Telecommunication Systems - Exercise sheets

Periodic and Aperiodic Signals

Exercise 2.1 Determine if the following signals are periodic or not.

1. $s_1(t) = 4\sqrt{\mathrm{W}}\sin\left(\sqrt{3}\mathrm{Hz} \cdot 2\pi t\right) + 2\sqrt{\mathrm{W}}\cos\left(\sqrt{12}\mathrm{Hz}\cdot 2\pi t\right)$

2. $s_1(t) = 4\sqrt{\mathrm{W}}\sin\left(\sqrt{3}\mathrm{Hz} \cdot 2\pi t\right) + 2\sqrt{\mathrm{W}}\cos\left(\sqrt{12}\mathrm{Hz}\cdot 2\pi t\right)$

3. $s_2(t) = 4\sqrt{\mathrm{W}}\sin\left(\sqrt{3}\mathrm{Hz}\cdot 2\pi t\right) + 2\sqrt{\mathrm{W}}\cos\left(\sqrt{2}\mathrm{Hz}\cdot 2\pi t\right)$

Complex Phasor

Exercise 2.2 Consider the two signals $$\begin{aligned}s_1(t)&=3\sqrt\mathrm{W}\cos(\omega t),\\ s_2(t)&= 5\sqrt\mathrm{W}\sin\left(\omega t+ \frac{\pi}{6}\right).\end{aligned}$$

1. Calculate the complex phasor $s_{\mathrm{P}1}$ that corresponds to the signal $s_1(t)$.

2. Calculate the complex phasor $s_{\mathrm{P}2}$ that corresponds to the signal $s_2(t)$.

3. Calculate the complex phasor $s_{\mathrm{P}3}$ that corresponds to the signal $s_3(t)=s_1(t)+s_2(t)$.

4. Assume $\omega=100\pi\,\frac{1}{\mathrm{s}}$ to be the angular frequency of both signals $s_1(t)$ and $s_2(t)$ and calculate $s_3(t)$ for $t=2.73\,\mathrm{s}$.

Signal Classification in a Communication System

Exercise 2.3 Consider the digital communication system of Figure 1.2.

1. Determine for each of the signals in the block diagram, viz., $x(t)$, $x[n]$, $b[k]$, $u[k]$, $s(t)$, $r(t)$, $\hat{u}[k]$, $\hat{b}[k]$, $\hat{x}[n]$, and $\hat{x}(t)$, if the signal is continuous-time, discrete-time, continuous-value, discrete-value, analog and/or digital.

2. Explain why a digital communication system finally transmits the digital signal in an analog way? Why don’t we just transmit the analog signal directly over the channel?

Signal Power

Exercise 2.4 Compute the power $P_u$ of the unit step signal $$u(t)= \begin{cases} 1\sqrt{\mathrm{W}}, & t\geq 0, \\ 0, & t < 0. \end{cases}$$

Signal Type

Exercise 2.5 Determine whether the signal $x(t) = e^{-0.5t}\cos(t)$ is of energy-type, power-type or neither of the two.

Hint: Use the following integral: $$\int e^{ax} \cos ^2x dx = \frac{e^{ax}}{4+a^2}\left(a\cos ^2 x + \sin2x + \frac{2}{a}\right).$$

Ensemble Average Mean

Exercise 2.6 Assume the random process $X(t)$ defined as $X(t)=A+Bt$, with $A$ and $B$ being statistically independent random variables. $A$ is uniformly distributed on $[1,3]$, i.e., $A\sim{\cal U}(1,3)$, and $B$ is uniformly distributed on $[-1,1]$, i.e., $B\sim{\cal U}(-1,1)$. Calculate the ensemble average mean $m_X(t)$.

Autocorrelation Function

Exercise 2.7 Assume the same random process as known from Exercise 2.6, i.e., $X(t)=A+Bt$ with $A\sim{\cal U}(1,3)$ and $B\sim{\cal U}(-1,1)$ being statistically independent random variables. Calculate the autocorrelation function $r^\prime_X(t_1,t_2)$ of the random process $X(t)$.

