MIMO Channel Modelling

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Keysight TechnologiesMIMO Channel Modeling and

Emulation Test Challenges

Application Note


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Introuction ...............................................................................................................3

Rviwing MIMO Tchnologis ............................................................................ 4

Multipl antnna tchniqus .............................................................................. 5

MIMO in wirlss stanars.............................................................................12

Channl corrlation ffcts on MIMO prformanc ....................................13

Challngs in mulating MIMO channls ......................................................14

MIMO Channl Ovrviw .....................................................................................16

Wirlss propagation charactristics ..............................................................17

Macroscopic (slow) faing ................................................................................18

MIMO Channl Corrlation ..................................................................................35

Spatial corrlation ............................................................................................... 35

Antnna polarization corrlation ......................................................................37

Combin spatial an antnna polarization corrlation .............................. 40

Pr-path corrlation vrsus pr-channl corrlation ....................................44

Thortical MIMO channl capacity ...............................................................45

Configuring th channl mulator to achiv th sir corrlation .....46

Applying SNR to MIMO channls ....................................................................48

Configuring Stanar-Compliant MIMO Channls using th PXB ............... 52

Rlat Litratur ..................................................................................................54

Appnix A: Thortical Mol for MIMO Channl Capacity .......................55

Appnix B: SNR for Uncorrlat an Corrlat MIMO Channls .............58

Table of Contents

This application not bgins with a rviw of MIMO tchnologis an

th basic proprtis of wirlss channls an gos on to introuc th

concpts of spatial corrlation an its ffcts on MIMO prformanc. It

also inclus a monstration of moling th spatial charactristics of

MIMO channls an describes how these complex channels can be emulated

using commrcially availabl instrumntation such as th Kysight

Tchnologis, Inc. N5106A PXB basban gnrator an signal mulator.


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Multipl Input Multipl Output (MIMO) tchnology hols th promis of highr

ata rats with incras spctral fficincy. Du to th potntial improvmnt

in systm prformanc an avancs in igital signal procssing, many wir-

lss systms, incluing th IEEE 802.11n wirlss LAN, IEEE 802.16-bas

Mobil WiMAX Wav 2 an th Long-Trm Evolution (LTE) mobil wirlss

systm, hav rcntly aopt th us of MIMO an multipl antnna tchnol-

ogis. All of ths commrcial wirlss systms oprat in high multipathnvironmnts an it is th bnfit of multipath that provis th prformanc

improvmnt whn using multipl antnna configurations.

Whil MIMO offrs th potntial for incras signal robustnss an capacity

improvmnt whn oprating in rich multipath nvironmnts, vloping an

tsting MIMO componnts an systms rquirs avanc channl mulation

tools that ar asily configur an provi an accurat rprsntation of

ralistic wirlss channls an conitions.

This application not bgins with a rviw of MIMO tchnologis an th

basic proprtis of wirlss channls an gos on to introuc th concpts

of spatial corrlation an its ffcts on MIMO prformanc. It also inclus a

monstration of moling th spatial charactristics of MIMO channls anscribs how ths complx channls can b mulat using commrcially

availabl instrumntation such as th Kysight N5106A PXB basban gnra-

tor an signal mulator which will b rfrr to throughout th rmainr of

this ocumnt as th PXB.

Introduction


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Multipl antnnas plac at th transmittr an/or rcivr in wirlss

communication systms can b us to substantially improv systm prfor-

manc by lvraging th spatial charactristics of th wirlss channl.

Ths systms, now wily trm as Multipl Input Multipl Output (MIMO),

rquir two or mor antnnas plac at th transmittr an at th rcivr. In

MIMO trminology, th Input an Output ar rfrnc to th wirlss

channl. In ths systms, prformanc gains ar achiv as multipltransmittrs simultanously input thir signal into th wirlss channl an

thn combinations of ths signals simultanously output from th wirlss

channl into th multipl rcivrs. In a practical systm for th ownlink

communication, a singl Basstation (BS) woul contain multipl transmittrs

connct to multipl antnnas an a singl Mobil Station (MS) woul con-

tain multipl antnnas connct to multipl rcivrs. This sam configuration

may b us in th uplink. Figur 1 shows svral basic block iagrams for

conncting ach transmittr to ach rcivr in a wirlss systm using

multipl antnnas. Each arrow rprsnts th combination of all signal paths

btwn two antnnas that inclu th irct Lin of Sight (LOS) path, shoul

on xist, an th numrous multipath signals crat from rflction, scattring

an iffraction from th surrouning nvironmnt. For xampl, Singl Input

Singl Output (SISO) is th traitional configuration for raio an tlvision

broacast an arly 1st gnration cllular. This singl channl inclus th

LOS path an all multipaths prsnt ovr th wirlss link. Th Singl Input

Multipl Output (SIMO) an Multipl Input Singl Output (MISO) configurations

rquir th us of a singl antnna at ithr th transmittr or th rcivr. Th

SIMO cas may b usful whn transmitting uplink ata from a mobil vic,

which has a singl antnna, to a cllular bas station or WLAN accss point

containing two or mor antnnas. Altrnatly, th MISO cas may rprsnt

th configuration for th ownlink transmission of ata with transmit ivrsity.

Figur 1 also shows a 2x2 MIMO configuration whr two antnnas ar plac

at th transmittr which has two sparat transmit channls an two antnnas

at th rcivr which has two sparat rciv channls. This configuration

will b iscuss as th primary xampl in this application not. Thr arobviously numrous othr MIMO configurations using othr combinations of

multipl antnna pairs, such as 3x3 an 4x4. MIMO opration os not rquir

an qual numbr of antnnas at th transmittr an rcivr as thr may b

mor antnnas at on location than anothr, such as an M x N configuration

whr M os not qual N an M quals th numbr of transmit antnnas an

N quals th numbr of rciv antnnas.

Figur 1. Antnna an channl configurations for SISO, SIMO, MISO an MIMO (2x2) systms.

Reviewing MIMOTechnologies


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Multiple antenna techniques

Multipl antnna systms tak avantag of th spatial ivrsity obtain by

placing sparat antnnas in a ns multipath scattring nvironmnt. Ths

systms may b implmnt in a numbr of iffrnt ways to obtain ithr

a ivrsity gain to combat signal faing or to obtain a capacity improvmnt.

Gnrally, thr ar thr catgoris of multipl antnna tchniqus. Th first

on aims to improv th powr fficincy by maximizing spatial ivrsity. Such

tchniqus inclu lay ivrsity, spac-tim block cos (STBC), an spac-

tim trllis cos (STTC). Th scon typ uss spatial multiplxing, fin as

MIMO, whr unr rich scattring nvironmnts, inpnnt ata strams

ar simultanously transmitt ovr iffrnt antnnas to incras th ffctiv

ata rat. Th thir typ of multipl antnna systm xploits knowlg of th

channl at th transmittr, also trm as bamforming. It utilizs th channl

information to buil th bamforming matrics as pr- an post-filtrs at th

transmittr an rcivr to achiv capacity gain.

Spatial diversity

Signal powr in a wirlss channl fluctuats rapily ovr tim an istancu to th rich multipath nvironmnt. Whn th signal powr rops signifi-

cantly at th rcivr, th channl is sai to b in a multipath fa. Divrsity is

oftn us in wirlss channls to combat this faing ffct. Antnna ivrsity

combats faing by combining signals from two or mor inpnntly fa

channls. For xampl, in a SIMO systm, rciv antnna ivrsity will

improv systm prformanc whn th rcivr optimally combins signals

from sparat antnnas so that th rsultant signal xhibits a ruc ampli-

tu variation whn compar to th signal amplitu from any on antnna.

