Parseval's theorem

Parseval’s theorem states that if a signal $x$ has finite energy, i.e., $x\in {\mathsf{L}}_2^{\text{per}}$, then:

$$\frac{1}{T_0}{\int }_0^{T_0}|x(t){|}^{2}dt=\sum _{-\infty }^{\infty }|{\alpha }_k{|}^2$$ $$\frac{1}{T_0}\Vert x{\Vert }_2^2=\Vert \alpha {\Vert }_2^2$$

where $\alpha$ are the continuous-time Fourier series coefficients. Additionally, the following properties will also be satisfied:

• ${\hat{x}}_K$ converges to $x$ in energy.
• The CTFS coefficients $\{\alpha {\}}_{k=-\infty }^{\infty }$ have finite energy, i.e., $\alpha \in l_2$.

For an aperiodic signal, if it has finite energy, i.e., $x\in {\mathsf{L}}_2$, then:

$${\int }_{-\infty }^{\infty }|x(t){|}^{2}dt=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }|X(j\omega ){|}^{2}d\omega$$

where $X(j\omega )$ is the CT Fourier transform.