Advanced Signals and Systems - Deterministic Systems

 

19. System Classification.

Task

Let a discrete system be described by \(y(n) = a \cdot v(n-1)+b \cdot v^2(n)\). Is the system linear, shift-invariant, causal, dynamic, stable?

Amount and difficulty

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

Solution

A one-dimensional, discrete system \(S\) is given and is described by

\begin{equation*} S\{v(n)\}=y(n)=a\cdot v(n-1)+b\cdot v^2(n) \text{ .} \end{equation*}

  • Linearity:

    The system is linear if the following relation is fulfilled:

    \begin{align*} \alpha_1 \: y_1(n)+\alpha_2 \: y_2(n) &= \alpha_1 \: S\{v_1(n)\}+\alpha_2 \: S\{v_2(n)\}\\ &= S\{\alpha_1 \: v_1(n)+\alpha_2 \: v_2(n)\} \end{align*}

    \(\longrightarrow\) The given system is not linear

  • Shift-invariance:

    The system is shift-invariant if

    \begin{equation*} y(n-k) = S\{v(n-k)\} \end{equation*}

    holds true.

    \(\longrightarrow\) The given system is shift-invariant.

  • Causality:

    The system is causal if

    \begin{equation*} y(n_0) = S\{v(n)\} |_{n\leq n_0} \end{equation*}

    holds true.

    \(\longrightarrow\) The given system is causal.

  • Dynamic:

    A system is dynamic if \(y(n_0)\) does not only depend on \(v(n_0)\), but also on \(v(n\neq n_0)\).

    \(\longrightarrow\) The given system is dynamic.

  • Stability:

    A system is said to be BIBO-stable if

    \begin{align*} |v(n)| \leq M_1 < \infty, \ \ \ \forall \ n \\ |y(n)| \leq M_2 < \infty, \ \ \ \forall \ n \end{align*}

    \(\longrightarrow\) The given system is BIBO-stable.

 

20. Impulse Response, Transfer Function.

Task

Determine the impulse response, step response, and transfer function of the following systems:

  1. \(y(n) = \sum \limits_{k=-\infty}^{n}v(k)\)
  2. \(y(n) = \frac{1}{m} \sum \limits_{k=0}^{m-1}v(n-k)\)

Amount and difficulty

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

Solution

The general affiliations between the different functions are defined in chapter VI of the lecture.

      • Transfer function:

        The transfer function can be found by applying a z transform and the following equation.

        \begin{equation*} H(z) = \frac{Y(z)}{V(z)} = \frac{z}{z-1} \end{equation*}

      • Impulse response:

        \begin{equation*} H(z) \ \ \ \bullet-\circ \ \ \ h_0(n) = \gamma_{-1} (n) \end{equation*}

      • Step response:

        \begin{equation*} h_{-1}(n) = \sum \limits_{k=-\infty}^{n} h_0(k) = \begin{cases} 0 , &n<0\\ n+1, &n\geq 0 \end{cases}\end{equation*}

      • Impulse response:

        \begin{equation*} h_0(n) = \frac{1}{m} \sum \limits_{k=0}^{m-1} \gamma_{0} (n) = \frac{1}{m} \left( \gamma_{-1}(n) - \gamma_{-1}(n-m) \right)\end{equation*}

      • Step response:

        \begin{equation*} h_{-1}(n) = \sum \limits_{k=-\infty}^{n} h_0(k) = \begin{cases} 0 , &n<0\\ \frac{n+1}{m}, & 0\leq n < m-1\\ 1 , & n\geq m-1\end{cases}\end{equation*}

      • Transfer function:

        \begin{align*} h_0(n) \ \ \ \bullet-\circ \ \ \ H(z) &= \frac{1}{m} \left[\frac{z}{z-1}-\frac{z}{z-1} z^{-m}\right] \\ &= \frac{1}{m} \left[ \frac{z-z^{-(m-1)}}{z-1} \right] \end{align*}

 

21. Linear, Time-Invariant System.

Task

When the input to a LTI system is

\begin{equation} v(n) = 5 \cdot \gamma_{-1}(n) \nonumber \end{equation}

the output is

\begin{equation} y(n) = (2\cdot 0.5^{n} + 3 \cdot (-0.75)^{n}) \cdot \gamma_{-1}(n) \text{ .} \nonumber \end{equation}

      1. Find the transfer function \(H(z)\) of the system. Plot the zeros and the poles of \(H(z)\) and indicate the region of convergence. Is the system stable?
      2. Find the impulse response \(h_0(n)\) of the system for all values of \(n\).
      3. Define the difference equation that characterizes the system.

Amount and difficulty

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

Solution

  1. The transfer function is defined by

    \begin{equation*}H(z) = \frac{Y(z)}{V(z)} \end{equation*}

    Transforming the input and output sequences to the z domain and inserting this to the given equation yield to the desired transfer function

    \begin{equation*}H(z) = \frac{z(z-1)}{(z-0,5)(z+0,75)} \ \ \ |z|>0,75 \text{.} \end{equation*}

    The Figure shows the zeros and poles of the system. As the poles lie within the unit circle the system is stable.

  2. Using the partial fraction expantion for \(\frac{H(z)}{z}\) and transforming this back to time domain the desired impulse response can befound.

    \begin{equation*}h_0(n) = -0,4\cdot (0,5)^n\cdot \gamma_{-1}(n) + 1,4 \cdot (-0,75)^n \cdot \gamma_{-1}(n)\text{.}\end{equation*}

  3. The difference equation can be found by rearranging the transfer function and applying an inverse z transform.

    \begin{align*} \frac{Y(z)}{V(z)} &= \frac{1-z^{-1}}{1+\frac{1}{4}z^{-1}-\frac{3}{8}z^{-2}} \\ \rightarrow &y(n) = v(n) -v(n-1)-\frac{1}{4}y(n-1)+\frac{3}{8}y(n-2) \end{align*}

Website News

03.12.2017: Added pictures from our Sylt meeting.

01.10.2017: Started with a Tips and Tricks section for KiRAT.

01.10.2017: Talks from Jonas Sauter (Nuance) and Vasudev Kandade Rajan (Harman/Samsung) added.

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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

Jugend Forscht

On November 24th, one of our DSS team members, Owe Wisch, took part in the "Jugend forscht Perspektivforum" at the CAU. Thirty young students from the "Jugend forscht" project came to Kiel and participated in three different workshops focusing on career paths in maritime climate protection. Owe Wisch from our chair lead one of the workshops and presented his research topics, beamforming ...


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