Lecture 9 homework

1. [1 point] Calculation.  This is like lecture 8 exercise 1, except that you should pick 3 systems and use an estimate of their (convex-hull) surface area to calculate the holographic bound on their entropy (given any amount of energy in them).  Give the number of distinguishable states and the average entropy density per cubic Angstrom of the object's volume.  Supposing that the surface area were reformed into the shape of a sphere, calculate the sphere's radius.  Then, using the formula relating the mass of a black hole to its area (green book sec. 2.2.1, p. 33), calculate how much mass you would have to stuff into your surface to actually achieve the maximum entropy.
2. [6 points] Analysis.  Due Feb. 6.  Suppose you have a total of 100 W of power to burn on wave-based communications signals, and a noise floor of 1 W in any given channel you might use, and each channel uses a frequency band having 2 GHz of bandwidth.
• (a) According to the Hartley-Shannon law, what is the maximum bit-rate capacity if you communicate using only a single channel?
• (b) Supposing that you divide your signal power evenly among n parallel channels (each 2 GHz), what is the optimum number of channels to maximize your total bit-rate?  (Or if there is no optimal solution, say so.)
• (c) Use the Smith entropy-flux bound to calculate the maximum possible bit-rate, using that power level, from a 100 cm² surface.
• (d) How large a frequency band would you need to achieve the bit-rate in part (c) using only a single channel?
• (e) If you use the optimal number of channels for the given noise level and available power, from part (b), how wide a frequency band would each channel need to have in order to attain the Smith limit?
• (f) What is the average energy density of the radiation traveling outwards from our surface in the part (c) scenario?  (Hint: You only need the power, area, and velocity.)
• (g) Based on (f), what temperature is the blackbody radiation field used in part (c)?  Use equation 2.10 from green book sec. 2.2.3, p.36.
• (h) The peak power of the blackbody radiation spectrum at temperature T occurs at photon energies of 2.82 k_BT.  The frequency of a photon is its energy divided by h-bar (Planck's constant over 2 pi, tough to render in html).  Using these facts, what is the highest-power frequency of the spectrum used in part (c)?
• (i) If you used a total signal bandwidth in each channel that was equal to the peak frequency from part (h), how many channels would you need to achieve the part (c) bit rate?
• (j) If each channel requires a surface area roughly equal to the squared wavelength at its peak frequency, how many such channels can you fit in our surface?
• (k) Finally, what signal-to-noise ratio is required in order for the Shannon bound to coincide with the Smith bound using the number of channels from part (j) and the per-channel bandwidth from part (i)?
• (l) What do you think is the significance of the noise level obtained from part (k)?