Match List I with List II: Choose the correct answer from the options given below: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
(A) - (II), (B) - (I), (C) - (IV), (D) - (III)
Understanding Key Theorems: Shannon, Bayes, and Parseval Matching
This question asks us to match fundamental theorems and concepts from various fields of engineering and mathematics to their associated definitions or applications. Let's analyze each pairing based on standard definitions and principles.
Analyzing Each Pairing
- (A) Shannon's theorem: There are several theorems attributed to Claude Shannon. One of the most fundamental is the Source Coding Theorem, which establishes the theoretical limit on the minimum average number of bits per symbol required to encode the output of a discrete memoryless source for reliable transmission. This limit is the entropy of the source. Another perspective is the Rate Distortion Theorem, dealing with the minimum rate required to transmit information from a source subject to a specified level of distortion. Both relate to the concept of the efficiency or rate of information representation or transmission. Looking at the options in List II, "(II) Rate of Information" aligns best with the core ideas behind Shannon's source coding theorems, which quantify and provide limits on the rate at which information can be compressed or transmitted from a source.
- (B) Shannon-Hartley theorem: This theorem, an application of Shannon's channel coding theorem, specifies the maximum rate at which information can be transmitted over a communications channel of a specified bandwidth in the presence of noise, assuming the noise is a Gaussian process with known power. This maximum rate is known as the channel capacity. List II offers "(I) Capacity of Gaussian Noise channel", which is exactly what the Shannon-Hartley theorem quantifies. The formula is given by:
\(C = B \log_2(1 + S/N)\)
Where \(C\) is the channel capacity in bits per second, \(B\) is the bandwidth of the channel in hertz, and \(S/N\) is the signal-to-noise power ratio.
- (C) Bayes theorem: This is a fundamental theorem in probability theory. It describes how to update the probability of a hypothesis based on new evidence. It mathematically relates conditional probabilities. Given two events A and B, Bayes theorem states:
\(P(A|B) = \frac{P(B|A) P(A)}{P(B)}\)
Here, \(P(A|B)\) is the posterior probability of A given B, \(P(B|A)\) is the likelihood of B given A, \(P(A)\) is the prior probability of A, and \(P(B)\) is the probability of B. This theorem is directly about relating and calculating conditional probabilities. List II option "(IV) Conditional probabilities" is the direct domain of Bayes theorem.
- (D) Parseval's theorem: This theorem relates the energy of a signal in the time domain to the energy of its Fourier transform in the frequency domain. It states that the total energy of a signal is the same whether calculated in the time domain or the frequency domain. For a continuous-time signal \(x(t)\) with Fourier transform \(X(\omega)\):
\(E = \int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(\omega)|^2 d\omega\)
For a discrete-time signal \(x[n]\) with Discrete-Time Fourier Transform \(X(e^{j\omega})\):
\(E = \sum_{n=-\infty}^{\infty} |x[n]|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(e^{j\omega})|^2 d\omega\)
Parseval's theorem is explicitly concerned with the energy of a signal, making List II option "(III) Energy of a signal" the correct match.
Summary of Matchings
Based on the analysis:
- (A) Shannon's theorem → (II) Rate of Information
- (B) Shannon-Hartley theorem → (I) Capacity of Gaussian Noise channel
- (C) Bayes theorem → (IV) Conditional probabilities
- (D) Parseval's theorem → (III) Energy of a signal
Let's put this into a table format for clarity.
| List I | List II | Matching Concept |
|---|---|---|
| (A) Shannon's theorem | (II) Rate of Information | Deals with the minimum bits per symbol for source encoding. |
| (B) Shannon-Hartley theorem | (I) Capacity of Gaussian Noise channel | Calculates the maximum reliable data rate over a noisy channel. |
| (C) Bayes theorem | (IV) Conditional probabilities | Provides a way to update probabilities based on evidence. |
| (D) Parseval's theorem | (III) Energy of a signal | Relates signal energy in time and frequency domains. |
Comparing our derived matchings with the given options, the correct option corresponds to: (A) - (II), (B) - (I), (C) - (IV), (D) - (III).
Revision Table: Key Theorems and Concepts
| Theorem/Concept | Field | Primary Application/Definition |
|---|---|---|
| Shannon's Theorem (Source Coding) | Information Theory | Minimum Rate of Information / Data Compression Limit (Entropy) |
| Shannon-Hartley Theorem | Information Theory / Communications | Channel Capacity (Max reliable rate) for AWGN channels |
| Bayes Theorem | Probability Theory / Statistics | Updating Conditional Probabilities / Bayesian Inference |
| Parseval's Theorem | Signal Processing / Fourier Analysis | Energy Conservation between Time and Frequency Domains |
Additional Information: Expanding on Related Concepts
These theorems are cornerstones in their respective fields.
- Information Theory: Founded by Claude Shannon, this field provides a mathematical framework for quantifying, storing, and communicating information. Key concepts include entropy (uncertainty or information content), channel capacity (maximum reliable transmission rate), and coding theorems (limits on compression and reliable transmission).
- Probability and Statistics: Bayes theorem is fundamental for understanding how probabilities change as new data becomes available. It's widely used in machine learning, medical diagnosis, and various scientific fields for probabilistic inference.
- Signal Processing: Parseval's theorem is crucial in Fourier analysis and signal processing. It allows engineers to calculate the total energy of a signal by analyzing its frequency components, which is often easier or provides different insights than working in the time domain. It underlies concepts like power spectral density.
Important Questions from Channel Capacity
Noise factor of a system is defined as:
The information capacity (bits/sec) of a channel with bandwidth C and transmission time T is given by
The capacity of band-limited additive white Gaussian Noise (AWGN) channel is given by \(C = W{\log _2}\left[ {1 + \frac{P}{{{\sigma ^2}w}}} \right]\) bits per second (bps), where W is the channel Bandwidth, P is the average power received and σ2 is the one-sided power spectral density of the AWGN.
For a fixed \(\frac{P}{{{\sigma ^2}}} = 1000\), the channel capacity (in kbps) with infinite Bandwidth (W → ∞) is approximately
Consider an additive white Gaussian noise (AWGN) channel with bandwidth W and noise power spectral density $\frac{N_0}{2}$. Let $P_{av}$ denote the average transmit power constraint. Which one of the following plots illustrates the dependence of the channel capacity C on the bandwidth W (keeping $P_{av}$ and $N_0$ fixed)?
A voice-grade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of 4.0 kHz and two-sided noise power spectral density $ \frac{\eta}{2} = 2.5\times10^{-5} $ Watt per Hz. If information at the rate of 52 kbps is to be transmitted over this channel with arbitrarily small bit error rate, then the minimum bit-energy $E_b$ (in mJ/bit) necessary is ____________
