Majorization and Matrix-Monotone Functions in Wireless Communications

Majorization and Matrix-Monotone Functions in Wireless

Communications

Eduard A. Jorswieck

Signal Processing,School of Electrical EngineeringRoyal Institute of Technology

Osquldas vag 10, 100 44 Stockholm, Sweden

Fall 2007

Majorization and Matrix-Monotone Functions Eduard A. Jorswieck

5th Lecture

Majorization - Applications II

1


User distributions

➮ In the signal model, there are K mobile users who are going to receivedata from one base station.

➮ The single-antenna quasi-static block flat-fading channels h1, ..., hK

between the mobiles and the base are modeled as constant for a block ofcoherence length T and from block to block as zero-mean independentcomplex Gaussian distributed (CN(0, ck)).

➮ The variance is ci = E (h∗i hi) for 1 ≤ i ≤ K.

➮ In order to guarantee a fair comparison, we constrain the sum varianceto be equal to the number of users, i.e.

∑Kk=1 ck = K.

➮ Definition 1. [More spread out user distribution] A userdistribution c1 is called more spread out than a user distributionc2 if the fading variance vector c1 is majorized by c2, i.e. c1 � c2. 2


Symmetric scenario c1 = (1, 1, ..., 1)

3


One mobile moves to the base and another to the cell edge. The sum oftheir fading variances stays constant. c2 = (1 + α, 1, 1, ..., 1 − α)

4


All but one mobile at the cell edge c3 = (8, 0, ..., 0)

5


➮ In the figures, the symmetric scenario is majorized by all other scenariosand scenario three majorizes all other scenarios i.e.

c1 � c2 � c3.

➮ Note, that the measure of user distribution and the measure of spatialcorrelation can be combined for e.g. multiuser MIMO systems. Thechannel of a user k can be modelled for all 1 ≤ k ≤ K under theKronecker model assumption as

Hk = ckR1/2R,kWkR

1/2T,k

with normalized transmit and receive correlation matrix as well asnormalized random matrix, i.e. for all 1 ≤ k ≤ K tr RT,k = nT ,tr RR,k = nR. Then, the long-term fading is captured by ck, the spatialcorrelation by RT and RR and the rich multi-path environment by W.

6


Average sum rate in BC

➮ Lemma 1. The sum rate with perfect CSI at the base station isachieved by TDMA. The optimal power allocation is to transmit intodirection of the best user l with ||hl||

2 > ||hk||2 for all 1 ≤ k ≤ K and

l 6= k. The ergodic sum capacity is then given by

CpCSI(ρ, c) = E(

log[

1 + ρmax(

||h1||2, ..., ||hK||2

)])

. (1)

The optimal transmit strategy to achieve the average sum capacity withlong-term CSI is TDMA. Only the user with highest channel variance ck

is allowed to transmit. The achievable average sum capacity is given by

CcCSI(ρ, c) = E log(1 + ρc1w1). (2)

For no CSI at the base, the most robust transmit strategy against worstcase user distribution is equal power allocation and the ergodic sum rate1

1We do not allow time division within a block.7


is given by

CnoCSI(ρ, c) = E log

(

1 + ρ

K∑

k=1

ckwk

)

. (3)

➮ Theorem 1. Assume perfect CSI at the mobiles. For perfect CSI atthe base, the ergodic sum capacity in (1) is a Schur-convex functionw.r.t. the fading variance vector c. For a base which knows the fadingvariances, the ergodic sum capacity in (2) is a Schur-convex functionw.r.t. the fading variance vector c. For an uninformed base station, theergodic sum rate in (3) is a Schur-concave function w.r.t. the fadingvariance vector c.

8


Average sum rate in opportunistic scheduling

➮ Consider TDMA-based opportunistic scheduling first. Only a single useris scheduled.

➮

CT (ρ) = E

[

log

(

1 + ρ max1≤k≤K

max1≤i≤nT

|hHk wi|

2

)]

(4)

=

∫ ∞

0

log(1 + ρx)p(x)dx (5)

with pdf p(x) of the random variable max1≤k≤K max1≤i≤nTckλisk,i.