Fourier Transform

Exercise 2.8 Compute the Fourier transform $X(f)$ of the following signals and draw the signals as well as the absolute value of their spectra, i.e., $|X(f)|$, in a diagram by using a Computer Algebra System (CAS) of your choice. If you don’t have a CAS at your hand, a qualitative plot is sufficient.

1. $x(t) = A\mathop{\mathrm{sinc}}(t f_\mathrm{s})$

2. $x(t) = A\cos(2\pi f_0 t)\mathop{\mathrm{rect}}(tf_0/4)$

3. $x(t) = A\mathop{\mathrm{rect}}(t/T_\mathrm{s}-1/2)$

Power Spectral Density

Exercise 2.9 Consider the random process $X(t)=Y\cos(2\pi f_0 t) + Z\sin(2\pi f_0 t)$ with $Y$ and $Z$ being zero-mean statistically independent Gaussian random variables with the variance $\sigma^2_Y$ and $\sigma^2_Z$, respectively. For now let’s assume that $\sigma^2_Y = \sigma^2_Z$ holds.

1. Calculate the ensemble average mean $m_X(t)$.

2. Calculate the autocorrelation function $r^\prime_X(t_1,t_2)$.

3. Calculate the power spectral density $R_x(f)$.

4. Answer subtasks a. and b. for the case that $\sigma^2_Y \neq \sigma^2_Z$.

Hint: Use the product-to-sum identities of sine and cosine: $$\begin{aligned} \cos(\alpha)\cdot\cos(\beta) &= \frac{1}{2} (\cos (\alpha+\beta) + \cos (\alpha-\beta) ) \\ \sin(\alpha)\cdot\sin(\beta) &= \frac{1}{2} (\cos (\alpha-\beta) - \cos (\alpha+\beta) ) \end{aligned}$$

System Response

Exercise 2.10 Consider an LTI system whose response to the input signal $x(t) = e^{-\alpha t}u(t)$ is $y(t) = te^{-\alpha t}u(t)$, $\alpha>0$, and $u(t)$ being the unit step signal as defined in Exercise 2.4. Determine this system’s output $y^\prime(t)$ to an input function $x^\prime(t) = e^{-\alpha t } \cos (\beta t)u(t)$.

Hint: Use the following integral $$\int xe^{ax} = \left(\frac{x}{a}-\frac{1}{a^2}\right)e^{ax},$$ and the following Fourier pair $$x(t)=e^{-\alpha t}\sin(\omega_0t)u(t) \leftrightarrow X(f)=\frac{\omega_0}{\left(\alpha+j\omega\right)^2+\omega_0^2},\ \alpha>0.$$

Exercise 3:

Sampling

Exercise 3.1 The normalised output signal of a microphone, i.e., $$x(t) = 1\,\sqrt{\mathrm{W}}\cdot\sin\frac{2\pi t}{\mathrm{ms}} +\frac{1\,\sqrt{\mathrm{W}}}{2} \sin\frac{4\pi t}{\mathrm{ms}},$$ is sampled with the sampling rate $f_\mathrm{s}=8\,\mathrm{kHz}$.

1. Plot the signal $x(t)$ for $t\in[0,2\,\mathrm{ms}]$ using a CAS of your choice.

2. Compute the ideally sampled signal $x_\mathrm{s}(t)$ and add it to the figure from above by hand.

3. Add to this figure also the sampled signals $x_\mathrm{ns}(t)$ and $x_\mathrm{sh}(t)$ in case of natural sampling and sampling based on the sample-and-hold method, respectively, assuming $\Delta t= T_\mathrm{s}/2$ in case of natural sampling.

4. Compute the spectrum $X(f)$ of the original signal $x(t)$ and the spectrum $X_\mathrm{s}(f)$ of the sampled signal $x_\mathrm{s}(t)$. Draw the results in a diagram for $f\in[-3\,\mathrm{kHz},20\,\mathrm{kHz}]$.

5. Explain in words how the spectra $X_\mathrm{ns}(f)$ and $X_\mathrm{sh}(f)$ in case of natural sampling and sampling based on the sample-and-hold method, respectively, differ from the one of $X_\mathrm{s}(f)$.