Divrsity is charactriz by th numbr of inpnntly faing channls, also

known as ivrsity orr, an is qual to th numbr of rciv antnnas in a

SIMO configur systm. It is important to not that if th faing channls ar

not inpnnt, or in othr wors corrlat, thn antnna ivrsity may not

improv th systm prformanc.

Transmit ivrsity is applicabl to MISO channls an has bcom an activ

ara of rsarch. If th channls from ach transmit antnna to th singl

rciv antnna hav inpnnt faing charactristics, thn th ivrsity

orr is qual to th numbr of transmit antnnas. If th transmittr os not

hav prior knowlg of th channl charactristics thn a suitabl sign of

th transmitt signal is rquir to achiv ivrsity gain at th rcivr. On

vry popular transmit ivrsity tchniqu that has rcntly gain much attn-

tion is Spac Tim Coing (STC). This tchniqu sns th sam usr ata to

both transmit antnnas, but at iffrnt tims, for improving th probability of

succssfully rcovring th sir ata. Th STC procss ffctivly ncos

th ata in both spac an tim.


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A simplifi block iagram using Alamouti STC is shown in Figur 2. In this

systm, two iffrnt symbols ar simultanously transmitt from th two

antnnas uring any symbol prio. During th first tim prio, th first

symbol in th squnc, s0, is transmitt from th uppr antnna #1 whilth scon symbol, s1, is simultanously transmitt from th lowr antnna#2. During th nxt symbol tim th signal -s1*is transmitt from th uppr

antnna an th signal s0* is transmitt from lowr antnna. Not that ( )*is th complx conjugat opration. Kp in min that th ata symbols ar

complx numbrs rlating to th slct moulation schm, for xampl,

whn using QPSK moulation, th ata symbols ar rprsntativ of th four

constllation points in th IQ vctor iagram. At th rcivr, a singl antnna

rcivs a combination of th two transmitt signals aftr transmission

through th multipath nvironmnt. Th channl cofficint, h0, rprsnts th

magnitu an phas of th transmission path btwn transmit antnna #1

an th rciv antnna. Th channl cofficint, h1, rprsnts th path

btwn transmit antnna #2 an th rciv antnna. Not that th channl

cofficints, h0 an h1, ar complx numbrs that rprsnt th total amplitu

an phas of thir rspctiv channls incluing all multipath ffcts.

Figur 2. Simplifi Alamouti Spac Tim Coing (STC) block iagram.

During th first symbol tim shown in Figur 2, th rciv signal, r0, is

th combination of both symbols, s0an s1, but is moifi by th channlcofficints, h0an h1. During th nxt symbol prio, th rcivr masursr1which contain moifi vrsions of s0an s1. Th rciv signals, r0anr1, as a function of th transmitt signals an channls cofficints can brprsnt as

Equation 1

Equation 2

r0= h

0s

0+ h

1+ s

1

1= h

1s

0*+ h

0(s

1*)


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In orr to rcovr th actual transmitt symbols, s0an s1, th rcivrrquirs knowlg of th channl cofficints, h0an h1. Ths channlcofficints ar oftn stimat at th rcivr by masuring known signals

mb in th transmitt wavforms. For xampl, in a WiMAX Wav 2

signal, th OFDM wavform is sign such that pilot subcarrirs transmitt

on on transmittr channl o not ovrlap in tim with pilot subcarrirs on th

othr transmittr channl. If th pilot wavforms ar known at th rcivr,thn th channl cofficints can b stimat from th associat rcivr

masurmnts. Onc th channl cofficints ar accuratly known by th

rcivr, Equations 1 an 2 can b rarrang in trms of th sir ata, s0an s1. In this cas th rcivr can proprly co th sir symbols usingth rciv signals, r0an r1, masur ovr two conscutiv symbol timsusing th following quations.

Equation 3

Equation 4

Equation 5

It shoul b not that this ivrsity tchniqu os not improv th systm

ata rat but rathr improvs th signal quality. Th squnc shown in

Figur 2 uss ncoing prform in spac an tim (spactim coing). Th

ncoing may also b on ovr th spac an frquncy omains. In this

cas, insta of two conscutiv symbol prios transmitt from two sparat

antnnas, two frquncy carrirs may b us (spacfrquncy coing).

Utilization of ivrsity in MIMO channls rquirs a combination of th transmit

an rciv ivrsity scrib abov. Th ivrsity orr woul thn b qual

to th prouct of th numbr of transmit an rciv antnnas if th channlbtwn ach transmit-rciv antnna pair fas inpnntly.

0 0* *

* *

0 0 1( )s A h r h r = +

11 0 0 1( )s A h s h r =

2 2

0 1

1

whrA h h= +


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Spatial Multiplexing

Spatial multiplxing can offr an incras in th transmission rat, whil using

th sam banwith an powr as in a traitional SISO systm. Th thortical

incras in capacity is linarly rlat to th numbr of transmit/rciv

antnna pairs a to th MIMO systm. A MIMO systm can also b config-

ur with an unqual numbr of antnnas at th transmittr an th rcivr,

such as an MxN cas whr M transmit antnnas os not qual N rciv

antnnas. In this configuration, th capacity improvmnt is proportional to

th smallr numbr, M or N.

Figur 3 shows a simpl spatial multiplxing systm using a 2x2 MIMO

configuration. This concpt can asily b xtn to mor gnral MxN MIMO

systms. In this xampl, th first ata symbol, s0, is transmitt from thuppr transmit antnna, Tx0, an th scon ata symbol, s1, is transmittfrom lowr antnna transmit antnna, Tx1. Th transmission of ths two atasymbols occurs simultanously uring th first symbol tim. During th nxt

symbol tim, ata symbols s2an s3ar simultanously transmitt. In thisprocss, th ata rat is oubl as altrnat symbols ar transmitt from

ach antnna an ach symbol is only transmitt onc. This tchniqu isiffrnt from STC whr ata symbols ar rpat ovr two symbol tims

across th two antnnas.

Transmission of th signal from transmit antnna Tx0to th rciv antnnaRx0 occurs ovr th wirlss channls with a complx channl cofficint h00.Transmission from antnna Tx0 to th antnna Rx1occurs ovr th wirlsschannl with a complx channl cofficint h10. By proprly placing th antn-nas it can b assum that ths two channl cofficints ar iffrnt. Thr

is a similar rlationship btwn Tx1an th two rciv antnnas rsulting ina total of four potntially uniqu channl cofficints, h00, h10, h01an h11.

Figur 3. Simplifi 2x2 MIMO block iagram.


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Aftr transmission through th channl, th rcivr masurs th signal, r0, atth uppr antnna, Rx0, as a combination of th s0 an s1 incluing channlffcts, h00an h01. At th sam tim, th lowr antnna masurs r1as acombination of s0 an s1 moifi by th channl ffcts, h10an h11rspc-tivly. Th quations for r0an r1as a function of transmitt symbols anchannl cofficints can b rprsnt as

Equation 6

Equation 7

Unr favorabl channl conitions, th spatial signaturs of th two signals,

r0an r1, ar wll sparat. Th rcivr, having knowlg of th channlcofficints, can iffrntiat an rcovr symbols, s0an s1. Th quationsfor calculating s0an s1bas on masurmnts of r0an r1an th channlcofficints ar

Equation 8

Equation 9

Equation 10

Aftr coing, th sub-strams ar multiplx into th original symbol

stram. Spatial multiplxing incrass transmission rats proportionally with

th numbr of transmit-rciv antnna pairs.

