Note Technique: CSI estimation in massive MIMO systems and pilot contamination for MMSE receiver architecture

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Note Technique: CSI estimation in massive MIMO systems and pilot contamination for MMSE receiver

architecture

Jean-Pierre Cances, B Sissokho, Vahid Meghdadi, Ahmed D. Kora

To cite this version:

Jean-Pierre Cances, B Sissokho, Vahid Meghdadi, Ahmed D. Kora. Note Technique: CSI estimation in massive MIMO systems and pilot contamination for MMSE receiver architecture. 2016. �hal-01400495�

I. Note Technique: CSI estimation in massive MIMO systems and pilot contamination forMMSE receiver architecture

JP Cances, B Sissokho, V Meghdadi, A Kora Xlim Limoges UMR 7252

1- Context and System:

We consider L cells, where each cell contains one base station equipped with M antennas and K single-antenna users. Assume that the L base station share the same frequency band. We consider uplink transmission, where the lth base station receives signals from all users in all cells (Fig.1). Then, the M × 1 received vector at the lth base station is given by:

l i L

i l i u

l

p x n

y   

1



,

(1)

Where 

_i,l

represents the M × K channel matrix between the lth base station and the K users in the ith cell, i.e.,

_,

i l m k,

 

   is the channel coefficient between the mth antenna of the lth base station and the kth user in the ith cell. p

_u

x

_i

is the K × 1 transmitted vector of K users in the ith cell (the average power used by each user is p

_u

) and n

_l

contains M × 1 additive white Gaussian noise (AWGN) samples. We assume that the elements of n

_l

are Gaussian distributed with zero mean and variance:

²

l H

M

l l



 

  

E n n _nI

and

²

l H

l l M



 

  

E n n _n

.

Fig. 1: Uplink transmission in multicell MU-MIMO system 2- Physical Channel Model:

Here, we introduce the finite-dimensional channel model that will be used throughout this technical note. The angular domain is divided into a large but finite number of directions P.

L-th cell 1-th cell

i-th cell

l-th cell

Pis fixed regardless of the number of base station antennas (P<M). Each direction, corresponding to the angle: ^{ }

_k

^,

_k

^{ }  ^{ / 2, / 2 , }  ^{k  1,..., P} ,is associated with an M × 1 array steering vector a ( ) 

_k

which is given by:



^{jf jf jf}



^T

k

e

^k

e

^k

e

^{M k}

P

) ( )

( )

(

, ,...,

) 1

(  

^{ 1  2  }

a (2)

Where f

_i

( )  is some function of _ . Typically, for a regular phased array, we have:

( ) 2 ( 1) sin ( )

i

f   d i 

  

The channel vector from kth user in the ith cell to the lth base station is then a linear combination of the steering vectors as follows: ( )

1

m P

m

g

ilkm

a 





, where g

_ilkm

is the propagation coefficient from the kth user of the ith cell to the lth base station, associated with the physical direction m (direction of arrival 

_m

). Let G

_{i l},

  g

_il1

, ..., g

_ilK

 be a P × K matrix with



_{ilk ilkP}



^T

ilk

 g

₁

,..., g

g that contains the path gains from the kth user in the ith cell to the lth base station. The elements of G

_i,l

are assumed to be independent. Then, theM × K channel matrix between the lth base station and the K users in the ith cell is:

, ,

i l

 AG

i l

 (3)

Where A 

^_

a   

1

,..., a   

_P ^_

is a full rank M × Pmatrix. The propagation channel G

_i,l

models independent fast fading, geometric attenuation, and log-normal shadow fading. Its elements g

_{il k m}

are given by:

P m

h

g

_ilkm



_ilkm



_ilk

 1 , 2 ,..., (4)

Where h

_ilkm

is a fast fading coefficient assumed to be zero mean and have unit variance.



ilk

models the path loss and shadowing which are assumed to be independent of the direction m and to be constant and known a priori. This assumption is reasonable since the value 

_ilk

changes very slowly with time. Then, we have:

2 / 1

, ,

,l il il

i

H D

G  (5)

