Advanced Signals and Systems - Indealized Linear, Shift-invariant Systems

 

24. Idealized linear, shift-invariant systems.

Task

Given an one-sided ideal bandpass filter with center-frequency \(\Omega_m\), bandwidth \(2\cdot \Delta\Omega\) and linear phase \(\Omega n_0\).

  1. Determine the frequency response \(H^{(1)}(e^{j{\Omega}})\).
  2. Find the impulse response \(h_0^{(1)}(n)\).
  3. Find the frequency response \(H^{(2)}(e^{j\Omega})\) of the corresponding two-sided bandpass filter.
  4. What is the impulse response \(h_0^{(2)}(n)\) of the two-sided bandpass?

Consider now a different linear-phase system with cosinusoidal attenuation ripples

\begin{equation}\nonumber H^{(3)}\left(e^{j\Omega}\right) = A \left( 1+\alpha \cos\left(2\pi \frac{\Omega}{\Omega_0}\right)\right) e^{-j\Omega n_o}, \qquad \Omega_0=\frac{2\pi}{m},m\in \mathbb{N}. \end{equation}

  1. Find the impulse response \(h_0^{(3)}(n)\), assume that \(m=2\).
  2. The output signal of the above system is filtered by an ideal low-pass filter with cut-off frequency \(\Omega_{c}\). Determine the impulse response \(h_0^{(4)}(n)\) of the overall system.

Amount and difficulty

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

Solution

      1. The frequency response \(H^{(1)}(e^{j{\Omega}})\) can be derived by using the given information. It follows that

        \begin{equation*} H^{(1)}(e^{j{\Omega}}) = \begin{cases} A \cdot e^{-j \Omega n_0},& \Omega_m - \Delta \Omega \leq (\Omega - 2 \pi \lambda) \leq \Omega_m - \Delta \Omega \ \ \ \lambda \in \mathbb{Z} \\ 0 ,& \text{ otherwise.} \end{cases} \end{equation*}

      2. The impulse response \(h_0^{(1)}(n)\) can be derived by the inverse Fourier transform of \(H^{(1)}(e^{j{\Omega}})\). In addition one can use a rectangular prototype \(R_{\Delta \Omega} (e^{j{\Omega}})\) where the inverse transform is known.

        \begin{align*} h^{(1)}(n) &= \mathcal{F}^{-1} \left\{ A \cdot R_{\Delta \Omega} (e^{j{\Omega-\Omega_m}}) \cdot e^{-j \Omega n_0} \right\} \\ &= A \cdot \frac{\Delta \Omega}{\pi} \ \text{si}\left(\Delta \Omega(n-n_0)\right) \cdot e^{+j \Omega_m(n-n_0)} \end{align*}

      3. The frequency response \(H^{(2)}(e^{j\Omega})\) of the corresponding two-sided bandpass filter is derived similar to part (a).

        \begin{equation*} H^{(2)}(e^{j{\Omega}}) = A \cdot \left[ R_{\Delta \Omega} (e^{j{\Omega+\Omega_m}}) + R_{\Delta \Omega} (e^{j{\Omega-\Omega_m}}) \right]\cdot e^{-j \Omega n_0} \end{equation*}

      4. The impulse response \(h_0^{(2)}(n)\) of the two-sided bandpass is again calculated by the inverse Fourier transform.

        \begin{align*} h^{(2)}(n) &= \mathcal{F}^{-1} \left\{ H^{(2)}(e^{j{\Omega}}) \right\} \\ &= A \cdot \frac{\Omega_m + \Delta\Omega}{\pi} \ \text{si}\bigg((\Omega_m +\Delta \Omega)(n-n_0)\bigg) \cdots \\ & \ \ \ - A \cdot \frac{\Omega_m - \Delta\Omega}{\pi} \ \text{si}\bigg((\Omega_m -\Delta \Omega)(n-n_0)\bigg) \end{align*}

      5. In order to find the impulse response \(h_0^{(3)}(n)\) the cosine of the frequency response is expanded and the a inverse Fourier transform is applied. The result is given by

        \begin{align*} h^{(3)}(n) &= \mathcal{F}^{-1} \left\{ H^{(3)}(e^{j{\Omega}}) \right\} \\ &= A \ \gamma_0(n-n_0) + A \frac{\alpha}{2} \ \gamma_0(n-n_0+\frac{2\pi}{\Omega_0})+ A \frac{\alpha}{2} \ \gamma_0(n-n_0-\frac{2\pi}{\Omega_0}) \end{align*}

      6. Cascaded filters can be combined to one filter by the multiplication of their two freqeuncy responses. Hence,

        \begin{equation*} H^{(3)}(e^{j{\Omega}}) = A \cdot \left( 1+\alpha \ \cos\left(2\pi\frac{\Omega}{\Omega_0}\right) \right) e^{-j\Omega n_0} \cdot R_{\Omega_c} (e^{j{\Omega}}) \end{equation*}

        and the inverse Fourier transform leads to the impulse response

        \begin{align*} h^{(4)}(n) &= \mathcal{F}^{-1} \left\{ H^{(3)}(e^{j{\Omega}}) \cdot R_{\Omega_c} (e^{j{\Omega}} )\right\} \\ &= h^{(3)}(n) * \frac{\Omega_c}{\pi} \ \text{si} (\Omega_c n) \\ &= A \frac{\Omega_c}{\pi} \ \text{si} \bigg(\Omega_c (n-n_0)\bigg) + A \frac{\alpha}{2} \frac{\Omega_c}{\pi} \ \text{si} \bigg(\Omega_c (n-n_0+m)\bigg) \cdots \\ & \ \ \ + A \frac{\alpha}{2} \frac{\Omega_c}{\pi} \ \text{si} \bigg(\Omega_c (n-n_0-m)\bigg) \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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