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*}

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.

13.08.2017: New Gas e.V. sections (e.g. pictures or prices) added.

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 ...


Read more ...