text: # 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.
explanation: The document contains a multiple-choice question from an exam, specifically the UGC NET Paper 2: Electronic Science. The question asks to match items from List I with items from List II and provides four options for the correct matches. The document also includes a detailed solution explaining the concepts related to the correct answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a multiple-choice question from an exam, specifically the UGC NET Paper 2: Electronic Science. The question asks to match items from List I with items from List II and provides four options for the correct matches. The document also includes a detailed solution explaining the concepts related to the correct answer.
  
  Label: multiple-question-and-answer