# Advanced Signals and Systems - Linear and Cyclic Convolution

### 17. Linear convolution of sequences.

Find the convolution sum $$v(n) = v_1(n) \ast v_2(n)$$ of the following sequences $$v_1(n)$$ and $$v_2(n)$$

\begin{align} v_1(n) =& \rho _1^n \cdot \gamma_{-1}(n) \nonumber \\ v_2(n) =& \rho _2^n \cdot \gamma_{-1}(n)\nonumber \end{align}

where $$0 < \rho_1$$ and $$\rho_2 < 1$$.

## Amount and difficulty

• Working time: approx. xx minutes
• Difficulty: xx

## Solution

\begin{align*} v(n) &= \sum \limits_{k=-\infty}^{\infty} v_1(k) \cdot v_2(n-k) \\ &= \sum \limits_{k=-\infty}^{\infty} \rho_1^k \cdot \gamma_{-1}(k) \cdot \rho_2^{n-k} \cdot \gamma_{-1}(n-k) \\ &= \cdots\\ &= \begin{cases} \frac{\rho_1^{n+1}-\rho_2^{n+1}}{\rho_1-\rho_2}\cdot \gamma_{-1}(n) &, \rho_1\neq \rho_2\\ \rho_2^n\cdot(n+1)\cdot \gamma_{-1}(n) &, \rho_1 = \rho_2 \end{cases}\end{align*}

### 18. Linear and cyclic convolution.

Given two sequences $$v_1(n)$$ and $$v_2(n)$$ of length $$M=5$$:

\begin{align} v_1(n) =& [5,4,3,2,1] \nonumber \\ v_2(n) =& [1,2,3,4,5]\nonumber \end{align}

Determine the linear convolution $$v_3(n)$$ and the cyclic convolution $$v_4(n)$$ of the sequences. Give a method to calculate the linear convolution.

## Amount and difficulty

• Working time: approx. xx minutes
• Difficulty: xx

Linear convolution

Cyclic convolution

Calculating the linear convolution by cyclic convolution:

## Solution

• Linear convolution

\begin{align*} v_3(n) &= v_1(n) * v_2(n) = \sum \limits_{k=-\infty}^{\infty} v_1(k) \cdot v_2(n-k)\\ &\cdots\\ v_3(n) &= [5, 14, 26, 40, 55, 40, 26, 14, 5] \ \ \ \text{ for } \ \ 0\leq n \leq M-1 \end{align*}

• Cyclic convolution

\begin{align*} v_4(n) &= \sum \limits_{k=-\infty}^{\infty} v_1(k) \cdot v_2(\text{mod}(n-k,M))\\ &\cdots\\ v_4(n) &= [45, 40, 40, 45, 55] \end{align*}

• Modified cyclic convolution to get correct result

The cyclic convolution of $$v'_1(n)$$ and $$v'_2(n)$$ is calculated, where $$v'_1(n)$$ and $$v'_2(n)$$ denote the sequences $$v_1(n)$$ and $$v_2(n)$$ padded with $$M-M_2$$ and $$M-M_1$$ zeros, where $$M$$ is the length of the resulting sequence $$v_5(n)$$ and $$M=M_1+M_2+1$$.

\begin{align*} v_5(n) &= \sum \limits_{k=-\infty}^{\infty} v'_1(k) \cdot v'_2(\text{mod}(n-k,M))\\ &\cdots\\ v_5(n) &= [5, 14, 26, 40, 55, 40, 26, 14, 5] \ \ \ \text{ for } \ \ 0\leq n \leq M-1 \end{align*}

### Website News

13.08.2017: New Gas e.V. sections (e.g. pictures or prices) added.

05.08.2017: The first "slide carousel" added.

03.08.2017: Started with the RED project. Will be ready in a few years ...

30.07.2017: List of PhD theses updated and extended.

### Recent Publications

P. Durdaut, J. Reermann, S. Zabel, Ch. Kirchhof, E. Quandt, F. Faupel, G. Schmidt, R. Knöchel, and M. Höft: Modeling and Analysis of Noise Sources for Thin-Film Magnetoelectric Sensors Based on the Delta-E Effect, IEEE Transactions on Instrumentation and Measurement, published online, 2017

P. Durdaut, S. Salzer, J. Reermann, V. Röbisch, J. McCord, D. Meyners, E. Quandt, G. Schmidt, R. Knöchel, and M. Höft: Improved Magnetic Frequency Conversion Approach for Magnetoelectric Sensors, IEEE Sensors Letters, published online, 2017

### Contact

Prof. Dr.-Ing. Gerhard Schmidt

E-Mail: gus@tf.uni-kiel.de

Christian-Albrechts-Universität zu Kiel
Faculty of Engineering
Institute for Electrical Engineering and Information Engineering
Digital Signal Processing and System Theory

Kaiserstr. 2
24143 Kiel, Germany

## Recent News

Jens Reermann Defended his Dissertation with Distinction

On Friday, 21st of June, Jens Reermann defended his research on signals processing for magnetoelectric sensor systems very successfully. After 90 minutes of talk and question time he finished his PhD with distinction. Congratulations, Jens, from the entire DSS team.

Jens worked for about three and a half years - as part of the collaborative research center (SFB) 1261 - on all kinds of signal ...