2.16  Exercises

2.1. An FSK modem uses eight distinct frequencies for signaling. How many symbols per second (in bauds) this modem must transmit to reach a 2.4 kbps bit rate?
2.2. Assume a digital binary signal $m(t)$ in which the duration of 1 bit is equal to 1 second, modulating a sinusoidal carrier $c(t)$ of frequency 6 Hz and peak amplitude of 1 V. For $m(t)$ representing a sequence of five bits {1 1 0 0 1} using rectangular pulses, draw carefully the output waveforms $m(t)c(t)$ of two modulators: (a) ASK with $m(t)$ having amplitudes 1 and 0, (b) PSK with $m(t)$ having amplitudes 1 and $-1$.
2.3. Being tired of symmetric constellations, a student designed $\{-5,1,3,4\}$ as its PAM constellation. Considering that the shaping pulse $p(t)$ has unitary energy and the symbols, transmitted at 4 bauds, are independent and uniformly distributed: a) what is the average energy of a transmit symbol? b) what is the transmit signal power? c) can you criticize the chosen constellation? If yes, in what aspects?
2.4. A binary communication system uses antipodal signals $p(t)$ and $-p(t)$, with $p(t)=A\text{sin}<!--\; FUNCTION\; APPLICATION\; -->(\mathit{\pi t}∕T_{\mathrm{\text{sym}}})$ with $0\le t\le T_{\mathrm{\text{sym}}}$ or 0 otherwise, and $A\in \{\pm 1\}$. Find a reasonable value for $T_{\mathrm{\text{sym}}}$ assuming the symbol rate must be 400 bauds. For the chosen value of $T$, sketch the transmit waveform corresponding to the information bits [0 1 0 0 1 0], where bit 1 corresponds to $A=-1$ and bit 0 to $A=1$.
2.5. The generation of a 4-PAM discrete time signal $s[n]$ is modeled by the convolution between the symbols $m[n\prime ]$ upsampled by a factor of $L$ and a shaping pulse $p[n]$. Assume $p[n]=3\delta [n]+4\delta [n-1]+5\delta [n-2]$, L=4, and symbols $\{-3,-1,1,3\}$ with Gray mapping to bits $\{00,01,11,10\}$, respectively. Draw carefully the signal $s[n]$ for the transmit bitstream 11 01 10 11 00.
2.6. Table 2.1 and the associated Figure 2.8 and Figure 2.7 contrast Gray and natural coding for 8-PAM. Assume the noise at the receiver has a relatively low power and that in all errors, a symbol is confused by one of its neighbor symbols. The symbols are equiprobable and the probability of symbol error is $P_e=0.01$. Estimate the bit error probability $P_b$ for each constellation.
2.7. Assume the 8-PAM constellation of Figure 2.8 is in volts and the symbols 7 and $-7$ have probability 0.3 each, while the remaining symbols are equiprobable. The basis function $p(t)$ has unitary energy and the transmitted signal $s(t)$ is obtained by multiplying the constellation symbols by $p(t)$. The symbol period is $T_{\mathrm{\text{sym}}}=2$ s. Calculate: (a) the average power of $s(t)$ and (b) the number of bits per second.
2.8. What are the theoretical power values that should be obtained with Listing 2.3 (consider an infinite number N of PAM symbols) for the sequences: a) m, b) m_upsampled and c) s.
2.9. An M-PAM system uses $R_{\mathrm{\text{sym}}}=500$ bauds and a shaping pulse $p(t)$ with energy $E_p$ such that the transmit PAM signal power $P$ coincides with the average energy constellation ${\overline{E}}_c$. Design this constellation to maximize the bit rate constrained to use a transmit power of 25 dBm and a modulation order $M$ that is a power of 2. The symbols must be uniformly spaced by a distance $d$. The path loss is 15 dB and the noise power at the receiver is such that the separation between neighboring PAM symbols at the receiver must be at least $\mathrm{\Delta }=60$ mV to achieve the desired BER of $10^{-3}$. Inform: a) the number $M$ of symbols, b) the respective bit rate, c) the corresponding ${\overline{E}}_c$ and d) the value of each symbol in volts. You may find useful that the M-PAM constellation $\pm \frac{d}{2},\pm \frac{3d}{2},\pm \frac{5d}{2},\dots ,\pm \frac{(M-1)d}{2}$ has an average energy ${\overline{E}}_c$ given by Eq. (2.10).
2.10. Sketch the block diagram of a binary FSK digital communication system using only analog signal processing (do not use DSP, ADC nor DAC). Provide a high-level description using a block diagram with e. g. filters and amplifiers.
2.11. Using unsigned bytes (i. e., from 0 to 255): a) inform the bitstream corresponding to unpacking a 16-PAM organized as the bytes sequence 0, 255, 15, 1.
2.12. Create a constellation for 16-PAM with zero-mean and average energy of $E_c=4.5$ J. You can list the code or the obtained symbols.
2.13. Modify the code of Application 2.5 to use $b=8$ bits per symbol and include an AWGN channel in your simulation. Pick a file for testing, with size larger than 5 MB. Assuming $R_{\mathrm{\text{sym}}}=100$ bauds, what is the minimum SNR that you still achieve error-free transmission? (if you do not have diff, transmit a zip file and check if you can uncompress the received version).
2.14. Assume that a binary FSK system must be implemented with synchronous modulation using digital signal processing. The receiver has the following specifications: two frequencies in the uplink are 980 and 1180 Hz, the sampling rate is $F_s=9.6$ kHz and $R_{\mathrm{\text{sym}}}=3000$ bauds, such that three samples represent each basis function. Assume the code below is used to obtain the two corresponding basis functions and explain: a) if they are orthogonal or not, b) the Octave/Matlab commands to normalize them for having unitary energy and check if they are orthonormal.

1T_sample = 1/Fs; t = 0:T_sample:1/Rsym-T_sample; %discretized time 
2basis0=cos(2*pi*980*t); basis1=cos(2*pi*1180*t); %basis functions