Signals and Systems – Parseval’s Power Theorem

Signals and SystemsElectronics & ElectricalDigital Electronics

Average Power

The average power of a signal is defined as the mean power dissipated by the signal such as voltage or current in a unit resistance over a period. Mathematically, the average power is given by,

$$\mathit{P}\:\mathrm{=}\:\lim_{T \rightarrow \infty}\frac{1}{\mathit{T}}\int_{\mathrm{-(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T}/\mathrm{2})}}|\mathit{x}\mathrm{(\mathit{t})}|^\mathrm{2}\:\mathit{dt}$$

Parseval's Power Theorem

Statement − Parseval's power theorem states that the power of a signal is equal to the sum of square of the magnitudes of various harmonic components present in the discrete spectrum.

Mathematically, the Parseval's power theorem is defined as −

$$\mathit{P}\:\mathrm{=}\:\displaystyle\sum\limits_{n=-\infty}^\infty |\mathit{C}_\mathit{n}|^2$$

Proof

Consider a function $\mathit{x}\mathrm{(\mathit{t})}$. Then, the average power of the signal $\mathit{x}\mathrm{(\mathit{t})}$ over one complete cycle is given by,

$$\mathit{P}\:\mathrm{=}\:\frac{1}{\mathit{T}}\int_{\mathrm{-(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T}/\mathrm{2})}}|\mathit{x}\mathrm{(\mathit{t})}|^\mathrm{2}\:\mathit{dt}$$ $$\because|\mathit{x}\mathrm{(\mathit{t})}|^\mathrm{2}\:\mathrm{=}\: \mathit{x}\mathrm{(\mathit{t})}\mathit{x}^{*}\mathrm{(\mathit{t})}$$ $$\therefore\mathit{P}\:\mathrm{=}\:\frac{1}{\mathit{T}}\int_{\mathrm{-(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T}/\mathrm{2})}}\mathit{x}\mathrm{(\mathit{t})}\mathit{x}^{*}\mathrm{(\mathit{t})}\mathit{dt}\:\dotsm(1)$$

But, from the definition of exponential Fourier series, we have,

$$\mathit{x}\mathrm{(\mathit{t})}\:\mathrm{=}\:\displaystyle\sum\limits_{n=-\infty}^\infty \mathit{C}_\mathit{n}\mathit{e}^\mathit{jn\omega t}\:\dotsm(2)$$

Hence, from eqns. (1) & (2), we get,

$$\mathit{P}\:\mathrm{=}\:\frac{1}{\mathit{T}}\int_{\mathrm{-(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T}/\mathrm{2})}}  \begin{bmatrix}\displaystyle\sum\limits_{n=-\infty}^\infty \mathit{C}_\mathit{n}\mathit{e}^\mathit{jn\omega t} \end{bmatrix}
\mathit{x}^{*}\mathrm{(\mathit{t})}\:\mathit{dt}$$

Now, by interchanging the order of integration and summation, we get,

$$\mathit{P}\:\mathrm{=}\:\displaystyle\sum\limits_{n=-\infty}^\infty \mathit{C}_\mathit{n} \begin{bmatrix} \frac{1}{\mathit{T}}\int_{\mathrm{-(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T}/\mathrm{2})}}\mathit{x}^{*}\mathrm{(\mathit{t})}\mathit{e}^\mathit{jn\omega t}\:\mathit{dt}\end{bmatrix}  \:\mathrm{=}\:\displaystyle\sum\limits_{n=-\infty}^\infty\mathit{C}_\mathit{n}\mathit{C}_\mathit{n}^{*}$$ $$\therefore\mathit{P}\:\mathrm{=}\:\displaystyle\sum\limits_{n=-\infty}^\infty|\mathit{C}_\mathit{n}|^{2}\:\dotsm(3)$$

The expression in eq. (3) is called the Parseval's power theorem. Hence, it is clear that the Parseval's power theorem defines the power of a signal in terms of its Fourier series coefficients or in other words, in terms of harmonics present in the signal

raja
Published on 07-Jan-2022 11:49:42
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