6. For reconstruction of the original signal, an ideal low-pass filter with the cut-off frequency $f_\mathrm{c}$ is applied. Formulate the impulse response $h(t)$ as well as the frequency response $H(f)$ of this reconstruction filter!

7. What values of the cut-off frequency $f_\mathrm{c}$ guarantee an ideal reconstruction?

8. What is the minimum value of the sampling frequency $f_\mathrm{s}$ for which ideal reconstruction is possible?

9. Plot the spectrum $X_\mathrm{s}(f)$ of the sampled signal $x_\mathrm{s}(t)$ if a sampling frequency of $f_\mathrm{s}=3.5\,\mathrm{kHz}$ is applied. Why is no ideal reconstruction possible in this case? Compute and plot the reconstructed signal if an ideal low-pass filter with the cut-off frequency $f_\mathrm{c}=3\,\mathrm{kHz}$ is used as the reconstruction filter.

Quantisation Levels and Thresholds

Exercise 3.2 The sampled values $x[n]$ of a signal are given in Table 3.1.

Table 3.1: Sampled values $x[n]$ of a signal

$n$	0	1	2	3	4	5	6
$\frac{x[n]}{\sqrt{\mathrm{W}}}$	0.42	1.92	–1.5	4.19	2.96	–2.12	1.49

In this exercise, the sampled values of Table 3.1 are quantised with a linear mid-riser quantiser assuming $N_\mathrm{q}=4$ quantisation levels and an maximum quantisation level of $A=3\,\sqrt{\mathrm{W}}$.

1. Compute the step size $\Delta$ as well as the quantisation thresholds and levels.

2. Compute and plot the quantised values $x_\mathrm{q}[n]$.

3. Assuming that the sampled values $x[n]$ follow a uniform distribution between $-3\,\sqrt{\mathrm{W}}$ and $5\,\sqrt{\mathrm{W}}$, what transformation would improve the quantisation result?

Signal-to-Quantisation-Noise Power Ratio

Exercise 3.3 The samples $x[n]=x(nT_\mathrm{s})$, $n\in\mathbb{Z}$, $T_\mathrm{s}=1/f_\mathrm{s}$, of the function $x(t)$ in Exercise 3.1 are linearly quantised with $m=8\,\mathrm{bit}$.

1. Compute the SQNR $\gamma_\mathrm{q}$ in this case if the maximum level of the quantiser $A$ has been chosen to cover the peak value $\hat{x}$ of the signal, i.e., $A=\hat{x}=\max_t x(t)$. What is the value of $A$?

2. Explain, why nonlinear quantisation has no benefits when applied to the given signal $x(t)$.

3. Compute $\gamma_\mathrm{q}$ if the maximum level of the quantiser has been chosen to be $A^\prime=3\,\sqrt{\mathrm{W}}$.

PCM Code

Exercise 3.4 Compute the PCM code to the sampled value $$x[0]=-0.2013487456\,\sqrt{\mathrm{W}},$$ according to ITU-T G.711.

Exercise 4:

Time-Limited Pulse Shaping Filter

Exercise 4.1 Consider the following causal pulse shaping filters:

• unipolar (NRZ) code (no symbol mapping is applied, i.e., $d_\mathrm{uni}[\ell]=Du[\ell]\in\{0,D\}$,
• polar NRZ code ($d[\ell]\in\{-D,D\}$) and
• Manchester code.

Based on these filters, the binary PCM stream $u[\ell]$ as given in Table 4.2 shall be transmitted in the baseband.

Table 4.2: Bits $u[\ell]$ to be transmitted

$\ell$	0	1	2	3	4	5	6	7	8	9
$u[\ell]$	1	1	0	1	0	1	0	0	1	0

1. What specific characteristic is the same for all pulse shaping filters? In what kind of systems are they mainly used therefore?

2. Plot for all three filters the baseband signal $s(t)=T\sum_{\ell=0}^9 d[\ell] p(t-\ell T)$ of the data stream given in Table @ref(tab:timelimitedpulse) in the region $t\in[0,10T]$ and with $D=1\,\sqrt{\mathrm{W}}$. Compute all values for $d_\mathrm{uni}[\ell]$ and $d[\ell]$ (unipolar and polar symbols) first.

