A problem listed for a given section requires an understanding of that section and earlier sections, but not of later sections. Within each section, problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.
• A space probe sends a picture containing 500 × 500 elements, each containing a brightness scale with 256 possible levels. This scale requires eight binary digits. Thus, altogether 8 × 500 × 500 = 2 × 106 pulses are required to encode the picture for transmission. Suppose that the transmitter uses a bandwidth of 1000 Hz. (For the faint signals from distant space probes the bandwidth must be kept small to reduce the effects of electronic noise that is present at all frequencies.) Roughly how long is needed to send one picture? Note that the center-to-center separation of adjacent pulses must be at least 2 Δt (the total width of any one pulse).
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