Exercises: Communications, Navigation, and Data Handling

Work these with a calculator. Use $c = 2.998\times10^8\ \text{m/s}$, Boltzmann's constant $k = 1.38\times10^{-23}\ \text{J/K}$ (equivalently $-228.6\ \text{dBW/K/Hz}$), $1\ \text{AU} = 1.496\times10^{11}\ \text{m}$, and dish efficiency $\eta = 0.6$ for spacecraft antennas and $0.7$ for DSN ground antennas unless told otherwise. Work link budgets in decibels. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in appendices/answers-to-selected.md — try each cold first. For the "implement it" problems, do not run the code: hand-trace it and write the result in an # Expected output: comment, as the chapter does.

Part A — Warm-ups: decibels, wavelengths, gains (⭐)

26.1 † A spacecraft transmitter radiates $20\ \text{W}$. Express this power in dBW.

26.2 Find the gain (in dBi) of a $3\ \text{m}$ dish at X-band ($f = 8.4\ \text{GHz}$).

26.3 † What is the wavelength of a Ka-band signal at $f = 32\ \text{GHz}$? Give it in millimeters.

26.4 Compute the half-power beamwidth of a $70\ \text{m}$ DSN dish at X-band ($8.4\ \text{GHz}$). Comment on what such a beamwidth implies for how accurately the dish must be pointed.

26.5 † Compute the free-space path loss (in dB) for a signal from the Moon ($d = 3.84\times10^8\ \text{m}$) at S-band, $f = 2.2\ \text{GHz}$.

26.6 † Express a received power of $2\times10^{-19}\ \text{W}$ in dBW.

26.7 † A Mars orbiter downlinks at X-band ($8.4\ \text{GHz}$) with $P_t = 15\ \text{W}$ and a $2.0\ \text{m}$ high-gain dish, to a DSN $70\ \text{m}$ antenna, at a worst-case range of $2.5\ \text{AU}$, with $L_{\text{other}} = 2\ \text{dB}$. Compute the EIRP, the free-space path loss, the received power $P_r$ (dBW), and the ratio $P_r/N_0$ (dB-Hz) assuming a system noise temperature $T_s = 30\ \text{K}$.

26.8 Using the $P_r/N_0$ from Problem 26.7, find the maximum data rate (bit/s) the link can support if the coded receiver needs $E_b/N_0 = 2.3\ \text{dB}$. Does the link close at $256\ \text{kbit/s}$?

26.9 † Starting from the same Mars-orbiter link, you double both dish diameters (spacecraft and ground). By how many decibels does the received power change, and by what numerical factor? (Reason in dB; do not recompute the whole budget.)

26.10 Compute the figure of merit $G/T$ (in dB/K) for a DSN $34\ \text{m}$ dish at X-band ($8.4\ \text{GHz}$, $\eta = 0.7$) with system noise temperature $T_s = 25\ \text{K}$.

Part C — Implement it in Python (⭐⭐)

Write each function, hand-trace it for the given inputs, and record the result in an # Expected output: comment. Do not run it. Use $C = 2.998\times10^8$ and the formulas of the chapter.

26.11 † Write fspl_db(dist_m, freq_hz) returning the free-space path loss in dB. Trace it for fspl_db(3.84e8, 2.2e9) (the Earth–Moon S-band case).

26.12 Write dish_gain_db(diameter_m, freq_hz, efficiency=0.6). Trace it for dish_gain_db(70.0, 8.4e9, 0.7) (a DSN 70 m at X-band).

26.13 † Write link_budget(pt_w, gt_db, gr_db, freq_hz, dist_m, extra_loss_db=0.0) returning the received power in dBW. Trace it for link_budget(15, 42.7, 74.2, 8.4e9, 3.74e11, 2.0) (the Mars orbiter of Problem 26.7).

Part D — Find the error (⭐⭐)

26.14 † A student computes free-space path loss as $L_{\text{fs}} = 10\log_{10}(4\pi d/\lambda)$ and gets $152\ \text{dB}$ for a deep-space link, concluding the signal is far stronger than the textbook claims. What did they get wrong, and what is the correct expression?