Spatial multiplxing also can b appli in a multiusr format, also known as

Spac Division Multipl Accss (SDMA). Consir two mobil usrs transmit-

ting thir iniviual signals ovr th sam wirlss channl that arriv at a

bas-station quipp with two antnnas. Th bas-station can sparat th

two signals using th spatial multiplxing tchniqu scrib abov. Th

incras in capacity is proportional to th numbr of antnnas at th bas-

station or th numbr of mobil usrs, whichvr numbr is smallr. This

tchniqu has bn fin in th WiMAX Wav 2 stanar an is trm

Uplink Collaborativ Spatial Multiplxing (UL-CSM).

0 00 0 01 1h s h s= +

1 10 0 11 1r h s h s= +

0 11 0 01 1( )s B h r h r =

1 10 0 00 1( )s B h r h r = +

00 11 01 10

1whr B

h h h h=


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It is important to not that spatial multiplxing can only incras transmission

rats whn th wirlss nvironmnt is vry rich in multipath. Th rich multipa-

th will rsult in low corrlations btwn th channls, making ata rcovry

possibl at th rcivr. Whn th channls ar highly corrlat, th spatial

multiplxing prformanc rapily gras. In mathmatical trms, Equations 6

an 7 abov can b writtn in matrix form as

Equation 11

Equation 12

In orr to corrctly rcovr th ata symbols at th rcivr, Equation 12 is

rarrang in matrix form as

Equation 13

Th channl cofficint matrix [H] ns to b invrt in orr to rtriv th

ata from th rciv signals. If th channl cofficints in [H] ar highlycorrlat, matrix invrsion bcoms ifficult an th matrix is consir

ill-conition. In this tchniqu, an ill-conition [H] matrix causs th

calculation of s0an s1to bcom vry snsitiv to small changs in thvalus of th calculat channl cofficints an masur valus of r0an r1.Thrfor, any nois in th systm may gratly affct th rcovry of s0an s1.

Beamforming

In a traitional bamforming application, th sam signal, or ata symbol, is

simultanously transmitt from ach antnna lmnt aftr a complx wight

(magnitu an/or phas) is appli to ach signal path in orr to str

th antnna array for optimal SNR ovr th wirlss link. In a bamformr

optimiz for spatial ivrsity or spatial multiplxing, ach antnna lmnt

simultanously transmits a wight combination of two ata symbols. This

bamforming tchniqu rquirs knowlg of th channl charactristics

at th transmittr, which was not a rquirmnt for th spatial ivrsity an

spatial multiplxing tchniqus prviously iscuss. In this cas, it may b

rquir to masur th channl at th rcivr an sn information back to

th transmittr. Th channl knowlg at th transmittr can b full or partial.

Full channl knowlg implis that th channl matrix [H] is known to th

transmittr. Partial knowlg might rfr to som paramtrs of th instanta-

nous channl, such as th channl matrixs conition numbr or a statistical

proprty rlat to th transmit an/or rciv corrlation charactristics. Th

conition numbr is th ratio of th largst singular valu ovr th smallst

singular valu. It provis an inication of th accuracy in th matrix invrsion,which trmins th suitability for MIMO multiplxing. A conition numbr

nar 1 (0 B) inicats a wll-conition matrix whras a valu largr than

6 B inicats a poorly fin channl matrix. Signal analyzrs such as th

Kysight 89600-sris Vctor Signal Analyzr can irctly masur th MIMO

conition numbr.

00 01 00

10 11 11

h hr s

h hr s=

[R] [H][S]

[S] [H]1[R]


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A pr-coing framwork for xploiting channl knowlg at th transmittr is

shown in Figur 4. Th symbols to b transmitt, s0, s1, s2, s3, , ar multipliby a wighting function that can b intrprt as th bamformr. Aftr apply-

ing th pr-coing wights, two sparat ata strams ar simultanously

transmitt from two transmit antnnas as spatial multiplxing. As shown in

Figur 4, uring th first symbol tim, th ata, x0, transmitt from th uppr

antnna is a linar combination of th first two ata symbols, s0an s1.During this sam tim, th lowr antnna transmits ata x1that rprsnts aiffrnt combination of ths two symbols, thus ffctivly oubling th ata

rat. Hr, th transmitt ata is rlat to th input symbols by th following

quations.

Equation 14

Equation 15

Dnot th 2x2 pr-coing matrix as [W], an thn in matrix form, th

transmitt signals ar rlat by

Equation 16

Equation 17

For this pr-coing schm, th transmission rat also incrass proportionally

with th numbr of transmit-rciv antnna pairs, as was th cas for spatial

multiplxing iscuss abov, but th aitional flxibility for optimizing th

signal transmission into th wirlss channl at th transmittr may also

improv th rlativ systm prformanc.

Figur 4. Bamforming transmit ncor.

x0= w

00s

00+ w

1s

1

x1= w

10s

0+ w

11s

1

00 01 00

10 11 11

w wx sw wx s

=

[X] = [W][S]


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MIMO in wireless standards

MIMO tchnologis hol th promis of highr ata rats with incras

spctral fficincy. Du to th larg potntial improvmnt in wirlss systm

prformanc, many stanars committs hav rcntly aopt or ar consi-

ring th us of MIMO an multipl antnna tchnologis. For instanc, th

Intrnational Tlcommunications Union (ITU) working group has intgrat

MIMO tchniqus into th high-sp ownlink packt accss (HSDPA)channl,1, 2which is a part of th Univrsal Mobil Tlcommunications Systm

(UMTS) stanar. In WLAN systms, MIMO applications hav bn fin in

th IEEE 802.11n stanar.3,4In mobil broaban wirlss accss (BWA),

MIMO has also bn aopt into th IEEE 802.16 stanar that is th basis for

Mobil WiMAX,5, 6which is th stanar on which Mobil WiMAX7,8 Wav 2

profils ar bas. Lastly, th volving LTE stanar9, 10has inclu MIMO

into th currnt roamap. All of ths commrcial wirlss systms oprat in

high multipath nvironmnts an it is th bnfit of rich multipath charactris-

tics that provis th prformanc improvmnt whn using multipl antnna

systms.

1. Kysight Application Not, Concepts of High Speed Downlink Packet Access: Bringing Increased Throughput

and Efficiency to W-CDMA, Litratur numbr 5989-2365EN, January 18, 2007.

2. Aitional information about HSDPA can b foun at www.keysight.com/find/HSDPA.

3. Kysight Application Not 1509, MIMO Wireless LAN PHY Layer [RF] Operation & Measurement,Litratur numbr 5989-3443EN, Sptmbr 16, 2005.

4. Aitional information about 802.11n WLAN can b foun at www.keysight.com/find/WLAN.5. Aitional information about th IEEE 802.16 spcification an working group can b foun at

www.ieee802.org/16/.6. Kysight Application Not 1578, IEEE 802.16e WiMAX OFDMA Signal Measurements and Troubleshooting,

Litratur numbr 5989-2382EN, Jun 6, 2006.

