State and prove the Parseval’s theorem for power.
The parseval’s theorem in analogous to the parseval’s power theorem. It states that the total energy of the signal(t) is equal to the sum of energies of the individual spectral components in the frequency domain.
The total normalized energy of a signal x(t) is given by
\(E=\int_{-\infty}^{\infty}\left|x\left(t\right)\right|^2dt\)
According to the Parseval’s energy theorem, we have
Total normalized energy
\(E=\int_{-\infty}^{\infty}\left|x\left(f\right)\right|^2df\) … (1)
Hence, the total normalized energy is equal to the area under the signal corresponding to the square of the amplitude spectrum \(\left|X\left(f\right)\right|\) of the signal.
Proof:
We know that the normalized energy of the signal x(t) is given by
\(E=\int_{-\infty}^{\infty}\left|x\left(t\right)\right|^2dt=\int_{-\infty}^{\infty}\left|x\left(t\right)x^\left(t\right)\right|dt\) … (2)
Using the definition of inverse Fourier transform (IFT), we have
\(x\left(t\right)=\int_{-\infty}^{\infty}X\left(f\right)e^{j2\pi ft}df\)
Hence, the complex conjugate of x(t) can be obtained as under
\(x^\left(-1\right)\left(t\right)=\left[\int_{-\infty}^{\infty}X\left(f\right)e^{j2\pi ft}df\right]^\left(-1\right)=\int_{-\infty}^{\infty}{X\left(f\right)e^{-j2\pi ft}df}\)
Substituting the value of \(x^\left(-1\right)\left(t\right)\) into equation (2), we have
\(E=\int_{-\infty}^{\infty}x\left(t\right)\left[\int_{-\infty}^{\infty}{X\left(f\right)e^{-j2\pi ft}df}\right]dt\)
\(E=\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}{X\left(t\right)e^{-j2\pi ft}df}\right]X\left(f\right)df\) … (3)
In equation (3), the term inside the square braket is the Fourier transform X(f) of the signal x(t).
Therefore total energy
\(E=\int_{-\infty}^{\infty}{{X\left(f\right)X}\left(f\right)df}\)
\(E=\int_{-\infty}^{\infty}\left|X\left(f\right)\right|^2df\)
Hence proved
State Channel capacity theorem
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