A binary symmetric channel (BSC) has a transition probability of $$\frac{1}{8}.$$  If the binary transmit symbol X is such that $$P\left( {X = 0} \right) = \frac{9}{{10}},$$    then the probability of error for an optimum receiver will be

Choose the correct one from among the alternatives a, b, c, dafter matching an item from Group 1 with the most appropriate item in Group 2.
Group 1 Group 2
1. FM P. Slope overload
2. DM Q. H-law
3. PSK R. Envelope detector
4. PCM S. Capture effect
T. Hilbert transform
U. Matched filter

A source emits bit 0 with probability $$\frac{1}{3}$$  and bit 1 with probability $$\frac{2}{3}.$$  The emitted bits are communicated to the receiver. The receiver decides for either 0 or 1 based on the received value R. It is given that the conditional density functions of R as
\[\begin{gathered} {f_{R|0}}\left( r \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{4},}&{ - 3 \leqslant r \leqslant 1} \\ {0,}&{{\text{otherwise;}}} \end{array}} \right. \hfill \\ {f_{R|1}}\left( r \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{6},}&{ - 1 \leqslant r \leqslant 5} \\ {0,}&{{\text{otherwise;}}} \end{array}} \right. \hfill \\ \end{gathered} \]
The minimum decision error probability is

The minimum sampling frequency (in samples/sec) required to reconstruct the following signal from its samples without distortion
$$x\left( t \right) = 5{\left( {\frac{{\sin 2\pi 1000t}}{{\pi t}}} \right)^3} + 7{\left( {\frac{{\sin 2\pi 1000t}}{{\pi t}}} \right)^2}$$
would be

The number of bits in a binary PCM system is increased from n to n + 1. As a result, the signal to quantization noise ratio will improve by a factor

A signal is sampled at 8 kHz and is quantized using 8-bit uniform quantizer. Assuming SNRq for a sinusoidal signal, the correct statement for PCM signal with a bit rate of R is

The bit stream 01001 is differentially encoded using 'Delay and Ex-OR' scheme for DPSK transmission. Assuming the reference bit as a '1' and assigning phases of '0' and 'π' for 1's and 0's respectively, in the encoded sequence, the transmitted phase sequence becomes

For a given data rate, the bandwidth Bp of a BPSK signal and the bandwidth B0 of the OOK signal are related as

A sinusoidal signal with peak-to-peak amplitude of 1.536 V is quantized into 128 levels using a mid-rise uniform quantizer. The quantization noise power is