7. For mor information about WiMAX, visit www.wimaxforum.org.8. For mor information about tst solutions for WiMAX, visit www.keysight.com/find/wimax.9. For mor information about th 3GPP an LTE spcifications visit th 3GPP hom pag at www.3gpp.org/.10. For mor information about Kysight sign an tst proucts for LTE visit www.keysight.com/find/LTE.
http://www.keysight.com/find/HSDPAhttp://www.keysight.com/find/WLANhttp://www.ieee802.org/16/http://www.wimaxforum.org/http://www.wimaxforum.org/http://www.keysight.com/find/wimaxhttp://www.3gpp.org/http://www.keysight.com/find/LTEhttp://www.keysight.com/find/LTEhttp://www.3gpp.org/http://www.keysight.com/find/wimaxhttp://www.wimaxforum.org/http://www.ieee802.org/16/http://www.keysight.com/find/WLANhttp://www.keysight.com/find/HSDPA


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Channel correlation effects on MIMO performance

For wirlss communication systms, th wirlss channl is th ky factor

that trmins systm prformanc. Channl ffcts, such as path loss an

multipath faing, rsult in th attnuation of th signal amplitu at th rcivr.

Multipath may also inuc intr-symbol intrfrnc if th lay spra is

longr than th cyclic prfix in an OFDM signal. Spatial ivrsity an spatial

multiplxing hav bn shown, both thortically an xprimntally, tosubstantially improv prformanc an ovrcom th unsir ffcts of

multipath but only if th spatial imnsion is proprly configur to lvrag

th richnss of th multipath nvironmnt.

As introuc abov, th ivrsity gain achivabl using STC is pnnt on

th channl ivrsity orr. Only whn th channls btwn ach transmit-

rciv antnna pair fa inpnntly will th channl ivrsity orr b

qual to th prouct of th numbr of transmit an rciv antnnas.

Altrnatly, if th channls btwn transmit-rciv antnna pairs ar highly

corrlat, thn th achivabl ivrsity gain is vry limit.

Low corrlation channls ar also rquir in spatial multiplxing MIMO

applications. Th iffrnt spatial signal strams can b wll sparat only

unr favorabl channl conitions. This oftn rquirs propr positioning of

th transmit an rciv antnnas in orr to provi low channl-to-channl

corrlations btwn th antnna pairs.

As a masurmnt xampl, Figur 5 shows th 2x2 MIMO channl coffi-

cints, h00, h10, h01, an h11,for two iffrnt faing channls, on withrlativly high channl-to-channl corrlations an th othr with low

corrlations. Ths masurmnts wr ma using an Kysight ual-channl

89600-sris vctor signal analyzr (VSA) on a WiMAX OFDMA signal that was

fa using th PXB. Th plot on th uppr lft shows th four channl coffi-

cints as a function of subcarrir frquncy for th high corrlation cas. It can

b obsrv that th magnitu of th cofficints hav a similar frquncyrspons rsulting from th high gr of corrlation btwn som of th

paths. Th lowr plot isplays th masur constllation for th moulat

symbols which shows a high lvl of signal corruption. As a comparison, th

figur on th uppr right shows th cofficints for low channl-to-channl

corrlations. In this cas, th frquncy rsponss of th cofficints ar

issimilar, rsulting in an improvmnt in th MIMO symbol rcovry, as shown

by th masur constllation in th lowr right of Figur 5.

Figur 5. Masur channl cofficints an moulat constllations for a 2x2 MIMO wavform.


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Challenges in emulating MIMO channels

Tsting MIMO rcivrs an systms unr ralistic channl nvironmnts

can oftn b challnging u to th larg numbr of transmit-rciv channl

combinations. For xampl, in a 2x2 MIMO configuration, using two sparat

SISO channl mulators is not aquat to mol th four sparat channls

that xist btwn th pairs of transmit an rciv antnnas. In aition, SISOchannl mulators o not provi any corrlation btwn channls, which

was prviously shown to b an important charactristic whn tsting systm

prformanc. Tsting irctly in a ral wirlss nvironmnt is not an ffc-

tiv mtho, spcially uring th sign an valiation stags, as th channl

is vry snsitiv, not controllabl, an not rpatabl. Also, tsting in a ral

channl is not practical whn iffrnt nvironmnts ar rquir an whn

mobility tsting is also ncssary.

Crating ralistic MIMO channls using softwar tools is anothr option but is

oftn tim-consuming an proucs rsults that ar not ral-tim. For xampl,

aftr crating th channl faing cofficints in softwar, th convolution of

ths cofficints with th transmitt signals is a rlativly long procss

prvnting ral-tim prformanc. In som typs of softwar-bas tstsystms, th moulat ata an fa signals ar us to crat complx

(I/Q) wavforms that ar ownloa into th mmory of an arbitrary wav-

form gnrator (ARB) for playback. Th ARBs may b intrnal to th RF signal

gnrator, such as thos in th Kysight E4438C ESG signal gnrators, or

xtrnal to th RF signal gnrator, such as th Kysight N6030A-sris arbi-

trary wavform gnrators. Thr ar many softwar tools that can acclrat

th cration of fa wavforms, such as Kysight Signal Stuio, Mathworks

MATLAB an Kysight Avanc Dsign Systm (ADS), but ths tools ar

oftn limit to traitional faing mols. In aition, th arbitrary wavform

gnrators hav limit playback mmory rsulting in rlativly short

wavforms that rpat ovr tim. Thrfor, spcializ instrumntation that

mulats ralistic MIMO channls provis th bst solution for ths

challnging tst conitions.


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A channl mulator, such as th PXB, that rplicats ral-worl MIMO coni-

tions using powrful igital signal procssing tchnology will mak it possibl

to rapily isolat prformanc issus arly in th sign, vlopmnt an vri-

fication cycl, an provi th quickst path for troublshooting avanc raio

componnts an systms. Th channl mulator also has th avantags that

it can gnrat ralistic faing scnarios incluing path an channl corrla-

tions, an has a lowr implmntation cost an a fastr calibration procss.Th PXB provis up to 4 basban gnrators an 8 fars usful for tsting

an troublshooting up to 4x2 MIMO systms. Figur 6 shows a simplifi

configuration iagram for tsting a 2x2 MIMO rcivr using th PXB connct

with two RF signal gnrators for signal upconvrsion. Th PXB intrnal bas-

ban gnrators crat th stanars-compliant wavforms such as WiMAX,

LTE an WLAN signals. Ths basban gnrators ar asily connct to th

channl fars through a softwar GUI. Each far can b inpnntly

configur with a stanars-compliant faing mol, such as a WiMAX ITU

Pstrian B, or custom configur mol using a varity of path an faing

conitions.

Figur 6. Simplifi block iagram for tsting a 2x2 MIMO rcivr using th PXB.

N5106A PXB baseband generatorand signal emulator

ESG or MXGsignal generator


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16

A signal propagating through a wirlss channl arrivs at th stination along

a numbr of iffrnt paths, rfrr to as multipath. Figur 7 is a iagram of a

typical mobil subscribr riving along a roaway. It picts thr of th many

signal paths from th transmittr to rcivr. Ths paths aris from scattring,

rflction an iffraction of th raiat nrgy by objcts in th nvironmnt or

rfraction in th mium. Th various propagation mchanisms influnc path

loss an faing mols iffrntly.

Figur 7. Typical multipath faing scnario.

Variations in th rciv signal powr ar u to thr ffcts: man propaga-

tion (path) loss, macroscopic (larg scal or slow) faing an microscopic

(small scal or fast) faing, which ar monstrat in Figur 8. Th man

propagation loss is rang pnnt an rsults from absorption by watr an

foliag an th ffct of groun rflction. Macroscopic faing rsults from th

shaowing ffct by builings an natural faturs. Microscopic faing rsults

from th constructiv an structiv combination of multipath an is also

known as fast faing sinc amplitu fluctuations ar rapi whn compar to

macroscopic faing.

