--- title: "FT of periodic signals; rect pulse and constant" subject: "Signals and Systems" chapter: 3 key: "semester_4/signals_and_systems:ft-periodic-rect-constant" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-periodic-rect-constant site: BCT Engineering --- # FT of periodic signals; rect pulse and constant Signals and Systems · Chapter 3: Fourier Transform - Asked in: 2074 Bhadra, 5+5 marks - Video: [7.6: The Continuous-Time Fourier Transform of Periodic Signals](https://www.youtube.com/watch?v=chfVJux_7tk) — ECE Done Better - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-periodic-rect-constant - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3 ## Question How do you find the Fourier transform of periodic signals? Find the Fourier transform of continuous time rectangular pulse and constant amplitude A and explain the result. ## Short answer For a periodic signal, X(jω) = Σ 2π a_k δ(ω - kω0), using its Fourier series. A rectangular pulse of width 2T1 has transform 2T1 sinc(ωT1/π); a constant A has transform 2πA δ(ω). ## Answer #### Fourier transform of periodic signals A periodic signal has infinite energy, so its FT is found through its Fourier series. If $x(t)$ has period $T_0$ and $\omega_0 = 2\pi/T_0$: $$ x(t) = \sum_{k=-\infty}^{\infty} a_k\,e^{jk\omega_0 t} $$ Using the pair $e^{jk\omega_0 t} \leftrightarrow 2\pi\,\delta(\omega - k\omega_0)$ (because the inverse FT of $2\pi\delta(\omega-k\omega_0)$ is $e^{jk\omega_0 t}$) and linearity: $$ X(j\omega) = \sum_{k=-\infty}^{\infty} 2\pi\,a_k\,\delta(\omega - k\omega_0) $$ So the FT of a periodic signal is a train of impulses at the harmonics $k\omega_0$, with strengths $2\pi a_k$. Example: $\cos\omega_0 t \leftrightarrow \pi[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$. #### FT of a rectangular pulse Let $x(t) = 1$ for $|t| < T_1$ and $0$ otherwise (width $2T_1$). $$ \begin{aligned} X(j\omega) &= \int_{-T_1}^{T_1} e^{-j\omega t}dt = \frac{e^{j\omega T_1} - e^{-j\omega T_1}}{j\omega} \\ &= \frac{2\sin(\omega T_1)}{\omega} = 2T_1\,\text{sinc}\left(\frac{\omega T_1}{\pi}\right) \end{aligned} $$ **Explanation:** The spectrum is a real sinc function with peak $2T_1$ (the pulse area) at $\omega = 0$ and zeros at $\omega = \pm\pi/T_1, \pm 2\pi/T_1, \dots$. A time-limited pulse has an infinitely wide spectrum. Most of the energy lies in the main lobe $|\omega| < \pi/T_1$, and a narrower pulse gives a wider main lobe (inverse time–bandwidth relation). ```text x(t) X(jw) 1 +-----+ 2T1 ^ | | .|. ---+--+--+---> t . | . -T1 0 T1 . . . | . . . ----'-----+-----'-----> w -pi/T1 pi/T1 ``` #### FT of a constant amplitude $A$ $x(t) = A$ for all $t$ is not absolutely integrable. Take $X(j\omega) = 2\pi A\,\delta(\omega)$; its inverse is $\frac{1}{2\pi}\int 2\pi A\,\delta(\omega)e^{j\omega t}d\omega = A$. Therefore $$ A \;\longleftrightarrow\; 2\pi A\,\delta(\omega) $$ **Explanation:** A constant has no variation, so all its content is at $\omega = 0$: a single impulse. This also follows as the limit of the rectangular pulse as $T_1 \to \infty$: the sinc becomes taller ($2T_1A$) and narrower, approaching $2\pi A\,\delta(\omega)$. Together, the two results show the inverse relation between time and frequency: a signal that is infinitely wide in time is infinitely narrow in frequency, and a narrow pulse has a wide spectrum. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the FT of a periodic signal obtained? A. By integrating over one period only B. By sampling the pulse FT at unit intervals C. It does not exist D. Impulses of area 2π a_k at kω0, from the Fourier series coefficients Answer: D. Using e^(jkω0t) <-> 2πδ(ω - kω0) and linearity on the Fourier series. (Key point: FT of periodic signal) 2. For x(t) = 1 for |t| < T1, what is X(jω)? A. 2 sin(ωT1)/ω B. sin(ωT1)/ω C. 2 sin(ω)/T1 D. 2T1 sin(ωT1) Answer: A. ∫ from -T1 to T1 of e^(-jωt) dt = 2 sin(ωT1)/ω, with peak 2T1 at ω = 0. (Key point: Rect pulse FT) 3. What happens to the main lobe when the pulse gets narrower? A. It gets narrower B. It gets wider (zeros at π/T1 move out) C. It stays the same D. It becomes an impulse Answer: B. Time and bandwidth are inversely related. (Key point: Rect pulse width relation) 4. What is the FT of the constant A? A. A δ(ω) B. A/(jω) C. 2πA δ(ω) D. 2A sinc(ω) Answer: C. Its inverse (1/2π)∫2πAδ(ω)e^(jωt)dω = A. (Key point: FT of constant) ## Related in Signals and Systems - [FT of rectangular pulse and constant; duality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-rect-pulse-constant-duality) — Signals and Systems, chapter 3 - [FT of periodic signals; pulse example](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-periodic-pulse-example) — Signals and Systems, chapter 3, asked 2× - [Fourier transform of CT periodic signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-periodic-signal-derivation) — Signals and Systems, chapter 3, asked 3× - [FT of unit impulse and rectangular pulse](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-impulse-rectangular-pulse) — Signals and Systems, chapter 3 - [FT of periodic signals; e^(−2|t|)sin t](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-periodic-damped-sine) — Signals and Systems, chapter 3, asked 2× - [Fourier transform of periodic sequences](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-of-periodic-sequence) — Signals and Systems, chapter 3, asked 3× ## Related in other subjects - [Properties of continuous-time Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-properties) — Data Communication, chapter 3 - [Fourier transform of a ramp pulse](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-ramp-signal) — Data Communication, chapter 3 - [Fourier transform property: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1 - [Fourier integral of rectangular pulse](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-integral-rectangular-pulse) — Applied Mathematics, chapter 4