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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp. 13742-13754

© Research India Publications. http://www.ripublication.com

the Future

Electrical & Information Engineering Department, Covenant University, Ota Ogun State Nigeria.

*Orcid: 0000-0002-7958-1121

One of the technologies aimed to provide large increase in

data rate, enhanced spectral efficiency, transmit power

efficiency, high sum rates, and increase link reliability for the

fifth generation network (5G) is the massive multiple input

multiple output (MIMO) antenna system. The projected

benefits of massive MIMO depend on the propagation

environment. However, due to the non wide-sense stationarity

properties of massive MIMO, small scale characterization

(SSC) is not enough for modeling its propagation channel as

the spatial domain is also required. Giving consideration to

the dynamic adaptation of the elevation angles which is not

captured in 2D channel models will open up new possibilities

for 3D beamforming which will introduce considerable

performance gains for 5G network capacity enhancement. In

this paper therefore, we review the various non wide-sense

stationary channel parameters for characterizing massive

MIMO channel particularly in the 3D plane and their methods

of measurement, All through the discussion, we identified

outstanding research challenges in these areas and their future

directions.

Channel model, massive MIMO.

COST 2100: Cooperation in Science and Technology

2100

MF: Matched Filter

MIMO: Multiple-Input Multiple-Output

SAGE: Space-Alternating Generalized Expectation-

SSC: Small Scale Characterization

TDD: Time Division Duplexing

VNA: Vector Network Analyzer

WSS: Wide-Sense stationary

scatterer

In large scale or massive MU-MIMO, tens or hundreds of

antennas at the BS concurrently serve fewer numbers of users

in the same time-frequency resource. According to [1, 2], as

the number of BS antennas in an massive MIMO approaches a

very large number, the effect of uncorrelated noise, small-

scale fading and the required transmitted energy/bit tends to

zero, thermal noise is averaged out and the system is largely

restricted by interfering symbols from other transmitters.

Also, simple linear signal processing approaches like the zero

forcing (ZF) and the matched filter (MF) pre-coding/detection

can be used to achieve these advantages. Other advantages or

benefits of massive MIMO include reduced latency, large

spectral efficiencies, simplification of the MAC layer, the use

of low powered inexpensive components, cell edge

performance improvement as well as robustness to intentional

jamming [3]. The performance of MIMO systems generally

depend on the propagation environment and the properties of

the antenna arrays. [4] According to [5, 6], it is therefore

essential for massive MIMO operation to obtain channel state

information (CSI) at the BS for full achievement of its

benefits. In spite of its many advantages which have presented

very encouraging unique feat based on theoretical studies,

many questions are yet to be answered in the practical

application of massive MIMO. In theoretical massive MIMO

with transmit antennas NT approaching infinity, independent

and identically distributed (i.i.d) Rayleigh channels are always

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp. 13742-13754

© Research India Publications. http://www.ripublication.com

transmission with optimal performance using linear pre-

coding and near optimal detection schemes as earlier

enumerated. However, the number of antennas cannot tend to

infinity neither is propagation channels hardly i.i.d Rayleigh

in real propagation environment [5, 7, 8]. Thus we need to

carry out real propagation environment channel measurement

to ascertain what can be harness from massive MIMO

practically while massive MIMO channel characterization for

high speed train, crowded scenarios and hotspot environment

are highly sort research areas.

Unlike in conventional MIMO, massive MIMO antenna array

are arranged in a large spatial format making the small scale

characterization (SSC) assumptions inapplicable, as a result of

this, the propagation channel parameters such as the Azimuth

angle of arrival (AAoA), Azimuth angle of departure (AAoD),

birth/death processes of multipath clusters etc as observed by

the different antenna elements making the array fluctuate due

to their spatial displacement leading to the non-stationarity

property of massive MIMO channels [9]. Massive MIMO

channel non-stationarity properties have been investigated in

the literature. For example in [10, 11, 12] where

measurements at 2-8GHz, 1GHz and 5.6GHz frequency bands

were performed using the virtual linear array and the virtual

2D array for both LOS and NLOS scenarios using antenna

array up to 128 elements where parameters such as the rms

delay spread, cluster number, power delay profile and the k-

factor were studied.