3. Let $T=10\,\mu\mathrm{s}$. What is the minimum necessary transmission bandwidth for all three filters
   Hint: Since the considered filters are not band-limited, use the first root in the spectrum as the definition of the bandwidth.

4. List advantages and drawbacks for each of the filters.

Raised-Cosine Filter

Exercise 4.2 A raised-cosine filter has the cut-off frequencies $f_1 = (1 - \alpha ) /(2T) = 12.4\,\mathrm{MHz}$ and $f_2 = (1 + \alpha ) /(2T) = 31.6\,\mathrm{MHz}$. Compute the roll-off factor $\alpha$, the data rate $1/T$ and the (onesided) bandwidth $B$ of the filter.

Nyquist Spectra and Signals

Exercise 4.3 Plot for each of the filters in the Example 4.8 the spectrum $$P_\text{s}(f)=\sum_{k=-\infty}^\infty G\left(f-\frac{k}{T}\right),$$ as well as the baseband signal $$s(t)= T\sum_{\ell=-\infty}^\infty d[\ell] p(t-\ell T),$$ assuming a symbol duration of $T=1\,\mu\mathrm{s}$ and the symbol sequence $$\begin{aligned}d[\ell] &= \delta[\ell] +\delta[\ell-1] - \delta[\ell-2]- \delta[\ell-3] -\delta[\ell-4]\\ &\phantom{=}+\delta[\ell-5]-\delta[\ell-6].\end{aligned}$$ Use Python if you are interested in exact drawings, otherwise a qualitative sketch of the graphs is sufficient.

Nyquist Criterion

Exercise 4.4 Which of the following pulse shaping filters fulfill the first Nyquist criterion and what is the largest symbol rate in this case?

1. Rectangular filter of length $T$,

2. Manchester code of length $T$,

3. Sinc filter with first zero at $t=T$,

4. Raised-cosine filter with first zero at $t=T$,

5. Gaussian filter,

6. Filter with the trapezoidal spectrum as depicted in Figure 4.22 and

7. Filter with the parabola spectrum as depicted in Figure 4.23.

Figure 4.22: Trapezoidal spectrum

Figure 4.23: Parabola spectrum

Eye Diagram of a Triangular Filter

Exercise 4.5 Consider the following triangular filter for pulse shaping: $$p(t) = \begin{cases} \frac{1}{T}\left(1-\frac{|t|}{T}\right), & |t|<T, \\ 0, & \text{otherwise}. \end{cases}$$ In other words, the triangular pulse shaping filter $p(t)$ is used to transmit the binary data $u[\ell]\in\{0,1\}$ based on the baseband signal $s(t)=p(t)*d(t)$ with $$d(t) = T\sum_{-\infty}^{\infty} d[\ell]\delta(t-\ell T)\ \text{and}\ d[\ell] = \begin{cases} +D, & u[\ell]=0,\\ -D, & u[\ell]=1. \end{cases}$$ The data rate is $1/T=1\,\mathrm{MHz}$, $D=1\,\sqrt{\mathrm{W}}$ and the transmission channel is ideal. Plot the resulting eye diagram and determine $A_\mathrm{v}$ and $T_\mathrm{h}$! Do we observe ISI in this case?

Eye Diagram of a Sawtooth Filter

Exercise 4.6 The digital data stream $\dots$, $u[-1]$, $u[0]$, $u[1]$, $\dots$ shall be transmitted over a baseband system. To do so, we define the following data signal $$d(t) = T\sum_{-\infty}^{\infty} d[\ell]\delta(t-\ell T)\ \text{and}\ d[\ell] = \begin{cases} +D, & u[\ell]=0,\\ -D, & u[\ell]=1, \end{cases}$$ and apply the sawtooth pulse shaping filter defined as $$p(t) = \begin{cases} \frac{1}{T}\left(1-\frac{|t|}{aT}\right), & |t|\leq \frac{3aT}{2}, \\ \frac{1}{T}\left(\frac{|t|}{aT}-2\right), & \frac{3aT}{2}<|t|\leq 2aT, \\ 0, & \text{otherwise}. \end{cases}$$

The resulting baseband signal is then $s(t)=p(t) * d(t)$. The data rate is $1/T$, $D=1\,\sqrt{\mathrm{W}}$ and the transmission channel is ideal.