Figur 8. Signal powr fluctuation vrsus rang in wirlss channl.

MIMO ChannelOverview


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Multipath propagation rsults in th spraing of th signal ovr tim an

ths tim lays or lay spra caus frquncy slctiv faing.

Multipath is charactriz by th channl impuls rspons an is mol

using a tapp lay lin implmntation. Th charactristic of th tap variability

is charactriz by th Dopplr spctrum. In aition to lay spra an

Dopplr spra, angular or angl spra is anothr important charactristic

of th wirlss channl. Angl spra at th rcivr rfrs to th spra inAngls of Arrival (AoA) of th multipath componnts at th rciv antnna

array. Similarly, angl spra at th transmittr rfrs to th spra in Angls

of Dpartur (AoD) for thos multipath signals that finally rach th rcivr.

Angl spra causs spatial slctiv faing which mans that signal amplitu

pns on th spatial location of th transmit an rciv antnnas. Whn

multipl antnnas ar appli to a wirlss communication systm, th various

transmit-rciv antnna pairs may hav iffrnt channl impuls rsponss

u to th spatial ffcts caus by angl spra, antnna raiation pattrn

an th surrouning nvironmnt. As MIMO opration rquirs low channl-

to-channl corrlation, it is important to unrstan how ths spatial charac-

tristics may influnc systm prformanc. In th nxt fw sctions of this

application not thr is a rviw of th basic charactristics foun in any

wirlss channl, such as lay spra an Dopplr spra, an in aition,th spatial ffcts will also b introuc as a mans to crat improv

mols for high prformanc channl mulators.

Wireless propagation characteristics

Mean propagation loss

Th ovrall man loss in signal strngth as a function of istanc will follow a1/dn law, whr is th istanc btwn th transmittr an th rcivr an

n is th slop inx ranging from a valu of 2 to 6 pning on th

nvironmnt. For xampl, in fr spac, n 2 rsulting in a 20 B/ca

slop. In a trrstrial nvironmnt, a typical valu of n 4 rsults in a 40 B/

ca signal loss as a function of istanc. In this trrstrial stting, changingth istanc from 100 ft to 1000 ft (on ca) woul rsult in an avrag

signal rop of 40 B. Svral mpirically bas path loss mols hav bn

vlop for iffrnt propagation nvironmnts such as th mols in

COST-2311an ITU-R M.12252.

1. COST 231 TD (973) 119-REV 2(WG2). Urban transmission loss models for mobile radio in the 900- and

1800-MHz bands, Sptmbr 1991.

2. IEEE P802.11 WirelessLANs TGn channel models , May 2004.


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Macroscopic (slow) fading

Macroscopic or slow faing is caus by th shaowing ffcts of builings or

natural faturs an is trmin by a local man of th rciv signal ovr a

istanc of approximatly 20 wavlngths. Th macroscopic faing istribution

is influnc by antnna hights, th oprating frquncy an th spcific typ

of nvironmnt. Th viation of slow faing about th man propagation loss

is trat as a ranom variabl that approachs a normal istribution whn

xprss in cibls (B) an is consir to b log-normal as scrib by

th following Probability Dnsity Function (PDF).

Equation 18

In th abov quation,x(in B), is a ranom variabl rprsnting th largscal signal powr lvl fluctuation. Th variabls, an , ar th man anstanar viation ofx, rspctivly. Both an ar xprss in B. Thman valu, , is qual to th man propagation loss iscuss in th prvious

sction. Th stanar viation, , may hav valus as high as 8 B for somurban nvironmnts.

Microscopic (fast) fading

Microscopic or fast faing rsults from th constructiv an structiv intr-

frnc of numrous multipath signals rciv from th surrouning nviron-

mnt. Rapi changs in rciv signal strngth may occur whn th istanc

is vari by approximatly on-half wavlngth, thus giving this charactristic

th nam fast faing. Whn xamining th faing statistics in th rciv

powr ovr a rlativly short istanc of approximatly 20 wavlngths, th

in-phas (I) an quaratur (Q) componnts of th suprimpos signal can b

mol as an inpnnt zro-man Gaussian procss. This mol assums

that th numbr of scattr componnts is vry larg an inpnnt. Thvoltag amplitu nvlop of this rciv signal woul thn hav a Rayligh

istribution with a PDF givn by

Equation 19

whrxis a ranom variabl takn hr as th rciv voltag amplitu anis th stanar viation. A similar rspons woul also b foun for astationary subscriber as a function of time due to the relative motion of scatterers

in th local vicinity of th subscribr. Th rlativ chang in powr lvl

btwn a pak to null is typically 15-20 B but can b as high as 50 B unrsom channl conitions.

( ) ( )22xe

2

1xf

=

( )2 22

20

0

xx e xf x

=

x < 0


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19

If there is a direct path present between transmitter and receiver, the signalenvelope is no longer Rayleigh and the statistics of the signal amplitude followa Rician distribution. Rician fading is formed by the sum of a Rayleigh distributedsignal and a direct or Line-Of-Sight (LOS) signal. A fading environment associ-ated with Rician statistics has one strong direct path reaching the receiver atroughly the same time delay as multipath from the local scatterers. The voltage

amplitude envelope for a Rician distribution has a PDF given by

Equation 20

wherex is a random variable taken here as the received voltage amplitude and is the standard deviation. The term I0( )is the modified Bessel function ofthe first kind, order zero. Since I0( ) = 1, the Rician distribution reduces to theRayleigh distribution when K= 0. The Rician distribution is defined in terms ofthis Kfactor which for wireless environments is defined as the ratio of the

power in the LOS component to the power in the scattered components.

As a measurement example showing the amplitude variation as a function oftime for two independent channels in a SIMO system, the PXB was configuredto create two independently Rayleigh-faded signals. Figure 9 shows the PXBmeasurement configuration screen of two parallel baseband generators that areindependently faded using a Rayleigh distribution and the faded waveforms areconnected to external RF signal generators for upconversion. As the channelsuse independent fading statistics, it is expected that their amplitude levelswould be uncorrelated over time. Figure 10 shows the measurements of theamplitude for the two faded signals as a function of time. These measurementswere obtained using an Keysight E4440A PSA-series spectrum analyzer set toZero-Span mode. As shown in the figure, the two channels appear uncor-

related with each having separate fading nulls, some as deep as 45 dB.

Figure 9. PXB setup screen for configuring two independent Rayleigh-faded channels using two signalgenerators.

Figure 10. Received signal power as a function of time for two independent Rayleigh-faded channels.

( )( )( ) ( )

2 2 2 22

020

0 0

x Kx xKe I xf x

x

+

=

2fd, the maximum power spectral density S(f)shall be less than S(fc) by at least 30 dB.

3) Simulated Doppler frequency, fd, shall be computed from the measuredDoppler power spectrum. The tolerance on Doppler shall be 5%.

1. 3GPP2 standard for Recommended Minimum Performance Standards for cdma2000 High RatePacket Data Access Network. More information available atwww.3gpp2.org/Public_html/specs/C.S0032-A_v1.0_051230.pdf.
http://www.3gpp2.org/Public_html/specs/C.S0032-A_v1.0_051230.pdfhttp://www.3gpp2.org/Public_html/specs/C.S0032-A_v1.0_051230.pdf


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The theoretical and measured Doppler power spectrum is shown in Figure 18.Here the Doppler frequency on the PXB was set to 120 Hz. The measuredresults demonstrate that the emulated Doppler spectrum performance caneasily satisfy the recommended requirements. The computed Doppler frequencyfrom the measured Doppler power spectrum is 121.23 Hz, resulting in ameasurement error of 1.025%, which is well under the recommended 5%

tolerance.