Where H

_i,l

is the P × K matrix of fast fading coefficients between the K users in the ith cell

and the lth base station, the columns H

_{i l,}

(:, ) k are independent and we have:

2 2

, ,

1 1

(:, ) (:, ) ( , ) 1

P P

H

i l i l il ilkm

m m

H ilk ilk

k k h m k h

 

   

 

         

         

E

^{H H} E h h

E E

Furthermore within one given column we consider the elements: h m k

_{i l}

( , ) independent, i.e.:



hilkmhilkn^*



^0

E

if m ≠ n. D

_i,l

is a K × K diagonal matrix whose diagonal elements are given by  

il kk

 

ilk

, ,

D . Therefore, (1) can be written as:

l L

i l i l i u

l L

i l i u

l

p A G x n p A H D x n

y

i i







  



 1

2 / 1

, , 1

,

(6)

In the following sections we will use the following intermediate parameters:

1 K

jl jlk

k

 



  ^,

1 L

iln ln

i

 



  ^{, and}

1

ˆ

^L

l i l

i

 



  ^.

3- Channel Estimation:

Channel estimation is performed by using training sequences received on the uplink. A part of the coherence interval is used for the uplink training. All users in all cells simultaneously transmit pilot sequences of length  symbols. The assumption on synchronized transmission represents the worst case from the pilot contamination point of view, but it makes no fundamental difference to assume unsynchronized transmission. We assume that the same set of pilot sequences is used in all L cells. Therefore, the pilot sequences used in the lth cell can be represented by a K   matrix p

_p



_l

 p

_p

 ( ≥ K), which satisfies  

^H

 

_Κ

where:

u

p

p

p   . From (6), the received pilot matrix at the lth base can be written as:

1/2

, , ,

1 L

p l p i l i l l

i

p



  ^ 

Y A H D N (7)

N

l

is the resulting noise matrix of size M   and of course we have: E   N

l

 0

M 2

l

H

l l

 

M

  

 

E

N N

n

I and

²

l

H

l l

M 

_

  

 

E

N N

n

I .

3.1. Minimum Mean-Square Error (MMSE) Estimation:

We assume that the base station uses here MMSE estimation. The received pilot matrix Y

_p,l

is multiplied by 

^H

at the right side to obtain: Y 

_{p l,}

 Y

_{p l,}



^H

:

, ,

1/ 2

, ,

1

1/ 2

, ,

1 H

p l p l

L

H

p i l i l l

L

p i l i l l

p p







 

 





 



i

i

Y Y

A H D N A H D W

(8)

With: W

_l

 N

_l



^H

. Since we consider here M-PSK modulation format for the transmit pilot matrixwe can denote  as:

1 1

11 12

2 2

21 22

1 2

1 2

1

p

p

m m mp m

Kp K

K K

j j

j j

j j

j j

j j j j

j j

j j

e e e e

e e e e

e e e e

e e e e









 

 

 

 

   

 

 



 

 

 

 

 

  

 

 

 

 

 

 

     

 

     

 



Hence, we have:

_,

1

^,

( , ) k m a

_{k m}

e

^{jk m}

  

 and

_,

1

^,

( , )

^{j k n}

H

n k b

n k

e

^



 



 . The matrix product

H l



N yields:

,

1 1

( , ) ( , ) ( , ) 1 ( , )

^{j m k}

H H

l l l

k k

n m n n k k m n n k e

 







 

   

 

N

In the same way we obtain:

* * ,

1 1

( , ) ( , ) ( , ) 1 ( , )

^{jn k}

H

l l l

k k

n m n k n m k n m k e

 











   

 N

The matrix product: W W = N

_{l l}^H _l



^H

N

_l

 

^H

N

_l^H

is thus equal to:

, ,

, ,

1

*

1 1 1

( )

*

1 1 1

( , ) ( , ) ( , )

1 1

( , ) ( , )

1 ( , ) ( , )

k s k r

k r k s

K

H H H H

l l l l

k K

j j

l l

k s r

K

j

l l

k s r

n m n k k m

n n s e n m r e n n s n m r e

 

 

 

 

 







  



  









  



