1

16.548 Coding and Information Theory

Lecture 15: Space Time Coding and MIMO:

2

Credits

3

Wireless Channels

4

Signal Level in Wireless Transmission

5

Classification of Wireless Channels

6

Space time Fading, narrow beam

7

Space Time Fading: Wide Beam

8

Introduction to the MIMO Channel

9

Capacity of MIMO Channels

10

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Single Input- Single Output systems (SISO)

y(t) = g • x(t) + n(t)

x(t): transmitted signaly(t): received signalg(t): channel transfer functionn(t): noise (AWGN, 2)

Signal to noise ratio :

Capacity : C = log2(1+)

x(t)y(t)

g

2x2

σ

Eρ g

12

Single Input- Multiple Output (SIMO)

Multiple Input- Single Output (MISO)

• Principle of diversity systems (transmitter/ receiver)• +: Higher average signal to noise ratio

Robustness• - : Process of diminishing return

Benefit reduces in the presence of correlation• Maximal ratio combining >

Equal gain combining > Selection combining

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Idea behind diversity systems

• Use more than one copy of the same signal• If one copy is in a fade, it is unlikely that all the others

will be too.

• C1xN>C1x1

• C1xN more robust than C1x1

1

N

)N1(log2N1 xC

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Background of Diversity Techniques

• Variety of Diversity techniques are proposed to combat Time-Varying Multipath fading channel in wireless communication– Time Diversity– Frequency Diversity– Space Diversity (mostly multiple receive antennas)

• Main intuitions of Diversity:– Probability of all the signals suffer fading is less then probability of single

signal suffer fading– Provide the receiver a multiple versions of the same Tx signals over

independent channels• Time Diversity

– Use different time slots separated by an interval longer than the coherence time of the channel.

– Example: Channel coding + interleaving– Short Coming: Introduce large delays when the channel is in slow fading

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Diversity Techniques

• Improve the performance in a fading environment– Space Diversity

• Spacing is important! (coherent distance)

– Polarization Diversity• Using antennas with different polarizations for

reception/transmission.

– Frequency Diversity• RAKE receiver, OFDM, equalization, and etc.• Not effective over frequency-flat channel.

– Time Diversity• Using channel coding and interleaving.• Not effective over slow fading channels.

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RX Diversity in Wireless

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Receive Diversity

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Selection and Switch Diversity

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Linear Diversity

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Receive Diversity Performance

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Transmit Diversity

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Transmit Diversity with Feedback

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TX diversity with frequency weighting

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TX Diversity with antenna hopping

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TX Diversity with channel coding

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Transmit diversity via delay diversity

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Transmit Diversity Options

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MIMO Wireless Communications: Combining TX and RX Diversity

• Transmission over Multiple Input Multiple Output (MIMO) radio channels

• Advantages: Improved Space Diversity and Channel Capacity

• Disadvantages: More complex, more radio stations and required channel estimation

Space-TimeEncoder

Datasymbolsd

N DataSymbols

Pilo

tsy

mbo

ls

P

L_tTransmitantennas

Wireless Channel(What a Big Cloud!)

Space-TimeDecoder

L_rReceiveantennas

Datasymbols

Pilo

tsy

mbo

ls

P

d_hat

29

MIMO Model

• Matrix Representation

– For a fixed T

TNTMMNTN WXHY

T: Time index

W: Noise

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Part II: Space Time Coding

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Multiple Input- Multiple Output systems (MIMO)

H

1

M

1

N

Nx1Mx1NxMNx1nxy H

22

totalP

• Average gain

• Average signal to noise ratio

H11

HN1

H1M

HNM

HH

1,H

22 ijE

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Shannon capacity

K= rank(H): what is its range of values?Parameters that affect the system capacity• Signal to noise ratio • Distribution of eigenvalues (u) of H

H

2

H22

T2

H2x

2

M

ρdetlog

Mσ

Pdetlog

σ

EdetlogC

HHI

HHIHHI g

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Interpretation I: The parallel channels approach

• “Proof” of capacity formula

• Singular value decomposition of H: H = S·U·VH

• S, V: unitary matrices (VHV=I, SSH =I)

U : = diag(uk), uk singular values of H

• V/ S: input/output eigenvectors of H• Any input along vi will be multiplied by ui and will appear as an output along si

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Vector analysis of the signals

1. The input vector x gets projected onto the vi’s

2. Each projection gets multiplied by a different gain ui.

3. Each appears along a different si.

u1

u2

uK

<x,v1> · v1

<x,v2> · v2

<x,vK> · vK<x,vK> uK sK

<x,v1> u1 s1

<x,v2> u2 s2

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Capacity = sum of capacities