According to [9, 13, 14, 15] the non-stationarity property of

massive MIMO arises as a result of the smaller than Rayleigh

distance connecting some clusters and the antenna array when

the number of antenna is large leading to the non applicability

of the far-field propagation assumption, therefore the

wavefront that is expected to be plane become spherical

causing variations in AAoAs, AAoDs etc of multipath signals

along the array. Also the closer two antennas in an array is the

more common clusters they share [9] given rise to two sets of

clusters, those wholly visible to the entire array and those that

are visible to a part of the array elements only as a result of

their shapes, direction, sizes etc otherwise called partially

visible clusters. According to the work in [14] massive MIMO

channel capacity is increased by the non-stationarity

behaviour of partially visible clusters eliciting interest in

research work in spatially non-stationarity of massive MIMO.

Channel propagation models are employed to predict the radio

signal propagation characteristics within the wireless

environment of a particular geographical location for efficient

network planning, coverage and deployment. The use of

suitable channel propagation models is critical not only for the

performance assessment of diverse candidate 5G technologies,

but also for the advancement of new algorithms and products

exploiting the large scale antenna system [12]. These models

which could be deterministic, stochastic or empirical channel

models are required to be only as complex as necessary and

can thus neglect propagation effects that do not have

considerable impact on the system performance [8] therefore

each model is limited to the parameters that characterize them.

Wide-sense stationarity and uncorrelated scattering (WSSUS)

stochastic process, where -WSS- means that the mean and

variance of the distribution are independent of time and -US-

means that the path gains resulting from various delays are

uncorrelated, and based on these assumptions we characterize

the channel by second order statistics with the channel

statistics believed to be stationary (does not change) in time

and frequency within a specific period during which the

statistics can be used for channel estimation, data detection etc

purposes [16]. According to [17], WSSUS are not sufficient

for MIMO channels anymore as the spatial domain

characterization is also required particularly for massive

MIMO where we have the effect of non-stationarity property

playing out. Such effects include variation in the directions of

arrival of signals at different parts of the array as well as

variation in the average received energy at each antenna [14].

Since we have the phenomenon of near-field and non-WSS

effects in massive MIMO as against conventional MIMO,

therefore WINNER II and COST 2100 which are state of the

art MIMO channel models and other such MIMO channels

models [8, 18] which fail to capture these features are

unsuitable for direct use as massive MIMO channel models

[19]. Again, massive MIMO is expected to play a key role in

the architecture of the new 5G network, including network

backhauling in ultra-dense heterogeneous network

environment as well as 3D beamforming for users in elevated

high-rise positions. This will require characterization of

multipath channel parameters in the elevation plane as against

the current 2D model where characterization were done in the

Azimuth plane. In the establishment of 2D models, estimates

of the channel multipath component (MPC) parameters are

extracted with the aid of parameter estimation algorithms such

as the Space-Alternating Generalized Expectation-

maximization (SAGE) procedure and the RiMAX technique

[9] where the constant channel or stationary channel is

assumed in the Azimuth plane. Since it is not possible to use

these channel models in the large scale antenna array MIMO

scenarios with satisfactory accuracy, channel sounding for

massive MIMO channel characterization is therefore required.

The remainder of this paper is structured as follows. In section

II we review channel characteristics of massive MIMO

system, while section III investigate various measurable non-

WSS channel parameters and their evaluation metrics as well

as the challenges and future direction for MIMO channel

measurements, while in section IV various MIMO channel

models are studied, including extension to 3D models.

Challenges and future direction for massive MIMO channel

modeling were also considered. Finally we conclude the paper

in section V.

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp. 13742-13754

© Research India Publications. http://www.ripublication.com

Hardening

is assumed where ''favorable'' propagation is interpreted as a

sufficiently complex scattering environment such that as the

BS antennas increases, user channels become pair wise

orthogonal which is as a result of the asymptotic of random

matrix theory setting in with many consequences as effects

that were random previously, begins to appear deterministic

such as, the allocation of the singular values of the

propagation channel matrix which tends towards a

deterministic function. Another important condition is the

channel hardening phenomenon where tall/wide matrices

begin to be very well conditioned. As the antenna array size

become large, some matrix operations such as inversions can

be achieve faster, by the use of series expansion methods (this

is what makes linear algorithms like ZF and MMSE which

requires matrix inversion operation to be near optimal in

performance) [4].

expressed mathematically as:

and reduce the effect of small scale fading as M → ∞

(hi Hhj) = {0, i, j = 1, 2….., k, and i j} and

(hi Hhi) = {||hk||2 ≠ 0, k = 1, 2,…..K}

We also say that the channel offers asymptotically favorable

propagation if

where k ≠ j and M is the BS transmit antenna.