   

N N N N

Taking the expectation, we obtain:

, ,

( )

*

1 1 1

( , ) 1 ( , ) ( , )

^{k r k s}

K j

H H

l l l l

k s r

n m n n s n m r e

 

 





  

    

 ^{N   N}    

E E

Clearly, ( , )

^*

( , )

²

l l l

n n s n m r 

  

 

ⁿ

E

if and only if : m n  and r s  . In these conditions we obtain:

2 2

1

( , ) 1

l l

K

H H

l l

k

n n   K 





   

 ^{N   N}  

^{n n}

E

We eventually obtain :

²

l

H H H

l l l l

K 

M

     

 W W   N   N 

_n

I

E E

Concerning the other quantity

E

^ _ W W

_l^H _l

^ _ , we can obtain directly:

2

l

H H H H H

l l l l l l

M 

K

        

 W W    N N     N N  

_n

I

E E E

Remark : If we only consider one column (typically the nth column) of W

_l

denoted as w

_ln

, we have :

 

²l

H

ln ln

 

M

E

w w

n

I and  

²l

H

ln ln

 M 

E

w w

n

We are now coming back to the channel estimation problem. Since H

_l,l

has independent columns, we can estimate each column of H

_l,l

independently. Let ~ y

_pln

be the nth column of

l p,

Y ~ . Then:

ln /

iln iln p

/ lln lln p ln

p

p Ah p A h w

y

L

l i





 



2 1 2

~ 

1

 (9)

Where h

_iln

and w

_ln

are the nth column of H

_i,l

and W

_l

respectively. Denote by:

ln /

iln iln p

ln

p A h w

z

L

l i



 



2



1

, then the MMSE estimate of h

_iln

is given by :

ln p H

p lln H p lln

lln

p A p AA R

_ln

y

h ˆ  

^1/2

(  

_z

)

^1

~ (10)

With:  

²

1

ln l

H H

ln ln

p

p



iln



M

 

  

^L



z n

i ,i l

R E z z AA I (11)

Reporting (11) into (10), we obtain:

1

1/2 2

1 1

1/2 2

1

ˆ

l

l

-

H H H

lln lln p p lln p iln M pln

H H

lln p p iln M pln

p p p

p p

   

  

 





 

    

 

 

   

 









L

n i ,i l

L

n i

h A AA AA I y

A AA I y

We can rewrite ˆ

h

lln

in the following form :

1/ 2 1

2 2

1 1/ 2 1

2 2 2

1 1

1/ 2 2

2 2 2

1 1

1

ˆ

l l

l l l

l

l l l

H

lln p p H

lln iln M p ln

lln p H p p H

iln iln M p ln

lln p p H p p

iln iln

p iln

p p

p p p

p p p p

p

 

 

  

  

 

 

   







 

 



 

       

 

 

   

 

 

 

 

  

 

 



 

 







L

n n i

L L

i i

n n n

L L

n L

i i

n n n

i

h A AA I y

A A A I y

A A

1

2 1

1 1/ 2

2 2 2

1 1 1

2 1

l

l l l

l

H

iln M p ln

p H p p H

lln

iln iln iln M p ln

iln

p p p

 

   

  

 







  



 

  

 

 

 

   

   

             



  







L

n i

L L L

L n i n i n i

n i

A I y

A A A I y

The former equation can be written: h ˆ

_lln

  B

^H

( BB

^H

 I

_M

)

^1

y 

_pln

, with:

₂

l 1 L p

iln i

p 



_

 

n

B A _,

and

1/ 2

2 1

l

lln L

iln i

 

 







n

. Using the matrix inversion lemma, we have:



m



¹



n



¹

 

  

P I QP I PQ P

With the following sizes for P and Q:     Q

_mn

, P

_nm

. We use this lemma with: P  B

^H

,

B

Q

, we obtain: B BB

^H

(

^H

 I

_M

)

^1

 ( I

_P

 B B

^H

)

^1

B

^H

, so we eventually obtain:

1/ 2 1

2 2

1 1

2 1 1/ 2 1

2 2

1

1

1/ 2 2

1

ˆ

ˆ ˆ

l l

l

l l

l

L

p H p H

lln

lln L iln P iln p ln

i iln

i

p lln p H H

lln iln P p ln

H H

lln p lln p iln P p ln

p p

p p

p p

  

 

 

 

 

  



 











 

       

 

       

 

   

 

 













L

n i n

n

L

n n i

L

n i

h AA I A y

h AA I A y

h AA I A y

(12)

The kth diagonal element of 

 L

i

AA

1 iln H

p

p

 in (12) equals: 

 L

i 1 iln p

P

Mp  . Since the uplink is

typically interference-limited we have:

²

1

l

p iln

Mp

P  





^L



ⁿ

i

, therefore, h ˆ

_lln

_{can be} approximated as:

ln p H iln H p p lln

lln

p p AA A y

h

L

i

ˆ ~

1

1 1/2





 

 



 ^   ^{ (13)}

Thus, the MMSE estimate of H

_l,l

is:

2 / 1 1 , 1

2 / 1 ,

) ~ ˆ (

ll l l p H H

p l

l

p A A A Y D D

H 

^{  }

(14)

Where: 





^L

i il l

1

D

D . Then, the estimate of the physical channel matrix between the lth base station and the K users in the lth cell is given by:

l ll l p p

ll l l l

l_,

A H ˆ

_,

D

^1/2

p

^1/2 _A

Y ~

_,

D

¹

D

ˆ  

^



^

 (15)

Where

A

^A  ^A

^H

^A  ^A

^H

1



 is the orthogonal projection onto A. We can see that since post- multiplication of Y

_p,l

with 

^H

means just multiplication with the pseudo-inverse  ^{ }

^H

^{ I}

^K

 ^,

l p,

Y ~ is the conventional least-squares channel estimate. The MMSE channel estimator that we derived thus performs conventional channel estimation and then projects the estimate onto the physical (beamspace) model for the array.

3.2. Bayesian estimator with MAP rule:

We start from equation (7):

1/2

, , , ,

1 L H

p l p l p i l i l l

i

p



   

 

Y Y A H D W

Let ~ y

_pln

be the nth column of ~

_p,l

Y . Then:

ln l

i

iln p

lln p

ln

p

p Ah p A h w

y

L i

iln

lln

 

 



1,

2 / 1

~ 

1/2

 (16)

Where h

_iln

and w

_ln

are the nth column of H

_i,l

and W

_l

respectively. Denote by:

ln /

iln iln p

ln

p A h w

z

L

l i



 



2



1

, we can write:

1/ 2 p ln

 p

p lln



lln



ln

y



Ah z (17)

Applying Bayes’ rule, the conditional distribution of the channel h

_lln

, given the observed received training signal ~ y

_pln

, is:

~ ) ( ) ) (

(~

~ ) ( ) ) (

( ~ _{lln pln lln}

ln p

ln l ln p ln l ln

p ln

l p p

p p

p p h y h

y h y y h

h  

(18)

We use the Gaussian probability density function (PDF) of the random vector h

_lln

and assume its elements: h

_lln

(1),..., h i

_lln

( ),..., h P

_lln

( ) , are mutually independent, giving the joint pdf :

 

exp[ 1 ]

( )

2 det

lln

lln

H

lln lln

lln P

p



 

 ^h

h

h R h h

R

(19)

Since the elements: h

_lln

(1),..., h i

_lln

( ),..., h P

_lln

( ) are mutually independent this reduces to:

exp[   ]

( )

2

H lln lln

lln P

p 

  h h

h (20)

We also have:

1/2 1 1/2

exp[ ( ) ( )]

( )

det

ln

ln

H

pln p lln pln p lln

pln lln P

p p

p  



 





  



^{lln z lln}

z

y Ah R y Ah

y h

R (21)

Combining (20) and (21) and reporting into (18) we obtain:

ln P

P ln l ln

p ln l

p f

Rz

y h

h ( 2 ) det

)) ( ) exp(

( ~





 

(22)

With: ₍

^{1 2}

₎₍

^{1 2}

₎

ln

/ / H

p iln iln ln p iln iln ln

p  p 

 

