Fourier transform properties; Rayleigh energy theorem

The Fourier transform pair is $G(f)=\int_{-\infty}^{\infty} g(t)e^{-j2\pi ft}dt$ and $g(t)=\int_{-\infty}^{\infty} G(f)e^{j2\pi ft}df$, written $g(t)\leftrightarrow G(f)$.

Four properties of the Fourier transform

• Linearity: the transform of a weighted sum is the weighted sum of the transforms.
• Time shifting: a delay changes only the phase spectrum (linear phase), not the amplitude spectrum.
• Scaling: compressing a signal in time expands its spectrum, and vice versa.
• Modulation: multiplying by a carrier shifts the spectrum to $\pm f_c$; this is the basis of AM.

Rayleigh energy theorem

Statement: the total energy of an energy signal is the same whether calculated in the time domain or in the frequency domain:

$$E=\int_{-\infty}^{\infty}\lvert g(t)\rvert^2dt=\int_{-\infty}^{\infty}\lvert G(f)\rvert^2df$$

Derivation: write $\lvert g(t)\rvert^2=g(t)g^*(t)$ and express $g^*(t)$ by the inverse transform:

$$g^*(t)=\left[\int_{-\infty}^{\infty}G(f)e^{j2\pi ft}df\right]^*=\int_{-\infty}^{\infty}G^*(f)e^{-j2\pi ft}df$$

Substitute into the energy integral and change the order of integration:

$$\begin{aligned} E&=\int_{-\infty}^{\infty}g(t)\left[\int_{-\infty}^{\infty}G^*(f)e^{-j2\pi ft}df\right]dt\\ &=\int_{-\infty}^{\infty}G^*(f)\left[\int_{-\infty}^{\infty}g(t)e^{-j2\pi ft}dt\right]df\\ &=\int_{-\infty}^{\infty}G^*(f)\,G(f)\,df\\ &=\int_{-\infty}^{\infty}\lvert G(f)\rvert^2df \end{aligned}$$

The inner bracket in the second line is just $G(f)$ by definition. Hence the theorem is proved.

Meaning: $\lvert G(f)\rvert^2$ is the energy spectral density $\Psi_g(f)$ (unit J/Hz); integrating it over all frequencies gives the total energy. Example: for $g(t)=A\,\text{rect}(t/T)$, time-domain energy is $A^2T$, and integrating $A^2T^2\text{sinc}^2(fT)$ over $f$ also gives $A^2T$.