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.

Website News

20.01.2017: Talk from Dr. Sander-Thömmes added.

12.01.2018: New RED section on Trend Removal added.

29.12.2017: Section Years in Review added.

28.12.2017: Update of our SONAR section.

03.12.2017: Added pictures from our Sylt meeting.

Recent Publications

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

Recent News

New PhDs in the DSS Team

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 ...


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