Advanced Signals and Systems - Z Transform / Pole-zero Plot

 

12. Symmetry relations of Z-transform.

Task

Given the sequence \(v(n)\) and its Z-transform \(V(z)\), determine the Z-transform of the following sequences:

  1. \(v_1(n) = v(-n)\)
  2. \(v_2(n) = v^*(n)\)
  3. \(v_3(n) = v(n-k)\)

Amount and difficulty

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

Solution

  1. \(v_1(n) = v(-n)\) \begin{equation*} V_1(z) = \sum \limits_{n=-\infty}^\infty v_1(n) z^{-n} = \cdots = \sum \limits_{n=-\infty}^\infty v(n) \left( z^{-1} \right)^{-n} = V(z^{-1}) \end{equation*}

  2. \(v_2(n) = v^*(n)\) \begin{equation*} V_2(z) = \sum \limits_{n=-\infty}^\infty v(n)^* z^{-n} = \cdots = V^*(z^{*}) \end{equation*}

  3. \(v_3(n) = v(n-k)\) \begin{equation*} V_3(z) = \sum \limits_{n=-\infty}^\infty v(n-k)^* z^{-n} = \cdots = z^{-k} \cdot V^*(z^{*}) \end{equation*}

 

13. Z and inverse Z-transform.

Task

Given the sequence \(v(n)\) and its Z-transform \(V(z)\), determine the Z-transform of the following sequences:

  1. Find the z transform of the sequence \(v(n)\) \begin{equation} v(n) = \begin{cases} 1 & ,|n|\leq N\\ 0 & , \text{else} \;\;\;\;\; \text{ .} \nonumber \end{cases} \nonumber \end{equation}
  2. Given the z transform \(Y(z)\) of a sequence \(y(n)\) \begin{equation} Y(z) = \frac{23z^3 - 34z^2 - 28z + 56}{z^5 - 5z^4 + 6z^3 + 4z^2 - 8z} \text{ ,} \nonumber \end{equation} find \(y(n)\) using partial fraction expansion.

Amount and difficulty

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

Solution

  1. The Z-transform can be derived by applying the definition and using a geometric series. \begin{align*} V(z) &= \sum \limits_{n=-\infty}^\infty v(n) z^{-n} = \sum \limits_{n=0}^N z^{n} + \sum \limits_{n=0}^N z^{-n} - 1 = \cdots \\ &= \frac{z^{-N}-z^{N+1}}{1-z} \end{align*}

  2. Any ratio of polynomials \(V(z) = \frac{V_1(z)}{V_2(z)}\) can be expressed by partial fraction expansion \begin{equation*} V(z) = \sum \limits_{\nu=1}^{N_0}\sum \limits_{\mu=1}^{N_\nu} B_{\nu\mu} \frac{1}{(z-z_{\infty\nu})^\mu} \end{equation*} where

    • \(N_0=\) number of distinct pole
    • \(N_\nu=\) order of the \(\nu\)th pole
    • \(\nu=\) index of the pole
    • \(\mu=\) index within a value \(\leq N_\nu\)

    But the inverse transform of \( \frac{1}{(z-z_{\infty\nu})^\mu}\) is not listed in transform tables, therefore, the addend of the partial fraction is extended by \(\frac{1}{z}\). Hence, the partial fraction is done with \(\frac{Y(z)}{z}\) where the following poles and coefficients can be found: \begin{align*} z_{\infty 0} &= 0 & z_{\infty 1} &= 0 & z_{\infty 2} &= -1 & z_{\infty 3} &= 2 & z_{\infty 4} &= 2 & z_{\infty 5} &= 2 \\ B_{01} &= 0 & B_{02} &= -7 & B_{11} &= -1 & B_{21} &= 1 & B_{22} &= 4 & B_{23} &= 4 \end{align*} Transforming \(Y(z)\) by using a transformation table leads to \begin{equation*} y(n) = -7\gamma_0(n-1) + \left[ (-1)^{n+1} +2^{n-1}(n^2+3n+2) \right] \gamma_{-1}(n) \end{equation*}

 

14. Pole-zero plot.

Task

Given is the following pole-zero plot, belonging to the z transform \(X(z)\) of a causal sequence \(x(n)\):

In the following it is assumed that \begin{equation} y(n) = \left( \frac{1}{2} \right)^n \cdot x(n) \nonumber \end{equation}

  1. Define the poles and the zeros of \(Y(z)\).
  2. Sketch the pole-zero plot of \(Y(z)\) and the region of convergence (ROC).

Amount and difficulty

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

Solution

  1. The following z transform can be extracted from the pole-zero plot: \begin{equation*} X(z) = K \cdot \frac{(z+j)(z-j)}{z-\frac{1}{2}} = K \cdot \frac{z^2+1}{z- \frac{1}{2}} \end{equation*} Additionally we do know that \begin{equation*} z_0^n f(n) \ \ \ \circ-\bullet \ \ \ F \left(\frac{z}{z_0}\right) \end{equation*} so it follows \begin{equation*} Y(z) = K \cdot \frac{(2z)^2+1}{2z- \frac{1}{2}} \text{ .} \end{equation*} Hence the poles and zeros are \begin{equation*} z_{\infty 1} = \frac{1}{4} \text{ and } z_{0 1} = \frac{j}{2}, z_{0 2} = - \frac{j}{2} \text{ .} \end{equation*}

  2. As we have a causal sequence \(|z|>\frac{1}{4}\) has to hold true for the region of convergence.

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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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