--- title: "Time-scaling property of CTFS" subject: "Signals and Systems" chapter: 2 key: "semester_4/signals_and_systems:ctfs-time-scaling" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctfs-time-scaling site: BCT Engineering --- # Time-scaling property of CTFS Signals and Systems · Chapter 2: Fourier Series - Asked in: 2081 Chaitra, 5 marks; 2076 Baisakh, 4 marks (asked 2 times) - Video: [8. Continuous Time Fourier Series: Properties 2 (Time Reversal and Scaling)](https://www.youtube.com/watch?v=nnx4RcRmlPE) — Momin Uppal - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctfs-time-scaling - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2 ## Question State and prove the time-scaling property for continuous-time (exponential) Fourier series pair. ## Short answer If x(t) has coefficients a_k and period T, then x(αt) (α>0) has coefficients a_k unchanged, period T/α, and fundamental frequency αω0. Proof: direct substitution in synthesis gives ∑ a_k e^(jkαω0 t); the analysis integral with change of variable τ=αt also yields a_k. ## Answer #### Statement If $x(t)$ is periodic with period $T$ (fundamental frequency $\omega_0 = 2\pi/T$) and $$ x(t) = \sum_{k=-\infty}^{\infty}a_k e^{jk\omega_0 t}, $$ then for $\alpha > 0$, the time-scaled signal $x(\alpha t)$ is periodic with period $T/\alpha$ and fundamental frequency $\alpha\omega_0$, and its Fourier series coefficients are **the same** $a_k$: $$ x(\alpha t) = \sum_{k=-\infty}^{\infty}a_k\, e^{jk(\alpha\omega_0)t} $$ Time scaling does not change the coefficients; it only changes the spacing of the spectral lines from $\omega_0$ to $\alpha\omega_0$. #### Proof **Synthesis route:** replace $t$ by $\alpha t$ in the synthesis equation: $$ x(\alpha t) = \sum_{k}a_k e^{jk\omega_0(\alpha t)} = \sum_k a_k e^{jk(\alpha\omega_0)t} $$ This is a Fourier series with fundamental frequency $\alpha\omega_0$ and coefficients $a_k$. **Analysis route (check):** let $y(t) = x(\alpha t)$, period $T' = T/\alpha$, $\omega_0' = \alpha\omega_0$: $$ \begin{aligned} b_k &= \frac{1}{T'}\int_0^{T'} x(\alpha t)e^{-jk\alpha\omega_0 t}dt \qquad (\text{put } \tau = \alpha t,\ dt = d\tau/\alpha) \\ &= \frac{\alpha}{T}\int_0^{T}x(\tau)e^{-jk\omega_0\tau}\frac{d\tau}{\alpha} \\ &= \frac{1}{T}\int_0^T x(\tau)e^{-jk\omega_0\tau}d\tau = a_k \end{aligned} $$ Hence $b_k = a_k$. #### Interpretation - Compressing in time ($\alpha > 1$) spreads the spectral lines further apart in frequency; expanding ($\alpha < 1$) brings them closer. - The amplitude and phase of each harmonic are unchanged, only its frequency $k\omega_0$ becomes $k\alpha\omega_0$. **Example:** $x(t) = \cos(2t)$ has $a_{\pm1} = 1/2$ at $\omega_0 = 2$. Then $x(3t) = \cos(6t)$ still has $a_{\pm1} = 1/2$, now at $\omega_0' = 6$ rad/s. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What happens to the Fourier coefficients under x(alpha t), alpha > 0? A. They are multiplied by alpha B. They are divided by alpha C. They stay a_k D. They become a_(k/alpha) Answer: C. Coefficients are unchanged; only the fundamental frequency changes. (Key point: Statement) 2. What is the period of x(alpha t)? A. alpha T B. T/alpha C. T D. T alpha^2 Answer: B. Compressing time by alpha divides the period by alpha. (Key point: New period) 3. Which substitution is used in the analysis-route proof? A. tau = t - t0 B. tau = alpha t, dt = d tau / alpha C. tau = -t D. tau = t/alpha, dt = alpha d tau Answer: B. The factor alpha from dt cancels the 1/T' = alpha/T, leaving a_k. (Key point: Proof substitution) 4. What happens to the spectral lines when alpha > 1? A. They move closer together B. They spread further apart at spacing alpha w0 C. Their heights increase D. Their phases flip Answer: B. Harmonics move to k alpha w0, so lines are further apart while heights stay. (Key point: Effect on spectrum) ## Close variants set in other exams - [Time-scaling property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-time-scaling) — Signals and Systems, chapter 2, asked 6× ## Related in Signals and Systems - [CTFS time shifting and time scaling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-shift-scale-properties) — Signals and Systems, chapter 2, asked 2× - [CTFS time reversal, scaling, frequency shift](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-reversal-scaling-shift) — Signals and Systems, chapter 2 - [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× - [Exponential Fourier series of A sin ω₀n](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aexponential-fs-sine) — Signals and Systems, chapter 2 - [Convolution property of CTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-convolution-property) — Signals and Systems, chapter 2 - [CTFS shifting, scaling and conjugation](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-shift-scale-conjugation) — Signals and Systems, chapter 2 ## 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 property: Scaling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-scaling) — Communication Systems, chapter 1 - [Exponential Fourier series of sawtooth wave](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Afourier-exponential-form-sawtooth) — Electrical Circuit Theory, chapter 6 - [Fourier transform property: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1