Advanced Signals and Systems - Hilbert Transform

 

25. Hilbert-transform and single side band modulation.

Task

In the first part of this problem some fundamentals about the Hilbert-transform will be repeated and afterwards an example of use will be discussed.

  1. Give the definition of the Hilbert-transform. Give both, the frequency response and the impulse response. How is the so-called analytic signal \(v_a(n)\) defined?
  2. Is the Hilbert-transformer causal and bandlimited?
  3. Give a realization of the ideal single side band modulator (SSB) with a Hilbert-transformer as a block diagram. Use therefore the definition of a single side band modulation.

Amount and difficulty

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

Solution

  1. The definition of the Hilbert-transform is given by the frequency response,

    \begin{equation*} H(e^{j\Omega}) = -j \cdot \text{sign}(\Omega) \text{ , } \end{equation*}

    and the impulse response ,

    \begin{equation*} h_0(n) = \begin{cases} \frac{1-(-1)^n}{\pi n} &, \text{if \(n\) odd}\\ 0&, \text{otherwise} \end{cases}\text{ .} \end{equation*}

    The analytic signal respectively analytic spectrum is defined by

    \begin{align*} v_a(n) &= v(n) + j \tilde{v}(n) \ \ \ \ \text{ where } \tilde{v}(n) = v(n) * h_0(n) \text{ ,}\\ V_a(e^{j\Omega}) &= V(e^{j\Omega}) + j \tilde{V}(e^{j\Omega}) \text{ .} \end{align*}

  2. The Hilbert-transformer is non-causal (due to its impulse response) and not bandlimited (see frequency response)?

  3. The definition of a single side band modulation is given by the following resulting signal,

    \begin{align*} y(n) &= \mathcal{F}^{-1} \left\{ V _a ^L \left(e^{j(\Omega + \Omega_0)}\right) + V_a^R \left(e^{j(\Omega - \Omega_0)}\right) \right\} \\ &= e^{-j\Omega_0 n} \left[ \mathcal{F}^{-1} \left\{ V(e^{j\Omega}) \right\} - j \mathcal{F}^{-1} \left\{ \tilde{V}(e^{j\Omega}) \right\} \right] + e^{j\Omega_0 n} \left[ \mathcal{F}^{-1} \left\{ V(e^{j\Omega}) \right\} + j \mathcal{F}^{-1} \left\{ \tilde{V}(e^{j\Omega}) \right\} \right] \\ &= 2 \cdot \left[ v(n) \cos(\Omega_0 n) - \tilde{v}(n) \sin(\Omega_0 n) \right] \text{ .} \end{align*}

    Blockdiagramm

 

26. Hilbert-transform of a bandpass signal.

Task

Given the signal

\begin{equation}\nonumber v(n) = \frac{\Omega_c}{\pi} \ \frac{\sin(\Omega_c n)}{\Omega_c n}\cdot \cos(\Omega_0 n)\text{ ,} \end{equation}

determine

  1. the Hilbert-transform \(\tilde{v}(n)=\mathcal{H}\left\{v(n)\right\}\) and
  2. the analytic signal \(v_a(n)\).
  3. Find the instantaneous envelope \(e(n)\), phase \(\varphi (n)\), and frequency \(\Omega (n)\).

Amount and difficulty

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

Solution

  1. The Hilbert-transform \(\tilde{v}(n)=\mathcal{H}\left\{v(n)\right\}\) can be derived by

    \begin{align*} \tilde{v}(n) &= \mathcal{F}^{-1} \left\{ V(e^{j\Omega}) \cdot \left[ -j \cdot \text{sign}(\Omega) \right] \right\} \\ &= j \cdot \frac{\Omega_c}{2\pi} \ \frac{\sin(\Omega_c n)}{\Omega_c n}\cdot e^{-j\Omega_0 n} + (-j) \cdot \frac{\Omega_c}{2\pi} \ \frac{\sin(\Omega_c n)}{\Omega_c n}\cdot e^{j\Omega_0 n} \\ &= \cdots\\ &= \frac{\Omega_c}{\pi} \ \frac{\sin(\Omega_c n)}{\Omega_c n}\cdot \sin(\Omega_0 n) \end{align*}

  2. The analytic signal

    \begin{align*} v_a(n) &= v(n) + j \tilde{v}(n)\\ &= \cdots \\ &= \frac{\Omega_c}{\pi} \ \frac{\sin(\Omega_c n)}{\Omega_c n}\cdot e^{j\Omega_0 n} \end{align*}

  3. Instantaneous envelope \(e(n)\)

    \begin{align*} e(n) &= \sqrt{v^2(n) + \tilde{v}^2(n)}\\ &= \cdots \\ &= \frac{\Omega_c}{\pi} \ \left|\frac{\sin(\Omega_c n)}{\Omega_c n}\right| \end{align*}

    Instantaneous phase \(\varphi(n)\)

    \begin{align*} \varphi(n) &= \arctan \left( \frac{\tilde{v}(n)}{v(n)} \right)\\ &= \cdots \\ &= \Omega_0 n \end{align*}

    Instantaneous frequency \(\Omega(n)\)

    \begin{align*} \Omega(n) &= \varphi(n) - \varphi(n-1) \\ &= \cdots \\ &= \Omega_0 \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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