Parseval's equality and theorem?

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In summary, Parseval's equality and theorem is a mathematical concept that links the energy or power of a signal to its frequency content. It is important because it allows for analysis of signals in both the time and frequency domains, and has applications in signal processing, communications, and control systems. It is derived from the Fourier transform of a signal and is used for both continuous-time and discrete-time signals. The applications of Parseval's equality and theorem are diverse, including audio signal analysis, communication signal power calculation, and control system stability analysis. It is also used in fields such as image processing, data compression, and medical imaging.
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car202
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It's kind of dumb question..
But, I just wanted to make sure.
Are Parseval's equality and Parseval's theorem same thing? (In terms of Fourier series)
i.e. do both mean [tex]\frac{1}{L}\int_c^{c+2L}|f(x)|^{2}dx = \frac{a_0^2}{2}+\sum_{n=1}^{\infty}[|a_n|^{2}+|b_n|^{2}][/tex]
 
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Basically, yes. Parceval's equality is the equality you give. Parseval's thereom is the statement that that equality holds, under given hypotheses, of course. Perhaps the crucial difference is the hypotheses. Parceval's equality doesn't make sense without specifying what "f", "an", "bn", etc. are.
 

FAQ: Parseval's equality and theorem?

What is Parseval's equality and theorem?

Parseval's equality and theorem is a mathematical concept that relates the energy or power of a signal to its frequency content. It states that the total energy of a signal in the time domain is equal to the sum of the squared magnitudes of its frequency components in the frequency domain.

What is the importance of Parseval's equality and theorem?

Parseval's equality and theorem is important because it allows us to analyze signals in both the time and frequency domains, providing valuable insights into the characteristics of a signal. It is also used in various applications such as signal processing, communications, and control systems.

How is Parseval's equality and theorem derived?

Parseval's equality and theorem can be derived from the Fourier transform of a signal. By taking the complex conjugate of the Fourier transform and multiplying it with the original signal, the result can be integrated over all time to obtain the total energy of the signal. This can then be equated to the sum of the squared magnitudes of the Fourier coefficients, giving us Parseval's equality.

What is the difference between Parseval's equality and Parseval's theorem?

Parseval's equality and Parseval's theorem are essentially the same concept, with the only difference being the type of signals they are applied to. Parseval's equality is used for continuous-time signals, while Parseval's theorem is used for discrete-time signals.

What are the applications of Parseval's equality and theorem?

The applications of Parseval's equality and theorem are vast and diverse. Some examples include analyzing the quality of audio signals, calculating the power of a communication signal, and determining the stability of a control system. It is also used in various fields such as image processing, data compression, and medical imaging.

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