Channel hardening; in which the off-diagonal components of

the channel gain matrix become progressively more weaker

with respect to the diagonal components as the size of the

channel gain matrix increases [20] that is as the transmitter

antenna size increases making matrix inversion operation

simpler and the use of linear detector optimal.

(1/M)||hk||2 = (1/M)tr(R) → 0

M → ∞

are key properties of the radio channel exploited in achieving

Massive MIMO benefits.

Field Effect

According to [14], the complex random sequence hk is wide

sense stationary (WSS) if its expectation E[hk] is a constant

that does not depend on k, neither the covariance ρkl =

E[h* khl] depend on the values of k and l but exclusively on

k−l, otherwise it is not WSS. The difference between Massive

MIMO channels and the conventional MIMO channels is that

the massive MIMO antennas are widely distributed in a large

spatial region that makes the small scale characterization

(SSC) assumptions inapplicable. The SSC is based on the

wide-sense stationarity and uncorrelated scatterers for

characterizing radio channels where the channel statistics are

believed to be stationary in time and frequency within a

coherent period. Resulting from the above, the propagation

paths parameters observed through various antennas in the

massive MIMO array fluctuate due to the spatial displacement

of these antennas, here various base station antennas detect

diverse groups of clusters at dissimilar time slots, which is

described as the cluster birth and death process [12]. The

channel exhibit spatial non-stationarity [9], see figure 1 below

where cluster 1 is visible to the upper antenna array elements

while cluster 4 and 5 are visible to the lower last antenna

element. It is therefore necessary to determine and estimate

the non-WSS channel parameters and investigate their

influence on the performance of massive MIMO.

Figure 1: Near-field effect and the non-WSS

phenomenon [12]

Again, as the number of antenna array increases to a large

figure with several antenna elements, the space between the

transmitter, receiver and or a cluster can become less than the

Rayleigh distance given as 2D2/λ (where D is the antenna

array dimension and λ is the carrier wavelength), and the far-

field and plane wavefront assumptions for SSC no longer

holds for massive MIMO. See figure 2 below.

Figure 2: Near Field effect and the Plane-wave Assumption

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp. 13742-13754

© Research India Publications. http://www.ripublication.com

favorable propagation condition, the non-stationarity property

of massive MIMO channel can be measured by a MIMO

correlation matrix based metric. Called the correlation matrix

distance (CMD), it estimate the distance between the

correlation matrices measured at various times to describe

how strong the spatial formation of the channel has changed

[21, 22]. This was used to measure and demonstrate the

variation in the direction of arrival (DoA) in [21]. With a

value ranging from 0 to 1, the CMD was used in [22] to

investigate the non-WSS property of the channel gain while

the non-WSSUS for vehicular channel was characterized in

[23] at the speed of 90km/hr using 5GHz center frequency and

240MHz bandwidth. [24] highlighted some challenges with

the use of CMD as a measuring metric for non-stationarity in

MIMO system and proposed two new metrics called the

normalized correlation matrix distance (NCMD) and the

distance between equi-dimentional subspaces (DES)

algorithm.

carried out to describe or characterize the physical properties

of a wireless channel where the measured data is collected

using an equipment called channel sounder. In channel

sounding, electromagnetic waves are transmitted to excite (or

sound) the channel and the channel output are recorded at the

receiver. Different sounding methods are used depending on

whether the channel of interest is narrowband or wideband,

SISO or MIMO channels. In the case of MIMO channels, the

channel impulse responses (CIRs) between all combinations

of the transmit and receive antenna branches are recorded.

Here three different types of array architectures can be used

which are : a) real-array, where each antenna element has its

own Radio Frequency (RF) chain such that they can transmit

or receive concurrently. However, the main difficulties here

are the cost and calibration procedure which are expensive

and complex, b) switched array architecture, where there is

only one RF chain for all transmit and receive branches.

Therefore only one antenna transmit and only one receive at a

time. This architecture has a number of advantages including

low cost and low complexity. Also, antenna arrays of any size

can be used at both link ends, where the maximum size of the

array is a function of coherence bandwidth and the speed of

the RF switch [25], finally c) virtual arrays, where there is but

one antenna element connected to a single RF chain at both

link end, such that the antennas are electronically moved to

predefined locations and the channel is thus sounded one after

the other for each location. The main disadvantage of this

architecture is that it allows very limited temporal variations

in the channel. From above, we see that the switched array

architecture is frequently the most suitable one for MIMO

measurements in fast fluctuating, time-variant channels.

