# Match List I with List II: List I List II (A) Shannon's theorem (I) Capacity of Gaussian Noise channel (B) Shannon-Hartley theorem (II) Rate of Information (C) Bayes theorem (III) Energy of a signal (D) Parseval's theorem (IV) Conditional probabilities Choose the correct answer from the options given below:

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1. (A) - (II), (B) - (I), (C) - (III), (D) - (IV)
2. (A) - (II), (B) - (I), (C) - (IV), (D) - (III)
3. (A) - (III), (B) - (II), (C) - (IV), (D) - (I)
4. (A) - (II), (B) - (III), (C) - (I), (D) - (IV)

Option 2 : (A) - (II), (B) - (I), (C) - (IV), (D) - (III)
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## Detailed Solution

Shannon–Hartley theorem:

It states the channel capacity C, i.e. the theoretical highest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power S through an analog communication channel that is subject to additive white Gaussian noise (AWGN) of power N.

Mathematically, it is defined as:

$${\rm{C}} = {\rm{B\;lo}}{{\rm{g}}_2}\left( {{\rm{\;}}1 + \frac{{\rm{S}}}{{\rm{N}}}{\rm{\;}}} \right)$$

C = Channel capacity

B = Bandwidth of the channel

S = Signal power

N = Noise power

Bayes’ theorem

It states that the conditional probability of an event, based on the occurrence of another event, is equal to the likelihood of the second event given the first event multiplied by the probability of the first event.

Parseval's Theorem:

For continuous-time, periodic signal, the energy is given by:

$$\frac{1}{T}\int\limits_T {{{\left| {x(t)} \right|}^2}dt = {{\sum\limits_{k = - \infty }^{ + \infty } {\left| {{a_k}} \right|} }^2}}$$

Where ak is the Fourier series coefficient of x(t), and T is the period of the signal.

For average power in one period of the periodic signal x(t), we write:

$$\frac{1}{T}\int\limits_T {{{\left| {{a_k}{e^{jk{\omega _o}t}}} \right|}^2}dt = \frac{1}{T}{{\int\limits_T {\left| {{a_k}} \right|} }^2}} dt = {\left| {{a_k}} \right|^2}$$

$${\left| {{a_k}} \right|^2}$$ is the average power in the kth harmonic of x(t).

∴ Parseval's relation states that the total average power in a periodic signal equals the sum of the average powers in all of its harmonic components.
The text passage contains a multiple-choice question that requires matching items from List I to items in List II. The passage also provides a detailed explanation of the theorems and concepts mentioned in the question.
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the passage does not contain a mathematical question that requires a calculation or a word problem. instead, it provides a multiple-choice question and detailed explanations of the theorems and concepts.
Given a text passage, determine:
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    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
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# Match List I with List II: List I List II (A) Shannon's theorem (I) Capacity of Gaussian Noise channel (B) Shannon-Hartley theorem (II) Rate of Information (C) Bayes theorem (III) Energy of a signal (D) Parseval's theorem (IV) Conditional probabilities Choose the correct answer from the options given below:

This question was previously asked in
UGC NET Paper 2: Electronic Science 29 Oct 2022 Shift 1
View all UGC NET Papers >
1. (A) - (II), (B) - (I), (C) - (III), (D) - (IV)
2. (A) - (II), (B) - (I), (C) - (IV), (D) - (III)
3. (A) - (III), (B) - (II), (C) - (IV), (D) - (I)
4. (A) - (II), (B) - (III), (C) - (I), (D) - (IV)

Option 2 : (A) - (II), (B) - (I), (C) - (IV), (D) - (III)
Free
UGC NET Paper 1: Mock Test
11.3 K Users
50 Questions 100 Marks 60 Mins

## Detailed Solution

Shannon–Hartley theorem:

It states the channel capacity C, i.e. the theoretical highest upper bound on the information rate of data that can be communicated at an arbitrarily low error rate using an average received signal power S through an analog communication channel that is subject to additive white Gaussian noise (AWGN) of power N.

Mathematically, it is defined as:

$${\rm{C}} = {\rm{B\;lo}}{{\rm{g}}_2}\left( {{\rm{\;}}1 + \frac{{\rm{S}}}{{\rm{N}}}{\rm{\;}}} \right)$$

C = Channel capacity

B = Bandwidth of the channel

S = Signal power

N = Noise power

Bayes’ theorem

It states that the conditional probability of an event, based on the occurrence of another event, is equal to the likelihood of the second event given the first event multiplied by the probability of the first event.

Parseval's Theorem:

For continuous-time, periodic signal, the energy is given by:

$$\frac{1}{T}\int\limits_T {{{\left| {x(t)} \right|}^2}dt = {{\sum\limits_{k = - \infty }^{ + \infty } {\left| {{a_k}} \right|} }^2}}$$

Where ak is the Fourier series coefficient of x(t), and T is the period of the signal.

For average power in one period of the periodic signal x(t), we write:

$$\frac{1}{T}\int\limits_T {{{\left| {{a_k}{e^{jk{\omega _o}t}}} \right|}^2}dt = \frac{1}{T}{{\int\limits_T {\left| {{a_k}} \right|} }^2}} dt = {\left| {{a_k}} \right|^2}$$

$${\left| {{a_k}} \right|^2}$$ is the average power in the kth harmonic of x(t).

∴ Parseval's relation states that the total average power in a periodic signal equals the sum of the average powers in all of its harmonic components.
user
The text passage contains a multiple-choice question that requires matching items from List I to items in List II. The passage also provides a detailed explanation of the theorems and concepts mentioned in the question.

TYPE: NO_QUESTION

The passage does not contain a mathematical question that requires a calculation or a word problem. Instead, it provides a multiple-choice question and detailed explanations of the theorems and concepts.