--- title: "Fourier transform of CT periodic signals" subject: "Signals and Systems" chapter: 3 key: "semester_4/signals_and_systems:ft-periodic-signal-derivation" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ft-periodic-signal-derivation site: BCT Engineering --- # Fourier transform of CT periodic signals Signals and Systems · Chapter 3: Fourier Transform - Asked in: 2072 Magh, 5 marks - Video: [Fourier Transform of Periodic Signals](https://www.youtube.com/watch?v=zkbFHDmOA1o) — Salaar Khan - 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-signal-derivation - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-3 ## Question Derive the expression for Fourier transform of continuous time periodic signal. ## Short answer Expand the periodic signal in a Fourier series: x(t)=∑ a_k e^{jkω0t}. Since e^{jkω0t} ↔ 2π δ(ω−kω0), linearity gives X(jω)=∑ 2π a_k δ(ω−kω0). Thus the FT is a train of impulses at harmonic frequencies with area 2π a_k. ## Answer A periodic signal is not absolutely integrable, so its ordinary Fourier integral does not converge. Its Fourier transform is obtained from its Fourier series using impulses. **Step 1: Fourier series.** Let $x(t)$ be periodic with period $T_0$, $\omega_0 = 2\pi/T_0$: $$ x(t) = \sum_{k=-\infty}^{\infty} a_k\,e^{jk\omega_0 t}, \qquad a_k = \frac{1}{T_0}\int_{T_0} x(t)\,e^{-jk\omega_0 t}\,dt $$ **Step 2: FT of one complex exponential.** Consider $X(j\omega) = 2\pi\,\delta(\omega - \omega_0)$. Its inverse FT is $$ \frac{1}{2\pi}\int_{-\infty}^{\infty} 2\pi\,\delta(\omega - \omega_0)\,e^{j\omega t}\,d\omega = e^{j\omega_0 t} $$ by the sifting property. Hence $$ e^{j\omega_0 t} \;\longleftrightarrow\; 2\pi\,\delta(\omega - \omega_0), \qquad e^{jk\omega_0 t} \;\longleftrightarrow\; 2\pi\,\delta(\omega - k\omega_0) $$ **Step 3: apply linearity** to the Fourier series term by term: $$ X(j\omega) = \sum_{k=-\infty}^{\infty} 2\pi\,a_k\,\delta(\omega - k\omega_0) $$ **Result:** The FT of a periodic signal is a train of impulses located at the harmonic frequencies $k\omega_0$, with area $2\pi a_k$. **Link with the FT of one period:** If $X_0(j\omega)$ is the FT of one period of $x(t)$, then $a_k = \frac{1}{T_0}X_0(jk\omega_0)$, so $X(j\omega) = \omega_0\sum_k X_0(jk\omega_0)\delta(\omega-k\omega_0)$. **Examples:** - $\cos\omega_0 t$: $a_{\pm1} = \frac12$, so $X(j\omega) = \pi[\delta(\omega-\omega_0) + \delta(\omega+\omega_0)]$. - $\sin\omega_0 t$: $a_{\pm1} = \pm\frac{1}{2j}$, so $X(j\omega) = \frac{\pi}{j}[\delta(\omega-\omega_0) - \delta(\omega+\omega_0)]$. - Impulse train $\sum_n\delta(t - nT)$: $a_k = \frac1T$, so $X(j\omega) = \frac{2\pi}{T}\sum_k\delta\left(\omega - \frac{2\pi k}{T}\right)$. ```text X(jw) 2*pi*a_k impulses ^ ^ | ^ ^ | | | ^ --+--+----+----+--+--> w -2w0 -w0 0 w0 2w0 ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why can't a periodic signal's FT be found by the ordinary integral? A. It is non-causal B. It has no Fourier series C. It is always complex D. It is not absolutely integrable, so the integral does not converge Answer: D. Periodic signals have infinite energy, so impulses are needed. (Key point: Why not Fourier integral) 2. Which pair is used for each harmonic? A. e^(jkω0t) <-> 2π δ(ω - kω0) B. e^(jkω0t) <-> δ(ω - kω0) C. e^(jkω0t) <-> 2π δ(ω + kω0) D. e^(jkω0t) <-> π δ(ω - kω0) Answer: A. Verified by the sifting property in the inverse FT. (Key point: Exponential pair) 3. What is X(jω) for a periodic signal with coefficients a_k? A. Σ a_k δ(ω - kω0) B. Σ 2π a_k δ(ω - kω0) C. Σ a_k/(2π) δ(ω - kω0) D. Σ 2π a_k δ(ω - k) Answer: B. Linearity applied term by term to the Fourier series. (Key point: Result) 4. What is the FT of cos ω0 t? A. 2π[δ(ω-ω0) + δ(ω+ω0)] B. π/2 [δ(ω-ω0) + δ(ω+ω0)] C. π[δ(ω-ω0) + δ(ω+ω0)] D. π[δ(ω-ω0) - δ(ω+ω0)] Answer: C. a_{±1} = 1/2, so each impulse has area 2π(1/2) = π. (Key point: Cosine example) ## Close variants set in other exams - [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× - [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× ## Related in Signals and Systems - [Derive the CT Fourier transform pair](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-derivation) — Signals and Systems, chapter 3, asked 3× - [Fourier transform of a CT signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-unspecified-ct-signal) — Signals and Systems, chapter 3 - [Derive exponential Fourier series (CT)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-exponential-derivation) — Signals and Systems, chapter 2, asked 15× - [Derive trigonometric Fourier series (CT)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-trigonometric-derivation) — Signals and Systems, chapter 2, asked 3× - [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× - [Parseval's relation for CT periodic signals](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-parseval) — Signals and Systems, chapter 2, asked 4× ## Related in other subjects - [Z-transform of ramp-and-hold continuous signal](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Awaveform-ramp-to-1-at-6) — Digital Control System, chapter 2 - [Fourier transform of a DC signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-dc-signal) — Data Communication, chapter 3 - [Properties of continuous-time Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-properties) — Data Communication, chapter 3 - [Derive time shifting property of z-transform](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Az-time-shifting-derive) — Digital Signal Analysis and Processing, chapter 2