1. Plot $p(t)$.

2. What values of $a>0$ do not result in ISI? How is the underlying criterion called?

3. How large is the vertical eye size $A_\text{v}$ in this case?

4. In the following, it holds that $a=1$. Which four critical bit sequences are relevant for the inner eye? Mark the central bit which lies in the middle of the eye. Plot the baseband signal $s(t)$ for all four critical bit sequences.
   Hint: Plot first the bit sequence relevant for the inner eye in the first quadrant and then make use of symmetry!

5. How large is the horizontal eye size $T_\text{h}$ in this case?

Spectral Efficiency

Exercise 4.7 Compute the spectral efficiency $\eta$ of the filter defined in Exercise 4.2.

Exercise 5:

AWGN

Exercise 5.1 The signal $s(t)$ is superposed by additive white Gaussian noise $n(t)$ with the PSD $R_n(f) = N_0/2$. The noisy signal is filtered by an ideal passband filter with the center frequency $f_0=100\,\mathrm{kHz}$ and the bandwidth $B=3\,\mathrm{kHz}$. We measure the signal power at the output of the filter. If $s(t)$ is sinusoidal with amplitude $\hat{s}$ and frequency $f_1=100\,\mathrm{kHz}$, we measure a power of $P_1=1\,\mathrm{mW}$. Changing the center frequency of the passband filter such that the sinusoidal signal lies outside the passband, reduces the power to $P_2=0.2\,\mathrm{mW}$.

1. Compute the noise power density $N_0$.

2. Compute the SNR $\gamma=P_s/P_n$ of the first measurement. What is the value $10\lg \gamma$ in dB?

3. Finally, compute the amplitude $\hat{s}$ of the signal $s(t)$.

BEP

Exercise 5.2 A binary communication system transmits a unipolar NRZ signal with a rectangular pulse shaping filter over an AWGN channel. The symbols $d[\ell]\in\{0,1\sqrt{\mathrm{W}}\}$, i.e., $D_\mathrm{uni}=1\sqrt{\mathrm{W}}$, and the symbol duration is $T$. The communication channel has an attenuation of $25\,\mathrm{dB}$. At the input of the receiver, the noise power is measured as $50\,\mu\mathrm{W}$ and the noise figure of the receiver is given as $6\,\mathrm{dB}$.

1. Compute the resulting BEP?

2. What is the BEP if we choose polar symbol mapping with $d[\ell]\in\{-1\sqrt{\text{W}},1\sqrt{\text{W}}\}$, i.e., $D=1\sqrt{\text{W}}$?

MF and BEP

Exercise 5.3 The binary data $u[\ell]\in\{0,1\}$ of a source shall be wirelessly transmitted. To do so, the data is pulse shaped using the triangular function $$p(t) = \begin{cases} \frac{\sqrt{3}}{T}\left(1-\frac{t}{T}\right), & 0 \leq t \leq T, \\ 0, & \text{elsewhere}, \end{cases}$$ with the normalisation $\int_{-\infty}^\infty|p(t)|^2 dt=1/T$ and the symbol period $T$. The resulting baseband signal reads as $$s(t) = T\sum_{\ell=-\infty}^{\infty} d[\ell] p(t- \ell T),$$ with $$d[\ell] = \begin{cases} -D, & u[\ell] = 1, \\ D, & u[\ell] = 0, \end{cases}$$ and is perturbed by AWGN $n(t)$ with the power density spectrum $R_n(f) = N_0/2$. At the receiver, we apply the filter with the impulse response $g(t)$ which is at first a simple integrator (unmatched filter).

1. Compute and plot the signal part $y_s(t)=g(t) * s(t)$ of the signal $y(t)$ at the output of the receive filter for $0\leq t \leq T$ if $u[0]=0$ has been transmitted.

