--- title: "Parseval's relation; applications of FT" subject: "Signals and Systems" chapter: 3 key: "semester_4/signals_and_systems:parseval-ft-applications" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/parseval-ft-applications site: BCT Engineering --- # Parseval's relation; applications of FT Signals and Systems · Chapter 3: Fourier Transform - Asked in: 2078 Poush, 6+3 marks - Video: [Parseval's Theorem](https://www.youtube.com/watch?v=ML0eYMyhqOs) — Steve Brunton - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/parseval-ft-applications - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3 ## Question State and prove Parseval's relation for finite energy signal. Describe any three applications of Fourier Transform. ## Short answer Parseval's relation: ∫|x(t)|² dt = (1/2π)∫|X(ω)|² dω. Proof substitutes x*(t) via inverse FT, swaps integrals and identifies X(ω). Applications: 1) spectrum analysis and filter design; 2) LTI system analysis via Y(ω)=X(ω)H(ω); 3) communication (modulation, sampling). ## Answer #### 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Parseval's relation for a finite-energy signal states: A. ∫|x|^2 dt = (1/2π)∫|X(jω)|^2 dω B. ∫|x|^2 dt = ∫|X(jω)|^2 dω C. ∫|x|^2 dt = 2π∫|X(jω)|^2 dω D. ∫x dt = (1/2π)∫X dω Answer: A. Energy in time equals (1/2π) times the integral of |X|^2 over ω. (Key point: Parseval statement) 2. In the proof, what is substituted for x*(t)? A. Its inverse-transform expression (1/2π)∫X*(jω)e^(-jωt)dω B. Its Laplace transform C. The convolution of x with itself D. Its derivative Answer: A. Substituting x*(t) via the inverse FT and swapping integrals makes the inner integral equal X(jω). (Key point: Proof idea) 3. |X(jω)|^2 is called: A. Power spectral density B. Energy spectral density C. Phase spectrum D. Impulse response Answer: B. It shows how the energy of the signal is distributed over frequency. (Key point: Meaning of |X|^2) 4. Why does the FT simplify LTI system analysis? A. Convolution in time becomes multiplication, Y = X H B. Multiplication in time becomes convolution C. Differentiation becomes integration D. It removes the need for impulse response Answer: A. Y(jω) = X(jω)H(jω), which is simpler than convolving in time. 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