Determining the statistical properties of the channel require

that sounding be done either in the time realm or in the

frequency realm. The time-variant channel impulse response

(CIR) h(t,τ) for the time realm/domain measurements are

obtained at the receiver by exciting the channel with

intermittent pulses on a PN-sequence at the transmitter. In the

case of the frequency realm measurements, the time-variant

channel transfer function H(t,f) can be obtained through

sounding the channel with chirp-like multi-tone signals. The

channel sounding of the time-invariant and band-limited

channels can be done as long as the channel is sampled at

least at the Nyquist rate. However, for the channel sounding

of time-variant channels, it must to be ascertained that the

channel fulfills a two-dimensional Nyquist criterion [25].

All channel sounders measures (, ) or its equivalent. For

multiple antenna systems, the channel impulse response of the

radio channel from each of the transmit antenna elements to

each of the receive antenna elements is represented as:

, = ( ()

− (1)

− (1)

< > denote the inner product

Is the location of the transmitter

Is the location of the receiver

Is the Direction of Departure (DoD) containing both the

Azimuth and the Elevation angles

Ψ Is the direction of Arrival (DoA) containing both the

azimuth and the Elevation angle

τ Is the delay

In determining the degree of favorable propagation of a

channel, the channel condition number is used. This

evaluation metric is the singular value spreads of channel

matrices, where on performing singular value decomposition

(SVD) of the K×M normalized channel matrix denoted by H,

we have

H = UΣVH

where U and V are unitary matrices that contains the left and

right singular vectors, we obtain the singular values σ1,σ2,...,σk

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp. 13742-13754

© Research India Publications. http://www.ripublication.com

13746

on the diagonal of the matrix Σ. The singular value spread is

the ratio

κ contains information about how orthogonal the user channel

vectors are and when κ = 1, all user vectors are orthogonal to

each other. In this case, all users can be served simultaneously

without inter-user interference. The value of κ gets large when

user orthogonality is poor. If κ→∞, it means that some user

vectors are aligned [8]. The condition number of the channel

can be used to investigate the MIMO channel capacity under

various circumstances and to explore the MIMO beamforming

performance. When compared with correlation coefficient

[21, 22], a metric used for evaluating the orthogonality of

channel vectors of two users, the channel condition number is

better suitable in reflecting the channel harden phenomenon

and the orthogonality of multi-user channel vectors. [12].

B) Distance from Favorable Propagation

According to Erik Larsson and Thomas Marzetta in [26], the

channel condition number is not good enough as a metric for

favorable propagation condition whenever the various channel

vector's norms are not equal, a situation that plays out in

practice when the UEs have different locations. In [4, 8, 27]

favorable propagation in massive MIMO was discussed,

where the channel matrix condition number was used as a

metric for measuring how favorable the channel is. The

channels in those papers were considered only as i.i.d.

Rayleigh fading. However, in practice, due to the situations

where the UEs have different locations, [26] says the norms of

the channels are not identical and as such the condition

number is not a good metric for whether or not we have

favorable propagation, rather it proposed the “distance from

favorable propagation” measure, (Δc), explaining it as the

relative difference between the sum-capacity and the utmost

capacity achieved under favorable propagation condition.

In the uplink of a single cell central antenna system, where K

single antenna terminals simultaneously and independently

transmit data to the base station having M antennas, figure 3

below. If the terminals transmit K symbols x1, x2…….xk

where E[|xk|2] = 1, then the M x 1 received vector at the BS is

written as;

= √ +

Where x = [x1, x2, …….xk]T and G = [g1, g2,…….gk] is our

channel vector linking the kth terminal and the base station. n

is the i.i.d ~(0, 1) random variable noise vector and is

the normalized transmit signal to noise ratio (SNR). Here gk

include the effects of large-scale fading and small-scale fading

i.e. , k and

= √k where k = 1, 2…., K and m =

1, 2…., M

The sum capacity of the system with channel state information

at the base station is given by;

C = log2 (1 + )

Using Hadamard inequality.

C = log2|1+| ≤ log2( ∏ |1 + =1 |k,k)

= ∑ log2 =1 (|1+|k,k) .…1

= ∑ log2 =1 (1+…

© Research India Publications. http://www.ripublication.com

the Future

Electrical & Information Engineering Department, Covenant University, Ota Ogun State Nigeria.

