David versus Goliath:Small Cells versus Massive...

David versus Goliath:SmallCells versus Massive MIMO

Jakob Hoydis and Mérouane Debbah

1948: Cybernetics and Theory of Communications

• ”A Mathematical Theory of Communication”, Bell System Technical Journal, 1948, C. E. Shannon

• ”Cybernetics, or Control and Communication in the Animal and the Machine”, Herman et Cie/The Technology Press, 1948, N. Wiener

60 years later... MIMO Flexible Networks

We must learn and control the black box

• within a fraction of time• with finite energy.

In many cases, the number of inputs/outputs (the dimensionality of the system) is of the same order as the time scale changes of the box.

“David vs Goliath“ or ”Small Cells vs Massive MIMO“

How to densify: “More antennas or more BSs?”

Questions:

I Should we install more base stations or simply more antennas per base?

I How can massively many antennas be efficiently used?

I Can massive MIMO simplify the signal processing?

4 / 25

Vision

Bell Labs lightradio antenna module – the next generation small cell (picture from www.washingtonpost.com)

A thought experiment

Consider an infinite large network of randomly uniformly distributed basestations and user terminals.

What would be better?

A 2 × more base stations

B 2 × more antennas per base station

Stochastic geometry can provide an answer.

5 / 25

A thought experiment

Consider an infinite large network of randomly uniformly distributed basestations and user terminals.

What would be better?

A 2 × more base stations

B 2 × more antennas per base station

Stochastic geometry can provide an answer.

5 / 25

System model: Downlink

Received signal at a tagged UT at the origin:

rα/20

hH0 x0︸︷︷︸

desired signal

+∞∑i=1

rα/2i

hHi xi︸︷︷︸

interference

I hi ∼ CN (0, IN ): fast fading channel vectors

I ri : distance to ith closest BS

I P = E[xH

]: average transmit power constraint per BS

Assumptions:

I infinitely large network of uniformly randomly distributed BSs and UTswith densities λBS and λUT, respectively

I single-antenna UTs, N antennas per BS

I each UT is served by its closest BS

I distance-based path loss model with path loss exponent α > 2

I total bandwidth W , re-used in each cell

6 / 25

Received signal at a tagged UT at the origin:

rα/20

hH0 x0︸︷︷︸

desired signal

+∞∑i=1

rα/2i

hHi xi︸︷︷︸

interference

I hi ∼ CN (0, IN ): fast fading channel vectors

I ri : distance to ith closest BS

I P = E[xH

]: average transmit power constraint per BS

Assumptions:

I infinitely large network of uniformly randomly distributed BSs and UTswith densities λBS and λUT, respectively

I single-antenna UTs, N antennas per BS

I each UT is served by its closest BS

I distance-based path loss model with path loss exponent α > 2

I total bandwidth W , re-used in each cell

6 / 25

Transmission strategy: Zero-forcing

Assumptions:

I K = λUTλBS

UTs need to be served by each BS on average

I total bandwidth W divided into L ≥ 1 sub-bands

I K = K/L ≤ N UTs are simultaneously served on each sub-band

Transmit vector of BS i :

K∑k=1

wi,k si,k

I si,k ∼ CN (0, 1): message determined for UT k from BS i

I wi,k ∈ CN×1: ZF-beamforming vectors

7 / 25

Transmission strategy: Zero-forcing

Assumptions:

I K = λUTλBS

UTs need to be served by each BS on average

I total bandwidth W divided into L ≥ 1 sub-bands

I K = K/L ≤ N UTs are simultaneously served on each sub-band

Transmit vector of BS i :

K∑k=1

wi,k si,k

I si,k ∼ CN (0, 1): message determined for UT k from BS i

I wi,k ∈ CN×1: ZF-beamforming vectors

7 / 25

Performance metric: Average throughputReceived SINR at tagged UT:

γ =r−α0

∣∣hH0 w0,1

∣∣2∑∞i=1 r−αi

∑Kk=1

∣∣hHi wi,k

∣∣2 + KP

=r−α0 S∑∞

i=1 r−αi gi + KP

Coverage probability:

Pcov(T ) = P (γ ≥ T )

Average throughput per UT:

L× E [log(1 + γ)] =

L×∫ ∞

Pcov (ez − 1) dz

Remarks:

I expectation with respect to fading and BSs locations

I S =∣∣hH

0 w0,1

∣∣2 ∼ Γ(N − K + 1, 1), gi =∑K

∣∣hHi wi,k

∣∣2 ∼ Γ(K , 1)

I K impacts the interference distribution, N impacts the desired signal

I for P →∞, the SINR becomes independent of λBS

8 / 25

Performance metric: Average throughputReceived SINR at tagged UT:

γ =r−α0

∣∣hH0 w0,1

∣∣2∑∞i=1 r−αi

∑Kk=1

∣∣hHi wi,k

∣∣2 + KP

=r−α0 S∑∞

i=1 r−αi gi + KP

Coverage probability:

Pcov(T ) = P (γ ≥ T )

Average throughput per UT:

L× E [log(1 + γ)] =

L×∫ ∞

Pcov (ez − 1) dz

Remarks:

I expectation with respect to fading and BSs locations

I S =∣∣hH

0 w0,1

∣∣2 ∼ Γ(N − K + 1, 1), gi =∑K

∣∣hHi wi,k

∣∣2 ∼ Γ(K , 1)

I K impacts the interference distribution, N impacts the desired signal

I for P →∞, the SINR becomes independent of λBS

8 / 25

A closed-form result

Theorem (Combination of Baccelli’09, Andrews’10)

Pcov(T ) =

∫r0>0

∫ ∞−∞LIr0

(i2πrα0 Ts) exp

i2πrα0 TK

)LS (−i2πs)− 1

i2πsfr0 (r0)dsdr0

LIr0(s) = exp

(−2πλBS

∫ ∞r0

(1 + sv−α)K

LS (s) =

)N−K+1

fr0 (r0) = 2πλBSr0e−λBSπr20

The computation of Pcov(T ) requires in general three numerical integrals.

J. G. Andrews, F. Baccelli, R. K. Ganti, “A Tractable Approach to Coverage and Rate in Cellular Networks” IEEE

Trans. Wireless Commun., submitted 2010.

F. Baccelli, B. B laszczyszyn, P. Muhlethaler, “Stochastic Analysis of Spatial and Opportunistic Aloha” Journal on

Selected Areas in Communications, 2009

9 / 25

Example

I Density of UTs: λUT = 16

I Constant transmit power density: P × λBS = 10

I Number of BS-antennas: N = λUT/λBS

I Path loss exponent: α = 4

I UT simultaneously served on each band: K = λUT/(λBS × L)

⇒ Only two parameters: λBS and L

Table: Average spectral efficiency C/W in (bits/s/Hz)

sub-bands L λBS = 1 λBS = 2 λBS = 4 λBS = 8 λBS = 16

1 0.6209 0.8188 1.1964 1.5215 2.1456

2 1.1723 1.2414 1.3404 1.5068 x

4 0.8882 0.8973 1.1964 x x

8 0.5689 0.5952 x x

16 0.3532 x x x x

Fully distributing the antennas gives highest throughput gains!

10 / 25

Example

I Density of UTs: λUT = 16

I Constant transmit power density: P × λBS = 10

I Number of BS-antennas: N = λUT/λBS

I Path loss exponent: α = 4

I UT simultaneously served on each band: K = λUT/(λBS × L)

⇒ Only two parameters: λBS and L

Table: Average spectral efficiency C/W in (bits/s/Hz)

sub-bands L λBS = 1 λBS = 2 λBS = 4 λBS = 8 λBS = 16

1 0.6209 0.8188 1.1964 1.5215 2.1456

2 1.1723 1.2414 1.3404 1.5068 x

4 0.8882 0.8973 1.1964 x x

8 0.5689 0.5952 x x

16 0.3532 x x x x

Fully distributing the antennas gives highest throughput gains!

10 / 25

First conclusions

I Distributed network densification is preferable over massive MIMO if theaverage throughput per UT should be increased.

I More antennas increase the coverage probability, but more BSs lead to alinear increase in area spectral efficiency (with constant total transmitpower).

I If we use other metrics such as coverage probability or goodput, thepicture might change.

What happens if massively many antennas are used?

11 / 25

Beyond LTE: The 400-Antenna Base Station

Thomas L. Marzetta

Bell Laboratories

Alcatel-Lucent

28 May, 2010

Large Excess of Base Station Antennas Over Terminals Yields Energy Efficiency + Reliably High Throughput

M~400 base station antennas serve K~40 terminals via multi-user MIMO

Doubling M permits a reduction in total transmit power by factor-of-two

Extra base station antennas always help (even with noisy CSI) Eventually produce inter-cellular interference-limited operation: everybody can

now reduce power arbitrarily! reduce effects of uncorrelated noise and fast fading

compensate for poor-quality channel-state information

Multiple Cells: No Cooperation

If we could assign an orthogonal pilot sequence to every terminal in every cell then nothing bad would happen! Ever greater numbers of base station antennas would eventually defeat all

noise, and eliminate both intra- and inter-cell interference

But there aren’t enough orthogonal pilot sequences for everyone! Pilot sequences have to be re-used

Pilot contamination: the base station inadvertently learns the channel to mobiles in other cells Forward link: base station transmits interference to mobiles in other cells Reverse link: base station processing enhances his reception of

transmission from mobiles in other cells

Inter-cell interference due to pilot contamination persists, even with an infinite number of antennas! This is the only remaining impairment

Limiting Case: Infinite Number of Antennas

•Greatly simplifies multi-cellular analysis: all effects accounted for near-analytically• Acquisition of CSI• Imperfections in CSI• Inter-cellular interference• Propagation

• Fast (either line-of-sight, or independent Rayleigh, or something intermediate)

• Slow (geometric, log-normal shadow)

•Far-reaching and comprehensive conclusions ensue•Indicates a new direction in which the macro-cellular world can go: vastly improved energy efficiency and throughput compared with LTE

Summary of Limit Analysis

Multi-cellular TDD scenario, 42 terminals served per cell

500 sec coherence interval (7 OFDM symbols): 3 reverse-link pilots, 1 idle, 3 data

OFDM: 20 MHz bandwidth, cyclic prefix 4.76 sec

Fading: Fast + log-normal shadow (8 dB) + geometric (3.8 power)

No inter-cellular cooperation

Net downlink throughput (comparable uplink) for frequency re-use 7

mean– 730 Mbits/sec/cell– 17 Mbits/sec/terminal

95% likely: 3.6 Mbits/sec/terminal spectral efficiency constant with respect to bandwidth

throughput constant with respect to cell-size

number of terminals per cell proportional to coherence interval

performance independent of power

Cells Operate Independently, Each Serving Single-Antenna Terminals via Multi-User MIMO: TDD Only!

Maximum number of terminals limited by the time that it takes to send reverse pilots: pilot-interval divided by the channel delay-spread

Coherence interval: 500 sec (7 LTE OFDM symbols) – TGV speeds! 3 symbols for reverse-link pilots

3 symbols for data

1 symbol for computations and dead time

42 terminals per cell served simultaneously

Infinitely Many Antennas: Forward-Link Capacity For 20 MHz Bandwidth, 42 Terminals per Cell, 500 sec Slot

Frequency Reuse .95-Likely SIR (dB)

.95-Likely Capacity per

Terminal (Mbits/s)

Mean Capacity per Terminal

(Mbits/s)

Mean Capacity per Cell (Mbits/s)

1 -29 .016 44 1800

3 -5.8 .89 28 1200

7 8.9 3.6 17 730

Interference-limited: energy-per-bit can be made arbitrarily small!

LTE Advanced(>= Release 10)

Infinitely Many Antennas: Forward-Link Capacity For 20 MHz Bandwidth, 42 Terminals per Cell, 500 sec Slot

Frequency Reuse .95-Likely SIR (dB)

.95-Likely Capacity per

Terminal (Mbits/s)

Mean Capacity per Terminal

(Mbits/s)

1 -29 .016 44 1800

3 -5.8 .89 28 1200

7 8.9 3.6 17 730

Interference-limited: energy-per-bit can be made arbitrarily small!

LTE Advanced(>= Release 10)

Motivation of massive MIMO

Consider a N × K MIMO MAC:

y =K∑

hkxk + n

where hk , n are i.i.d. with zero mean and unit variance.

By the strong law of large numbers:

Hya.s.−−−−−−−−−−→

N→∞, K=const.xm

With an unlimited number of antennas,

uncorrelated interference and noise vanish,

the matched filter is optimal,

the transmit power can be made arbitrarily small.

T. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas” IEEE Trans. Wireless Commun.,

vol. 9, no. 11, pp. 35903600, Nov. 2010.

Jakob Hoydis (Supelec) Massive MIMO: How many antennas do we need? 4 / 30

y =K∑

hkxk + n

Hya.s.−−−−−−−−−−→

N→∞, K=const.xm

vol. 9, no. 11, pp. 35903600, Nov. 2010.

y =K∑

hkxk + n

Hya.s.−−−−−−−−−−→

N→∞, K=const.xm

vol. 9, no. 11, pp. 35903600, Nov. 2010.

About some fundamental assumptions

The receiver has perfect channel state information (CSI).What happens if the channel must be estimated?

The number of interferers K is small compared to N.What does small mean?

The channel provides infinite diversity, i.e., each antenna gives an independent lookon the transmitted signal.

What if the degrees of freedom are limited?

The received energy grows without bounds as N →∞.Clearly wrong, but might hold up to very large antenna arrays if theaperture scales with N.

On channel estimation and pilot contamination

1 The receiver estimates the channels based on pilot sequences.

2 The number of orthogonal sequences is limited by the coherence time.

3 Thus, the pilot sequences must be reused.

Assume that transmitter m and j use the same pilot sequence:

hm = hm + hj︸︷︷︸pilot contamination

+ nm︸︷︷︸estimation noise

Thus,1

Hya.s−−−−−−−−−→

N→∞,K=const.xm + xj

uncorrelated interference, noise and estimation errors vanish,

the transmit power can be made arbitrarily small (∼ 1/√N [Ngo’11]),

but the performance is limited by pilot contamination.

vol. 9, no. 11, pp. 35903600, Nov. 2010.

Thus,1

Hya.s−−−−−−−−−→

vol. 9, no. 11, pp. 35903600, Nov. 2010.

Thus,1

Hya.s−−−−−−−−−→

vol. 9, no. 11, pp. 35903600, Nov. 2010.

Thus,1

Hya.s−−−−−−−−−→

vol. 9, no. 11, pp. 35903600, Nov. 2010.

Uplink

System model and channel estimation

Uplink: L BSs with N antennas, K UTs per cell. Received signal at BS j :

yj =√ρ

L∑l=1

Hjlxl + nj

The columns of Hjl (N × K) are modeled as

hjlk = R12jlkwjlk , wjlk ∼ CN (0, IN)

Channel estimation:

yτjk = hjjk +∑l 6=j

hjlk +1√ρτ

MMSE estimate: hjjk = hjjk + hjjk

hjjk ∼ CN (0,Φjjk) , hjjk ∼ CN (0,Rjjk −Φjjk)

Φjlk = RjjkQjkRjlk , Qjk =

ρτIN +

yj =√ρ

L∑l=1

Hjlxl + nj

Channel estimation:

hjlk +1√ρτ

ρτIN +

yj =√ρ

L∑l=1

Hjlxl + nj

Channel estimation:

hjlk +1√ρτ

ρτIN +

yj =√ρ

L∑l=1

Hjlxl + nj

Channel estimation:

hjlk +1√ρτ

ρτIN +

yj =√ρ

L∑l=1

Hjlxl + nj

Channel estimation:

hjlk +1√ρτ

ρτIN +

Achievable rates with linear detectorsErgodic achievable rate of UT m in cell j :

Rjm = EHjj[log2 (1 + γjm)]

γjm =

∣∣∣rHjmhjjm

∣∣∣2E[rHjm

IN + hjjmhHjjm − hjjmhH

jjm +∑

l HjlHHjl

)rjm∣∣∣Hjj

]with an arbitrary receive filter rjm.

Two specific linear detectors rjm:

rMFjm = hjjm

rMMSEjm =

Hjj + Zj + NλIN

where λ > 0 is a design parameter and

Zj = E

HjjHHjj +

∑l 6=j

HjlHjl

(Rjjk −Φjjk) +∑l 6=j

Rjlk .

B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” IEEE Trans. Inf. Theory., vol.

49, no. 4, pp. 951–963, Nov. 2003.

Achievable rates with linear detectorsErgodic achievable rate of UT m in cell j :

Rjm = EHjj[log2 (1 + γjm)]

γjm =

∣∣∣rHjmhjjm

∣∣∣2E[rHjm

IN + hjjmhHjjm − hjjmhH

jjm +∑

l HjlHHjl

)rjm∣∣∣Hjj

]with an arbitrary receive filter rjm.

Two specific linear detectors rjm:

rMFjm = hjjm

rMMSEjm =

Hjj + Zj + NλIN

where λ > 0 is a design parameter and

Zj = E

HjjHHjj +

∑l 6=j

HjlHjl

(Rjjk −Φjjk) +∑l 6=j

Rjlk .

B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” IEEE Trans. Inf. Theory., vol.

49, no. 4, pp. 951–963, Nov. 2003.

Large system analysis based on random matrix theory

Assume N,K →∞ at the same speed. Then,

γjm − γjma.s.−−→ 0

Rjm − log2 (1 + γjm)a.s.−−→ 0

γMFjm =

tr Φjjm

1ρN2 tr Φjjm + 1

∑l,k

tr RjlkΦjjm +∑

∣∣ 1N

tr Φjlm

∣∣2

γMMSEjm =

1ρN2 tr ΦjjmT′j + 1

∑l,k µjlkm +

∑l 6=j |ϑjlm|2

and δjm, µjlkm, θjlm, T′j can be calculated numerically.

Rjm − log2 (1 + γjm)a.s.−−→ 0

γMFjm =

tr Φjjm

1ρN2 tr Φjjm + 1

∑l,k

tr RjlkΦjjm +∑

∣∣ 1N

tr Φjlm

∣∣2

γMMSEjm =

1ρN2 tr ΦjjmT′j + 1

∑l,k µjlkm +

∑l 6=j |ϑjlm|2

and δjm, µjlkm, θjlm, T′j can be calculated numerically.

A simple multi-cell scenario

intercell interference factor α ∈ [0, 1]transmit power per UT: ρHjl = [hjl1 · · · hjlK ] =

√N/PAWjl

A ∈ CN×Pcomposed of P ≤ N columns of a unitary matrix

Wij ∈ CP×Khave i.i.d. elements with zero mean and unit variance

Assumptions:

P channel degrees of freedom, i.e., rank (Hjl) = min(P,K) [Ngo’11]energy scales linearly with N, i.e., E

[tr HjlH

only pilot contamination, i.e., no estimation noise:

hjjk = hjjk +√α∑l 6=j

√N/PAWjl

Assumptions:

[tr HjlH

√N/PAWjl

Assumptions:

[tr HjlH

Asymptotic performance of the matched filter

Assume that N, K and P grow infinitely large at the same speed:

SINRMF ≈ 1

ρN︸︷︷︸noise

PL2︸︷︷︸

multi-user interference

+ α(L− 1)︸︷︷︸pilot contamination

where L = 1 + α(L− 1).

Observations:

The effective SNR ρN increases linearly with N.

The multiuser interference depends on P/K and not on N.

Ultimate performance limit:

SINRMF a.s−−−−−−−−−−−→N,P→∞, K=const.

SINR∞ =1

α(L− 1)

J. Hoydis, S. ten Brink, and M. Debbah, “Massive MIMO: How many antennas do we need”, Allerton Conference,

Urbana-Champaing, Illinois, US, Sep. 2011. [Online] http://arxiv.org/abs/1107.1709

Asymptotic performance of the matched filter

SINRMF ≈ 1

ρN︸︷︷︸noise

PL2︸︷︷︸

where L = 1 + α(L− 1).

Observations:

The effective SNR ρN increases linearly with N.

The multiuser interference depends on P/K and not on N.

Ultimate performance limit:

SINRMF a.s−−−−−−−−−−−→N,P→∞, K=const.

SINR∞ =1

α(L− 1)

Asymptotic performance of the MMSE detector

SINRMMSE ≈ 1

ρNX︸︷︷︸

PL2Y︸︷︷︸

where L = 1 + α(L− 1) and X ,Y are given in closed-form.

Observations:

As for the MF, the performance depends only on ρN and P/K .

The ultimate performance of MMSE and MF coincide:

SINRMMSE a.s−−−−−−−−−−−→N,P→∞, K=const.

SINR∞ =1

α(L− 1)

Asymptotic performance of the MMSE detector

SINRMMSE ≈ 1

ρNX︸︷︷︸

PL2Y︸︷︷︸

where L = 1 + α(L− 1) and X ,Y are given in closed-form.

Observations:

As for the MF, the performance depends only on ρN and P/K .

The ultimate performance of MMSE and MF coincide:

SINRMMSE a.s−−−−−−−−−−−→N,P→∞, K=const.

SINR∞ =1

α(L− 1)

Numerical results

0 100 200 300 4000

R∞ P = N

P = N/3

ρ = 1, K = 10, α = 0.1, L = 4

Number of antennas N

MF approx.MMSE approx.Simulations

Conclusions (I) - Uplink

Massive MIMO can be seen as a particular operating condition where

noise + interference� pilot contamination.

If this condition is satisfied depends on:

P/K : degrees of freedom per UT

ρN : effective SNR (transmit power × number of antennas)

α : path loss (or intercell interference)

Connection between N and P is crucial, but unclear for real channels.

As N →∞, MF and MMSE detector achieve identical performance. For finite N,the MMSE detector largely outperforms the MF.

The number of antennas needed for massive MIMO depends on all these parameters!

Downlink

L BSs with N antennas, K UTs per cell. Received signal at mth UT in cell j :

yjm =√ρ

L∑l=1

hHljmsl + qjm

sl =√λl

K∑m=1

wlmxlm =√λlWlxl

λl =1

tr WlWHl

=⇒ E[ρsH

Channel estimation through uplink pilots (as before):

hjjk = hjjk + hjjk

ρτIN +

yjm =√ρ

L∑l=1

hHljmsl + qjm

sl =√λl

K∑m=1

wlmxlm =√λlWlxl

λl =1

tr WlWHl

=⇒ E[ρsH

hjjk = hjjk + hjjk

ρτIN +

yjm =√ρ

L∑l=1

hHljmsl + qjm

sl =√λl

K∑m=1

wlmxlm =√λlWlxl

λl =1

tr WlWHl

=⇒ E[ρsH

hjjk = hjjk + hjjk

ρτIN +

Achievable rates with linear precoders

Ergodic achievable rate of UT m in cell j :

Rjm = log2 (1 + γjm)

γjm =

∣∣E [√λjhHjjmwjm

]∣∣21ρ

+ var[√

λjhHjjmwjm

(l,k)6=(j,m) E[∣∣∣√λlhH

ljmwlk

∣∣∣2] .

Two specific precoders Wj :

4= Hjj

HjjHHjj + Fj + NαIN

where α > 0 and Fj are design parameters.

J. Jose, A. Ashikhmin, T. Marzetta, and S. Vishwanath, “Pilot contamination and precoding in multi-cell TDD systems,” IEEE

Trans. Wireless Commun., no. 99, pp. 1–12, 2011.

Achievable rates with linear precoders

Ergodic achievable rate of UT m in cell j :

Rjm = log2 (1 + γjm)

γjm =

∣∣E [√λjhHjjmwjm

]∣∣21ρ

+ var[√

λjhHjjmwjm

(l,k)6=(j,m) E[∣∣∣√λlhH

ljmwlk

∣∣∣2] .

Two specific precoders Wj :

4= Hjj

HjjHHjj + Fj + NαIN

where α > 0 and Fj are design parameters.

J. Jose, A. Ashikhmin, T. Marzetta, and S. Vishwanath, “Pilot contamination and precoding in multi-cell TDD systems,” IEEE

Trans. Wireless Commun., no. 99, pp. 1–12, 2011.

Rjm − log2 (1 + γjm)a.s.−−→ 0

γBFjm =

tr Φjjm

∑l,k λl

tr RljmΦllk +∑

l 6=j λj

∣∣ 1N

tr Φljm

∣∣2γRZFjm =

λjδ2jm

(1 + δjm)2 + 1N

∑l,k λl

(1+δjm1+δlk

µljmk +∑

l 6=j λl

(1+δjm1+δlm

|ϑljm|2

and λj , δjm, µjlkm and ϑjlm can be calculated numerically.

J. Hoydis, S. ten Brink, M. Debbah, “Comparison of linear precoding schemes for downlink Massive MIMO”, ICC’12, 2011.

Rjm − log2 (1 + γjm)a.s.−−→ 0

γBFjm =

tr Φjjm

∑l,k λl

tr RljmΦllk +∑

l 6=j λj

∣∣ 1N

tr Φljm

∣∣2γRZFjm =

λjδ2jm

(1 + δjm)2 + 1N

∑l,k λl

(1+δjm1+δlk

µljmk +∑

l 6=j λl

(1+δjm1+δlm

|ϑljm|2

and λj , δjm, µjlkm and ϑjlm can be calculated numerically.

J. Hoydis, S. ten Brink, M. Debbah, “Comparison of linear precoding schemes for downlink Massive MIMO”, ICC’12, 2011.

Downlink: Numerical results

−3 −2 −1 0 1 2 3−3

Base stationsUser terminals

cell l

UT kdjlk2

cell j

7 cells, K = 10 UTs distributed on a circle of radius 3/4

path loss exponent β = 3.7, ρτ = 6 dB, ρ = 10 dB

Two channel models:

I No correlation

I Rjlk = d−β/2jlk [A 0N×N−P ], where A = [a(φ1) · · · a(φP)] ∈CN×P

a(φp) =1√P

[1, e−i2πc sin(φ), . . . , e−i2πc(N−1) sin(φ)

−3 −2 −1 0 1 2 3−3

cell l

UT kdjlk2

cell j

Two channel models:

I No correlation

a(φp) =1√P

−3 −2 −1 0 1 2 3−3

cell l

UT kdjlk2

cell j

Two channel models:

I No correlation

a(φp) =1√P

0 100 200 300 4000

12R∞ = 15.75 bits/s/Hz

Number of antennas N

No CorrelationPhysical ModelSimulations

P = N/2, α = 1/ρ,Fj = 0

Conclusions (II) - Downlink

For finite N, RZF is largely superior to BF:A matrix inversion can reduce the number of antennas by one order of magnitude!

Whether or not massive MIMO will show its theoretical gains in practice depends onthe validity of our channel models.

Reducing signal processing complexity by adding more antennas seems a bad idea.

Many antennas at the BS require TDD (FDD: overhead scales linearly with N)

Related work:

Overview paper: Rusek, et al., “Scaling up MIMO: Opportunities and Challengeswith Very Large Arrays”, IEEE Signal Processing Magazine, to appear.http://liu.diva-portal.org/smash/record.jsf?pid=diva2:450781

Constant-envelope precoding: S. Mohammed, E. Larsson, “Single-User Beamformingin Large-Scale MISO Systems with Per-Antenna Constant-Envelope Constraints:The Doughnut Channel”, http://arxiv.org/abs/1111.3752v1

Network MIMO TDD systems: Huh, Caire, et al., “Achieving “Massive MIMO”Spectral Efficiency with a Not-so-Large Number of Antennas”,http://arxiv.org/abs/1107.3862

Related work:

Related publications

I T. L. MarzettaNoncooperative cellular wireless with unlimited numbers of base station antennasIEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3590–3600, Nov. 2010.

I H. Q. Ngo, E. G. Larsson, T. L. MarzettaAnalysis of the pilot contamination effect in very large multicell multiuser MIMOsystems for physical channel modelsProc. IEEE SPAWC’11, Prague, Czech Repulic, May 2011.

I H. Q. Ngo, E. G. Larsson, T. L. MarzettaUplink power efficiency of multiuser MIMO with very large antenna arraysAllerton Conference, Urbana-Champaing, Illinois, US, Sep. 2011.

I J. Hoydis, S. ten Brink, M. DebbahMassive MIMO: How many antennas do we need?Allerton Conference, Urbana-Champaing, Illinois, US, Sep. 2011, [Online]http://arxiv.org/abs/1107.1709.

I J. Hoydis, S. ten Brink, M. DebbahComparison of linear precoding schemes for downlink Massive MIMOICC’12, submitted, 2010.

A two-tier network architecture

Massive MIMO base stations (BS) overlaid with many small cells (SCs)BSs ensure coverage and serve highly mobile UEsSCs drive the capacity (hot spots, indoor coverage)

Intra- and inter-tier interference is the main performance bottleneck.

There are many excess antennas in the network which should be exploited!

Jakob Hoydis (Bell Labs) Massive MIMO, Small Cells, and TDD CTW 2013 3 / 23

The essential role of TDD

A network-wide synchronized TDD protocol and the resulting channel reciprocity havethe following advantages:

The downlink channels can be estimated from uplink pilots.

→ Necessary for massive MIMO

Channel reciprocity holds for the desired and the interfering channels.

→ Knowledge about the interfering channels can be acquired for free.

TDD enables the use of excess antennas to reduce intra-/inter-tier interference.

An idea from cognitive radio

1 The secondary BS listens to the transmission from the primary UE:

y = hx + n

2 ...and computes the covariance matrix of the received signal:

E[yyH]

= hhH + SNR−1I

3 With the knowledge of the SNR, the secondary BS designs a precoder w which isorthogonal to the sub-space spanned by hhH.

4 The interference to the primary UE can be entirely eliminated without explicitknowledge of h.

3 With the knowledge of the SNR, the secondary BS designs a precoder w which isorthogonal to the sub-space spanned by hhH.

4 The interference to the primary UE can be entirely eliminated without explicitknowledge of h.

Translating this idea to HetNets

Every device estimates its received interference covariance matrix and precodes (partially)orthogonally to the dominating interference subspace.

Advantages

Reduces interference towards the directions from which most interference is received.

No feedback or data exchange between the devices is needed.

Every device relies only on locally available information.

The scheme is fully distributed and, thus, scalable.

Translating this idea to HetNets

Every device estimates its received interference covariance matrix and precodes (partially)orthogonally to the dominating interference subspace.

Advantages

Reduces interference towards the directions from which most interference is received.

No feedback or data exchange between the devices is needed.

Every device relies only on locally available information.

The scheme is fully distributed and, thus, scalable.

About the literature

Cognitive radioI R. Zhang, F. Gao, and Y. C. Liang, “Cognitive Beamforming Made Practical: Effective

Interference Channel and Learning-Throughput Tradeoff,” IEEE Trans. Commun.,2010.

I F. Gao, R. Zhang, Y.-C. Liang, X. Wang, “Design of Learning-Based MIMO CognitiveRadio Systems,” IEEE Trans. Veh. Tech., 2010.

I H. Yi, “Nullspace-Based Secondary Joint Transceiver Scheme for Cognitive RadioMIMO Networks Using Second-Order Statistics,” ICC, 2010.

TDD Cellular systemsI S. Lei and S. Roy, “Downlink multicell MIMO-OFDM: an architecture for next

generation wireless networks,” WCNC, 2005.

I B. O. Lee, H. W. Je, I. Sohn, O. S. Shin, and K. B. Lee, “Interference-awareDecentralized Precoding for Multicell MIMO TDD Systems,” Globecom. 2008.

Blind nullspace learningI Y. Noam and A. J. Goldsmith, “Exploiting spatial degrees of freedom in MIMO

cognitive radio systems,” ICC, 2012.

and many more...

System model and signaling

Each BS has N antennas and serves K single-antenna MUEs.

S SCs per BS with F antennas serving 1 single-antenna SUE each

The BSs and SCs have perfect CSI for the UEs they want to serve.

Every device knows perfectly its interference covariance matrix and the noise power.

Linear MMSE detection at all devices

The BSs and SCs use precoding vectors of the structure:

w ∼(PHHH + κQ + σ2I

I h channel vector to the targeted UEI H channel matrix to other UEs in the same cellI P, σ2: transmit and noise powersI Q interference covariance matrixI κ: regularization parameter (α for BSs, β for SCs)

About the regularization parameters

For α, β = 0, the BSs and SCs transmit as if they were in an isolated cell, i.e., MMSEprecoding (BSs) and maximum-ratio transmissions (SCs). By increasing α, β, theprecoding vectors become increasingly orthogonal to the interference subspace.

System model and signaling

Each BS has N antennas and serves K single-antenna MUEs.

S SCs per BS with F antennas serving 1 single-antenna SUE each

The BSs and SCs have perfect CSI for the UEs they want to serve.

Every device knows perfectly its interference covariance matrix and the noise power.

Linear MMSE detection at all devices

The BSs and SCs use precoding vectors of the structure:

w ∼(PHHH + κQ + σ2I

I h channel vector to the targeted UEI H channel matrix to other UEs in the same cellI P, σ2: transmit and noise powersI Q interference covariance matrixI κ: regularization parameter (α for BSs, β for SCs)

About the regularization parameters

For α, β = 0, the BSs and SCs transmit as if they were in an isolated cell, i.e., MMSEprecoding (BSs) and maximum-ratio transmissions (SCs). By increasing α, β, theprecoding vectors become increasingly orthogonal to the interference subspace.

Comparison of duplexing schemes and co-channel deployment

frequency

FDD TDD

frequency

co-channel TDD

frequency

co-channel reverse TDD

frequency

FDD: Channel reciprocity does not hold

TDD: Only intra-tier interference can be reduced

co-channel (reverse) TDD: Inter and intra-tier interference can be reduced

TDD versus reverse TDD (RTDD)

Order of UL/DL periods decides which devices interfere with each other.

The BS-SC channels change very slowly. Thus, the estimation of the covariancematrix becomes easier for RTDD.

Numerical results

1000 m

SC SUEMUEBS

3× 3 grid of BSs with wrap around

S = 81 SCs per cells on a regular grid

K = 20 MUEs randomly distributed

1 SUE per SC randomly distributed ona disc around each SC

3GPP channel model with path loss,shadowing and fast fading, N/LOS links

TX powers: 46 dBm (BS), 24 dBm(SC), 23 dBm (MUE/SUE)

20 MHz bandwidth @ 2 GHz

No user scheduling, power control

Averages over channel realizations andUE locations

TDD UL/DL cycles of equal length

Downlink spectral area efficiency regions

0 20 40 60 800

Macro DL area spectral efficiency(b/s/Hz/km2

ncy( b/

2) FDD (N = 20, F = 1)

0 20 40 60 800

FDD regionmore antennas

N = 20 → 100

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

0 20 40 60 800

FDD region

TDD region

less intra-tier interf.α = 0 → 1

more antennasN = 20 → 100

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

TDD (N = 100, F = 4, α = 1, β = 1)

0 20 40 60 800

FDD region

TDD region

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

TDD (N = 100, F = 4, α = 1, β = 1)

0 20 40 60 800

FDD region

TDD region

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

TDD (N = 100, F = 4, α = 1, β = 1)

0 20 40 60 800

FDD region

TDD region

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

TDD (N = 100, F = 4, α = 1, β = 1)

0 20 40 60 800

FDD region

TDD region

CoTDD region

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

TDD (N = 100, F = 4, α = 1, β = 1)

0 20 40 60 800

FDD region

TDD region

CoTDD region

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

TDD (N = 100, F = 4, α = 1, β = 1)

0 20 40 60 800

FDD region

TDD region

CoTDD region

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

TDD (N = 100, F = 4, α = 1, β = 1)

0 20 40 60 800

FDD region

TDD region

CoTDD region

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

TDD (N = 100, F = 4, α = 1, β = 1)

0 20 40 60 800

FDD region

TDD region

CoTDD region

CoRTDD region

ncy( b/

2) FDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

TDD (N = 100, F = 4, α = 1, β = 1)

Downlink SINR distribution

−20 −10 0 10 20 30 400

SINR (dB)

TDD Downlink SINR:

MUE SUEα = 0 α = 1 β = 0 β = 1

Mean 13.11 24.13 23.9 33.7895% 40.38 48.47 40 42.8750% 11.58 22.01 24.65 34.355% −8.48 7.86 6.02 22.62

−20 −10 0 10 20 30 400

SINR (dB)

Co-channel TDD Downlink SINR:

MUE SUEα, β = 0 α, β = 1 α, β = 0 α, β = 1

Mean −6.29 9.52 14.33 25.4595% 20.45 35.95 29.88 35.0150% −8.06 6.44 15.49 26.055% −26.64 −6.82 −6.51 13.6

Uplink spectral area efficiency regions

0 20 40 60 800

Macro UL sum-rate(b/s/Hz/km2

ate( b/

2)FDD/TDD (N = 20, F = 1)

FDD/TDD (N = 100, F = 4)

co-channel TDDco-channel reverse TDD

Observations

With the proposed precoding scheme, a TDD co-channel deployment of BSs andSCs leads to the highest area spectral efficiency (α = β = 1, 20 MHz BW):

Area throughput 7.63 Gb/s/km2 8.93 Gb/s/km2

Rate per MUE 38.2 Mb/s 25.4 Mb/sRate per SUE 84.8 Mb/s 104 Mb/s

Even a few “excess” antennas at the SCs lead to significant gains.

As the scheme is fully distributed and requires no data exchange between thedevices, the rates can be simply increased by adding more antennas to the BSs/SCsor increasing the SC-density.

Discussion

Channel reciprocity requires:

I Hardware calibrationI Scheduling of UEs on the same resource blocks in subsequent UL/DL cycles

The network-wide TDD protocol requires tight synchronization of all devices:I GPS (outdoor)I NTP/PTP (indoor)I BS reference signals

Channel estimation will suffer from interference and pilot contamination.

Covariance matrix estimation becomes difficult for large N.

We have considered a worst-case outdoor deployment scenario with fixed cellassociation, no power control or scheduling. Location-dependent user scheduling andinterference-temperature power control could further enhance the performance.

Massive MIMO for wireless backhaul

small cellwireless backhaul

wireless datawired backhaul

user equipment

massive MIMO base station

Core network

The unrestrained SC-deployment “where needed” rather than “where possible”requires a high-capacity and easily accessible backhaul network.

Already for most WiFi deployments, the backhaul capacity (10–100 Mbit/s) and notthe air interface (54–600 Mbit/s) is the bottleneck.

Why not provide wireless backhaul with massive MIMO?3

3T. L. Marzetta and H. Yang, “Dedicated LSAS for metro-cell wireless backhaul - Part I: Downlink,” Bell Laboratories, Alcatel-Lucent,

Tech. Rep., Dec. 2012.

Massive MIMO for wireless backhaul: Advantages

No standardization or backward-compatibility required

BS-SC channels change very slowly over time:

I Complex transmission/detection schemes (e.g., CoMP) can be easily implemented.

I Even FDD might be possible due to reduced CSI overhead.

Provide backhaul where needed:

I Adapt backhaul capacity to the load (support highly variable traffic)

I Statistical multiplexing opportunity to avoid over-provisioning of backhaul

I Enable user-centric small-cell clustering for virtual MIMO

SCs require only a power connection to be operational

Line-of-sight not necessary if operated at low frequencies

Massive MIMO for wireless backhaul: Is it feasible?

How many antennas are needed to satisfy the desired backhaul rates with a giventransmit power budget?

Assumptions:

Every BS knows the channels to all SCs.

The BSs can exchange some control information.

Full user data sharing between the BSs is not possible.

Single-antenna SCs, BSs with N antennas

TDD operation on a separate band (2/3 DL, 1/3 UL)

Same modeling assumptions as before

Find the smallest N such that the power minimization problem with target SINRconstraints for the multi-cell multi-antenna wireless system is feasible.4,5

4H. Dahrouj and W. Yu, “Coordinated beamforming for the multicell multi-antenna wireless system,” IEEE Trans. Wireless Commun.,

vol. 9, no. 5, pp. 1748–1759, May 2010.5

S. Lakshminarayana, J. Hoydis, M. Debbah, and M. Assaad, “Asymptotic analysis of distributed multi-cell beamforming, in IEEEInternational Symposium in Personal Indoor and Mobile Radio Communications (PIMRC), Istanbul, Turkey, Sep. 2010, pp. 2105–2110.

Massive MIMO for wireless backhaul: Numerical results

0 20 40 60 80 1000

Downlink backhaul rate (Mbit/s)

S = 81

S = 40

S = 20

0 10 20 30 40 50Uplink backhaul rate (Mbit/s)

Average minimum number of required BS-antennas N to serve S ∈ {20, 40, 81} randomly chosen

SCs with the same target backhaul rate.

Summary

Massive MIMO and SCs have distinct advantages which complement each other:

I Massive MIMO for coverage and mobility support

I SCs for capacity and indoor coverage

TDD and the resulting channel reciprocity allow every device to fully exploit itsavailable degrees of freedom for intra-/inter-tier interference mitigation.

A TDD co-channel deployment of massive MIMO BSs and SCs can achieve a veryattractive rate region.

Massive MIMO BSs can provide wireless backhaul to a large number of SCs. Theslowly time-varying nature of the BS-SC channels might allow for complex precodingand detection schemes.

For more details:

J. Hoydis, K. Hosseini, S. ten Brink, and M. Debbah, “Making Smart Use of Excess Antennas:Massive MIMO, Small Cells, and TDD,” Bell Labs Technical Journal, vol. 18, no. 2, Sep. 2013.

Thank you!

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