$s(t)$ is a multi-level (-1, 0, +1, +2) signal representing a long sequence of random symbols from the above source which is generating $10^4$ symbols per second.
Which of the following options is the correct value of equivalent Nyquist bandwidth of $s(t)$?
Nyquist Bandwidth Calculation for Digital Source
The problem asks for the equivalent Nyquist bandwidth of a signal $s(t)$ generated by a discrete memoryless source.
Key information provided:
- Source alphabet size: 4 symbols.
- Signal levels: {-1, 0, +1, +2}.
- Source symbol generation rate: $R_s = 10^4$ symbols per second.
Nyquist Bandwidth Concept
The Nyquist bandwidth is the minimum theoretical bandwidth required to transmit a digital signal without inter-symbol interference (ISI). It is directly related to the symbol rate ($R_s$) of the source.
The formula relating Nyquist bandwidth ($B_{Nyquist}$) and symbol rate ($R_s$) is:
$B_{Nyquist} = \frac{R_s}{2}$This formula indicates that the required bandwidth is half the symbol rate.
Bandwidth Calculation
Given the symbol rate $R_s = 10^4$ symbols/sec:
- Substitute the value of $R_s$ into the formula: $B_{Nyquist} = \frac{10^4 \text{ symbols/sec}}{2}$
- Calculate the result: $B_{Nyquist} = 5 \times 10^3 \text{ Hz}$
- Convert the bandwidth to kilohertz (kHz): $B_{Nyquist} = 5 \text{ kHz}$
The equivalent Nyquist bandwidth of the signal $s(t)$ is 5 kHz.
Conclusion
Comparing the calculated bandwidth with the given options:
- Option 1: 10 kHz
- Option 2: 64 kHz
- Option 3: 5 kHz
- Option 4: 20 kHz
The calculated value of 5 kHz matches Option 3.
Important Questions from Information Theory
For any binary (n, h) linear code with minimum distance (2t + 1) or greater \(n - h \ge {\log _2}\left[ {\mathop \sum \limits_{i = 0}^α \left( {\begin{array}{*{20}{c}} n\\ i \end{array}} \right)} \right]\) where α is:
The main processing functions of information system are given below. Arrange them in sequencing order:
(A) Process transaction
(B) Maintain master file
(C) Process enquiry
(D) Process report
(E) Process interactive supper applications
Choose the correct answer from the options given below:
The random variable $X$ takes values in $\{-1, 0, 1\}$ with probabilities $P(X = -1) = P(X = 1)$ and $\alpha$ and $P(X = 0) = 1 - 2\alpha$, where $0 < \alpha < \frac{1}{2}$. Let $g(\alpha)$ denote the entropy of $X$ (in bits), parameterized by $\alpha$. Which of the following statements is/are TRUE?
A source transmits symbols from an alphabet of size 16. The value of maximum achievable entropy (in bits) is _______
An analog baseband signal, bandlimited to 100 Hz, is sampled at the Nyquist rate. The samples are quantized into four message symbols that occur independently with probabilities $p_1 = p_4 = 0.125$ and $p_2 = p_3$. The information rate (bits/sec) of the message source is ____________