Figure 18. Rayleigh 6 dB theoretical spectral shape versus the measured spectral shape.

Dynamic fading

In mobile applications, the characteristics in the Power Delay Profile (PDP)would remain relatively constant over several meters. In this case the impulseresponse of a radio channel is averaged over this small distance to provide astatic or wide sense stationary view of the channel conditions. As a mobileterminal moves over a wider area, the shape and characteristics of the PDPchange dramatically as shown in the example in Figure 19.

Modern wireless communications systems must adapt to these dramaticchanges to continuously mitigate the impact of multipath delay spread. Toaccurately evaluate the performance over a time-varying PDP, a fading emulatormust be capable of emulating the time-varying changes in the paths delaycharacteristics. The sliding relative path delay and the Birth-Death time-varyingrelative path delay are two popularly employed models to emulate dynamicdelay spread.

Figure 19. Dynamic fading characteristic showing the time-varying PDP.


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Angle spread and Power Azimuth Spectrum

Traditional methods for modeling wireless channels, such as Power DelayProfile and Doppler spectrum, can accurately represent the multipath effects ina SISO system. The shortcoming of these traditional models is that they typicallydo not include spatial effects introduced by antenna position and polarizationwithin the multipath environment. They also do not include the antenna patterneffects on the system performance. For example, in the simple MIMO caseshown in Figure 20, the Tx0 transmit antenna, has two signal paths to the Rx0receive antenna, namely, the LOS and one multipath. The LOS path leaves Tx0with an angle of departure (AoD), d1, measured relative to the array boresightas shown. The array boresight is defined as the normal (perpendicular) direc-tion from the line of antenna array and it is primarily used as a reference pointto describe angular direction. As the transmitter and receiver array boresightdirections may not be pointing at each other, the received signals may arrivewith a different angle defined as the Angle of Arrival (AoA). In Figure 20, theLOS path from the transmit antenna Tx0 arrives at the receive antenna, Rx0,with AoA of a1. As shown in the figure, the AoD and AoD for the multipathbetween Tx0 and Rx0 are d2 and a2respectively. For the signal paths

connecting the Tx1 transmit antenna to Rx0, the associated AoDs and AoAsmay be different from the Tx0 to Rx0 angles depending on the spatial separa-tion of the Tx0 and Tx1 antennas. If the two transmit antennas are very closeto one another, then the AoA and AoD would be very similar and a high fadingcorrelation may exist between the antenna pairs (Tx0/Rx0 and Tx1/Rx0). Aspreviously discussed, high correlation between transmit-receive antenna pairsreduces the performance for MIMO and STC systems. Therefore it is importantfor any MIMO channel emulator to include a model for the spatial effects andresulting channel correlations for the antenna pairs.

Figure 20. Spatial diagram for a 2x2 MIMO system showing the Angle of Departures (AoD) and Angleof Arrivals (AoA) relative to the transmit and receive antenna array boresight directions.


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Rather than attempting to model each AoD and AoA in the channel emulator,an improved model for emulating the characteristics of a rich multipathenvironment can be achieved by including the spread of the AoDs and AoAsreferred to as angle spread. Angle spread causes spatial selective fading asthe received signal amplitude depends on the spatial location of the antennas.When utilizing multiple antennas at the transmitter or/and receiver, the differ-

ent transmit-receive antenna pairs may have different fading characteristicsdue to the antenna separations, the antenna radiation pattern and thesurrounding environment. In the example shown in Figure 21, the angle spreadfor a typical Base Station (BS) is very narrow due to the fact that most scatterersare positioned far from the BS antennas. In contrast, the Mobile Station (MS)contains a large number of local scatterers surrounding the MS thus resultingin a very wide angle spread. If the BS antennas are placed physically closetogether, the narrow angle spread will result in high channel correlation.Fortunately, a BS often has the area to place its antennas far apart reducingthe channel correlations. For MS with large angle spread, the antennas couldbe placed closer together while maintaining low channel correlations. Closeantenna spacing is ideal for a mobile handheld that requires the placement ofseveral antennas in a small package. Figure 21 also shows a tight grouping

of spatial angles around the BS referred to as a cluster. The cluster can bemodeled by a mean angle surrounded by an angular spread. This representationallows a statistical PDF model to be applied to the power received as a functionof angle.

Figure 21. Diagram of angle spread as a function of antenna placement in a multipath environment.

The angle spread is characterized by the Power Azimuth Spectrum (PAS).

Denotingthe AoA or AoD by , the PAS of a signal, s(t, ), represents theaverage power as a function of angle. Defined as PAS() = Et| |s(t,)2|

thedistribution is normalized to satisfy the probability density function requirement as

Equation 23( ) 1dPAS =


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29

Figure 22 shows three widely used PAS distribution models, Laplacian,Gaussian and Uniform, which are supported by the PXB. PAS distributions aretypically selected based on the desired propagation environment, for example,the Laplacian model is suited for outdoor propagation in urban and rural areas1, 2.Each cluster is assigned a PAS distribution that best estimates the measured ormodeled PAS for the wireless channel. The angle 0,kis the mean arrival/

departure angle of the kth cluster. As shown in the figure, the Laplacian andGaussian distributions are truncated to a value of 2kcentered around themean angle 0,k. Table 3 shows the multimodal distribution functions for theuniform, Gaussian and Laplacian models for PAS.

Figure 22. Power Azimuth Spectrum (PAS) distributions for modeling angular clusters.

1. K. I. Pedersen, P. E. Mogensen, and B. H. Fleury, Spatial channel characteristics in outdoor environments andtheir impact on BS antenna system performance, in Proc. IEEE Vehicular Technology Conf. (VTC) 1998, Ottawa,Canada, vol. 2, pp. 719723.

2. L. Schumacher, B. Raghothaman, Closed-form expressions for the correlation coefficient of directive antennasimpinged by a multimodal truncated Laplacian PAS, IEEE Transactions on Wireless Communications, Vol. 4,No. 4, July 2005, pp. 1351-1359.


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The value for Ncshown in Table 3 above is the number of clusters, 0,kwhichis the mean arrival/departure angle of the kth cluster, and the constant Qkisderived to fulfill the normalization requirement in Equation 23. The standarddeviation, k, in the Gaussian and Laplacian distributions are referred asAzimuth Spread (AS). The expression for S()is related to the truncation of

the distribution where the functions are only defined within a limited interval[0,k - k, 0,k + k]centered on the average angle 0,k. Defining U()as astep function, then the expression for S() in Table 3 is defined as

Equation 27

The notion of multimodal for the distributions in Table 3 refers to conditionswith more than one resolvable cluster, and whose spatial distribution can bemodeled by a specific PAS function. For example, Figure 23(a) shows the mea-sured PAS for a receiver operating in a relatively low multipath environment.The figure shows two high peaks representing two large clusters of multipathsignals occurring between the transmitter and the receiver. Each cluster can be

approximated by a PAS distribution using the best-fit to the actual distribution.For the example shown in Figure 23(a), the measured response is best approxi-mated by two truncated Laplacian distributions centered on the two clusterpeaks as shown in Figure 23(b).

