3.11  Exercises

3.1. Digital modulation schemes can be organized in two groups: orthogonal and phase/amplitude modulation. Describe each one and contrast the main characteristics of the two groups.
3.2. A constellation was created with three orthonormal basis functions ${\phi }_i(t),i=1,2,3$ and $M=8$ symbols located at the corners of a cube. a) Assume that $[0,0,0]$ and $[1,1,1]$ are two symbols of this constellation and find the other six. b) Calculate the average energy of this cubic constellation.
3.3. A QAM signal uses a carrier frequency ${\omega }_c=100$ rad/s and is given by

Find the expressions for its: a) IQ components $s_i(t)$ and $s_q(t)$, b) complex envelope $s_{\mathrm{\text{ce}}}(t)$ and c) associated analytic signal $s_{\mathrm{\text{+}}}(t)$. d) Show the graphs of their Fourier transforms and e) indicate the minimum ${\omega }_c$ value to allow obtaining $s_i(t)$ and $s_q(t)$ via QAM demodulation.
3.4. a) What is the minimum sampling frequency $F_s$ to represent the complex envelope of a passband signal of $\mathrm{\text{BW}}=20$ MHz centered at 1 GHz using quadrature sampling? b) In case a single ADC is used, what would be the minimum $F_s$?
3.5. BPSK can be interpreted as the multiplication of the carrier by a polar line code (or a PAM with constellation $\{\pm 1\}$) and a NRZ shaping pulse of amplitude $A$ over the symbol interval $T_{\mathrm{\text{sym}}}$. Hence, using Table 2.3, the PSD of the baseband BPSK (the BPSK complex envelope) is $S(f)=A^2{T}_{\mathrm{\text{sym}}}{\mathrm{\text{sinc}}}^2(f{T}_{\mathrm{\text{sym}}})$. Find the null-to-null bandwidth BW in terms of the bit rate $R$. Execute a Monte Carlo simulation to obtain an estimate of $S(f)$ and compare it with the theoretical expression, as done in Figure 2.21. But in this case, use the abscissa in kHz and $T_{\mathrm{\text{sym}}}=10^{-3}$. Choose appropriate values for the carrier $f_c$ and sampling $F_s$ frequencies. Change $T_{\mathrm{\text{sym}}}$ to confirm your findings on how BW depends on $R$.
3.6. Consider that the ADCs of a quadrature sampling hardware operate at $F_s=400$ MHz and the mixers have frequency $f_c=2$ GHz. The input RF signal is a sinusoid $x(t)=8\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(2\pi f_{0}t)$, where $f_0=2.14$ GHz, which is then frequency downconverted to 140 MHz and sampled by the ADCs. a) Show the Fourier transforms of the I and Q components, and of their complex envelope. b) How could you change this sampling scheme to pay less for the ADC(s)?
3.7. A modulation scheme uses four orthonormal basis functions ${\phi }_i(t),i=1,2,3,4$ and $M=256$ symbols. Each basis functions carries a 4-PAM with symbols separated by $d=2$. a) Calculate the average energy of this 256-symbols constellation. b) Carefully draw the block diagram of a correlative decoder for a receiver, trying to minimize its computational cost. For this diagram, consider “Pam demod” as the block that performs the decisions and maps PAM symbols into bits. c) Using inner products and the orthogonality property, prove that the correlative receiver can recover the symbol $[1,-3,-1,1]$ when it arrives at the receiver as $r(t)={\phi }_1(t)-3{\phi }_2(t)-{\phi }_3(t)+{\phi }_4(t)$.
3.8. Implement in Matlab/Octave the following transmission systems: a) BPSK, b) 8-PSK and c) $\pi ∕4$-DQPSK. Use coherent demodulation and, to simplify, assume that the carrier frequency is perfectly regenerated at the receiver. Choose the simulation parameters such that the bit rate $R$ is the same for all three cases. For each case, show the IQ data in a constellation format and the associated phase variations. Also, show the PSDs using theoretical expressions or Monte Carlo estimations.
3.9. Modify the matrix x in Listing 2.22 to use other transmit signals. For example, add the signals 1:D, D:-1:1, rand(1,D) and others. Then, execute the Gram-Schmidt procedure and observe how your choices with respect to the number $N$ of basis functions and constellation impact the PSD (signal power and bandwidth). Try to design a system with a relatively large number $M$ of input vectors and small $N$. Seek transmit signals that have relatively small bandwidth and limited power. Do you see advantages on using basis functions composed of sinusoids when compared to the ones obtained via a Gram-Schmidt procedure?
3.10. Assume a wireless channel is modeled with the two-ray impulse response $h(t)=\delta (t)-0.8\delta (t-\tau )$, where $\tau =2$ $\mu$s is the delay of the second path. The noise at the receiver has a one-sided PSD of $-140$ dBm/Hz. The transmit signal is a 16-QAM with $R_{\mathrm{\text{sym}}}=1.2$ Mbauds, which was shaped by a NRZ square pulse. The central (carrier) frequency is 900 MHz. Find: a) the baseband complex-valued channel model impulse response $h_{\mathit{bb}}(t)$ and its Fourier transform $H_{\mathit{bb}}(f)$, b) generate plots of both the baseband and passband channels within a bandwidth of $2R_{\mathrm{\text{sym}}}$, c) find the PSD of the baseband-equivalent noise, d) implement two Monte-Carlo simulations, one in passband and another using the baseband equivalent system. Properly scale the signals such that the SNRs are the same in both cases and evaluate the required $F_s$ and the computational cost of these two cases.
3.11. (Adapted from [Cio10]) A passband channel has a real-valued flat frequency response $H(f)$ with unitary gain from 75 to 175 MHz and zero otherwise. The QAM transmit signal cannot exceed 1 mW and its two-sided PSD $S_x(f)$ must obey a maximum level of $-83$ dBm/Hz. At the receiver, the signal is contaminated by WGN with a flat PSD level of $N_0∕2=-98$ dBm/Hz. The carrier frequency is $f_c=100$ MHz and the target symbol error probability is $P_e=10^{-6}$. Find: a) the baseband channel model $H_{\mathit{bb}}(f)$, b) the largest symbol rate that can be used with $f_c=100$ MHz, c) the maximum QAM signal power at the channel output, d) the maximum QAM data rate that can be achieved with the symbol rate of part b, e) a new carrier frequency value that maximizes the QAM data rate and f) the new data rate for part e.
3.12. Consider that the ADCs of a quadrature sampling hardware operate at $F_s=200$ MHz and the mixers have frequency $f_c=1$ GHz. The input RF signal is a sinusoid $x(t)=8\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(2\pi f_{0}t)$, where $f_0=1.16$ GHz, which is then frequency downconverted to 160 MHz and sampled by the ADCs. Show the Fourier transforms of the I and Q components, and of their complex envelope. Using the graphs, indicate the alias frequency that appears and its cancellation when one creates the complex envelope. You may find useful the discussion in [ url7iqs].