# 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*}

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T. O. Wisch, T. Kaak, A. Namenas, G. Schmidt: Spracherkennung in stark gestörten Unterwasserumgebungen, Proc. DAGA 2018

S. Graf, T. Herbig, M. Buck, G. Schmidt: Low-Complexity Pitch Estimation Based on Phase Differences Between Low-Resolution Spectra, Proc. Interspeech, pp. 2316 -2320, 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

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Since January this year we have two new PhD students in the team: Elke Warmerdam and Finn Spitz.

Elke is from Amsterdam and she works in the neurology department in the university hospital in the group of Prof. Maetzler. Her research topic is movement analysis of patients with neurologic disorders. Elke cooperates with us in signal processing related aspects of her research. Elke plays ...