*Orcid: 0000-0002-7958-1121

One of the technologies aimed to provide large increase in

data rate, enhanced spectral efficiency, transmit power

efficiency, high sum rates, and increase link reliability for the

fifth generation network (5G) is the massive multiple input

multiple output (MIMO) antenna system. The projected

benefits of massive MIMO depend on the propagation

environment. However, due to the non wide-sense stationarity

properties of massive MIMO, small scale characterization

(SSC) is not enough for modeling its propagation channel as

the spatial domain is also required. Giving consideration to

the dynamic adaptation of the elevation angles which is not

captured in 2D channel models will open up new possibilities

for 3D beamforming which will introduce considerable

performance gains for 5G network capacity enhancement. In

this paper therefore, we review the various non wide-sense

stationary channel parameters for characterizing massive

MIMO channel particularly in the 3D plane and their methods

of measurement, All through the discussion, we identified

outstanding research challenges in these areas and their future

directions.

Channel model, massive MIMO.

COST 2100: Cooperation in Science and Technology

2100

MF: Matched Filter

MIMO: Multiple-Input Multiple-Output

SAGE: Space-Alternating Generalized Expectation-

SSC: Small Scale Characterization

TDD: Time Division Duplexing

VNA: Vector Network Analyzer

WSS: Wide-Sense stationary

scatterer

In large scale or massive MU-MIMO, tens or hundreds of

antennas at the BS concurrently serve fewer numbers of users

in the same time-frequency resource. According to [1, 2], as

the number of BS antennas in an massive MIMO approaches a

very large number, the effect of uncorrelated noise, small-

scale fading and the required transmitted energy/bit tends to

zero, thermal noise is averaged out and the system is largely

restricted by interfering symbols from other transmitters.

Also, simple linear signal processing approaches like the zero

forcing (ZF) and the matched filter (MF) pre-coding/detection

can be used to achieve these advantages. Other advantages or

benefits of massive MIMO include reduced latency, large

spectral efficiencies, simplification of the MAC layer, the use

of low powered inexpensive components, cell edge

performance improvement as well as robustness to intentional

jamming [3]. The performance of MIMO systems generally

depend on the propagation environment and the properties of

the antenna arrays. [4] According to [5, 6], it is therefore

essential for massive MIMO operation to obtain channel state

information (CSI) at the BS for full achievement of its

benefits. In spite of its many advantages which have presented

very encouraging unique feat based on theoretical studies,

many questions are yet to be answered in the practical

application of massive MIMO. In theoretical massive MIMO

with transmit antennas NT approaching infinity, independent

and identically distributed (i.i.d) Rayleigh channels are always

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp. 13742-13754

© Research India Publications. http://www.ripublication.com

transmission with optimal performance using linear pre-

coding and near optimal detection schemes as earlier

enumerated. However, the number of antennas cannot tend to

infinity neither is propagation channels hardly i.i.d Rayleigh

in real propagation environment [5, 7, 8]. Thus we need to

carry out real propagation environment channel measurement

to ascertain what can be harness from massive MIMO

practically while massive MIMO channel characterization for

high speed train, crowded scenarios and hotspot environment

are highly sort research areas.

Unlike in conventional MIMO, massive MIMO antenna array

are arranged in a large spatial format making the small scale

characterization (SSC) assumptions inapplicable, as a result of

this, the propagation channel parameters such as the Azimuth

angle of arrival (AAoA), Azimuth angle of departure (AAoD),

birth/death processes of multipath clusters etc as observed by

the different antenna elements making the array fluctuate due

to their spatial displacement leading to the non-stationarity

property of massive MIMO channels [9]. Massive MIMO

channel non-stationarity properties have been investigated in

the literature. For example in [10, 11, 12] where

measurements at 2-8GHz, 1GHz and 5.6GHz frequency bands

were performed using the virtual linear array and the virtual

2D array for both LOS and NLOS scenarios using antenna

array up to 128 elements where parameters such as the rms

delay spread, cluster number, power delay profile and the k-

factor were studied.

