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

# CTFS time reversal, scaling, frequency shift

Signals and Systems · Chapter 2: Fourier Series

- Asked in: 2083 Baisakh (new course), 4 marks
- 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-reversal-scaling-shift
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2

## Question

Describe the continuous-time Fourier series' time reversal, time scaling, and frequency shifting properties.

## Short answer

Time reversal: x(-t) ↔ a_{-k}. Time scaling: x(αt) has same coefficients a_k but period T/α and fundamental frequency αω0. Frequency shifting: e^{jMω0 t}x(t) ↔ a_{k-M}, moving the spectrum M places to the right.

## Answer

Let $x(t)$ be periodic with period $T$, $\omega_0 = 2\pi/T$, and $x(t) \xleftrightarrow{FS} a_k$.

#### Time reversal
$$
x(-t) \xleftrightarrow{FS} a_{-k}
$$
Put $k = -m$ in the synthesis equation:
$$
x(-t) = \sum_k a_k e^{-jk\omega_0 t} = \sum_m a_{-m}e^{jm\omega_0 t}
$$
Reversing the signal in time reverses its coefficient sequence. If $x(t)$ is even, $a_{-k} = a_k$. If it is odd, $a_{-k} = -a_k$.

#### Time scaling
$$
x(\alpha t),\ \alpha > 0: \quad x(\alpha t) = \sum_k a_k e^{jk(\alpha\omega_0)t}
$$
$x(\alpha t)$ has period $T/\alpha$ and fundamental frequency $\alpha\omega_0$. The coefficients $a_k$ stay the same, and only the harmonic spacing changes from $\omega_0$ to $\alpha\omega_0$. Compressing the signal ($\alpha > 1$) spreads the spectral lines apart.

#### Frequency shifting
$$
e^{jM\omega_0 t}x(t) \xleftrightarrow{FS} a_{k-M}
$$
The coefficients of the product are
$$
\frac{1}{T}\int_T e^{jM\omega_0 t}x(t)e^{-jk\omega_0 t}dt = \frac{1}{T}\int_T x(t)e^{-j(k-M)\omega_0 t}dt = a_{k-M}
$$
Multiplying by a harmonic exponential moves the whole line spectrum $M$ places to the right. This is the dual of the time-shift property.

## Check yourself

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

1. If x(t) has coefficients a_k, then x(-t) has coefficients:
   A. a_k*
   B. -a_k
   C. a_{-k}
   D. a_{k+1}
   Answer: C. Putting k = -m in the synthesis equation shows x(-t) = Σ a_{-m} e^{jmω0 t}. (Key point: Time reversal)

2. From time reversal, for an odd signal x(t) we get:
   A. a_{-k} = a_k
   B. a_{-k} = -a_k
   C. a_k = 0 for all k
   D. a_{-k} = a_k*
   Answer: B. If x(-t) = -x(t) then a_{-k} = -a_k; for even signals a_{-k} = a_k. (Key point: Even and odd signals)

3. For α > 0, the period and fundamental frequency of x(αt) are:
   A. αT and ω0/α
   B. T/α and αω0
   C. T and ω0
   D. T/α and ω0
   Answer: B. x(αt) repeats every T/α, so its fundamental frequency is αω0, with coefficients unchanged. (Key point: Time scaling)

4. Multiplying x(t) by e^{jMω0 t} gives coefficients:
   A. a_{k+M}
   B. a_{k-M}
   C. M a_k
   D. a_k e^{jMω0}
   Answer: B. The product's coefficient integral becomes (1/T)∫x(t)e^{-j(k-M)ω0 t}dt = a_{k-M}, which moves the spectrum M places right. (Key point: Frequency shifting)

## 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 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 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×
- [DTFS frequency shifting and convolution](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-shifting-convolution) — Signals and Systems, chapter 2

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- [Fourier transform property: Time shifting](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aft-property-time-shifting) — Communication Systems, chapter 1
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