26.15 A student computes a spacecraft dish's gain as $G = (\pi D/\lambda)^2$ — omitting the aperture efficiency $\eta = 0.6$ — and uses it in a link budget. By how many decibels does this overstate the gain, and does it make the predicted link look better or worse than reality?

Part E — Design it (⭐⭐ / ⭐⭐⭐)

26.16 † You must return $500\ \text{kbit/s}$ from a probe at $5\ \text{AU}$ using a $30\ \text{W}$ X-band ($8.4\ \text{GHz}$) transmitter to a DSN $70\ \text{m}$ dish, with $T_s = 25\ \text{K}$, $L_{\text{other}} = 2\ \text{dB}$, and a required $E_b/N_0 = 2\ \text{dB}$. What spacecraft high-gain dish diameter do you need? (Solve the link budget for the required transmit gain, then invert the dish-gain formula.)

26.17 (Mission project) Write the communications note for your mission (Track A/B/C/D): the band, the required data rate and daily volume, the ground stations, and the antenna and transmitter power (draw the power figure from your Chapter 25 power budget). Save it to your MDR.

26.18 † Choose a frequency band (S, X, or Ka) for each mission and justify it in two sentences: (a) a lunar communications relay that must return $100\ \text{Mbit/s}$; (b) a probe in the Kuiper Belt at $45\ \text{AU}$ with only $\sim 200\ \text{W}$ of spacecraft power.

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

26.19 † Why does data rate fall as $1/d^2$? Explain the chain from received power to data rate in one or two sentences, then estimate: a probe returns $40\ \text{kbit/s}$ at Jupiter ($5\ \text{AU}$); what rate would the same system manage from the Kuiper Belt ($45\ \text{AU}$)?

26.20 Why can't you just put a $10\ \text{kW}$ transmitter on a Pluto probe to send data at megabits per second? Address both the power source and the waste heat. (Tie your answer to Chapter 25 and Chapter 24.)

26.21 † Estimate the round-trip light time to a rover on Mars at $1.5\ \text{AU}$. What does the number imply about "driving" the rover from Earth in real time?

Part G — Interleaved & synthesis (from earlier chapters) (⭐⭐/⭐⭐⭐)

26.22 (Ch. 9) A geostationary satellite is about $3.6\times10^7\ \text{m}$ away; a Pluto probe about $4.8\times10^{12}\ \text{m}$. By how many decibels does the free-space path loss differ (same frequency)? Use this to explain in one sentence why GEO links run at gigabits per second while Pluto runs at kilobits.

26.23 † (Ch. 25) A transmitter produces $25\ \text{W}$ of radio-frequency output through an amplifier that is $35\%$ efficient. How much electrical power does it draw from the bus? If the spacecraft's RTG supplies $200\ \text{W}$, what fraction of the power budget is the transmitter, and what does that imply for mission design?

26.24 (Ch. 13) Of the three radio observables (ranging, Doppler, delta-DOR), which one fixes the spacecraft's position across the sky? Explain, in two or three sentences, how orbit determination combines all three into a trajectory estimate.

26.25 † (Ch. 3, synthesis) A communications subsystem — high-gain dish, transponder, and amplifier — masses $40\ \text{kg}$. Explain, invoking the rocket equation and the theme "mass is the enemy," why a designer cannot simply keep enlarging the dish to buy more data rate. What is the tension?

26.26 (Ch. 24) A travelling-wave-tube amplifier is about $35\%$ efficient. For a $20\ \text{W}$ radio-frequency output, how much waste heat must be rejected, and why is dumping it uniquely hard in vacuum? (Recall how heat leaves a spacecraft.)

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "implement it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: a link budget laid out as a decibel column, explicit units on every quantity, and a sanity check on the final number (does the received power come out near $10^{-18}\ \text{W}$? does the data rate match the mission's distance?).