According to [9, 13, 14, 15] the non-stationarity property of

massive MIMO arises as a result of the smaller than Rayleigh

distance connecting some clusters and the antenna array when

the number of antenna is large leading to the non applicability

of the far-field propagation assumption, therefore the

wavefront that is expected to be plane become spherical

causing variations in AAoAs, AAoDs etc of multipath signals

along the array. Also the closer two antennas in an array is the

more common clusters they share [9] given rise to two sets of

clusters, those wholly visible to the entire array and those that

are visible to a part of the array elements only as a result of

their shapes, direction, sizes etc otherwise called partially

visible clusters. According to the work in [14] massive MIMO

channel capacity is increased by the non-stationarity

behaviour of partially visible clusters eliciting interest in

research work in spatially non-stationarity of massive MIMO.

Channel propagation models are employed to predict the radio

signal propagation characteristics within the wireless

environment of a particular geographical location for efficient

network planning, coverage and deployment. The use of

suitable channel propagation models is critical not only for the

performance assessment of diverse candidate 5G technologies,

but also for the advancement of new algorithms and products

exploiting the large scale antenna system [12]. These models

which could be deterministic, stochastic or empirical channel

models are required to be only as complex as necessary and

can thus neglect propagation effects that do not have

considerable impact on the system performance [8] therefore

each model is limited to the parameters that characterize them.

Wide-sense stationarity and uncorrelated scattering (WSSUS)

stochastic process, where -WSS- means that the mean and

variance of the distribution are independent of time and -US-

means that the path gains resulting from various delays are

uncorrelated, and based on these assumptions we characterize

the channel by second order statistics with the channel

statistics believed to be stationary (does not change) in time

and frequency within a specific period during which the

statistics can be used for channel estimation, data detection etc

purposes [16]. According to [17], WSSUS are not sufficient

for MIMO channels anymore as the spatial domain

characterization is also required particularly for massive

MIMO where we have the effect of non-stationarity property

playing out. Such effects include variation in the directions of

arrival of signals at different parts of the array as well as

variation in the average received energy at each antenna [14].

Since we have the phenomenon of near-field and non-WSS

effects in massive MIMO as against conventional MIMO,

therefore WINNER II and COST 2100 which are state of the

art MIMO channel models and other such MIMO channels

models [8, 18] which fail to capture these features are

unsuitable for direct use as massive MIMO channel models

[19]. Again, massive MIMO is expected to play a key role in

the architecture of the new 5G network, including network

backhauling in ultra-dense heterogeneous network

environment as well as 3D beamforming for users in elevated

high-rise positions. This will require characterization of

multipath channel parameters in the elevation plane as against

the current 2D model where characterization were done in the

Azimuth plane. In the establishment of 2D models, estimates

of the channel multipath component (MPC) parameters are

extracted with the aid of parameter estimation algorithms such

as the Space-Alternating Generalized Expectation-

maximization (SAGE) procedure and the RiMAX technique

[9] where the constant channel or stationary channel is

assumed in the Azimuth plane. Since it is not possible to use

these channel models in the large scale antenna array MIMO

scenarios with satisfactory accuracy, channel sounding for

massive MIMO channel characterization is therefore required.

The remainder of this paper is structured as follows. In section

II we review channel characteristics of massive MIMO

system, while section III investigate various measurable non-

WSS channel parameters and their evaluation metrics as well

as the challenges and future direction for MIMO channel

measurements, while in section IV various MIMO channel

models are studied, including extension to 3D models.

Challenges and future direction for massive MIMO channel

modeling were also considered. Finally we conclude the paper

in section V.

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp. 13742-13754

© Research India Publications. http://www.ripublication.com

Hardening

is assumed where ''favorable'' propagation is interpreted as a

sufficiently complex scattering environment such that as the

BS antennas increases, user channels become pair wise

orthogonal which is as a result of the asymptotic of random

matrix theory setting in with many consequences as effects

that were random previously, begins to appear deterministic

such as, the allocation of the singular values of the

propagation channel matrix which tends towards a

deterministic function. Another important condition is the

channel hardening phenomenon where tall/wide matrices

begin to be very well conditioned. As the antenna array size

become large, some matrix operations such as inversions can

be achieve faster, by the use of series expansion methods (this

is what makes linear algorithms like ZF and MMSE which

requires matrix inversion operation to be near optimal in

performance) [4].

expressed mathematically as:

and reduce the effect of small scale fading as M → ∞

(hi Hhj) = {0, i, j = 1, 2….., k, and i j} and

(hi Hhi) = {||hk||2 ≠ 0, k = 1, 2,…..K}

We also say that the channel offers asymptotically favorable

propagation if

where k ≠ j and M is the BS transmit antenna.

