DIGITAL COMMUNICATION (KEC601)
Section-wise Detailed Answers – University Exam Oriented
SECTION A
(Attempt all questions – descriptive explanations)
Q1(a) A die is thrown. Determine the probability that an even number comes up
When a fair die is thrown, the sample space consists of six equally likely outcomes, namely 1, 2, 3, 4, 5, and 6. Among these outcomes, the even numbers are 2, 4, and 6. Since there are three even outcomes out of a total of six possible outcomes, the probability of getting an even number is equal to three divided by six. Hence, the probability is 1/2.
Q1(b) Define random variable
A random variable is a numerical quantity whose value depends on the outcome of a random experiment. Instead of describing outcomes in words, a random variable assigns a real number to each possible outcome of the experiment. Random variables are mainly classified as discrete or continuous depending on whether they take countable values or values over a continuous range.
Q1(c) Explain inter-symbol interference
Inter-symbol interference, commonly known as ISI, occurs in digital communication systems when symbols overlap with each other due to channel distortion or limited bandwidth. Because of this overlap, the receiver finds it difficult to distinguish between consecutive symbols. ISI degrades system performance and increases the probability of error. It usually arises due to multipath propagation or improper pulse shaping.
Q1(d) NRZ-Unipolar waveform for 101101
In NRZ-unipolar coding, a binary ‘1’ is represented by a positive voltage level, while a binary ‘0’ is represented by zero voltage. For the data sequence 101101, the waveform alternates between high voltage for ‘1’ and zero voltage for ‘0’ without returning to zero between consecutive ones. This coding scheme is simple but suffers from DC component and synchronization problems.
Q1(e) Bandwidth of ideal binary ASK with data rate 64 kbps
In ideal binary Amplitude Shift Keying (ASK), the required bandwidth is approximately equal to the data rate. Given a data rate of 64 kbps, the bandwidth required is also approximately 64 kHz. This assumes ideal filtering and no excess bandwidth.
Q1(f) Applications of ASK modulation
ASK modulation is commonly used in optical communication systems, such as fiber-optic transmission, where the presence or absence of light represents digital data. It is also used in low-cost wireless applications like RFID systems and remote keyless entry systems. ASK is simple to implement but is highly sensitive to noise.
Q1(g) Property of a matched filter
A matched filter is designed to maximize the signal-to-noise ratio at the output in the presence of additive white Gaussian noise. One important property is that its impulse response is the time-reversed and delayed version of the input signal. This ensures optimal detection of signals in noisy environments.
Q1(h) Define bit error rate
Bit Error Rate (BER) is defined as the ratio of the number of erroneous bits received to the total number of bits transmitted over a communication channel. It is a key performance metric used to evaluate the quality and reliability of digital communication systems.
Q1(i) Explain information
Information is a measure of uncertainty reduction when a message is received. In digital communication, information quantifies how much knowledge is gained when a particular symbol is transmitted. The information content of a message is inversely proportional to its probability of occurrence. Rare events carry more information than common events.
Q1(j) Entropy of messages with probabilities 1/2, 1/4, 1/4
Entropy is the average information content of a source and is given by
H = − Σ p log₂ p
Substituting the given probabilities:
H = −[(1/2)log₂(1/2) + (1/4)log₂(1/4) + (1/4)log₂(1/4)]
This simplifies to:
H = (1/2)(1) + (1/4)(2) + (1/4)(2)
H = 0.5 + 0.5 + 0.5
H = 1.5 bits
Thus, the entropy of the source is 1.5 bits.
SECTION B
(Attempt any three – explained in detail)
Q2(a) Probability density function fx(x) = a e⁻ᵇ|x|
Since fx(x) is a probability density function, the total area under the curve must be equal to one. Integrating fx(x) over the range −∞ to +∞ and equating it to unity gives the relationship between constants a and b.
The autocorrelation function describes how a random variable correlates with itself at different time shifts. For symmetric exponential distributions, autocorrelation decreases exponentially with increasing lag. This type of distribution is commonly used to model noise processes.
Q2(c) PSK modulation and comparison between BPSK and DPSK
Phase Shift Keying (PSK) is a digital modulation technique in which the phase of the carrier signal is varied according to the digital data. In Binary PSK (BPSK), the carrier phase shifts between 0 and π radians depending on whether the transmitted bit is 1 or 0.
At the receiver, coherent detection is used to recover the data by comparing the received signal with a locally generated carrier.
Differential PSK (DPSK) avoids the need for carrier phase synchronization by encoding data as phase differences between successive symbols. While BPSK provides better error performance, DPSK offers simpler receiver design at the cost of slightly higher error probability.
Q2(d) Matched filter and its impulse response
A matched filter is an optimal linear filter used in digital receivers to maximize signal-to-noise ratio. Its impulse response is designed to match the shape of the transmitted signal. Mathematically, the impulse response is proportional to the time-reversed version of the signal delayed by the symbol duration.
This property ensures that the output peak occurs at the correct sampling instant, allowing accurate symbol detection even in noisy environments.
Q2(e) Entropy and mutual information
Entropy represents the uncertainty associated with a random variable, while mutual information measures the amount of information shared between two random variables.
The relationship
H(X,Y) = H(X|Y) + H(Y)
shows that joint entropy is equal to conditional entropy plus the entropy of the conditioning variable. This relationship is fundamental in information theory and helps in analyzing communication system efficiency.
SECTION C
(Attempt any one – detailed explanation)
Q3(a) Difference between WSS and SSS random processes
A Strict Sense Stationary (SSS) random process has statistical properties that are invariant under time shift. This means all joint probability distributions remain unchanged with time.
A Wide Sense Stationary (WSS) process is less restrictive. It requires only that the mean is constant and the autocorrelation depends only on time difference, not absolute time.
If two random variables are added and one has zero mean, the mean of the sum equals the mean of the other variable. The variance of the sum is equal to the sum of individual variances if the variables are uncorrelated.
Q3(b) Gaussian random process and central limit theorem
A Gaussian random process is one in which all random variables have Gaussian distributions. Such processes are important because noise in communication systems is often modeled as Gaussian.
According to the Central Limit Theorem, the sum of a large number of independent random variables tends toward a Gaussian distribution, regardless of their original distributions. This explains why thermal noise in communication systems follows a Gaussian model.
