---
title: "CTFS shifting properties; coefficients of A cos ω₀t"
subject: "Signals and Systems"
chapter: 2
key: "semester_4/signals_and_systems:ctfs-shifting-cosine-coefficients"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ctfs-shifting-cosine-coefficients
site: BCT Engineering
---

# CTFS shifting properties; coefficients of A cos ω₀t

Signals and Systems · Chapter 2: Fourier Series

- Asked in: 2082 Chaitra, 6 marks
- Video: [3.5.2 Time Shifting (Properties of CTFS)](https://www.youtube.com/watch?v=QPAvuBPSirg) — Junaid MIR
- 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-shifting-cosine-coefficients
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2

## Question

Describe the time shifting and frequency shifting properties of continuous time Fourier series. Find Fourier series coefficient of signal x(t) = A cos(ω₀t).

## Short answer

Time shift: x(t-t0) ↔ a_k e^{-j k ω0 t0}. Frequency shift: e^{j M ω0 t} x(t) ↔ a_{k-M}. For A cos(ω0 t), coefficients are a_1 = A/2, a_{-1} = A/2, all others zero.

## Answer

Let $x(t)$ have period $T$, $\omega_0 = 2\pi/T$, and $x(t) \overset{FS}{\longleftrightarrow} a_k$.

#### Time shifting property
$$
x(t - t_0) \overset{FS}{\longleftrightarrow} a_k e^{-jk\omega_0 t_0}
$$

Proof sketch: with $\tau = t - t_0$,

$$
b_k = \frac{1}{T}\int_T x(t - t_0)e^{-jk\omega_0 t}dt = e^{-jk\omega_0 t_0}\frac{1}{T}\int_T x(\tau)e^{-jk\omega_0\tau}d\tau = e^{-jk\omega_0 t_0}a_k
$$

A delay leaves $|a_k|$ unchanged and adds a phase $-k\omega_0 t_0$ that grows linearly with $k$.

#### Frequency shifting property
$$
e^{jM\omega_0 t}\,x(t) \overset{FS}{\longleftrightarrow} a_{k-M}
$$

Proof:

$$
b_k = \frac{1}{T}\int_T e^{jM\omega_0 t}x(t)e^{-jk\omega_0 t}dt = \frac{1}{T}\int_T x(t)e^{-j(k-M)\omega_0 t}dt = a_{k-M}
$$

Multiplying by a complex exponential at the $M$th harmonic shifts the whole set of coefficients by $M$ places (this is the basis of modulation).

#### Fourier series coefficients of $x(t) = A\cos(\omega_0 t)$
Using Euler's formula:

$$
x(t) = A\cos\omega_0 t = \frac{A}{2}e^{j\omega_0 t} + \frac{A}{2}e^{-j\omega_0 t}
$$

Comparing with $x(t) = \sum_k a_k e^{jk\omega_0 t}$:

$$
a_1 = a_{-1} = \frac{A}{2}, \qquad a_k = 0 \text{ for } k \ne \pm1
$$

Check with the analysis equation for $k = 1$:

$$
a_1 = \frac{1}{T}\int_0^T A\cos\omega_0 t\,e^{-j\omega_0 t}dt = \frac{A}{2T}\int_0^T\left(1 + e^{-j2\omega_0 t}\right)dt = \frac{A}{2}
$$

```text
 a_k
 A/2      |       |
      ----+---+---+----> k
         -1   0   1
```

**Answer:** $a_{\pm1} = A/2$ (real, zero phase), all other coefficients are zero.

## Check yourself

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

1. What is the effect of a delay t0?
   A. a_k multiplied by e^(-jk w0 t0), magnitude unchanged
   B. a_k multiplied by t0
   C. a_k shifted to a_(k - t0)
   D. a_k conjugated
   Answer: A. A delay adds linear phase -k w0 t0. (Key point: Time shift)

2. What is the FS of e^(jM w0 t) x(t)?
   A. a_(k+M)
   B. a_(k-M)
   C. M a_k
   D. a_k e^(jM)
   Answer: B. Multiplying by the Mth harmonic exponential shifts coefficients by M places. (Key point: Frequency shift)

3. What are the coefficients of A cos(w0 t)?
   A. a_1 = a_-1 = A/2
   B. a_1 = a_-1 = A
   C. a_1 = A/2, a_-1 = -A/2
   D. a_0 = A/2
   Answer: A. A cos = (A/2)(e^(j w0 t) + e^(-j w0 t)). (Key point: Cosine coefficients)

4. What are a_k for k not equal to +-1?
   A. A/2
   B. A
   C. 0
   D. Depends on k
   Answer: C. Only two exponentials are present. (Key point: Other coefficients)

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