The CDF is given as

P (x) =

K∏

k=1

nT∏

n=1

(

1 − exp

(

−x

ckλn

))

.

9


➮ For convenience, we collect all numbers ηk+(n−1)K = ckλn for 1 ≤k ≤ K and 1 ≤ n ≤ nT in one large vector η = [η1, ..., ηKnT

] andparameterize the throughput by η.

➮ By partial integration the average throughput can be written as

CT (ρ, η) =∣

∣

∣log(1 + ρx) [P (x) − 1]

∣

∣

∣

∞

0−

∫ ∞

0

ρ

1 + ρx[P (x) − 1] dx

=

∫ ∞

0

ρ

1 + ρx

1 −

KnT∏

k=1

(

1 − exp

(

−x

ηk

))

dx. (6)

➮ Theorem 2. The function CT (ρ, η) is Schur-convex with respect to η,i.e., η1 � η2 ⇒ CT (ρ, η1) ≥ CT (ρ, η2).

10


−10 −5 0 5 10 15 200

2

4

6

8

SNR [dB]

Avera

ge s

um

rate

[bit/c

u]

CT

max(ρ)

CT

min(ρ)

CT(ρ,η)

∆∞

➮ Average sum rate in bits per channel use (bit/cu) vs. SNR for TDMA-OB: Best-case, worst-case, and intermediate scenario. The high-SNRpower difference is 3.2 dB.

11


➮ In SDMA based opportunistic beamforming: The resulting averagethroughput can be upper bounded by [Sharif2005]

C(ρ, c,R) = nTE log

(

1 + max1≤k≤K

SINR k

)

(7)

with SINR k =||wH

i hk||2

σ2n+∑

l 6=i ||wHl

hk||2.

➮ Theorem 3. The average throughput of SDMA-OB is upper limited by

B(K, nT ,1) = limρ→∞

C(ρ,1, I) =(Ψ(K + 1) + γ)nT

(nT − 1) log(2)

=nT

(nT − 1) log(2)

K∑

k=1

1

k(8)

12


which is tight for asymptotically high SNR and Ψ(x) is the Psi-function.

➮ Theorem 4. The upper bound on the average sum capacity of SDMA-OB B(K, nT , c) is Schur-concave with respect to c, i.e., c1 � c2 ⇒B(K, nT , c1) ≤ B(K, nT , c2). The upper bound on the average sumrate of SDMA-OB is largest for the balanced scenario B(K, nT ,1) andsmallest for the completely unequally distributed scenario B(1, nT ) =B(K, nT , [K, 0, ..., 0]).

13


Including fairness

➮ Different types of scheduling:

➠ The first scheduler is called maximum throughput scheduler (MTS)and the achievable average sum rate is given by

RMTsum = E

[

log

(

1 + ρ max1≤k≤K

||hk||2

)]

. (9)

➠ The average sum rate of the round-robin scheduler (RRS) is given by

RRRsum = E

[

1

K

K∑

k=1

log(

1 + ρ||hk||2)

]

. (10)

➠ The proportional fair scheduler (PFS) has achievable sum rate given

14


by

RPFsum = E

[

log(

1 + ρ||hk∗||2)]

with (11)

k∗ = arg max1≤k≤K

||hk||2

ck.

Note that (11) can be rewritten as

RPFsum =

1

K

K∑

k=1

E

[

log

(

1 + ρck max1≤l≤K

wl

)]

(12)

=1

K

K∑

k=1

K∑

l=1

(−1)l−1

(

K

l

)

Ei

(

1,l

ρck

)

el

ρck

because the scheduling probability of all users is equal to 1K .