• The channel has been decomposed into K parallel subchannels

• Total capacity = sum of the subchannel capacities

• All transmitters send the same power:

Ex=Ek

2

k

2

k2

k

2

k

2

kk σ

Eu

s,nE

v,xEuρ

K

1ik2

K

1ik ρ1logCC

K

1i

2

22 1logC kk u

E

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Interpretation II: The directional approach

• Singular value decomposition of H: H = S·U·VH

• Eigenvectors correspond to spatial directions (beamforming)

1

M

1

N

(si)1

(si)N

37

Example of directional interpretation

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Space-Time Coding

• What is Space-Time Coding?– Space diversity at antenna– Time diversity to introduce

redundant data

• Alamouti-Scheme – Simple yet very effective– Space diversity at

transmitter end– Orthogonal block code

design

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Space Time Coded Modulation

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Space Time Channel Model

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STC Error Analysis

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STC Error Analysis

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STC Design Criteria

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STC 4-PSK Example

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STC 8-PSK Example

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STC 16-QAM Example

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STC Maximum Likelihood Decoder

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STC Performance with perfect CSI

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55

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Delay Diversity

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Delay Diversity ST code

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Space Time Block Codes (STBC)

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Decoding STBC

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62

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Block and Data Model• 1X(N+P) block of information symbols broadcast from transmit antenna: i

Si(d, t)

• 1X(N+P) block of received information symbols taken from antenna: j

Rj = hjiSi(d, t) + nj

• Matrix representation:

Function (mapping) S at antennai defines the Space-Timeencoding process

Assuming a single user andquasi-static and independentfading, n is AWGN

NHS

r

r

R

tL

.

.

.1

trrr

t

t

LLLL

L

L

hhh

hhh

hhh

H

...

....

....

....

...

...

21

22221

11211

),(

.

.

.

),(

),(

2

1

tdS

tdS

tdS

S

tL

64

Related IssuesSpace-Time

EncoderS_i(d,t)

i=1,2,…,K

Datasymbolsd

N DataSymbols

Pilo

tsy

mbo

ls

P

L_tTransmitantennas

• How to define Space-Time mapping Si(d,t) for diversity/channel capacity trade-off?

• What is the optimum sequence for pilot symbols?• How to get “best estimated” Channel State Information (CSI) from the

pilot symbols P?• How to design frame structure for Data symbols (Payload) and Pilot

symbols such that most optimum for FER and BER?

Wireless Channel(What a Big Cloud!)

Space-TimeDecoder

L_rReceiveantennas

Datasymbols

Pilo

tsy

mb

ols

P

d_hat

65

Specific Example of STBC: Alamouti’s Orthogonal Code

• Let’s consider two antenna i and i+1 at the transmitter side, at two consecutive time instants t and t+T:

• The above Space-Time mapping defines Alamouti’s Code[1].• A general frame design requires concatenation of blocks (each 2X2)

of Alamouti code,

d_0 d_1

-d_1*

Tim

e t

Tim

e

d_0*

Ant i+1

Tim

e t+

T

Ant. i

Space

...||

...||*

2*

3**

1

321

dddd

ddddD

o

o

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Estimated Channel State Information (CSI)

• Pilot Symbol Assisted Modulation (PSAM) [3] is used to obtain estimated Channel State Information (CSI)

• PSAM simply samples the channel at a rate greater than Nyquist rate,so that reconstruction is possible

• Here is how it works…

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Channel State Estimation

P 5P P 5D P 2P 3D P

1 7 13 19

5D P

25

2P 3D P

31

2P 3D P

37

5D …... 5D P 3D 2P P 3D 2P P …... P 5P

edge partuniform

partedge part

Frame size = 300

265 271 277 295

L_t = i

A typical slow fadingchannel

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Estimated CSI (cont.d) Block diagram of the receiver

+

n(t)

r(t) A/DConverter

Delay

Pilot SymbolExtractor

Channel StateEstimator

MLDecoder

Matched Filteru*(-t)

r_k

r_p_1, r_p_2, …, r_p_N

r_k+1,r_k+2,…,r_k+K

h_hat

D_hat

tL

ltlstlh

1)()()()(

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Channel State Estimation (cont.d)

P 5P P 5D P 2P 3D P

1 7 13 19

5D P

25

2P 3D P

31

2P 3D P

37

5D …... 5D P 3D 2P P 3D 2P P …... P 5P

edge partuniform

partedge part

Frame size = 300

265 271 277 295

L_t = i

• Pilot symbol insertion length, Pins=6. • The receiver uses N=12, nearest pilots to obtain estimated

CSI

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Channel State Estimation Cont.d• Pilot Symbols could be think of as redundant data symbols

