4.15  Exercises

4.1. Assume a binary digital communication system with equiprobable polar symbols ($-\; 1$ for bit 0 and +1 for bit 1). The shaping pulse is $g(t)\; =\; 4sin(\pi t∕T)$ from $t\; \in \; [0,T]$ or zero otherwise, with $T\; =\; 10\mu$s. Draw the transmitted signal $s(t)$ for $t\; \in \; [0,200]$ $\mu$s if the bit rate is 25 kbps and the bitstream repeats the pattern 1, 0, 1, 0, 1, 0, etc., with the first symbol 1 starting at time instant $t\; =\; 0$. Note that the symbol period $T_{\mathrm{\text{sym}}}$ may be different than $T$.
4.2. To study ISI, a discrete-time simulation with oversampling $L\; =\; 3$ is conceived. The symbols to be transmitted are $-\; 1$ and 1. However the shaping pulse is $p[n]\; =\; 1∕5\{\delta [n+4]+2\delta [n+3]+3\delta [n+2]+4\delta [n+1]+5\delta [n]+4\delta [n-1]+3\delta [n-2]+2\delta [n-3]+\delta [n-4]\}$. The symbols to be transmitted are (after upsampling by $L\; =\; 3$) $m[n]\; =\; \delta [n]\; -\; \delta [n\; -\; 3]\; -\; \delta [n\; -\; 6]\; +\; \delta [n\; -\; 9]$. a) Obtain the convolution $m[n]\ast p[n]$, b) the ISI parcel at $n\; =\; 3,6$, c) design a new pulse $p\prime [n]$ with 7 non-zero samples that leads to ISI free, d) assuming a channel with bandwidth 60 kHz, even not knowing about the spectrum of $p\prime [n]$, what is the maximum signaling rate for zero ISI? e) Why?
4.3. Draw the eye diagram of a PAM system when the pulse $p(t)$ has amplitude 1 V in the interval $[0,T_{\mathrm{\text{sym}}}[$, where $T_{\mathrm{\text{sym}}}=\; 1$ s is the symbol period. The symbols are $-\; 10$ and 10. The channel impulse response is $h(t)\; =\; \delta (t)\; -\; 0.2\delta (t\; -\; 2T_{\mathrm{\text{sym}}})$, where $x(t)$ is the PAM signal. Indicate the maximum horizontal opening in s and the maximum vertical opening in V.
4.4. A discrete-time scalar AWGN channel model assumes $\mathbf{\text{r}}=\mathbf{\text{s}}+\mathbf{\text{ν}}$ where $\mathbf{\text{s}}$ is a discrete-time sinusoid that corresponds to 3 Hz and power 10 dBm, while $\mathbf{\text{ν}}$ was obtained by filtering a WGN via an ideal lowpass filter of unit gain and cutoff frequency $F_s∕2$, where the sampling frequency is $F_s=\; 20$ Hz. Given that the SNR of $\mathbf{\text{r}}$ is 20 dB, what is the bilateral PSD constant value $N_0∕2$ of the continuous-time WGN that creates $\mathbf{\text{ν}}$?
4.5. A flat-fading channel has an impulse response given by $h(t)\; =\; 3e^{j0.2\pi }\delta (t\; -t_0)$, with $t_0=\; 4$ s. a) Find the impulse and frequency responses for its zero-forcing (ZF) equalizer. b) How does the ZF equalizer should be changed if $t_0$ is changed to 6 s?
4.6. The task is to estimate the receiver SNR of a flat-fading channel with impulse response $h(t)\; =\; \kappa e^{j\varphi }\delta (t\; -t_0)$, when these parameters are unknown. As depicted in Figure 4.2, discrete-time scalar WGN $\nu (t)$ is added at the receiver and it is uncorrelated with the signal $g(t)$. Known pilot symbols are used and after being properly decoded, allow to estimate the EVM at the receiver as 2 %. The transmitted $s(t$) and received $r(t)\; =\; g(t)\; +\; \nu (t)$ signals have power values of 6 dBm and 4 dBm, respectively. a) What is the SNR defined as $\mathrm{\text{SNR}}=\; 𝔼[|g(t){|}^2∕|\nu (t){|}^2]$? b) What is the estimated channel gain $\kappa$?