Figure 23. Measured PAS (a) and equivalent model (b) using Laplacian distribution.

( )1

( )Nc

U kk

PAS Q S =

=

( )( )

2

0,

2

1

( ) 22

Nck

kG

k kk

QSSAP

=

=

=

Uniform Equation 24

GaussianEquation 25

Equation 26Laplacian

0,

1

2( )

2

Nckk

L

k kk

QSAP

=

exp

exp ( )S

Table 3. Multimodal PAS distribution functions

( ) ( ) ( ),0,0 kkkkS U U = + [ ] ][


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The PXB can be used to easily define the cluster angles at the transmitter andreceiver for each active path in the MIMO channel model. As shown in Figure24, the PXB provides a table entry for the AoD, AoA and associated AzimuthSpread for each path within the selected channel. In this case, the PXB usestwo path definitions in the channel each having a unique spatial distribution.

Figure 24. PXB configuration table for entering values of AoA, AoD and associated Azimuth Spread formodeling PAS effects in the wireless path.

The Power Azimuth Spectrum is just one spatial characteristic that mayintroduce correlations between the various MIMO channels. These spatially-induced channel correlations may also be effected by antenna pattern, spacingand polarization. These topics will be discussed in the next few sections of thisapplication note and how they relate to the channel-to-channel correlation in a

MIMO system.


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Antenna gain and pattern

Antenna gain is a measure of the antennas ability to direct radiated power intoa particular direction. The antenna gain is typically quoted as a numeric valuerelative to a reference antenna where the reference antenna is usually taken asan ideal isotropic radiator that radiates equally in all directions. The antennapattern describes the radiated power as s function of three dimensional spacetypically taken in spherical coordinates using and . In general, one horizontalcut through the spherical coordinates will provide the Azimuth pattern as afunction of . This two-dimensional cut is typically displayed in either polar orrectangular coordinates. Antenna patterns typically fall into two categories -omni-directional and directive. The gain pattern for an omni-directional antennais uniform in all directions. For the case of a dipole antenna positioned vertically(vertical polarization), the gain pattern is uniform in the azimuth plane asshown in the polar plot in Figure 25. In this example, the azimuth gain isconstant for any angle from 0 degrees (boresight) to 180 degrees. In a mobileapplication, an omni-directional antenna is preferred so that the user is notrequired to position or point the antenna for optimal SNR performance. Incontrast, a directive antenna has a higher gain in the boresight direction as

more of the radiated power is focused into that direction. Figure 25 also showsthe gain pattern for a typical directive antenna. As shown in the figure, thedirective antenna has higher gain in the boresight direction as compared to theomni-directional antenna. Directive antennas are often used in base stationapplications to divide the area surrounding the BS into sectors for improvingcoverage and reducing interference within the system.

Figure 25. Typical gain pattern for omni-directional and directional antenna types.


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In general, antenna patterns are normalized to their maximum field strength sothat the displayed peak is set to 0 dB. The half-power or 3 dB beamwidth, 3dB,defines the angle for which the gain drops relative to the peak by 3 dB. For athree-sector base station antenna, the 3 dB beamwidth is typically equal to70 degrees. For a six-sector base station antenna, the 3 dB beamwidth is equalto 35 degrees. In many cellular standards, the sectorized gain pattern is defined as

Equation 28

Where is defined as the angle between the direction of interest and the bore-sight of the antenna. The value for mis defined as the maximum attenuationand is a constant. For the 3GPP standard1, is set to 12 dB. For a 3-sectorantenna, 3dB= 70, m = 20 dB, and the resulting gain pattern in rectangularcoordinates as a function of is shown in Figure 26. For a 6-sector antenna,3dB = 35, m = 20 dB and the antenna pattern is also shown in Figure 26.

Figure 26. Gain pattern as a function of azimuth angle for 3-sector and 6-sector cellular antennas.

( )

= m

2

dB3

,minG

1. Spatial channel model text description, SCM-077 SCM-Text v2.0, November 20, 2002, pp. 7-10.


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Antenna spacing

It can be shown that the antenna-to-antenna spacing at either the transmitterand/or receiver has a strong relationship to the overall spatial correlation. Asthe antenna spacing is reduced, one would expect that the channel-to-channelcorrelation would increase. In the extreme case, if the two transmit antennaswere co-located with the same polarization, it would be expected that thechannel characteristics to a single receive antenna would be identical. It istherefore important for the proper operation of a MIMO system that theantenna location be optimized for low channel-to-channel correlations. Forexample, Figure 27. shows two dipole antennas vertically oriented and spacedat a distance, d. Typically in a traditional phased array application, the antennaspacing is approximately /2 which is used to increase the gain of thecomposite array. In MIMO applications, the requirement is not for high arraygain but rather for low channel-to-channel correlation. In this case the antennaspacing may be much larger than /2 with the only limitation being the arearequired to space the individual elements. For example, the mobile may select aa /2 spacing due to space constraints in a handheld device while a basestation may use an antenna spacing of 4or more.

Figure 27. Dipole antenna placements with inter-element spacing equal to d.


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Spatial correlation

As MIMO systems require a rich multipath environment for proper operation, itis possible that the spatial positions of the multiple transmit antennas, relativeto each other and relative to their placement in the surrounding environment,may give rise to high fading correlation between the different MIMO channels.

The same conditions are also true for the antenna positions at the receiver. Itwill be shown in this section that inadequate antenna spacing leads to spatialcorrelation. The spatial correlation coefficient,

12, between two antenna

elements is a function of their spacing, the PAS, and the gain pattern of theindividual elements. It is assumed that the antenna elements are identical withthe same gain pattern. The correlation coefficient can be calculated using thefollowing equation.

Equation 29

PAS() is calculated using Equation 24, 25, or 26, dependent on the selection ofthe appropriate distribution, and d is the distance between antenna elements.The gain pattern,

G(), is calculated using Equation 28 and assumes that the

far field assumption holds and that the two antennas have exactly the sameradiation pattern and boresight direction.

Figure 28 shows the absolute value of the correlation coefficient as a functionof antenna spacing for several examples of antenna type and Azimuth Spread(AS). The antenna type was varied between omni-directional and directiveusing a 3-sector antenna. Each curve represents a different value for AS cov-ering 2, 5, 10, and 35 degrees. These curves assumed a single-modal Laplacianpower azimuth spectrum with mean AoA=200 degrees and = 180 degrees.

As expected, the correlation coefficient decreases for increasing normalizedspacing and for increasing AS. It is also worthwhile to remark that for a givenantenna spacing and large AS = 10 or 35, the directive antennas tend to beslightly more correlated than omni-directional ones.

Figure 28. Relationship between antenna spacing and correlation coefficient using a Laplacian PASwith AoA = 200 degrees and = 180 degrees.

