2}}} } \right.$$ du, and for v > 1,
use $$Q\,(v) \approx \,{e^{ - {v^2}/2}}$$
When $$\beta = - \,0.3,\,$$ the BER is closed to

Coherent orthogonal binary FSK modulation is used to transmit two equiprobable symbol waveforms $${s_1}\,(t)\, = \,\alpha \,\,\cos \,\,\,2\,\pi {f_1}\,t\,and\,\,{s_{2\,}}(t)\,\, = \,\alpha \,\,\cos \,\,\,2\,\pi {f_2}\,t$$, where $$\,\alpha = 4\,\,\,mV$$. Assume an AWGN channel with two-sided noise power spectral density $$\,{{{N_0}} \over 2} = 0.5\,\, \times \,{10^{ - 12}}$$ W/Hz. Using an optimal receiver and the relation $$Q(v) = {1 \over {\sqrt {2\,\pi } }}\,\int\limits_v^\infty {e{\,^{ - {u^2}/2}}} \,du$$, the bit error probability for a data rate of 500 kbps is

Let U and V be two independent zero mean Gaussian random variables of variances $${{1 \over 4}}$$ and $${{1 \over 9}}$$ respectively. The probability $$P(\,3V\, \ge \,\,2U)$$ is