LIST-I LIST-II A. Sinc $(\lambda)$ I. White Noise B. Power spectral density is independent of operating frequency II. Knowledge of a probabilistic model of the source C. M-ary PAM System III. 1 baud = $\log_2 M$ bits/sec D. Huffman code requires IV. $\frac{\sin (\pi\lambda)}{\pi\lambda}$
Choose the correct answer from the options given below:
Understanding the Matching Question: LIST-I and LIST-II
This question requires matching concepts from LIST-I with their corresponding descriptions or definitions in LIST-II. These concepts are related to digital signal processing and communication systems.
Analysis of LIST-I Items and Matching with LIST-II
A. Matching Sinc Function $(\lambda)$
The Sinc function is a fundamental function in signal processing, often defined mathematically. Let's examine the options in LIST-II.
- Option IV provides the mathematical expression: $\frac{\sin (\pi\lambda)}{\pi\lambda}$. This is the standard definition of the normalized sinc function.
Therefore, Sinc$(\lambda)$ correctly matches with Option IV.
B. Matching White Noise Property
LIST-I mentions a characteristic related to "Power spectral density is independent of operating frequency". This is a defining property of a specific type of noise.
- Option I defines White Noise. White noise is characterized by having a constant power spectral density across all frequencies, meaning it contains equal power within any frequency band of width equal to that of the noise power within another frequency band.
Thus, the property "Power spectral density is independent of operating frequency" matches with White Noise (Option I).
C. Matching M-ary PAM System
An M-ary PAM System is a type of digital modulation where symbols are represented by M distinct amplitude levels. The efficiency of such a system is related to how much information each symbol carries and the rate at which symbols are transmitted (baud rate).
- Option III states: 1 baud = $\log_2 M$ bits/sec. While not a direct definition of baud rate itself, this option relates the concept of an M-ary system to information content. In an M-ary system, each symbol carries $\log_2 M$ bits of information. If the symbol rate is measured in baud, then the bit rate is calculated as Baud Rate $\times \log_2 M$. This option likely refers to the information capacity per symbol transmission interval.
Therefore, M-ary PAM System is associated with the concept in Option III.
D. Matching Huffman Code Requirement
Huffman code is a widely used algorithm for lossless data compression.
- Option II states: Knowledge of a probabilistic model of the source. Huffman coding works by assigning shorter codes to more frequent symbols and longer codes to less frequent symbols. To do this effectively, the algorithm requires prior knowledge of the probabilities or frequencies of occurrence for each symbol in the source data.
Hence, Huffman code requires matches with Knowledge of a probabilistic model of the source (Option II).
Summary of Matches
Based on the analysis, the correct pairings are:
| LIST-I Item | LIST-II Item | Explanation |
| A. Sinc$(\lambda)$ | IV. $\frac{\sin (\pi\lambda)}{\pi\lambda}$ | Mathematical definition of the sinc function. |
| B. Power spectral density is independent of operating frequency | I. White Noise | Defining characteristic of white noise. |
| C. M-ary PAM System | III. 1 baud = $\log_2 M$ bits/sec | Relates symbol information capacity ($\log_2 M$ bits) in M-ary systems to transmission concepts. |
| D. Huffman code requires | II. Knowledge of a probabilistic model of the source | Essential input for constructing Huffman codes. |
Therefore, the correct combination is A-IV, B-I, C-III, D-II.
Important Questions from Probability Distribution
If the mean and variance of a binomial distribution are 5 and 4, respectively, then the value of n is:
For the distribution with unknown θ
\(f(x,\theta ) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{\theta };0 \le x \le \theta }\\ {0;elsewhere} \end{array}} \right.\)
We set the testing of hypothesis H 0 ∶ θ = 1 vs H 1 ∶ θ = 2. When the critical region X ≥ 0.4, the value of probability of type-II error is:
For the cumulative distribution function \(F(x) = \left\{ {\begin{array}{*{20}{c}} {0;x < - 1}\\ {\frac{1}{2}{{(x + 1)}^2}; - 1 \le x < 0}\\ {1 - \frac{{{{(1 - x)}^2}}}{2};0 \le x < 1}\\ {1.1 < x < \infty } \end{array}} \right.\)
the upper quartile point is
Let the joint probability density function of \( (X, Y) \) be
\[f(x, y) = \begin{cases} 6xy^2 & \text{if } 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases}\]
Then \( P\left(\frac{1}{2} < X < \frac{3}{4}\right) \) is:
Let X and Y have the joint p.m.f. f(x, y) = x + y / 21, where x = 1, 2, 3 and y = 1, 2. The marginal p.m.f. of X is:
