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

# DTFS linearity and time shift

Signals and Systems · Chapter 2: Fourier Series

- Asked in: 2072 Asoj, 6 marks
- 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-linearity-time-shift
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2

## Question

State and prove the linearity and time shift properties of discrete time Fourier series representation.

## Short answer

Linearity: for z[n]=A x[n]+B y[n], the DTFS coefficients are c_k = A a_k + B b_k. Time shift: x[n-n0] has coefficients a_k e^{-j k ω0 n0}, so magnitude remains unchanged, phase shifted linearly proportional to k and n0.

## Answer

Let $x[n]$ and $y[n]$ be periodic with period $N$ ($\omega_0 = 2\pi/N$), with $x[n] \xleftrightarrow{FS} a_k$ and $y[n] \xleftrightarrow{FS} b_k$, where
$$
a_k = \frac{1}{N}\sum_{n=\langle N\rangle} x[n]e^{-jk\omega_0 n}
$$

#### Linearity
**Statement:** for any constants $A$ and $B$,
$$
z[n] = A x[n] + B y[n] \xleftrightarrow{FS} c_k = A a_k + B b_k
$$
**Proof:** $z[n]$ is also periodic with period $N$. Then
$$
\begin{aligned}
c_k &= \frac{1}{N}\sum_{n=\langle N\rangle}\left(A x[n] + B y[n]\right)e^{-jk\omega_0 n} \\
&= A\cdot\frac{1}{N}\sum_{n=\langle N\rangle} x[n]e^{-jk\omega_0 n} + B\cdot\frac{1}{N}\sum_{n=\langle N\rangle} y[n]e^{-jk\omega_0 n} \\
&= A a_k + B b_k
\end{aligned}
$$
**Example:** if $x[n] = \cos(2\pi n/N)$ ($a_{\pm1} = 1/2$) and $y[n] = 1$ ($b_0 = 1$), then $3x[n] + 2y[n]$ has $c_0 = 2$ and $c_{\pm1} = 3/2$.

#### Time shifting
**Statement:** a delay of $n_0$ samples multiplies each coefficient by a linear-phase term:
$$
x[n - n_0] \xleftrightarrow{FS} a_k e^{-jk\omega_0 n_0} = a_k e^{-jk(2\pi/N)n_0}
$$
**Proof:** let $z[n] = x[n - n_0]$. Its coefficients are
$$
c_k = \frac{1}{N}\sum_{n=\langle N\rangle} x[n-n_0]e^{-jk\omega_0 n}
$$
Put $m = n - n_0$, so $n = m + n_0$. As $n$ runs over one period, so does $m$:
$$
\begin{aligned}
c_k &= \frac{1}{N}\sum_{m=\langle N\rangle} x[m]e^{-jk\omega_0 (m+n_0)} \\
&= e^{-jk\omega_0 n_0}\cdot\frac{1}{N}\sum_{m=\langle N\rangle} x[m]e^{-jk\omega_0 m} = e^{-jk\omega_0 n_0}a_k
\end{aligned}
$$

**Meaning:** $|c_k| = |a_k|$, so a time shift does not change the magnitude spectrum. It only adds a phase of $-k\omega_0 n_0$, which is linear in $k$.

## Check yourself

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

1. If z[n] = A x[n] + B y[n], then c_k equals:
   A. A a_k + B b_k
   B. AB a_k b_k
   C. A a_k b_k
   D. (A+B)a_k
   Answer: A. The analysis sum is linear. (Key point: Linearity)

2. For x[n] = cos(2 pi n/N) and y[n] = 1, what are the coefficients of 3x[n] + 2y[n]?
   A. c_0 = 3, c_{+-1} = 1
   B. c_0 = 2, c_{+-1} = 3/2
   C. c_0 = 5, c_{+-1} = 1/2
   D. c_0 = 2, c_{+-1} = 3
   Answer: B. c_0 = 2 b_0 = 2 and c_{+-1} = 3 a_{+-1} = 3/2. (Key point: Linearity example)

3. x[n - n0] has coefficients:
   A. a_{k-n0}
   B. a_k e^{-jk(2 pi/N)n0}
   C. a_k e^{jk(2 pi/N)n0}
   D. a_k n0
   Answer: B. Substituting m = n - n0 pulls out the factor e^{-jk w0 n0}. (Key point: Time shift result)

4. What does a time shift do to the magnitude and phase spectra?
   A. Changes the magnitude, not the phase
   B. Changes both
   C. Leaves the magnitude unchanged and adds a phase linear in k
   D. Leaves both unchanged
   Answer: C. |c_k| = |a_k|, with extra phase -k w0 n0 that is linear in k. (Key point: Effect on spectra)

## 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×
- [Four properties of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-four-properties) — Signals and Systems, chapter 2
- [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×
- [Time-shifting property of DTFS](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-time-shifting) — Signals and Systems, chapter 2, asked 5×
- [CTFT linearity and time shifting](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actft-linearity-time-shifting) — Signals and Systems, chapter 3
- [DTFT linearity and time shifting](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-linearity-time-shift) — Signals and Systems, chapter 3

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