Advanced Signals and Systems - Linear and Cyclic Convolution

 

17. Linear convolution of sequences.

Task

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

Additional material

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.

Task

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

Additional material

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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Recent Publications

J. Reermann, P. Durdaut, S. Salzer, T. Demming,A. Piorra, E. Quandt, N. Frey, M. Höft, and G. Schmidt: Evaluation of Magnetoelectric Sensor Systems for Cardiological Applications, Measurement (Elsevier), ISSN 0263-2241, https://doi.org/10.1016/j.measurement.2017.09.047, 2017

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

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 ...


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