# Advanced Signals and Systems - Discrete Signals

## Task

Let the signal $$v_0(t)$$ be sampled at the sampling frequency $$f_s$$.

1. Under which condition is the information of $$v_o(t)$$ fully preserved after sampling?
2. Explain the meaning of 'aliasing'.
3. How can the original continuous signal $$v_0(t)$$ be reconstructed from its sampled version?
4. Consider the signal $$v_0(t)=\mbox{cos}(2\pi\cdot 12000\cdot t)$$ sampled at $$f_s = 16000$$. Sketch and interpret the FOURIER spectrum.

## Amount and difficulty

• Working time: approx. 20 minutes
• Difficulty: easy

## Solution

1. The Sampling theorem has to be fulfilled.

If a continuous signal $$v_o(t)$$ has a band-limited FOURIER transform $$V_o(j\omega)$$, that is, $$|V_o(j\omega)| = 0$$ for $$|\omega| \geq 2\pi f_c$$, then $$v_o(t)$$ can be uniquely reconstructed without errors from equally spaced samples $$v_o(nT_s), -\infty <n < \infty$$, if \begin{equation*} f_s > 2\cdot f_c \end{equation*} where $$f_s = \frac{1}{T_s}$$ is the sampling frequency and $$f_c$$ is the cut-off frequency.

2. If the signal $$v_o(t)$$ is sampled below the Nyquist frequency ($$f_s=2f_c$$), then distortions due to the spectral fold over (aliasing) occur.
3. The original signal can be reconstructed by applying a low-pass filter whose cut-off frequency $$\omega_{\text{LP}}$$ lies between $$\omega_c<\omega_{\text{LP}}<\omega_s - \omega_c$$.
4. The following approach was used: \begin{equation*}V_o(j\omega) = \mathfrak{F}\{v_o(t)\} = \mathfrak{F}\{\cos(2\pi \omega t)\} = \pi \left[ \delta_0 (\omega -\omega_0) + \delta_0 (\omega +\omega_0) \right] \end{equation*} Due to the repetition of the spectra caused by the sampling, the FOURIER spectrum is given by: \begin{align*} V_s(j\omega) &= \sum \limits_{\mu =- \infty}^{\infty} V_o\left( j[\omega - \mu \omega_s] \right)\\ V_s(j\omega) &= \pi \sum \limits_{\mu =- \infty}^{\infty} \left[ \delta_0 (\omega -\omega_0- \mu \omega_s) + \delta_0 (\omega +\omega_0- \mu \omega_s) \right] \end{align*} The following result can be sketched and it can be concluded that the sampling theorem is not fulfilled.

## Task

Let the signal $$v_0(t)$$ be sampled at the sampling frequency $$f_s$$.

1. Consider the T-periodic signal $$v_o(t)$$ sampled at $$f_s=1/T_s$$. Determine $$\alpha = T/T_s$$ for which the discrete (sampled) signal $$v(n)$$ is periodic.
2. Assume the signal $$v_o(t) = \sin(\omega_0t)$$, where $$\omega_0 = \frac{2\pi}{T}$$, is sampled with $$T_s=T/\alpha$$. Show whether or not $$v(n)$$ is periodic and determine its period $$K$$ (if possible) for each of the following cases:
1. $$\alpha = 5,$$
2. $$\alpha = 5.5,$$
3. $$\alpha = \frac{16}{3},$$
4. $$\alpha = \pi,$$
5. $$\alpha = 1.$$

## Amount and difficulty

• Working time: approx. 25 minutes
• Difficulty: easy

## Solution

1. The Signal $$v_o(t)$$ is a continuous T-periodic signal, defined by \begin{equation*} v_o(t) = v_o(t+\mu T) \ \ \ ,\ \ \mu\in \mathbb{Z}, \ \ T\in \mathbb{R}^+ \text{ .} \end{equation*} The sequence $$v(n)$$ is defined by: \begin{equation*} v(n) = v_o(nT_s)=v_o(nT_s+\mu T) = v_o(n T_s + \mu \alpha T_s) = v_o((n+\mu \alpha)T_s) = v(n+\mu \alpha) \end{equation*}

The sequence is periodic if one value $$\mu$$ exists for which $$K = \mu\cdot \alpha \in \mathbb{N}$$ (natural, non-zero number) holds true.

The period $$K$$ is given by

$$K = \min\limits_{\mu} \left\{ \mu \cdot \alpha \ | \ \mu \cdot \alpha \in \mathbb{N} \right\}$$ and it is depended on $$\alpha$$. \begin{align*} 1) \ & \alpha \in \mathbb{N} : && \mu_{\text{min}} = 1 \ \rightarrow \ K=\alpha & \Longrightarrow v(n) \text{ is periodic}\\ 2) \ & \alpha \in \mathbb{Q}^+ : && \alpha = \frac{m}{n} \ \text{ where } m,n \in \mathbb{N} & \\ \ & && \mu_{\text{min}} = n \ \rightarrow \ K=m & \Longrightarrow v(n) \text{ is periodic}\\ 3) \ & \alpha \in \mathbb{R}^+ \setminus \mathbb{Q}^+ : && \text{there is no } \mu \in \mathbb{Z} \text{ for which} & \\ \ & && \mu \cdot \alpha \in \mathbb{N} & \Longrightarrow v(n) \text{ is non-periodic} \end{align*}
2. The period $$K$$ can be found by utilizing equation (1).
3. \begin{align*} (i) \ & \alpha = 5: && \mu \cdot \alpha = 5, \ \mu_{\text{min}} = 1, \ K=5 & \Longrightarrow v(n) \text{ is periodic}\\ (ii) \ & \alpha = 5.5:&& \mu \cdot \alpha = 11, \ \mu_{\text{min}} = 2, \ K=11 & \Longrightarrow v(n) \text{ is periodic}\\ (iii) \ & \alpha = \frac{16}{3}: && \mu \cdot \alpha = 16, \ \mu_{\text{min}} = 3, \ K=16 & \Longrightarrow v(n) \text{ is periodic}\\ (iv) \ & \alpha = \pi: && \text{there is no } \mu \cdot \alpha \in \mathbb{N} \text{ with } \mu \in \mathbb{Z} & \Longrightarrow v(n) \text{ is non-periodic}\\ (v) \ & \alpha = 1: && \mu \cdot \alpha = 1, \ \mu_{\text{min}} = 1, \ K=1& \Longrightarrow v(n) \text{ is periodic,} \\ & && & \text{but the sampling theorem is not fulfilled}\\ \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 ...