• Pilot symbol insertion length will not change the performance much, as long as we sample faster than fading rate of the channel

• If the channel is in higher fading rate, more pilots are expected to be inserted

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Estimated CSI, Space-time PSAM frame design

P 5P P 5D P 2P 3D P

1 7 13 19

5D P

25

2P 3D P

31

2P 3D P

37

5D …... 5D P 3D 2P P 3D 2P P …... P 5P

edge partuniform

partedge part

Frame size = 300

265 271 277 295

L_t = i

• The orthogonal pilot symbol (pilots chosen from QPSK constellation) matrix is, [4]

• Pilot symbol insertion length, Pins=6. • The receiver uses N=12, nearest pilots to obtain estimated CSI• Data = 228, Pilots = 72

P 5P P 5D P 2P 3D P

1 7 13 19

5D P

25

2P 3D P

31

2P 3D P

37

5D …... 5D P 3D 2P P 3D 2P P …... P 5P

edge partuniform

partedge part

Frame size = 300

265 271 277 295

L_t = i+1

11

11P

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Channel State Estimation (cont.d)MMSE estimation

• Use Wiener filtering, since it is a Minimum Mean Square Error (MMSE) estimator

• All random variables involved are jointly Gaussian, MMSE estimator becomes a linear minimum mean square estimator [2]:

• Wiener filter is defined as, .

• Note, and

1)(][ p

H

prCovhrEW

INhCovrCovo

H

pppp )()(

][)(H

ppphhEhCov

]|[ˆp

rhEh

ppp

H

pWrrrCovhrEh 1)(][ˆ

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Block diagram for MRRC scheme with two Tx and one Rx

+

channelestimator

combinerh_hat_0

h_hat_1

h_hat_0 h_hat_1 d_hat_0 d_hat_1

maximum likelihood detector

n_0n_1

Interference& noise

rx antenna

h_0(t)h_0(t+T)

h_1(t)h_1(t+T)

tx antenna 0 tx antenna 1

d_0-d_1*

d_1d_0*

d_0 d_1

-d_1*

Tim

e t

Tim

e

d_0*

Ant i+1

Tim

e t+

T

Ant. i

Space

74

Block diagram for MRRC scheme with two Tx and one Rx

• The received signals can then be expressed as,

• The combiner shown in the above graph builds the following two estimated signal

oooondthdthtrr

11)()()(

1

*

01

*

11)()()( ndTthdTthTtrr

o

*

1

*

1

*

000])()([ˆ rthrTthd

*

1

*

0

*

011])()([ˆ rthrTthd

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Maximum Likelihood Decoding Under QPSK Constellation

• Output of the combiner could be further simplified and could be expressed as follows:

• For example, under QPSK constellation decision are made according to the axis.

*

110

*

00

*

11

*

000)()()]()()()([ˆ nthnTthdTththTththd

*

100

*

11

*

11

*

001)()()]()()()([ˆ nthnTthdTththTththd

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Space-Time Alamouti Codes with Perfect CSI,BPSK Constellation

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Space-Time Alamouti Codes with PSAM under QPSK Constellation

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Space-Time Alamouti Codes with PSAM under QPSK Constellation

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Performance metrics• Eigenvalue distribution

• Shannon capacity– for constant SNR or

– for constant transmitted power

• Effective degrees of freedom(EDOF)

• Condition number

• Measures of comparison– Gaussian iid channel

– “ideal” channel

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Measures of comparisonGaussian Channel

Hij =xij+jyij : x,y i.i.d. Gaussian random variables

Problem: poutage

“Ideal” channel (max C)

rank(H) = min(M, N)

|u1 | = |u2 | = … = |uK |

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Eigenvalue distribution

Ideally:

As high gain as possible

As many eigenvectors as possible

As orthogonal as possible

H

1

M

1

N

Limits

Power constraints

System size

Correlation

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Example: Uncorrelated & correlated channels

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Shannon capacity

• Capacity for a reference SNR (only channel info)

• Capacity for constant transmitted power (channel + power roll-off info)

Href2 M

ρdetlogC HHI

H

2x

2 σ

EdetlogC HHI

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Building layout

RCVR(hall)

XMTR

RCVR(lab)

4m

6m

3.3m

3.3m

2m

0o

90o

270o

180o

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LOS conditions: Higher average SNR, High correlationNon-LOS conditions: Lower average SNR,More scattering

XMTR

RCVR(lab)

4m

6m

3.3m

3.3m

2m

0o

90o

270o

180o

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Example: C for reference SNR

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Example: C for constant transmit pwr

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Other metrics EDOF (Effective degrees of freedom)