2. Compute the SNR at the output of the receive filter and at the sampling time $t=T$. Why has a constant factor in the impulse response $g(t)$ no influence on the SNR?

3. Compute the BEP of the complete system.

In the following, the receive filter $g(t)$ is chosen to be the MF, i.e., $g(t)=g_\mathrm{MF}(t)$.

4. Determine $g_\mathrm{MF}(t)$ in this case and compute and plot $y_s(t)$ for $0\leq t\leq T$ if again $u[0]=0$ has been transmitted.

5. How much larger is the SNR compared to using an integrator as the receive filter?

6. How does this influence the BEP?

7. Assume that it is not synchronously sampled, i.e., $t=aT$ with $0\leq a \leq 1$. How does this change the BEP?

Exercise 6:

Special 4-QAM

Exercise 6.1 The bit stream $u[k]$, given by $$u[k] = \delta[k-3]+\delta[k-4]+\delta[k-6]+\delta[k-7],$$ with $k\in\{0,1,2,3,4,5,6,7\}$ shall be transmitted via a digital communication system. The symbol mapping is defined as follows: $$\frac{d[\ell]}{\sqrt{\mathrm{W}}}=\begin{cases} 0, & u[2\ell]=0 \wedge u[2\ell+1]=0,\\ j, & u[2\ell]=1 \wedge u[2\ell+1]=0,\\ 1, & u[2\ell]=0 \wedge u[2\ell+1]=1,\\ 1+j, & u[2\ell]=1 \wedge u[2\ell+1]=1, \end{cases}$$ where $\ell\in\{0,1,2,3\}$. The rectangular filter $p(t) = \frac{1}{T}\mathop{\mathrm{rect}}(t/T-1/2)$ is used as the pulse shaping filter. The symbol duration is $T=2/f_0$ assuming a carrier frequency of $f_0$. Derive and plot the resulting passband signal $s(t)$ as well as the inphase component $s_\mathrm{I}(t)$ and the quadradture component $s_\mathrm{Q}(t)$ for the time period $t\in[0,4T]$!

16-QAM

Exercise 6.2 Consider QAM with the symbols $$d[\ell]\in\{\pm 1 \pm \text{j}, \pm 2 \pm 2\text{j}, \pm 1 \pm 2\text{j}, \pm 2 \pm \text{j}\}.$$

1. Plot the symbols on the complex plane. How many symbols $M$ are available and how many bits $b$ per symbol can be transmitted?

2. Add the decision lines to the plot. What is special in this case?

3. Plot the block diagram of the corresponding modulator.

Satellite Communication System

Exercise 6.3 Consider a satellite system with a transponder bandwidth of $70\,\mathrm{MHz}$ and a maximum bit rate of $90\,\mathrm{Mbit}/\mathrm{s}$. The system uses QPSK modulation and root-raised-cosine pulse shaping.

1. What is the symbol rate $1/T$ of the transmission?

2. What is the maximum possible roll-off factor $\alpha$?

3. What is the spectral efficiency $\eta$?

4. Compute the BEP $P_\text{BE}$ at $10\lg(\bar{E}_b/N_0)=6\,\mathrm{dB}$.

5. Now, we use BPSK instead of QPSK. How much can we decrease the transmit power in order to achieve the same BEP?

Exercise 6:

OFDMA

Exercise 7.1 In Long Term Evolution (LTE), OFDM is used with a maximum bandwidth of $30.72\,\mathrm{MHz}$ and an FFT length of $2048$. The duration of an OFDM symbol is $66.67\,\mu\text{s}$ and the guard interval has a length of $16.67\,\mu\text{s}$.

1. Name one advantage and one disadvantage of OFDM.

2. Determine the subcarrier spacing $\Delta f$ and bandwidth efficiency $\beta$.

3. What is the maximum length of the channel (delay spread) so that no intersymbol interference occurs?

4. How many subcarriers must be switched off or assigned zeros at the edges of the OFDM spectrum so that a maximum bandwidth of $18\,\mathrm{MHz}$ is not exceeded?

5. Determine the maximum data rate that can be transmitted at this bandwidth if 64-QAM is used as the modulation alphabet.