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Parseval's Theorem is a mathematical theorem that 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 can be calculated by integrating the squared magnitude of its Fourier transform.
x(t) can be any continuous function in the time domain, as long as it satisfies certain conditions such as being absolutely integrable. This means that it must have a finite total energy and cannot have any infinite or undefined values.
Parseval's Theorem is used to analyze signals in both the time and frequency domains. It allows for the calculation of the energy of a signal without having to compute its entire Fourier transform. This is useful in applications such as filtering, noise reduction, and spectral analysis.
Yes, Parseval's Theorem is applicable to all types of signals as long as they meet the necessary conditions. This includes both continuous and discrete signals, as well as periodic and non-periodic signals.
One limitation of Parseval's Theorem is that it only applies to signals with finite energy. This means that it cannot be used for signals with infinite energy, such as impulse signals. Additionally, it assumes that the signal is time-invariant, which may not always be the case in real-world applications.