15


➠ The sum rate performance of the opportunistic round robin scheduler(ORS) is given by

RORsum =

1

K2

K∑

n=1

n

K∑

i=1

n−1∑

j=0

(

n − 1j

)

(−1)j

·e

1+jci

1 + jEi

(

1,1 + j

ci

)

. (13)

➠ Lemma 2. The average sum rate of the ORS (13) can be written as

∫ ∞

0

[

1 −1

K2

K∑

n=1

K∑

i=1

(

1 − e− t

ci

)n]

ρ

1 + ρtdt. (14)

➮ Analysis of the sum rate performance

16


➠ Theorem 5. The average sum rate of the MTS is Schur-convex withrespect to the vector of average user powers c, i.e.

c � d =⇒ RMTsum(c) ≥ RMT

sum(d). (15)

➠ Theorem 6. The average sum rate of the RRS is Schur-concavewith respect to the vector of average user powers c, i.e.

c � d =⇒ RRRsum(c) ≤ RRR

sum(d). (16)

➠ Theorem 7. The average sum rate of the PFS is Schur-concave withrespect to the vector of average user powers c, i.e.

c � d =⇒ RPFsum(c) ≤ RPF

sum(d). (17)

➠ Theorem 8. The average sum rate of the ORS is Schur-concave

17


with respect to the vector of average user power c, i.e.

c � d =⇒ RORsum(c) ≤ ROR

sum(d). (18)

➮ Fairness analysis: The average worst-case delay E[Dm,K] measures theaverage number of transmissions that are needed until all K users havebeen active at least m times. We define D1 = E[D1,K].

➮ Average worst-case delay of the deterministic schedulers

DRRS1 = DORS

1 = K. (19)

➮ Average worst-case of PFS

DPFS1 = K

∫ ∞

0

1 − (1 − exp(−x))Kdt (20)

18


Note that (20) can be written as

DPFS1 = K (Ψ(K + 1) + γ) (21)

with the Ψ-function and Euler’s constant γ.

➮ Average worst-case of MTS

DMTS1 = n

∫ ∞

0

(

1 −

K∏

k=1

(

1 −Γ(m, dkt)

Γ(m)

)

)

dt. (22)

➮ Theorem 9. The average worst-case delay E[D1,K] of MTS is Schur-convex with respect to d, i.e.

d1 � d2 −→ DMTS1 (d1) ≥ DMTS

1 (d2). (23)

19


➮ Lemma 3. The mapping from the vector of user distributions to thevector of service probabilities is not order preserving with respect to thepartial order Majorization.

➮ Scaling with number of users:

➮ Lemma 4. For symmetrically distributed users c = 1, the average sumrate of MTS, PFS, and ORS scale for large K with log(K), i.e.

limK→∞

RMTsum(K)

log(K)= lim

K→∞

RPFsum(K)

log(K)

= limK→∞

RORsum(K)

log(K)= 1. (24)

20


➮ Lemma 5. For symmetrically distributed users, the average worst-casedelay scales linearly with K for RRS and ORS. For MTS and PFS, itscales as K log(K), i.e.

limK→∞

DRRS1 (K)

K= lim

K→∞

DORS1 (K)

K= 1

limK→∞

DMTS1 (K)

K log(K)= lim

K→∞

DPFS1 (K)

K log(K)= 1. (25)

➮ Illustrations:

21


2 4 6 8 10 12 14 16 18 202.5

3

3.5

4

4.5

5

5.5Scaling with number of users [symmetrical distribution]

Number of users

Avera

ge s

um

rate

[bpcu]

RRSMTSPFSORS

2 4 6 8 10 12 14 16 18 202

2.5

3

3.5

4

4.5

5

5.5

6

Number of users

Avera

ge s

um

rate

[bpcu]

Scaling with number of users [assymetrical with t=0.2]

RRSMTSPFSORS

➮ Observation and main conclusion: Therefore, taking the tradeoff betweenfairness and average sum rate into account, the PFS and ORS performreasonable well. PFD is advantageous in symmetric scenarios whereasORS performs better in asymmetric scenarios.

22


References I

i) E. Jorswieck, P. Svedman, and B. Ottersten, ”On the Performance ofTDMA and SDMA based Opportunistic Beamforming”, submitted toIEEE Transactions on Wireless Communications, 2007.

ii) E. Jorswieck, A. Sezgin, and X. Zhang, ”Framework for analysis ofopportunistic schedulers: average sum rate vs. average fairness”, to besubmitted to RAWCOM 2008.