Condition number

Definition UMAX/ umin

+ Intuition Simplicity

- Dependence on reference SNR

No information on intermediate eigenvalue distribution

ref

outageref

ρ2

ρ2Clim

δ

p

δ

0δ

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From narrowband to wideband

• Wideband: delay spread >> symbol time

• -: Intersymbol interference

+: Frequency diversity

• SISO channel impulse response:

SISO capacity:

L

1lll τtδgg(t)

L

1l

2

l

2

2x

2

2 gg,σ

Eg1logC

90

Matrix formulation of wideband case

)(

)(

)(

)(

)(

)(

τtδhth

1

1

1111

L

1llij,lij

tn

tx

tx

ty

ty

tntxty

MNMN

M

N

HH

HH

H

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Equivalent treatment in the frequency domain

• Wideband channel = Many narrowband channels

H(t) H(f)

bandwidthbandwidthNB ff

fC H

0

x2WB

02H

2x

2NB

)()(N

)(EdetlogC

(BW)Nσ,σ

EdetlogC

HHI

HHIf

Noise level

92

Extensions

• Optimal power allocation

• Optimal rate allocation

• Space-time codes

• Distributed antenna systems

• Many, many, many more!

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Optimal power allocation

• IF the transmitter knows the channel, it can allocate power so as to maximize capacity

• Solution: Waterfilling

total

K

1ik

K

1i

2

k2k

2 PE,uσ

E1logC

2

2

kk

kk

σ

uλ

)λ

1(νE

94

Illustration of waterfilling algorithm

2

2

kk

kk

σ

uλ

)λ

1(νE

Stronger subchannels get the most power

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Discussion on waterfilling• Criterion: Shannon capacity maximization(All the SISO discussion on coding, constellation limitations etc is pertinent)

• Benefit depends on the channel, available power etc.Correlation, available power Benefit

• Limitations:– Waterfilling requires feedback link– FDD/ TDD– Channel state changes

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Optimal rate allocation

• Similar to optimal power allocation• Criterion: throughput (T) maximization

• Bk : bits per symbol (depends on constellation size)• Idea: for a given k, find maximum Bk for a target

probability of error Pe

(b/Hz)B T

(bps/Hz) ρ1logC

K

1kk

K

1kk2

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Discussion on optimal rate allocation

• Possible limits on constellation sizes!

• Constellation sizes are quantized!!!

• The answer is different for different target probabilities of error

• Optimal power AND rate allocation schemes possible, but complex

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Distributed antenna systems

• Idea: put your antennas in different places

• +: lower correlation

- : power imbalance, synchronization, coordination

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Practical considerations

• Coding

• Detection algorithms

• Channel estimation

• Interference

100

Detection algorithms• Maximum likelihood linear detector

y = H x + n xest = H+yH+ = (HH H)-1 HH : Pseudo inverse of H

• Problem: find nearest neighbor among QM points (Q: constellation size, M: number of transmitters)

• VERY high complexity!!!

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Solution: BLAST algorithm

• BLAST: Bell Labs lAyered Space Time

• Idea: NON-LINEAR DETECTOR– Step 1: H+ = (HH H)-1 HH

– Step 2: Find the strongest signal(Strongest = the one with the highest post detection SNR)

– Step 3: Detect it (Nearest neighbor among Q)– Step 4: Subtract it– Step 5: if not all yet detected, go to step 2

102

Discussion on the BLAST algorithm• It’s a non-linear detector!!!

• Two flavors– V-BLAST (easier)– D-BLAST (introduces space-time coding)

• Achieves 50-60% of Shannon capacity

• Error propagation possible • Very complicated for wideband case

103

Coding limitations• Capacity = Maximum achievable data rate that can be achieved over the channel with arbitrarily low probability of error

• SISO case: – Constellation limitations– Turbo- coding can get you close to Shannon!!!

• MIMO case:– Constellation limitations as well– Higher complexity– Space-time codes: very few!!!!

104

Channel estimation

• The channel is not perfectly estimated because– it is changing (environment, user movement)– there is noise DURING the estimation

• An error in the channel transfer characteristics can hurt you– in the decoding – in the water-filling

• Trade-off: Throughput vs. Estimation accuracy• What if interference (as noise) is not white????

105

Interference

• Generalization of other/ same cell interference for SISO case

• Example: cellular deployment of MIMO systems• Interference level depends on

– frequency/ code re-use scheme– cell size– uplink/ downlink perspective– deployment geometry– propagation conditions– antenna types

106

Summary and conclusions• MIMO systems are a promising technique for high

data rates

• Their efficiency depends on the channel between the transmitters and the receivers (power and correlation)

• Practical issues need to be resolved

• Open research questions need to be answered