ELEC5460 Assignment 2 Spring 2015 1) Show that the MMSE

ELEC5460
Assignment 2
Spring 2015
1) Show that the MMSE-SIC receiver is optimal, i.e., it achieves the best possible
sum rate in the following sense:
!!
log 1 + 𝑆𝐼𝑁𝑅! = log det 𝐼!! +
!!!
1
𝐻𝐾! 𝐻∗
𝑁!
where the SINR and rate for stream 𝑘 are 𝑆𝐼𝑁𝑅! and log 1 + 𝑆𝐼𝑁𝑅! ,
respectively.
Hint: You might need the following identity: log 𝐴 + 𝑥𝑥 ∗ − log 𝐴 = log 1 +
𝑥 ∗ 𝐴!! 𝑥 .
[15 points]
2) One important goal in this course is the ability to make system-level decisions
upon different “situations” based on the component-technologies we have learnt.
For example, the types of channels (fast vs slow fading, frequency flat vs
frequency selective fading) and the availability of channel state information (CSI)
at the transmitter in various communication components.
(a) Compare and contrast the following technologies in terms of their typical
operating fading channels and requirement of CSIT.
[15 points]
Space Time Coding (e.g. Alamouti scheme, orthogonal Space Time
Coding),
• BLAST type Layered Space Time Coding (e.g. VBLAST, HBLAST and
DBLAST),
• OFDM,
• Rate and/or power adaptation.
Explain briefly on your choice.
•
(b) Suppose you are the engineer to draft a new radio interface to support the
following requirement.
• Transmission BW = 20MHz.
• Maximum base station transmit power = 20dBm
• Peak bit rate of 20Mbps.
You have to support delay sensitive applications where the delay requirement is 5
ms. Time dimension is partitioned into frames with frame duration = 5 ms.
i.
Suppose both the base station and mobile station have single antenna only.
What should be the constellation size and symbol duration to realize the
target bit rate? (Assume BW = 1/Ts).
[5 points]
It is well-known that using MIMO technology could improve the system
performance. Suppose we have 2 antennas at both the base station and the
mobile side.
ii.
Using Alamouti code with the same constellation in (i), what are the bit
rate and the diversity order that could be realized in an uncoded system?
Explain your answer.
[5 points]
iii.
Using VBLAST architecture with the same constellation. What is the bit
rate and maximum diversity order that could be realized in an uncoded
system? Explain your answer.
[5 points]
iv.
Suppose we have convolutional coding (rate ½) with BPSK constellation.
Repeat (i) and (ii) again.
[5 points]
v.
In (iii), it seems we could double the bit rate (spatial multiplexing) and
achieve full diversity (spatial diversity) simultaneously using coded
VBLAST. Does that contradict with diversity-multiplexing tradeoff
results?
[5 points]
vi.
Suppose you use linear ZF receiver to detect coded VBLAST symbols.
What is the maximum diversity order you could achieve?
[5 points]
3) (Stochastic Subgradient Projection) Let 𝜋! be a random mapping from
𝑘 𝜖 {1,2, … ,7} to 𝜋! 𝜖 {1,2,3,4}. Assume that for any 𝑘, 𝜋! is independently
generated from a distribution over {1,2,3,4} with probability mass function
𝒑 = [𝑝! , … , 𝑝! ] (i.e., for any 𝑘, 𝜋! = 𝑙 with probability 𝑝! ). Consider the
following stochastic optimization problem:
min −𝐸[ min {𝑞!! }] ,
𝒒 ! !!
! !{!,!,…,!}
where 𝒒 = [𝑞! , … , 𝑞! ]! is the optimization variable and
!
𝐷! = 𝒒: 𝑞! 𝜖 0,1 , ∀𝑙, 𝑎𝑛𝑑 𝑞! ≤ 3 .
!!!
Note that in this stochastic optimization problem, the system state is 𝝅 =
{𝜋! , … , 𝜋! }.
(a) What type of stochastic optimization problem is it?
[5 points]
(b) Prove that this problem is convex.
[5 points]
(!)
(c) Let 𝝅(!) = {𝜋!! , … , 𝜋! } denote a realization of the system state at the 𝑖 !!
time slot. Prove that 𝒈(!) = [𝑔!! , … , 𝑔!! ]! is a noise unbiased subgradient of
(!)
the objective function at 𝒒, where 𝑔! = −1 𝑙 = 𝜋!!∗ , ∀𝑙 , and 𝑘 ∗ is any
index satisfying 𝑞!!∗ = min!!!!! {𝑞!! }.
[10 points]
(d) At time slot 𝑖, the stochastic Subgradient projection update for solving the
above problem is given by:
𝒒 !!! = 𝑎𝑟𝑔 min 𝒒 ! − 𝛼 ! 𝒈 ! 𝒒 !
𝒒 ! !!
How to choose the step size 𝛼 ! ?
− 𝒒 !
[5 points]
(e) Suppose the probability mass function 𝒑 is known. Hence, the original
problem reduces to a deterministic optimization problem.
[15 points]
i.
Calculate the closed-form expression for the objective function.
ii.
Is this deterministic problem convex?
iii.
Find the closed-form expression for the optimal solution of this
problem.