 



^E

  

^L

 

^L

  

z

i l i l

R A h w A h w and:

1/ 2 1 1/ 2

( ) ( ) ( )

ln

H H

lln lln lln p ln p lln p ln p lln

f h  h h  y



 p Ah

_lln

 R

_z^

y



 p Ah

_lln

 The ML estimation ˆ

h

lln

of h

_lln

is given by:

 

1

1

1/ 2 1

ˆ arg max exp( ( ))

ˆ arg min ( )

ˆ

P lln

P lln

ln

lln lln

lln lln

H H

lln lln p p lln p ln

f f

p p

 











 







C C h

h

z

h h

h h

h A A A + R y

(23)

We have clearly:

1 2 1 2

2

( )( )

ln

ln l

/ / H

p iln iln ln p iln iln ln

L

H

M p iln

i l

p p

p

 

 

 



 

      

 

 



E ^{L L}

z

i l i l

z n

R A h w A h w

R I A A

(24)

And we find again:

1

1/ 2 2

1

ˆ

l

H H

lln



lln

p

p

p

p



iln



M pln





 

   

 

^{L n}



^

i

h A AA I y (25)

To obtain (23) and (25) we have applied (39) and (40) (see Appendix) with: R

_h

 I ,



ln

n z

R R ,

1/ 2 p lln

p 



S A .

3.3. Bayesian estimation with vectorized model:

We start from equation (8):

l L

l

l i l i p

l l l l p l

p

l L

l i l i p

l L

l i l i p

l p l p

p p

p p

W D

H A D

AH Y

W D

H A N

D H A Y

Y

i

i i





























2 / 1

, , 2

/ 1

, , ,

1

2 / 1

, ,

* 1

2 / 1

, ,

* , ,

~

~  

Starting from this equation we begin by vectorizing the received matrix ~

_p,l

Y of size M  K ,

we obtain the vector:

, 1 , ,

,

)

~ (:, )

~ (:, ) 1

~ (:,

~





 

 









 

 









l MK p

l p

l p

l p

K k Y

Y Y y





(26)

With:

 

  ^{(:, ) (:, )}

)

~ (:,

) (:, )

(:, )

(:, )

~ (:,

2 / 1

, , ,

2 / 1

, , 2

/ 1

, , ,

k k

p k

k k

p k p

k

l l

l l l p l

p

l L

l

l i l i p

l l l l p l

p

Z D

AH Y

W D

H A D

AH Y

i





 



 



 

 



(27)

With: _{(:, k ) p}

_, ¹_,^/2

_{(:, k )}

_l

_{(:, k )}

L

l

l i l i p

l

A H D W

Z

i

 



 



  



Clearly, we have the property:

 

, ^1/2

2 / 1

,

,_{l il} (:, ) _il(:, ) _ilk

i D k AH k



AH 

(28)

Applying Bayes’ rule, the conditional distribution of the channel H

_{l l,}

(:, ) k given the observed received training signal ~ y

_p,l

is:

) ) (:, )

~ (:, ( )) (:, )) (

~ (:, (

) ) (:, )

~ (:, ( )) (:, (

) )

~ (:, ) (:, (

, ,

, ,

, ,

, , ,

k k

p k k p

p

k k

p k p

k k

p

l l l

p l

l l

p

l l l

p l

l

l p l

l

H Y

Y H

H Y

H Y H



 (29)

We use the Gaussian probability density function (pdf) of the random vector H

_l,l

(:, k ) and assume its elements: H

_l,l

( 1 , k ),..., H

_l,l

( 1 , k ),..., H

_l,l

( P , k ) are mutually independent, giving the joint pdf:

) (:, 1

) (:,

,

( 2 ) det

] ) (:, )

(:, exp[

)) (:, (

k P

H l k l, H

l l, l

l

l l, l l,

k k

k p

H H

R H R

H

H 





 (30)

Since the elements: H

_{l l,}

(1, ),..., k H m k

_{l l,}

( , ),..., H

_{l l,}

( , ) P k are mutually independent, this reduces

to: