Advanced Topics in Signal Processing Advanced Topics in Signal Processing for Wireless Communications for Wireless Communications

Narayan MandayamNarayan Mandayam

WINLABWINLAB, , Rutgers UniversityRutgers University

www.winlab.rutgers.edu/~narayanwww.winlab.rutgers.edu/~narayan

Page:2

IntroductionIntroduction

Wireless Data on the move is the primary driver for innovations in signal processing

Examples of situations include:

Cellular like networks for wireless data (Licensed)

Wireless access to the Internet: Wireless LANs (Unlicensed)

Infostations: Intermittent pockets of high bandwidth on the move (Unlicensed)

Wireless Data Communications characterized by Channel variations (time, frequency, space) due to mobility and propagation effectsMultiple Access Interference from known and unknown entities

Challenges in enabling wireless data communications

Mitigating or Exploiting channel variations

Mitigating Multiaccess interference

Page:3

Challenges in Enabling Wireless Data Challenges in Enabling Wireless Data

Exploiting Variations: Opportunistic CommunicationsOpportunities for transmission arise in time, frequency and space

Examples include:

MIMO, Space-Time Coding, Scheduling, Resource Allocation

Signal Processing challenges in opportunistic transmission strategies ?

Knowledge of temporal and spatial variations of wireless channels

Higher carrier frequencies, higher mobility, great no. of unknown parameters

Mitigating Interference: Multiuser Detection Exploit interference structure to design tailored receivers

Examples include:

Cellular 3G, Unlicensed band Wireless LANs

Signal Processing challenges in Multiuser Detection ?

Blind and Adaptive Techniques

Page:4

Topics Covered in this TalkTopics Covered in this Talk

Opportunistic CommunicationsPilot Assisted MIMO Channel Estimation

Multiuser Detection Blind Interference Cancellation Techniques for CDMA Systems

Subspace TechniquesSIR Estimation in CDMA Systems

Pilot Assisted Estimation of MIMO Pilot Assisted Estimation of MIMO Fading Channel Response and Fading Channel Response and

Achievable Data RatesAchievable Data Rates

Joint work with Dragan SamardzijaJoint work with Dragan Samardzija

Bell Labs, Lucent TechnologiesBell Labs, Lucent Technologies

Page:6

IntroductionIntroduction

Pilot assisted MIMO estimation and its impact on achievable rates

The effects of the estimation error are evaluated for Estimates being available at the receiver only: open loop Estimates are fed back to the transmitter allowing water

pouring optimization: closed loop

Results/Analysis may be interpreted as a study of mismatched receiver and transmitter algorithms in MIMO systems

Page:7

System AssumptionsSystem Assumptions

Multiple-input multiple-output (MIMO) wireless systems Frequency-flat time-varying wireless channel with additive white

Gaussian noise (AWGN), i.e., Block fading channel We consider two pilot based approaches for the estimation:

Single pilot symbol per block with variable (from data symbols) power

More than one symbol per block with same (as data symbols) power

Orthogonality between the pilots assigned to different transmit antennas

Maximum-likelihood estimate of the channel response

Page:8

Signal ModelSignal Model MIMO communication system that consists of M transmit and N receive

antennas Received spatial vector y

y(k) = H(k) x(k) + n(k) (1)

where y(k) in CN, x(k) in CM, n(k) in CN, H(k) in CN x M

x is transmitted vector, n is AWGN (E [n nH] = No INxN), and H is the MIMO channel response matrix, all corresponding to the time instance k

hnm (k) is the n-th row and m-th column element of the H(k)

corresponds to a SISO channel response between the transmit antenna m and receive antenna n

Page:9

Signal Model, contd.Signal Model, contd.

n-th component of the received spatial vectorspatial vector y(k)=[y1(k)…yN(k)]T (i.e., signal at the receive antenna n) is

(2)

gm (k) is the transmitted signal from the m-th transmit antenna, i.e., x(k)=[g1(k) … gM(k)]T .

The channel response H(k) is estimated using a pilot (training) signal that is a part of the transmitted data

Pilot is sent periodically, every K symbol periods

)()()()(1

k n kgkh k y n

M

mmnmn

Page:10

Signal Model, contd.Signal Model, contd. At transmitter m, the K-dimensional temporal vectortemporal vector

gm=[gm(1) … gm(K)]T (whose k-th component is gm(k) (in (2))) is

(3)

a dim=A and a p

im=Ap are amplitudes related as Ap= A d d

im is the unit-variance data, and |d pjm|2=1 is the pilot symbol

sdi and sp

im are temporal signaturestemporal signatures, all corresponding to the m-th transmitter;

L is the number of signal dimensions allocated to the pilot, per transmit antenna

Temporal signatures are mutually orthogonal and they could be: canonical waveform - a TDMA-like waveform K-dimensional Walsh sequence - a CDMA-like waveform

PILOT

L

j

pjm

pjm

pjm

DATA

LMK

i

di

dim

dimm

dada

11

ssg

Page:11

Signal Model, contd.Signal Model, contd. Rewrite spatial received signal vector as:

y(k) = H(k)(d(k) + p(k)) + n(k) (4)

d(k) =[d1(k) … dM(k)]T is the data bearingdata bearing transmitted

spatial signalspatial signal where

p(k) =[p1(k) … pM(k)]T is the pilot portionpilot portion of the transmitted

spatial signalspatial signal

DATA

LMK

i

di

dim

dimm ksdakd

1

)()(

PILOT

L

j

pjm

pjm

pjmm ksdakp

1

)()(

sd1

dd11

A

sdK-LM

ddK-LM 1

A

datadata

sp11

dp11

Ap

spL1

dpL 1

Ap

pilot pilot +

TX1

sd1

dd1M

A

sdK-LM

ddK-LM M

A

datadata

sp1M

dp1M

Ap

spLM

dpL M

Ap

pilotpilot +

TXM

X M•Pilots are orthogonal between the TxsPilots are orthogonal between the Txs

Data temporal signatures reused across TxsData temporal signatures reused across Txs

MIMO transmitter with M antennas

1g

Mg

ppp

Page:13

Model AssumptionsModel Assumptions

Block-fading channel model with channel coherence K Tsym, hnm(k)~hnm,

for k = 1,…, K, for all m and n The elements of H are iid random variables When applying different number of transmit antennas, the total average

transmitted power must be conserved. Per pilot period it is

(5)

Amount of transmitted energy that is allocated to the pilot (percentage wise) is

(6)

L

j

pjm

LMK

i

dimav aa

K

MP

1

2

1

2

[%]1002

2

LLMK

L

Page:14

Pilot Arrangements – case 1Pilot Arrangements – case 1 Two different pilot arrangements:

L=1 and Ap= A, single dimension taken by pilot, with different power from data symbols. The data symbol amplitude is

(7)

In SISO systems applied in CDMA wireless systems (e.g., IS-95 and WCDMA)

In MIMO systems, applied in narrowband MIMO implementations [Foschini, Valenzuela, Wolniansky]

Also wideband MIMO implementation based on 3G WCDMA.

M

P

MK

KA av

21

Page:15

Pilot Arrangements – case 2Pilot Arrangements – case 2

L > 1 and Ap= A ( = 1), multiple signal dimensions taken by pilot, with the same power as data. The data symbol amplitude is

(8)

Frequently used in SISO systems Wire-line modems Wireless standards (e.g., IS-136 and GSM). Not common practice in MIMO systems.

M

P

MLK

KA av

12

Page:16

Estimation of Channel ResponseEstimation of Channel Response Based on previously introduced assumption:

Pilot signatures maintain orthogonality elements of H are iid Background noise is AWGN

Sufficient to estimate hnm (for m=1,…, M, n = 1,…, N) independently

Identical to estimating a SISO channel response between the transmit antenna m and receive antenna n

The estimate of the channel response hnm

L

jn

pjm

pjm

pnm d

LAh

1

H1ˆ rs

Page:17

Estimation of Channel Response, contd.Estimation of Channel Response, contd.

(9)

The estimation error is

(10)

corresponds to sample of a white Gaussian random process The channel matrix H estimate is

(11) He is the estimation error matrix Each component of the error matrix He is independent identically

distributed random variable nenm

L

jn

pjm

pjm

pnmnm d

LAhh

1

H1ˆ ns

L

jn

pjm

pjm

p

enm d

LAn

1

H1ns

eHHH ˆ

Page:18

Detection and Effective Noise Detection and Effective Noise The sufficient statistics are obtained at the N receive antennas by projecting

the received signal vectors with the corresponding temporal signatures si, i=1,…K-LM

The sufficient statistic for ith signature can be written as

(12)

where E[ni niH] = No INxN

The effective noise vector is

(13)

Covariance matrix of the effective noise vector is

(14)

i

iii

iii AA

dHn

dHdHn

dHHz eeeˆ

i

ii

An dH

ne

H

2

H EE eeH|nHHInnΥΥ

A

NoA ii

i

Page:19

Open Loop CapacityOpen Loop Capacity Channel estimates are available to the receiver only Under the assumptions:

Estimate of H has to be stationary and ergodic The channel coding will span across great number of channel blocks Effective noise is treated as independent Gaussian interference

The lower bound for the open loop ergodic capacity is

(15)

(K-LM)/K because L temporal signatures per each transmit antenna allocated to the pilot

1H

x2ˆˆˆ1

detlogE ΥHHIH MK

LMKRC MM

Page:20

Comparison to SISO ResultsComparison to SISO Results SISO case [see Shamai, Biglieri, Proakis, IT’98], capacity lower bound

for mismatched decoding as

(16)

where h and are the SISO channel response and its estimate

PropositionFor M = 1, N = 1 (i.e., SISO) the rate R in (17) and R* in (18) are related as

(17)

where is obtained using the pilot assisted estimation Bound in (15) is an extension of the information theoretical bound in (16),

capturing the more specific pilot assisted estimation scheme and generalizing it to the MIMO case

NoPhhE

PhRC

hh

h )|ˆ(|

ˆ1logE

2ˆ|

2

2ˆ*

1x1

h

avPLLK

KPR

K

LKR

2*

)(for

h

Achievable open loop rates vs. power allocated to the pilot, SISO Achievable open loop rates vs. power allocated to the pilot, SISO system, SNR=4, 12, 20dB, K=10, Rayleigh channelsystem, SNR=4, 12, 20dB, K=10, Rayleigh channel

Page:22

Capacity Efficiency RatioCapacity Efficiency Ratio

Evaluate performance under optimum pilot power allocation ? For any given SNR, define the capacity efficiency ratio as,

(18)

Maximum rate R is maximized with respect to pilot power Ergodic capacity CMxN, with the ideal knowledge of the

channel response The index M and N correspond to number of transmit and

receive antennas, respectively

NMNM C

R

xx

max

Capacity efficiency ratio vs. channel coherence time length, SISO Capacity efficiency ratio vs. channel coherence time length, SISO system, SNR=4, 12, 20dB, K=10, 20, 40, 100, Rayleigh channelsystem, SNR=4, 12, 20dB, K=10, 20, 40, 100, Rayleigh channel

Pilot arrangement case 1 is more efficient compared to case 2

solid line: channel response estimation

dashed line: ideal channel knowledge

Pilot arrangement case 1

Open loop rates vs. power allocated to the pilot, MIMO system, Open loop rates vs. power allocated to the pilot, MIMO system, SNR=12dB, K=40, Rayleigh channelSNR=12dB, K=40, Rayleigh channel

Capacity efficiency ratio vs. channel coherence time length, Capacity efficiency ratio vs. channel coherence time length, MIMO system, SNR=12dB, K=10, 20, 40, 100, Rayleigh channelMIMO system, SNR=12dB, K=10, 20, 40, 100, Rayleigh channel

Pilot arrangement case 1

1x4 the most efficient

The efficiency is getting smaller as the number of TX antennas grow (for fixed number of received antennas)

Page:26

Closed Loop Rates: Mismatched Water PouringClosed Loop Rates: Mismatched Water Pouring H(i-1) and H(i) correspond to the consecutive block faded

channel responses Receiver feeds back the estimate Instead of H(i) , is used to perform the water

pouring transmitter optimization for the i-th block Singular value decomposition (SVD) is performed

For data vector d(k), at the transmitter

(19) S(i) is a diagonal matrix whose elements sjj (j=1, …, M) are

determined by the water pouring algorithm per singular value of

)1(ˆ iH

)1(ˆ)1(ˆ)1(ˆ)1(ˆ H iiii VΣUH

)()()1(ˆ)( kiik dSVd

)1(ˆ iH

)1(ˆ iH

Page:27

Mismatched Water Pouring, contd.Mismatched Water Pouring, contd. For diagonal element of (denoted as j = 1, …, M), the

diagonal element of S(i) is defined as

(20)

y0 is a cut-off value that depends on the channel fading statistics

such that the average transmit power stays the same Pav [Goldsmith 93]

Water pouring optimization is on information bearing portion of the signal d(k)

Pilot p(k) is not changed Receiver knows that the transformation in (19) is performed at the

transmitter

)1(ˆ iΣ )1(ˆ ijj

otherwise0

)1(ˆfor)1(ˆ

11

)(0

22

220

2yAi

Aiyisjj

jjjj

Page:28

Closed Loop Achievable RatesClosed Loop Achievable Rates Receiver performs estimation resulting in Error matrix Effective noise and its covariance are modified resulting in

(21)

Above application of the water pouring algorithm per eigen mode is suboptimal, i.e., it is mismatched ( is used instead of H(i))

Closed loop system capacity is lower bounded as,

(22)

Assumptions on estimates and effective noise same as before

)()1(ˆ)(ˆˆ iii SVHG

Hˆ2

E eeG|GGGIΥΥ

e

A

NoAWPWP

1H

x2ˆ )(ˆˆ1detlogE WP

MMWPWP

MK

LMKRC ΥGGI

G

)()1(ˆ)( iiiee SVHG

)1(ˆ iH

Ergodic capacity vs. SNR, MIMO system, ideal knowledge of the Ergodic capacity vs. SNR, MIMO system, ideal knowledge of the channel response, Rayleigh channelchannel response, Rayleigh channel

solid line: open loop capacity

dashed: closed loop capacity

Gap between closed loop and open loop is getting smaller for Higher SNR Larger ratio N/M

(number of RX vs. TX antennas)

CDF of capacity, MIMO system, ideal knowledge of the channel CDF of capacity, MIMO system, ideal knowledge of the channel response, Rayleigh channelresponse, Rayleigh channel

solid line: open loop capacity

dashed: closed loop capacity

Closed-loop rates vs. correlation between successive channel Closed-loop rates vs. correlation between successive channel responses, MIMO system, SNR=4dB, K=40, Rayleigh channelresponses, MIMO system, SNR=4dB, K=40, Rayleigh channel

solid line: channel response estimation

dashed: ideal channel response

In both cases delay (temporal mismatch) exists

Pilot arrangement case 1

Page:32

Summary of MIMO Pilot EstimationSummary of MIMO Pilot Estimation

Pilot Assisted Channel Estimation for Multiple-input multiple-output wireless systems

Open loop and closed loop ergodic capacity lower bounds are determined

Performance depends on: Percentage of the total power allocated to the pilot Background noise level Channel coherence time length Temporal correlation (for the water pouring)

First pilot-based approach is less sensitive to the fraction of power allocated to the pilot

As the number of transmit antenna increases, the capacity efficiency ratio is lowered (while keeping the same number of receive antennas)