Parseval’s Theorem
Sources:
- B. P. Lathi & Roger Green. (2021). Chapter 3: Signal Representation by Fourier Series. Signal Processing and Linear Systems (2nd ed., pp. 297-300). Oxford University Press.
Recall that (see my past post) the trigonometric Fourier series of a periodic real signal \(x(t)\) is given by \[ x(t)=C_0+\sum_{n=1}^{\infty} C_n \cos \left(n \omega_0 t+\theta_n\right) \]
where \(n > 0\).
Every term on the right-hand side of this equation is a power signal. As shown in before, the power of \(x(t)\) is equal to the sum of the powers of all the sinusoidal components on the right-hand side. \[ P_x=C_0{ }^2+\frac{1}{2} \sum_{n=1}^{\infty} C_n^2 \]
This result is one form of Parseval's theorem, as applied to power signals. It states that the power of a periodic signal is equal to the sum of the powers of its Fourier components.
We can apply the same argument to the exponential Fourier series (see Prob. 1.1-11). The power of a periodic signal \(x(t)\) can be expressed as a sum of the powers of its exponential components. In Eq. (1.4), we showed that the power of an exponential \(D e^{j \omega_0 t}\) is \(\left|D^2\right|\). We can use this result to express the power of a periodic signal \(x(t)\) in terms of its exponential Fourier series coefficients as \[ P_x=\sum_{n=-\infty}^{\infty}\left|D_n\right|^2 \]
For a real \(x(t),\left|D_{-n}\right|=\left|D_n\right|\). Therefore, \[ P_x=D_0{ }^2+2 \sum_{n=1}^{\infty}\left|D_n\right|^2 \]