Parseval's relation; applications of FT

Parseval's relation

Statement: For a finite-energy signal $x(t)$ with Fourier transform $X(j\omega)$, the total energy computed in the time domain equals the total energy computed in the frequency domain:

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

In terms of frequency $f$ (Hz), $E = \int_{-\infty}^{\infty}|X(f)|^2\,df$. So $|X(j\omega)|^2$ is the energy spectral density: it shows how the energy is distributed over frequency.

Proof: Start with the energy and write $|x(t)|^2 = x(t)\,x^*(t)$:

$$E = \int_{-\infty}^{\infty} x(t)\,x^*(t)\,dt$$

Express $x^*(t)$ using the inverse Fourier transform. Since $x(t) = \frac{1}{2\pi}\int X(j\omega)e^{j\omega t}d\omega$, taking the conjugate gives

$$x^*(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X^*(j\omega)\,e^{-j\omega t}\,d\omega$$

Substitute and change the order of integration:

$$\begin{aligned} E &= \int_{-\infty}^{\infty} x(t)\left[\frac{1}{2\pi}\int_{-\infty}^{\infty} X^*(j\omega)e^{-j\omega t}d\omega\right]dt \\ &= \frac{1}{2\pi}\int_{-\infty}^{\infty} X^*(j\omega)\left[\int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt\right]d\omega \\ &= \frac{1}{2\pi}\int_{-\infty}^{\infty} X^*(j\omega)\,X(j\omega)\,d\omega \\ &= \frac{1}{2\pi}\int_{-\infty}^{\infty} |X(j\omega)|^2\,d\omega \end{aligned}$$

The bracketed inner integral is exactly the definition of $X(j\omega)$. Hence the relation is proved.

Example: $x(t) = e^{-at}u(t)$ has energy $\int_0^\infty e^{-2at}dt = \frac{1}{2a}$. Its transform is $\frac{1}{a+j\omega}$, and $\frac{1}{2\pi}\int \frac{d\omega}{a^2+\omega^2} = \frac{1}{2\pi}\cdot\frac{\pi}{a} = \frac{1}{2a}$. Both sides agree.

Applications of Fourier transform

1. Spectrum analysis and filter design: The FT shows which frequencies a signal contains. Filters (low-pass, band-pass, etc.) are designed and analysed using the frequency response $H(j\omega)$, which is the FT of the impulse response.
2. LTI system analysis: Convolution in time becomes multiplication in frequency, $Y(j\omega) = X(j\omega)H(j\omega)$. This makes finding the output of LTI systems and solving linear differential equations much easier.
3. Communication (modulation and sampling): Amplitude modulation, frequency-division multiplexing and the sampling theorem are all explained with the FT (modulation property shifts the spectrum to $\omega \pm \omega_c$).

Other uses: image and audio compression (JPEG, MP3), bandwidth calculation, and energy/power spectral analysis.