1.10  Exercises

1.1. A signal $x(t)=\text{cos}<!--\; FUNCTION\; APPLICATION\; -->(2\pi f_{x}t)$ is mixed with a carrier $c(t)=\text{cos}<!--\; FUNCTION\; APPLICATION\; -->(2\pi f_{c}t)$, where $f_x=1\text{MHz}$ and $f_c=10\text{MHz}$. Assume ideal mixing. What frequencies will appear in the output signal $x(t)c(t)$, and what is the corresponding output PSD?
1.2. A signal $x(t)$ has a unilateral PSD occupying a bandwidth ${\text{BW}}_i=2\text{MHz}$, from 0 to 2 MHz. It is mixed with a carrier $c(t)=\text{cos}<!--\; FUNCTION\; APPLICATION\; -->(2\pi f_{c}t)$, where $f_c=20\text{MHz}$. What is the resulting range of frequencies in the output signal $x(t)c(t)$, and what shape best describes its PSD assuming $x(t)$ has a triangular shape?
1.3. Given that $\nu (t)$ is a WGN with amplitude in volts, sketch graphs of possible (arbitrary but consistent): (a) probability density function, (b) power spectral density and (c) time waveform. Inform the numerical values and units in each graph.
1.4. Assuming an AWGN channel model, the received signal $r(t)=s(t)+\nu (t)$ is contaminated by WGN $\nu (t)$ with a unilateral PSD $N_0=-140$ dBm/Hz. The $\mathrm{\text{SNR}}=\frac{𝔼[|s(t){|}^2]}{𝔼[|\nu (t){|}^2]}$ at the output of this AWGN channel must be at least 36 dB. The transmit power is always $-4$ dBm, independent on the receiver bandwidth BW. Find the maximum value of BW to have $\mathrm{\text{SNR}}\ge 36$ dB.
1.5. A discrete time signal $z[n]=s[n]+\nu [n]$ is the summation of a sinusoid $s[n]$ of power 10 dBm (corresponding to a continuous-time signal of frequency 3 Hz) and WGN $\nu [n]$. The signals are uncorrelated. Inform: a) the amplitude of $s[n]$, b) the power of $\nu [n]$ such that the SNR is 20 dB.
1.6. As oscillator has accuracy of 10 ppm when the frequency is 1.8 GHz. What are its minimum and maximum frequencies?
1.7. Using pages 10 to 14 from [ url5lbu] as starting point, generate plots of the link margin variation over frequency within the 1 to 5 GHz range for both directions: client to access point and vice versa.
1.8. The thermal noise at the input of an amplifier is modeled as WGN with a (unilateral) PSD with $N_0=-174$ dBm/Hz. Assuming the equipment has a gain of 60 dB and a noise figure of 5 dB, what is the noise PSD level at its output?
1.9. Modify the code of Example 1.12 to calculate the SNR at input and output assuming that the amplifiers have a bandwidth of: a) 10 MHz and b) 100 MHz. Comment on how the bandwidth impacts the noise figure of their cascade.
1.10. Motivated by Application 2.1, the goal here is to confirm the importance of the regeneration capability of digital communication systems. Assume the signal is transmitted from city A to city B, which are separated by 40 km. The signal transmit power is 80 dBm and the path loss is 12 dB/km. The transmission chain is composed by three intermediate nodes (or “stations”), uniformly spaced at intervals of 10 km. The input thermal noise power at all nodes (transmitter, receiver and intermediate stations) is $-70$ dBm and the receiver at city B has sensitivity $-54$ dBm and $\mathrm{\text{NF}}=4$ dB. Calculate the output SNR at each intermediate node, the link margin and input SNR at city B considering two alternatives for the stations: a) they perform signal regeneration before retransmission by implementing the full demodulation process or b) regeneration does not occur and the stations are repeaters that only implement analog amplification. A regeneration station of a) is modeled as equivalent to the transmitter of city A and receiver of city B, i. e., a receiver with $\mathrm{\text{NF}}=4$ dB and sensitivity $-54$ dBm, and transmitter with output power 80 dBm; while a repeater station of b) is modeled as an amplifier with $\mathrm{\text{NF}}=4$ dB and 120 dB gain. c) Then consider what would happen if the station gains of b) are only 100 dB.