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Spectral correlation in OFDM systems

➮ Consider the standard frequency selective channel model described in timedomain by a cannel impulse response h1, ..., hL with L taps probablycorrelated with channel correlation matrix R, i.e. Rk,l = E[hkh

∗l ].

➮ Ideal cyclic prefix OFDM is applied with N > L carriers.

➮ The fading on carrier k is given by Hk =∑L−1

l=0 hl exp(

−i2π kN

)

.

➮ Therefore, the power of carrier k is distributed according to an exponentialdistribution with mean

ak(R) = E[|Hk|2] = 2

L−2∑

n=0

L−1∑

m=n+1

Rm,n cos

(

2πk

N(m − n)

)

+

L∑

k=1

Rk,k

for all 1 ≤ k ≤ N .24


➮ N parallel flat-fading channels are obtained by ideal CP-OFDM yk =Hkxk + nk for 1 ≤ k ≤ N , transmit signals xk and AWGN nk.

➮ In the following, we focus on the average rate of the system as a functionof the power allocation p = [p1, ..., pN ], the inverse noise power ρ = 1

σ2n,

and the fading variances a = [a1, ..., aN ]

C(a, ρ) = E [R(H, ρ,p)] (26)

with instantaneous rate which depends on the channel vector H =[H1, ...,HN ] and the power allocation

R(H, ρ,p) =1

N

N∑

k=1

log (1 + ρHkpk) . (27)

25


➮ We assume that the receiver has perfect CSI and consider three differenttypes of CSI at the transmitter.

➮ At first, assume the transmitter does not have CSI at all. Then theoptimal power allocation is equal power allocation and the achievableaverage rate is given by

Cno(a, ρ) = E

[

1

N

N∑

k=1

log

(

1 +P

Nρak

)

]

. (28)

➮ If the transmitter has information about R it can adapt its powerallocation to a and the average rate is given by

Cpart(a, ρ) = maxpk≥0,

∑

pk=PE

[

1

N

N∑

k=1

log (1 + ρHkpk)

]

. (29)

26


➮ If the transmitter has perfect CSI, spectral power allocation is optimaland the average rate is given by

Cp(a, ρ) = E

[

maxpk≥0,

∑

pk=P

1

N

N∑

k=1

log (1 + ρHkpk)

]

. (30)

➮ Definition 2. The frequency selective channel described by R has morecorrelated taps than the channel described by S if

a(R) � a(S), (31)

i.e. a(R) majorizes a(S), i.e.

n∑

k=1

ak(R) ≥

n∑

k=1

ak(S) for 1 ≤ n ≤ N − 1 and

N∑

k=1

ak(R) ≥

N∑

k=1

ak(S).

27


➮ Consider for example two taps L = 2 with R(c) =

(

1 c

c 1

)

and N = 4

carriers. The average channel power of the four carriers is then given by

a(R(c)) = [2 + 2c, 2, 2, 2 − 2c]. (32)

For completely uncorrelated coefficients, a(R(0)) = [2, 2, 2, 2]. Forcompletely correlated coefficients, a(R(1)) = [4, 2, 2, 0]. From (32), itfollows that all correlation matrices R(c1),R(c2) are comparable.

➮ TBD-List:

➠ Is it possible to compare all channel correlation matrices R?➠ Are C(a, ρ) Schur-convex or Schur-concave? High SNR behavior

analysis.➠ Maximum Loss/Gain due to spectral correlation?

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References II

i) Kafedziski, V. Capacity of Frequency Selective Slowly Fading Channelswith Correlated Coefficients Proc. of SCC, will appear, 2008.

ii) Yao, Y. and Giannakis, G. B. Rate-Maximizing Power Allocation inOFDM Based on Partial Channel Knowledge IEEE Trans. on WirelessCommunications, 2005, 4, 1073-1083.

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Majorization and Matrix-Monotone Functions in Wireless Communications

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