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

# DTFS frequency shifting and convolution

Signals and Systems · Chapter 2: Fourier Series

- Asked in: 2073 Magh, 3+3 marks
- Video: [Frequency Shift | Properties of DTFS | EnggClasses](https://www.youtube.com/watch?v=50O1EuLXo4Q) — 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-shifting-convolution
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-2

## Question

State and prove the frequency shifting and convolution properties for discrete time Fourier series pair.

## Short answer

Frequency shifting: e^{jMω₀n} x[n] has DTFS coefficients a_{k-M}. Convolution property: periodic convolution Σ_{r=0}^{N-1} x[r] y[n-r] has coefficients N a_k b_k.

## Answer

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

#### Frequency shifting property
**Statement:** multiplying a signal by a harmonic exponential shifts its coefficients:
$$
e^{jM\omega_0 n}x[n] \xleftrightarrow{FS} a_{k-M}
$$
**Proof:** let $z[n] = e^{jM(2\pi/N)n}x[n]$. Its coefficients are
$$
\begin{aligned}
c_k &= \frac{1}{N}\sum_{n=\langle N\rangle} e^{jM\omega_0 n}x[n]e^{-jk\omega_0 n} \\
&= \frac{1}{N}\sum_{n=\langle N\rangle} x[n]e^{-j(k-M)\omega_0 n} = a_{k-M}
\end{aligned}
$$
So multiplying by $e^{jM\omega_0 n}$ moves the whole spectrum $M$ places to the right. This is the DT form of modulation.

#### Convolution property (periodic convolution)
**Statement:** periodic convolution in time corresponds to multiplication of the coefficients, scaled by $N$:
$$
\sum_{r=\langle N\rangle} x[r]\,y[n-r] \xleftrightarrow{FS} N a_k b_k
$$
**Proof:** let $z[n] = \sum_{r=\langle N\rangle} x[r]y[n-r]$, which is also periodic with period $N$. Then
$$
\begin{aligned}
c_k &= \frac{1}{N}\sum_{n=\langle N\rangle}\sum_{r=\langle N\rangle} x[r]\,y[n-r]\,e^{-jk\omega_0 n} \\
&= \frac{1}{N}\sum_{r=\langle N\rangle} x[r]e^{-jk\omega_0 r}\sum_{n=\langle N\rangle} y[n-r]e^{-jk\omega_0 (n-r)}
\end{aligned}
$$
Put $m = n - r$. Because $y[m]e^{-jk\omega_0 m}$ is periodic with period $N$, the inner sum over any $N$ consecutive $m$ is
$$
\sum_{m=\langle N\rangle} y[m]e^{-jk\omega_0 m} = N b_k
$$
Therefore
$$
c_k = \frac{1}{N}\cdot N b_k\sum_{r=\langle N\rangle} x[r]e^{-jk\omega_0 r} = N b_k\cdot a_k = N a_k b_k
$$
**Dual (multiplication) property:** $x[n]y[n] \xleftrightarrow{FS} \sum_{l=\langle N\rangle} a_l b_{k-l}$, which is periodic convolution of the coefficients.

## Check yourself

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

1. Multiplying x[n] by e^{jM w0 n} produces coefficients:
   A. a_{k+M}
   B. a_{k-M}
   C. a_k e^{jM}
   D. M a_k
   Answer: B. The exponent becomes -j(k-M)w0 n, giving a_{k-M}. (Key point: Frequency shifting)

2. The frequency-shift property moves the spectrum:
   A. M places to the left
   B. Without moving it
   C. To negative frequencies only
   D. M places to the right
   Answer: D. Replacing k by k-M moves the whole coefficient sequence M places to the right. (Key point: Direction of shift)

3. Periodic convolution of x and y in time corresponds to coefficients:
   A. a_k b_k
   B. N a_k b_k
   C. a_k + b_k
   D. a_k * b_k (convolution)
   Answer: B. c_k = N a_k b_k. (Key point: Convolution property)

4. In the convolution proof, the inner sum of y[m] e^{-jk w0 m} over a period equals:
   A. b_k
   B. N b_k
   C. b_k/N
   D. N^2 b_k
   Answer: B. By definition b_k has a 1/N factor, so the sum itself equals N b_k. (Key point: Proof step)

## Related in Signals and Systems

- [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×
- [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×
- [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×
- [DTFT conjugation and frequency shifting](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtft-conjugate-frequency-shift) — Signals and Systems, chapter 3
- [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×
- [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

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