4.7. Assuming a complex-valued shaping pulse $p(t)\; =\; u(t)\; -\; 2u(t\; -\; 3)\; +\; u(t\; -\; 6)\; +\; j[2u(t)\; -\; u(t\; -\; 1)\; -\; u(t\; -\; 2)]$, draw the graph of its corresponding matched filter impulse response.
4.8. Consider the correlative decoding block of Figure 4.10. Assuming the input signal is a white noise with bilateral PSD of constant value $N_0∕2$ = 2 mW/Hz, and the basis functions compose an orthonormal set with $D\; =\; 4$ basis, a) what is the average total power of a vector $\mathbf{\text{x}}$ with elements corresponding to the four outputs? b) what is the mean and variance of each element of $\mathbf{\text{x}}$? c) when a cosine of amplitude $A\; =\; 4$ V is added to the noise at the input, what is the power of the first element of $\mathbf{\text{x}}$ in case the inner product between the cosine and the corresponding basis function ${\phi }_1(t)$ over $T_{\mathrm{\text{sym}}}$ is 2?
4.9. a) Using the matched-filter (MF) bound, estimate the SNR at the output of a MF with a transmit shaping pulse $p(t)$ consisting of a NRZ pulse with duration of 1 ms and amplitude 2 V. The modulation is PAM with $M\; =\; 4$ symbols from the constellation $[-3,-1,1,3]$ V. The channel is AWGN with $N_0∕2\; =\; -60$ dBm/Hz being the noise PSD value. b) If instead of AWGN, the channel is now a frequency-selective Gaussian LTI, would you consider that the new SNR can eventually be larger than the previously calculated SNR? Why?
4.10. A digital communication system was designed to target a baseband channel with $\mathrm{\text{BW}}=\; 2$ MHz and have zero ISI. The adopted symbol rate was $R_{\mathrm{\text{sym}}}=\; 3$ Mbauds. An ADC operating at sampling frequency $F_s$ is used to digitize the received signal, such that the symbols are recovered using digital signal processing. Given that you know both the sampling theorem and the Nyquist criterion for zero ISI, what is the minimum value for $F_s$?
4.11. a) A complex-valued analytic signal has support in frequency domain ranging from 100 kHz to 800 kHz. Using IQ sampling (two ADCs), what is the minimum sampling frequency $F_s$ to represent this signal without doing any frequency downconversion? b) In case one can use frequency up or downconversion with $e^{\pm j{\omega }_{0}t}$, what is the new minimum $F_s$ and the value of ${\omega }_0$?
4.12. A digital baseband communication system achieves 16 Mbps using QAM with $M\; =\; 256$ symbols. It uses a raised cosine as the shaping pulse with roll-off $r\; =\; 1$. a) What is the required bandwidth? b) What is the symbol rate?
4.13. Before DSL technologies were adopted using $\mathrm{\text{BW}}>\; 1$ MHz over copper cables, the Internet access over POTS (“plain old telephony system”) was based on the ITU V.90 and V.34 standards in downstream (operator’s central office to user premises) and upstream, respectively. The V.90 downstream connection used a PAM signal with $R_{\mathrm{\text{sym}}}=\; 8$ kbauds and $b\; =\; 7$ bits per symbol to achieve 56 kbps. The V.34 upstream achieved 33.6 kbps adopting QAM. Some people thought the V.34 maximum bit rate was close to the channel capacity $C$, when estimated using the AWGN channel with $C\; =\mathrm{\text{BW}}{log}_2(1\; +\mathrm{\text{SNR}})$, with a passband from 300 to 3400 Hz and $\mathrm{\text{SNR}}\le \; 40$ dB. But the download speed of 56 kbps modems were a breakthrough. Read https://perswww.kuleuven.be/~u0068190/Onderwijs/Extra_info/Rockwell%2056.PDF and explain why the techniques used for downloading could not be used for uploading.
4.14. The overall impulse response of a communication channel is $h(t)\; =\; \delta (t)\; +\; \alpha \delta (t\; -T_{\mathrm{\text{sym}}})$, where $T_{\mathrm{\text{sym}}}$ is the symbol duration and . Find the impulse response $h_e(t)$ of the corresponding zero-forcing equalizer filter. Show that the concatenation $h(t)\ast h_e(t)$ mitigates the ISI.