---
title: "CTFS time shifting and time scaling"
subject: "Signals and Systems"
chapter: 2
key: "semester_4/signals_and_systems:ctfs-shift-scale-properties"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctfs-shift-scale-properties
site: BCT Engineering
---

# CTFS time shifting and time scaling

Signals and Systems · Chapter 2: Fourier Series

- Asked in: 2074 Bhadra, 4 marks; 2083 Bhadra (new course), 4 marks (asked 2 times)
- Video: [# 2 : Properties of CTFS : Time - Shift](https://www.youtube.com/watch?v=Q1MLloK6E3U) — Neeharika Academy
- 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-shift-scale-properties
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2

## Question

State and prove time shifting and time scaling properties of continuous time Fourier series.

## Short answer

Time shift: x(t - t0) ↔ a_k e^{-j k ω0 t0}, proof by substituting τ = t - t0. Time scaling: x(αt) for α>0 has period T/α and same coefficients a_k, proof via τ = αt.

## Answer

Let $x(t)$ be periodic with period $T$, $\omega_0 = 2\pi/T$, and $x(t) \overset{FS}{\longleftrightarrow} a_k$, where $a_k = \frac{1}{T}\int_T x(t)e^{-jk\omega_0 t}dt$.

#### Time shifting property
**Statement:**

$$
x(t - t_0) \overset{FS}{\longleftrightarrow} a_k\, e^{-jk\omega_0 t_0}
$$

**Proof:** $y(t) = x(t - t_0)$ has the same period $T$. Its coefficients are

$$
\begin{aligned}
b_k &= \frac{1}{T}\int_T x(t - t_0)e^{-jk\omega_0 t}dt \qquad (\tau = t - t_0) \\
&= \frac{1}{T}\int_T x(\tau)e^{-jk\omega_0(\tau + t_0)}d\tau \\
&= e^{-jk\omega_0 t_0}\,\frac{1}{T}\int_T x(\tau)e^{-jk\omega_0\tau}d\tau = e^{-jk\omega_0 t_0}\,a_k
\end{aligned}
$$

So $|b_k| = |a_k|$ (magnitude unchanged) and the phase changes by $-k\omega_0 t_0$.

#### Time scaling property
**Statement:** for $\alpha > 0$, $x(\alpha t)$ is periodic with period $T/\alpha$ and fundamental frequency $\alpha\omega_0$, and

$$
x(\alpha t) = \sum_{k=-\infty}^{\infty}a_k e^{jk(\alpha\omega_0)t}
$$

The coefficients stay $a_k$; only the fundamental frequency changes.

**Proof:** let $y(t) = x(\alpha t)$, $T' = T/\alpha$:

$$
\begin{aligned}
b_k &= \frac{1}{T'}\int_0^{T'}x(\alpha t)e^{-jk(\alpha\omega_0)t}dt \qquad (\tau = \alpha t,\ dt = d\tau/\alpha) \\
&= \frac{\alpha}{T}\int_0^{T}x(\tau)e^{-jk\omega_0\tau}\frac{d\tau}{\alpha} = a_k
\end{aligned}
$$

#### Summary
| Property | Signal | Coefficients | Fundamental |
|---|---|---|---|
| Time shift | $x(t - t_0)$ | $a_k e^{-jk\omega_0 t_0}$ | $\omega_0$ |
| Time scale | $x(\alpha t)$ | $a_k$ | $\alpha\omega_0$ |

Example: $x(t) = \cos\omega_0 t$ ($a_{\pm1} = 1/2$); $x(t - t_0)$ has $a_{\pm1} = \frac{1}{2}e^{\mp j\omega_0 t_0}$; $x(2t) = \cos 2\omega_0 t$ still has $a_{\pm1} = 1/2$ but at fundamental $2\omega_0$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What are the coefficients of x(t - t0)?
   A. a_k e^(jk w0 t0)
   B. a_k e^(-jk w0 t0)
   C. a_(k - t0)
   D. a_k / t0
   Answer: B. Delay gives the phase factor e^(-jk w0 t0). (Key point: Time shift)

2. What happens to |a_k| after a time shift?
   A. Unchanged
   B. Scaled by t0
   C. Reduced by 1/k
   D. Becomes zero for odd k
   Answer: A. The phase factor has unit magnitude. (Key point: Magnitude under shift)

3. What happens to a_k when x(alpha t) is formed?
   A. Divided by alpha
   B. Unchanged; the fundamental becomes alpha w0
   C. Multiplied by alpha
   D. Shifted by alpha places
   Answer: B. Coefficients are the same, only frequency spacing changes. (Key point: Time scaling)

4. Why do the coefficients stay a_k in the proof?
   A. The alpha from dt cancels with 1/T' = alpha/T
   B. The integral is zero
   C. w0 is unchanged
   D. The signal is even
   Answer: A. With tau = alpha t, alpha/T times 1/alpha gives 1/T. (Key point: Proof for scaling)

## Related in Signals and Systems

- [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
- [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
- [CTFS shifting properties; coefficients of A cos ω₀t](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-shifting-cosine-coefficients) — Signals and Systems, chapter 2
- [Time-scaling property of CTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-time-scaling) — Signals and Systems, chapter 2, asked 4×
- [Time shifting, scaling and inversion](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Atime-shift-scale-inversion) — Signals and Systems, chapter 1, asked 2×
- [FS coefficients; time scaling and conjugation](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-significance-scaling-conjugation) — Signals and Systems, chapter 2

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- [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: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1
- [Fourier transform property: Scaling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-scaling) — Communication Systems, chapter 1
- [State and prove Fourier convolution theorem](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aconvolution-theorem-fourier-prove) — Applied Mathematics, chapter 4, asked 3×