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k=1 K 2 in .NET Creator Code 3 of 9 in .NET k=1 K 2




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k=1 K 2 generate, create 3 of 9 none with .net projects BIRT Reporting Tools 1. Let h1 hK be i.i.d. (0,1) random variables. Show that h2 = 1 k k=1 (6.85). Hint: You might fi Code 3/9 for .NET nd it easier to prove the following stronger result (using induction):. h 2 has the same distribution as hk k (6.86). 2. Using the previ Code39 for .NET ous part, or directly, show that h2 1 loge K as K (6.

87). thus the mean of t he effective channel grows logarithmically with the number of users. 3. Now suppose h1 hK are i.

i.d. / 1 + 1/ 1 + (i.

e., Rician random variables with the ratio of specular path power to diffuse path power equal to ). Show that 1 h2 loge K 1+ as K (6.

88). i.e., the mean of .

net framework Code 39 the effective channel is now reduced by a factor 1 + compared to the Rayleigh fading case. Can you see this result intuitively as well Hint: You might find the following limit theorem (p. 261 of [28]) useful for this exercise.

Let h1 hK be i.i.d.

real random variables with a common cdf F and pdf f satisfying F h is less than 1 and is twice differentiable for all h, and is such that lim d 1 F h dh fh =0 (6.89). Then 1 k K max Kf lK hk lK converges in distr ibution to a limiting random variable with cdf exp e x In the above, lK is given by F lK = 1 1/K. This result states that the maximum of K such i.i.

d. random variables grows like lK ..

6.9 Exercises Exercise 6.22 (Sel ective feedback) The downlink of IS-856 has K users each experiencing i.i.

d. Rayleigh fading with average SNR of 0 dB. Each user selectively feeds back the requested rate only if its channel is greater than a threshold .

Suppose is chosen such that the probability that no one sends a requested rate is . Find the expected number of users that sends in a requested rate. Plot this number for K = 2 4 8 16 32 64 and for = 0 1 and = 0 01.

Is selective feedback effective Exercise 6.23 The discussions in Section 6.7.

2 about channel measurement, prediction and feedback are based on an FDD system. Discuss the analogous issues for a TDD system, both in the uplink and in the downlink. Exercise 6.

24 Consider the two-user downlink AWGN channel (cf. (6.16)): yk m = hk x m + zk m k=1 2 (6.

90). Here zk m are i.i. Code 3/9 for .

NET d. 0 N0 Gaussian processes marginally k = 1 2 . Let us take h1 > h2 for this problem.

1. Argue that the capacity region of this downlink channel does not depend on the correlation between the additive Gaussian noise processes z1 m and z2 m . Hint: Since the two users cannot cooperate, it should be intuitive that the error probability for user k depends only on the marginal distribution of zk m (for both k = 1 2).

2. Now consider the following specific correlation between the two additive noises of the users. The pair z1 m z2 m is i.

i.d. with time m with the distribution 0 Kz .

To preserve the marginals, the diagonal entries of the covariance matrix Kz have to be both equal to N0 . The only parameter that is free to be chosen is the off-diagonal element (denoted by N0 with 1): Kz = N0 N0 N0 N0. Let us now allow t he two users to cooperate, in essence creating a point-to-point AWGN channel with a single transmit but two receive antennas. Calculate the capacity C of this channel as a function of and show that if the rate pair R1 R2 is within the capacity region of the downlink AWGN channel, then R1 + R2 C (6.91).

3. We can now choo 3 of 9 barcode for .NET se the correlation to minimize the upper bound in (6.

91). Find the minimizing (denoted by min ) and show that the corresponding (minimal) C min is equal to log 1 + h1 2 P/N0 . 4.

The result of the calculation in the previous part is rather surprising: the rate log 1 + h1 2 P/N0 can be achieved by simply user 1 alone. This means that with a specific correlation min , cooperation among the users is not gainful. Show this formally by proving that for every time m with the correlation given by min , the sequence of random variables x m y1 m y2 m form a Markov chain (i.

e., conditioned on y1 m , the random variables x m and y2 m are independent). This technique is useful in characterizing the capacity region of more involved downlinks, such as when there are multiple antennas at the base station.

Exercise 6.25 Consider the rate vectors in the downlink AWGN channel (cf. (6.

16)) with superposition coding and orthogonal signaling as given in (6.22) and (6.23),.

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