A source transmits symbol  that takes values uniformly at random from the set . The receiver obtains , where  is a zero-mean Gaussian random variable independent of . The receiver uses the maximum likelihood decoder to estimate the transmitted symbol .

Suppose the probability of symbol estimation error  is expressed as follows:

where  denotes the probability that  exceeds 1.

What is the value of  ?

A source transmits a symbol , taken from  with equal probability, over an additive white Gaussian noise channel. The received noisy symbol  is given by , where the noise  is zero mean with variance 4 and is independent of .

Using , the optimum symbol error probability is_______.

A full scale sinusoidal signal is applied to a 10-bit ADC. The fundamental signal component in the ADC output has a normalized power of 1W, and the total noise and distortion normalized power is . The effective number of bits (rounded off to the nearest integer) of the ADC is ________.

A digital communication system transmits through a noiseless bandlimited channel . The received signal  at the output of the receiving filter is given by  where  are the symbols and  is the overall system response to a single symbol. The received signal is sampled at . The Fourier transform of  is . The Nyquist condition that  must satisfy for zero inter symbol interference at the receiver is _______.