Question

Question 3. 25 marks

This question is about the downlink of a two user system, with one base station (BS) sending signals to two users, denoted user 1 and user 2. The BS is equipped with an array of n antenna elements, and each user has a single antenna. The system is a flat fading scenario, with a single complex channel coefficient from each BS antenna to each user in the base-band channel representation. We denote the channel coefficients from the BS to user i by the row vector ⃗hi = (h1,i, h2,i, . . . , hn,i), i = 1, 2. The channel matrix representing the gains to both users is denote by H, and is the 2 × n matrix given as follows:

? h1,1 h2,1 ··· hn,1 ? H = h1,2 h2,2 ··· hn,2

There is a n × 2 precoding matrix, W , used by the BS for beamforming to the two users. The beamforming coefficients are given as follows:

 w1,1

w1,2 

 w2,1 W =  .

w2,2  .  . .

wn,1 wn,2
Denote the first column of W by w⃗1 and the second column of W by w⃗2. These are the

beamforming vectors that BS uses to beamform to users 1 and 2, respectively.

The complex valued data symbols for the two users, denoted by x1 and x2 are taken from a phase shift keying constellation, so that |xi| = 1 for i = 1, 2. We put these in a 2 × 1 column vector ⃗x. The components of this vector are independent. There is complex valued additive white Gaussian noise at the mobile receivers, denoted by zi, i = 1, 2. The zi are independent, identically distributed complex Gaussian random variables, with mean zero and variance σ2. We put these in a 2 × 1 column vector ⃗z. The overall multi-user channel is described by the matrix equation:

⃗y = H W ⃗x + ⃗z are the received samples at each user.

? y1 ?

where ⃗y = y
(a) Write down the effective (scalar) channel for user 1. In other words, write down an equation

2

for y1 in terms of the parameters ⃗h1,⃗h2,w⃗1,w⃗2,x1,x2 and z1.
You may use inner product notation <, > in your answer (for the inner product between two

column vectors) or you can use matrix multiplication instead.

Question 3 25 marks This question is about the downlink of a two user system, with one base station (BS) sending signals to two users, denoted user 1 and user 2. The BS is equipped with an array of n antenna elements, and each user has a single antenna. The system is a flat fading scenario, with a single complex channel coefficient from each BS antenna to each user in the base-band channel representation. We denote the channel coefficients from the BS to user i by the row vector h; = (h1, h2, 1..., hn,i), i = 1,2. The channel matrix representing the gains to both users is denote by H, and is the 2 x n matrix given as follows: h1,1 h2,1 hal hn.2 There is a n x 2 precoding matrix, W, used by the BS for beamforming to the two users. The beamforming coefficients are given as follows: w1.1 W12 2.1 12,2 1 = WW2 Denote the first column of W by w and the second column of W by us. These are the beamforming vectors that BS uses to beamform to users 1 and 2, respectively. The complex valued data symbols for the two users, denoted by and are taken from a phase shift keying constellation, so that K = 1 for i = 1,2. We put these in a 2 x 1 column vector i. The components of this vector are independent. There is complex valued additive white Gaussian noise at the mobile receivers, denoted by 21, i = 1,2. The zare independent, identically distributed complex Gaussian random variables, with mean zero and variance o?. We put these in a 2 x 1 column vector . The overall multi-user channel is described by the matrix equation: y = HW + 3 where 5 = (%) are the received samples at each user. (a) Write down the effective (scalar) channel for user 1. In other words, write down an equation for y, in terms of the parameters hi ha, 21, 22, 11, 1'2 and 21. You may use inner product notation <, > in your answer (for the inner product between two column vectors) or you can use matrix multiplication instead. (5 marks)

Answer 1

H = given ģ= () hi he W=( WJ W) Z= (24) and Z= Z Zz Y =HWF+ YA ha ( - # (ww.) (?) + (31 Wy> <h1, we 3) (2) ( ZI + Z2l () Khm, ws> <RT, ū> As hy and ha are row vectors but inner product is for column vectors 80, RI, FI used in inner products 91 ( ) + ZI Z Y2 KRI, Wi>X4+ <!,W2>X2 <5), W;> x4 + <ŘI, W, XX2 :.9=<FI, Ws >xy + <hi,W2>&2 +2, CS Scanned with CamScanner