Channel hardening; in which the off-diagonal components of

the channel gain matrix become progressively more weaker

with respect to the diagonal components as the size of the

channel gain matrix increases [20] that is as the transmitter

antenna size increases making matrix inversion operation

simpler and the use of linear detector optimal.

(1/M)||hk||2 = (1/M)tr(R) → 0

M → ∞

are key properties of the radio channel exploited in achieving

Massive MIMO benefits.

Field Effect

According to [14], the complex random sequence hk is wide

sense stationary (WSS) if its expectation E[hk] is a constant

that does not depend on k, neither the covariance ρkl =

E[h* khl] depend on the values of k and l but exclusively on

k−l, otherwise it is not WSS. The difference between Massive

MIMO channels and the conventional MIMO channels is that

the massive MIMO antennas are widely distributed in a large

spatial region that makes the small scale characterization

(SSC) assumptions inapplicable. The SSC is based on the

wide-sense stationarity and uncorrelated scatterers for

characterizing radio channels where the channel statistics are

believed to be stationary in time and frequency within a

coherent period. Resulting from the above, the propagation

paths parameters observed through various antennas in the

massive MIMO array fluctuate due to the spatial displacement

of these antennas, here various base station antennas detect

diverse groups of clusters at dissimilar time slots, which is

described as the cluster birth and death process [12]. The

channel exhibit spatial non-stationarity [9], see figure 1 below

where cluster 1 is visible to the upper antenna array elements

while cluster 4 and 5 are visible to the lower last antenna

element. It is therefore necessary to determine and estimate

the non-WSS channel parameters and investigate their

influence on the performance of massive MIMO.

Figure 1: Near-field effect and the non-WSS

phenomenon [12]

Again, as the number of antenna array increases to a large

figure with several antenna elements, the space between the

transmitter, receiver and or a cluster can become less than the

Rayleigh distance given as 2D2/λ (where D is the antenna

array dimension and λ is the carrier wavelength), and the far-

field and plane wavefront assumptions for SSC no longer

holds for massive MIMO. See figure 2 below.

Figure 2: Near Field effect and the Plane-wave Assumption

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp. 13742-13754

© Research India Publications. http://www.ripublication.com

favorable propagation condition, the non-stationarity property

of massive MIMO channel can be measured by a MIMO

correlation matrix based metric. Called the correlation matrix

distance (CMD), it estimate the distance between the

correlation matrices measured at various times to describe

how strong the spatial formation of the channel has changed

[21, 22]. This was used to measure and demonstrate the

variation in the direction of arrival (DoA) in [21]. With a

value ranging from 0 to 1, the CMD was used in [22] to

investigate the non-WSS property of the channel gain while

the non-WSSUS for vehicular channel was characterized in

[23] at the speed of 90km/hr using 5GHz center frequency and

240MHz bandwidth. [24] highlighted some challenges with

the use of CMD as a measuring metric for non-stationarity in

MIMO system and proposed two new metrics called the

normalized correlation matrix distance (NCMD) and the

distance between equi-dimentional subspaces (DES)

algorithm.

carried out to describe or characterize the physical properties

of a wireless channel where the measured data is collected

using an equipment called channel sounder. In channel

sounding, electromagnetic waves are transmitted to excite (or

sound) the channel and the channel output are recorded at the

receiver. Different sounding methods are used depending on

whether the channel of interest is narrowband or wideband,

SISO or MIMO channels. In the case of MIMO channels, the

channel impulse responses (CIRs) between all combinations

of the transmit and receive antenna branches are recorded.

Here three different types of array architectures can be used

which are : a) real-array, where each antenna element has its

own Radio Frequency (RF) chain such that they can transmit

or receive concurrently. However, the main difficulties here

are the cost and calibration procedure which are expensive

and complex, b) switched array architecture, where there is

only one RF chain for all transmit and receive branches.

Therefore only one antenna transmit and only one receive at a

time. This architecture has a number of advantages including

low cost and low complexity. Also, antenna arrays of any size

can be used at both link ends, where the maximum size of the

array is a function of coherence bandwidth and the speed of

the RF switch [25], finally c) virtual arrays, where there is but

one antenna element connected to a single RF chain at both

link end, such that the antennas are electronically moved to

predefined locations and the channel is thus sounded one after

the other for each location. The main disadvantage of this

architecture is that it allows very limited temporal variations

in the channel. From above, we see that the switched array

architecture is frequently the most suitable one for MIMO

measurements in fast fluctuating, time-variant channels.

