Parseval’s Theorem and Signal Energy Relations Quiz Quiz

Explore fundamental concepts of Parseval’s Theorem and energy relations in signals, including applications in the time and frequency domains. Enhance your understanding of energy calculations, continuous and discrete signals, and typical misconceptions in signal analysis.

1. Parseval's Theorem Statement

   Which of the following correctly expresses Parseval’s Theorem for a continuous-time signal x(t) with Fourier Transform X(f)?

   1. The total energy of x(t) equals the value of X(0).
   2. The total energy of x(t) equals the integral of |X(f)|^2 over all frequencies.
   3. The total energy of x(t) is the product of X(f) with its complex conjugate integrated over time.
   4. The total energy of x(t) equals the sum of X(f) over all frequencies.

   Explanation: Parseval’s Theorem states that the energy of a signal in the time domain is equal to the integral of the magnitude squared of its Fourier Transform in the frequency domain. The second option is incorrect because just summing X(f) does not give the energy. The third choice refers to multiplication in the time domain, not the correct integration in the frequency domain. The last option, using X(0), represents only the DC component, not the total energy.

2. Energy Calculation Example

   If a continuous-time signal x(t) has energy E in the time domain, what is its energy in the frequency domain according to Parseval’s Theorem?

   1. Exactly E, as given by the integral of |X(f)|^2.
   2. Zero, because energy is only defined in the time domain.
   3. E squared, as Parseval’s Theorem involves squaring energy.
   4. Twice E, since frequency domain doubles the value.

   Explanation: Parseval’s Theorem ensures that the total energy calculated in the frequency domain matches that of the time domain precisely, so the answer is exactly E. Doubling or squaring the energy (options two and four) are misunderstandings of the theorem’s equality. Option three is incorrect because energy can be calculated in both domains.

3. Discrete-Time Parseval’s Theorem

   For a discrete-time finite-length signal x[n], which mathematical operation does Parseval’s Theorem equate between the time and frequency domains?

   1. Sum of x[n] and X[k] without any squaring
   2. Sum of |x[n]|^2 and the sum of |X[k]|^2 (properly normalized)
   3. Difference between |x[n]|^2 and X[k]^2
   4. Product of x[n] and X[k]

   Explanation: In the discrete case, Parseval’s Theorem asserts that the sum of the squared magnitude of the signal samples equals the normalized sum of the squared magnitude of the DFT coefficients. The multiplication of the sequences is not what Parseval’s relates. Summing without squaring (option three) measures a different property. The difference (option four) is never used in Parseval’s context.

4. Physical Interpretation

   What does Parseval’s Theorem physically imply about energy-preserving transformations such as the Fourier Transform?

   1. Transformations always increase the energy of the signal.
   2. The total energy in a signal remains unchanged between the time domain and the frequency domain representations.
   3. Energy depends solely on the highest frequency present.
   4. Energy is only meaningful in the frequency domain after transformation.

   Explanation: Parseval’s Theorem reveals that the Fourier Transform is energy-preserving, meaning the energy calculation is consistent in both domains. Transformation does not increase energy, so option two is incorrect. Option three is wrong because energy is validly computed in both domains. Option four incorrectly claims energy is determined only by the highest frequency.

5. Application Limitation Scenario

   A student applies Parseval’s Theorem to a signal that does not have finite energy (such as a sinusoid continuing indefinitely). Why is this application invalid?

   1. Parseval’s Theorem applies to any signal, regardless of energy.
   2. Signals must be periodic to use Parseval’s Theorem.
   3. Sinusoids have no representation in the frequency domain.
   4. Parseval’s Theorem only applies to signals with finite energy (square-integrable signals).

   Explanation: Parseval’s Theorem requires the signal to have finite energy, meaning it must be square-integrable; sinusoids that extend infinitely do not meet this requirement. The claim that sinusoids have no frequency-domain representation is false because they do, but with infinite energy. The theorem does not apply to all signals or require periodicity, making options three and four incorrect.