DIGITAL COMMUNICATION – EEC601
B.Tech (SEM VI) | Section-wise Solved Answers
SECTION – A
(10 × 2 = 20 marks)
(a) Distribution function of random variable X
A trial consists of four successive draws of cards. Let X denote the number of kings appearing. Since there are 4 kings in 52 cards, X follows a binomial distribution. The probability mass function and cumulative distribution function Fx(x) can be obtained using binomial probability.
(b) Autocorrelation of white noise
The autocorrelation function of white noise is a delta function. This indicates that white noise samples are uncorrelated at different time instants.
(c) Condition for orthogonality of two signals
Two signals are orthogonal if the integral of their product over a given time interval is zero.
(d) Essential bandwidth of polar NRZ
The essential bandwidth of polar NRZ line code is approximately Rb/2, where Rb is the bit rate.
(e) Slow hopping vs fast hopping
In slow hopping, several symbols are transmitted per hop. In fast hopping, the hop rate is faster than symbol rate, and one symbol may span multiple hops.
(f) Spectrum occupancy of PSK and FSK
FSK requires larger bandwidth compared to PSK. PSK is spectrally more efficient.
(g) Bayes’ rule of probability
Bayes’ rule relates conditional probabilities and helps update the probability of a hypothesis based on new evidence.
(h) Advantages of Manchester coding
Manchester coding provides self-synchronization and eliminates DC component, making clock recovery easier.
(i) Manchester representation
Manchester coding represents logical 1 by low-to-high transition and logical 0 by high-to-low transition within a bit period.
(j) BPSK modulator
In BPSK, the carrier phase is shifted by 180° to represent binary symbols 0 and 1. It is robust against noise.
SECTION – B
(Attempt any five – 10 marks each)
(a) PSD of polar random signal
The power spectral density of a polar random signal is derived using autocorrelation function and Fourier transform, considering independent and equally likely bits.
(b) Frequency Hopping Spread Spectrum (FHSS)
FHSS changes carrier frequency according to a pseudo-random sequence.
Applications: Military communication, Bluetooth, secure wireless systems.
Processing gain (in dB) is calculated using hop parameters.
(c) Probability of error for BPSK
For BPSK in white noise, probability of bit error is
Pb=Q(2EbN0)P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)Pb=Q(N02Eb)
Using given data rate and noise PSD, transmission bandwidth and required signal power are calculated.
(d) Error probability in binary channel
The probability that no more than four errors occur in 100 received digits is calculated using binomial distribution.
(e) Source entropy and coding
Entropy is calculated using
H=−∑pilog2piH = -\sum p_i \log_2 p_iH=−∑pilog2pi
A compact binary code is designed, and average length, efficiency, and redundancy are evaluated.
(f) Convolution codes
Convolution codes add redundancy using shift registers.
Advantages: Error correction capability
Disadvantages: Complex decoding
(g) Chebyshev’s inequality
Chebyshev’s inequality provides bounds on probability that a random variable deviates from its mean.
(h) PN sequence
A PN sequence is generated using linear feedback shift register.
Its autocorrelation resembles impulse-like behavior.
Length, chip duration, and period are calculated from chip rate and number of stages.
SECTION – C
(Attempt any two – 15 marks each)
(3) Principle of OFDM
OFDM divides a high-rate data stream into multiple lower-rate streams transmitted on orthogonal subcarriers, reducing ISI and improving bandwidth efficiency.
(4)
(i) Matched Filter
A matched filter maximizes signal-to-noise ratio in the presence of noise.
(ii) Probability of error
Matched filter output is equivalent to correlator receiver. The probability of error is derived based on noise statistics.
(5) Short Notes
(i) Central Limit Theorem
States that sum of large number of independent random variables tends to follow normal distribution.
(ii) Error Correcting Codes
Used to detect and correct errors in transmitted data, improving reliability.
