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*}

Recent Publications

P. Durdaut, J. Reermann, S. Zabel, Ch. Kirchhof, E. Quandt, F. Faupel, G. Schmidt, R. Knöchel, and M. Höft: Modeling and Analysis of Noise Sources for Thin-Film Magnetoelectric Sensors Based on the Delta-E Effect, IEEE Transactions on Instrumentation and Measurement, published online, 2017

P. Durdaut, S. Salzer, J. Reermann, V. Röbisch, J. McCord, D. Meyners, E. Quandt, G. Schmidt, R. Knöchel, and M. Höft: Improved Magnetic Frequency Conversion Approach for Magnetoelectric Sensors, IEEE Sensors Letters, published online, 2017

 

Website News

18.06.2017: Page about KiRAT news added (also visible in KiRAT).

31.05.2017: Some pictures added.

23.04.2017: Time line for the lecture "Adaptive Filters" added.

13.04.2017: List of PhD theses added.

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

Alexej Namenas - A New Guy in the Team

In June Alexej Namenas started in the DSS Team. He will work on real-time tracking algorithms for SONAR applications. Alexej has done both theses (Bachelor and Master) with us. The Bachelor thesis in audio processing (beamforming) and the Master thesis in the medical field (real-time electro- and magnetocardiography). In addition, he has intership erperience in SONAR processing.

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