Extra Problems for Chapter 14

T.M. Cover and J.A. Thomas

1. Multiple-access channel

   Let the output Y of a multiple-access channel be given by

   

   where are both real and power limited,

   

   and

   Note that there is interference but no noise in this channel.

   1. Find the capacity region.
   2. Describe a coding scheme that achieves the capacity region.
2. Slepian Wolf

   Let (X,Y) have the joint pmf p(x,y)

   

   where . (Note: This is a joint, not a conditional, probability mass function.)

   1. Find the Slepian Wolf rate region for this source.
   2. What is in terms of ?
   3. What is the rate region if ?
   4. What is the rate region if ?
3. Square channel

   What is the capacity of the following multiple access channel?

   

   1. Find the capacity region.
   2. Describe achieving a point on the boundary of the capacity region.
4. Slepian-Wolf: Two senders know random variables and respectively. Let the random variables have the following joint distribution:

   

   where . Find the region of rates that would allow a common receiver to decode both random variables reliably.

5. Multiple access.
   1. Find the capacity region for the multiple access channel

      

      where

      

   2. Suppose the range of is . Is the capacity region decreased? Why or why not?
6. Broadcast Channel. Consider the following degraded broadcast channel.

   

   1. What is the capacity of the channel from X to ?
   2. From X to ?
   3. What is the capacity region of all achievable for this broadcast channel? Simplify and sketch.
7. Stereo. The sum and the difference of the right and left ear signals are to be individually compressed for a common receiver. Let be Bernoulli and be Bernoulli and suppose and are independent. Let , and .
   1. What is the Slepian Wolf rate region of achievable ? Again, simplify and sketch the rate region.

      

   2. Is this larger or smaller than the rate region of ? Why?

      

      There is a simple way to do this part.

8. Multiplicative multiple access channel. Find and sketch the capacity region of the multiplicative multiple access channel

   

   with , , and

9. Distributed data compression. Let be independent Bernoulli(p). Find the Slepian-Wolf rate region for the description of where

   

   

10. Noiseless multiple access channel Consider the following multiple access channel with two binary inputs and .
    1. Find the capacity region. Note that each sender can send at capacity.
    2. Now consider the cooperative capacity region, . Argue that the throughput does not increase, but the capacity region increases.
11. Infinite bandwidth multiple access channel Find the capacity region for the Gaussian multiple access channel with infinite bandwidth. Argue that all senders can send at their individual capacities.
12. A multiple access identity.
    Let denote the channel capacity of a Gaussian channel with signal to noise ratio x. Show

    

    This suggests that 2 independent users can send information as well as if they had pooled their power.

13. Square channel.
    What is the capacity of the following multiple access channel?

    

    1. Find the capacity region.
    2. Describe achieving a point on the boundary of the capacity region.
    3. What is the capacity if ?
14. FDMA. Maximize the throughput over to show that bandwidth should be proportional to transmited power for FDMA.

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