---
title: "Time-shifting property of DTFS"
subject: "Signals and Systems"
chapter: 2
key: "semester_4/signals_and_systems:dtfs-time-shifting"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/dtfs-time-shifting
site: BCT Engineering
---

# Time-shifting property of DTFS

Signals and Systems · Chapter 2: Fourier Series

- Asked in: 2078 Chaitra, 4 marks; 2075 Baisakh, 4 marks (asked 2 times)
- Video: [Time Shift | Properties of DTFS | EnggClasses](https://www.youtube.com/watch?v=8j1CEnWWS64) — EnggClasses
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/dtfs-time-shifting
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2

## Question

State and prove the time-shifting property for discrete-time Fourier series pair.

## Short answer

If x[n] ↔ a_k, then x[n - n0] ↔ a_k e^{-j k (2π/N) n0}. Proof: substitute m = n - n0 in the analysis summation. Magnitudes stay same, phase changes linearly.

## Answer

#### Statement
If $x[n]$ is periodic with period $N$ and $x[n] \overset{FS}{\longleftrightarrow} a_k$, then

$$
x[n - n_0] \overset{FS}{\longleftrightarrow} a_k\, e^{-jk(2\pi/N)n_0}
$$

A shift in time multiplies each coefficient by a linear phase term; the magnitudes $|a_k|$ do not change.

#### Proof
$x[n - n_0]$ is also periodic with period $N$. Its coefficients are

$$
b_k = \frac{1}{N}\sum_{n=\langle N\rangle}x[n - n_0]\,e^{-jk(2\pi/N)n}
$$

Put $m = n - n_0$, so $n = m + n_0$. As $n$ runs over one period, $m$ also runs over one period:

$$
\begin{aligned}
b_k &= \frac{1}{N}\sum_{m=\langle N\rangle}x[m]\,e^{-jk(2\pi/N)(m + n_0)} \\
&= e^{-jk(2\pi/N)n_0}\left[\frac{1}{N}\sum_{m=\langle N\rangle}x[m]e^{-jk(2\pi/N)m}\right] \\
&= e^{-jk(2\pi/N)n_0}\,a_k
\end{aligned}
$$

#### Interpretation
- $|b_k| = |a_k|$: the magnitude spectrum is unchanged by a shift.
- $\angle b_k = \angle a_k - k\frac{2\pi}{N}n_0$: the phase changes linearly with $k$.

**Example:** $x[n] = \cos(2\pi n/N)$ has $a_{\pm1} = 1/2$. Then $x[n-1]$ has $b_{1} = \frac{1}{2}e^{-j2\pi/N}$ and $b_{-1} = \frac{1}{2}e^{j2\pi/N}$, still of magnitude $1/2$.

## Check yourself

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

1. What is the DTFS of x[n - n0]?
   A. a_k e^(jk(2pi/N)n0)
   B. a_k e^(-jk(2pi/N)n0)
   C. a_(k-n0)
   D. a_k n0
   Answer: B. A delay multiplies a_k by the linear-phase factor e^(-jk(2pi/N)n0). (Key point: Statement)

2. What happens to |a_k| under a shift?
   A. It is unchanged
   B. It is multiplied by n0
   C. It is halved
   D. It becomes zero
   Answer: A. The multiplying factor has magnitude 1. (Key point: Magnitude)

3. How does the phase change?
   A. Constant shift
   B. Changes linearly with k by -k(2pi/N)n0
   C. Doubles
   D. Unchanged
   Answer: B. Angle b_k = angle a_k - k(2pi/N)n0. (Key point: Phase)

4. What substitution is used?
   A. m = n - n0, so n = m + n0, and m also covers one period
   B. m = 2n
   C. m = n/n0
   D. m = -n
   Answer: A. Since sum covers a period, the shifted index still covers one period, and the exponential splits. (Key point: Proof step)

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

## Related in Signals and Systems

- [Derive the discrete-time Fourier series](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-derivation) — Signals and Systems, chapter 2, asked 15×
- [Time shifting property of CTFT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-time-shifting) — Signals and Systems, chapter 3, asked 5×
- [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
- [Frequency shifting property of CTFT](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-frequency-shifting) — Signals and Systems, chapter 3, asked 3×
- [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×
- [Convolution property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-convolution-property) — Signals and Systems, chapter 2

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