Determining the statistical properties of the channel require

that sounding be done either in the time realm or in the

frequency realm. The time-variant channel impulse response

(CIR) h(t,τ) for the time realm/domain measurements are

obtained at the receiver by exciting the channel with

intermittent pulses on a PN-sequence at the transmitter. In the

case of the frequency realm measurements, the time-variant

channel transfer function H(t,f) can be obtained through

sounding the channel with chirp-like multi-tone signals. The

channel sounding of the time-invariant and band-limited

channels can be done as long as the channel is sampled at

least at the Nyquist rate. However, for the channel sounding

of time-variant channels, it must to be ascertained that the

channel fulfills a two-dimensional Nyquist criterion [25].

All channel sounders measures (, ) or its equivalent. For

multiple antenna systems, the channel impulse response of the

radio channel from each of the transmit antenna elements to

each of the receive antenna elements is represented as:

, = ( ()

− (1)

− (1)

< > denote the inner product

Is the location of the transmitter

Is the location of the receiver

Is the Direction of Departure (DoD) containing both the

Azimuth and the Elevation angles

Ψ Is the direction of Arrival (DoA) containing both the

azimuth and the Elevation angle

τ Is the delay

In determining the degree of favorable propagation of a

channel, the channel condition number is used. This

evaluation metric is the singular value spreads of channel

matrices, where on performing singular value decomposition

(SVD) of the K×M normalized channel matrix denoted by H,

we have

H = UΣVH

where U and V are unitary matrices that contains the left and

right singular vectors, we obtain the singular values σ1,σ2,...,σk

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 12, Number 23 (2017) pp. 13742-13754

© Research India Publications. http://www.ripublication.com

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on the diagonal of the matrix Σ. The singular value spread is

the ratio

κ contains information about how orthogonal the user channel

vectors are and when κ = 1, all user vectors are orthogonal to

each other. In this case, all users can be served simultaneously

without inter-user interference. The value of κ gets large when

user orthogonality is poor. If κ→∞, it means that some user

vectors are aligned [8]. The condition number of the channel

can be used to investigate the MIMO channel capacity under

various circumstances and to explore the MIMO beamforming

performance. When compared with correlation coefficient

[21, 22], a metric used for evaluating the orthogonality of

channel vectors of two users, the channel condition number is

better suitable in reflecting the channel harden phenomenon

and the orthogonality of multi-user channel vectors. [12].

B) Distance from Favorable Propagation

According to Erik Larsson and Thomas Marzetta in [26], the

channel condition number is not good enough as a metric for

favorable propagation condition whenever the various channel

vector's norms are not equal, a situation that plays out in

practice when the UEs have different locations. In [4, 8, 27]

favorable propagation in massive MIMO was discussed,

where the channel matrix condition number was used as a

metric for measuring how favorable the channel is. The

channels in those papers were considered only as i.i.d.

Rayleigh fading. However, in practice, due to the situations

where the UEs have different locations, [26] says the norms of

the channels are not identical and as such the condition

number is not a good metric for whether or not we have

favorable propagation, rather it proposed the “distance from

favorable propagation” measure, (Δc), explaining it as the

relative difference between the sum-capacity and the utmost

capacity achieved under favorable propagation condition.

In the uplink of a single cell central antenna system, where K

single antenna terminals simultaneously and independently

transmit data to the base station having M antennas, figure 3

below. If the terminals transmit K symbols x1, x2…….xk

where E[|xk|2] = 1, then the M x 1 received vector at the BS is

written as;

= √ +

Where x = [x1, x2, …….xk]T and G = [g1, g2,…….gk] is our

channel vector linking the kth terminal and the base station. n

is the i.i.d ~(0, 1) random variable noise vector and is

the normalized transmit signal to noise ratio (SNR). Here gk

include the effects of large-scale fading and small-scale fading

i.e. , k and

= √k where k = 1, 2…., K and m =

1, 2…., M

The sum capacity of the system with channel state information

at the base station is given by;

C = log2 (1 + )

Using Hadamard inequality.

C = log2|1+| ≤ log2( ∏ |1 + =1 |k,k)

= ∑ log2 =1 (|1+|k,k) .…1

= ∑ log2 =1 (1+…