MIMO ChannelCorrelation

=

dGPAS

dGPASed

j

)()(

)()()sin(2

12


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The spatial correlation matrix for the complete system can be calculated usingEquation 29 and forming the individual spatial correlation matrices at the BS andthe MS. For example, given a 2x2 MIMO system, assume the factors and represent the correlation coefficients, calculated using Equation 29, for the BSand MS antenna pairs, respectively. The correlation matrices for the BS and theMS are represented as

Equation 30

Equation 31

The system spatial correlation matrix for the downlink channel can be calculatedusing the Kronecker product

Equation 32

Equation 33

=1

1

BSR

=1

1

MSR

*

*

S BS MSR R R= W

1

1

1

1SR

*

* *

*

*

*

**

=


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Antenna polarization correlation

In the previous section, it was shown that systems operating with a narrowrange of angular spread may require antennas physically placed far apart inorder to achieve low spatial correlation. Unfortunately some wireless devicestend to be physically small, thus limiting the antenna separation to under awavelength depending on the frequency of operation. In some cases, analternate solution is required to achieve the low channel-to-channel fadingcorrelation required for MIMO operation. One technique to reduce the spatialcorrelation between two antennas is to cross polarize the antennas. In otherwords, position the antenna polarizations in orthogonal or near orthogonalorientations. As shown in Figure 29, two closely-spaced vertically-polarized(0/0) dipole antennas would have a high spatial correlation while orthogonallypolarized (0/90) antennas, one vertical and one horizontal, would have a muchlower correlation coefficient.

Figure 29. Diagram showing the effects of relative antenna polarization on the correlation characteristics.

The use of antennas with different polarizations at the transmitter and/orreceiver may lead to power and correlation imbalances between the variousMIMO channels. The antenna polarization matrix1 is defined at the transmitter

or receiver as Equation 34

where the index vrepresents vertical polarization and hhorizontal polarization.The first index denotes the polarization at the transmitter and the seconddenotes the polarization at the receiver. Correlation between polarized antennascan be quantified using the Cross Polarization Ratio (XPR). The XPR is thepower ratio between a pair of cross polarized antennas (v-horh-v) to that of aco-polarized (v-v or h-h) case. Assume that the XPR = 8 dB, then

Equation 35

=hhhv

vhvv

ss

ssS

( )2 2

2 2 8 0.1585vh hv

vv hh

s s dBs s

= = =

1. MIMO channel model for TWG RCT Ad-Hoc Proposal, V16, pp. 6-8.


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For example, consider a 2x2 MIMO system. The BS polarization matrix withpolarization angles 12 is

Equation 36

The MS polarization matrix with polarization angles 1 2is

Equation 37

For the downlink case, the total channel polarization matrix is the matrixproduct of the BS polarization, channel polarization and MS polarization.

Equation 38

Lastly, the polarization correlation matrix is defined as

Equation 39

For specified polarization angles, such as +45/45, 0/90 and 0/0, the diagonalelements of have the same value, which means there is no power imbalancebetween different channels. For arbitrary polarization angles, the diagonalelements of are not equal, which means that polarization leads to anundesired power imbalance between the channels.

Normalization of is required so that the diagonal elements reflect the channels

power. In this case the correlation matrix then becomes

Equation 40

This power normalization process is based on the assumption that the overallchannel power is K = NrNt for a MIMO system with Nt transmit antennas andNrreceive antennas. The correlation matrix

Rwill properly reflect the channel

imbalance due to polarization, and the diagonal elements in R relate to therelative power in each channel.

=

)sin()sin(

)cos()cos(P

21

21BS

)sin()sin(

)cos()cos(P

21

21

MS

BST

MS SPPQ =

{ }H)(vec)(vecE QQ =

=

=

K

1i

i,i

R K


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39

Using a 2x2 MIMO channel as an example, assume that the XPR= 8 dB andthe system uses cross-polarized MS antennas (0/90) and slant-polarized BSantennas (+45/45). The resulting polarization correlation matrix is

Equation 41

The diagonal elements of this polarization correlation matrix are all ones, whichshow that the selected polarization angles do not result in a power imbalanceamong the different MIMO channels. The other elements in the matrix relate tothe correlation between different channels. For example, in the first row, thismatrix shows that channel 1 is only correlated to channel 3 with a coefficientof 0.7264. The second row shows that channel 2 is only correlated withchannel 4. It can be shown that the use of antennas with differing polarizations

at the transmit and receive leads to polarization diversity.

As another example, consider a case where the correlation matrix results ina power imbalance. Here, assume that the antenna polarization angles are

10/80 at MS antenna and +30/60 at BS. The resulting polarizationcorrelation matrix is

Equation 42

For this matrix, the diagonal elements are not equal and they demonstrate thatusing this combination of polarization angles leads to a power imbalanceamong the different channels.

=

107264.00

0107264.0

7264.0010

07264.001

R

=

3413.11242.05911.02151.0

1242.06587.02151.05911.0

5911.02151.06587.01242.0

2151.05911.01242.03413.1

R


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Combined spatial and antenna polarization correlation

It is known that spatial and polarization correlation effects in compoundantenna systems are independent and multiplicative. In this case, thecorresponding spatial and polarization correlation matrices can be derivedseparately and combined by an element-wise matrix product. The combinedspatial-polarization correlation matrix is then defined as

Equation 43

where Rs is the system spatial correlation matrix calculated using Equation 33and Ris the polarization correlation matrix calculated using Equation 40.

Orthogonal antenna positions (0/90) provide the lowest spatial correlation butmay not always be required or practical under all conditions. For example, thediagrams in Figure 30 show two possible 2x2 MIMO configurations for basestation and mobile antenna positioning. In one case, as shown in Figure 30(a),all the antennas are vertically polarized resulting in a potentially high level ofspatial correlation. To overcome this problem, the BS antennas are spaced at

4to improve the correlation for this typically narrow angle spread condition.The spacing at the MS is fixed at /2. This first case is the reference antennaconfiguration for the high correlation channel model as specified in the WiMAXstandard. For this high correlation condition, all the antennas have the samepolarization angle, which does not provide any polarization diversity. Thereforethe polarization matrix is defined as

Equation 44

By applying Equation 43 to this high correlation antenna configuration, thecombined spatial-polarization correlation matrix is the same as the spatialcorrelation matrix previously defined in Equation 33.

Equation 45

RSRR =

=

1111

1111

1111

1111

R

R = RS R=

1

1

1

1

*

* *

*

*

*

**


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41

The second case, as shown in Figure 30(b), has the antennas at the BS polar-ized at 45-degree orientations while the mobile uses orthogonal polarization(0/90). This second combination can greatly reduce the channel-to-channelcorrelation required for the proper operation of the MIMO system. This combi-nation is the reference antenna configuration for the low correlation channelmodel in the WiMAX standard. For this configuration, the polarization matrix

was defined in Equation 41 and the final correlation matrix for this antennaconfiguration is defined as

Equation 46

where = 0.7264.

Comparing the two correlation matrices found in Equations 45 and 46, it can be

concluded that introducing different polarization angles at the BS and MS willlower the channel-to-channel correlations.

Figure 30. BS and MS antenna configurations for (a) high and (b) low channel correlations.

=

100

010

010

001

*

*R


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Conguring the channel emulator for spatial correlation

It is often desired to improve the process of entering the correlation matricesinto a wireless channel emulator while minimizing the mathematical complexityand still be capable of modeling realistic wireless channels. The PXB greatlyimproves the process of MIMO channel emulation by eliminating the need tocalculate complex correlation matrices and allowing the user to enter thephysical antenna characteristics directly in the instrument. For example,Figure 31 shows the PXB user interface for entering the receive channel (Rx)spatial parameters including antenna type and spacing. A similar table is usedto enter the transmit antenna parameters using the same menu. The PXBuses this spatial information along with the AoAs and AoDs entered in thefading paths table, shown in Figure 24, to automatically calculate the spa-tial-polarization correlation matrix. This simple entry table eliminates the bur-den for calculating the correlation matrices and manually entering the coeffi-cients into the emulator.

Figure 31. PXB antenna parameter setup screen.


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Correlation property validation

The correlation property among different channels is a unique characteristic forMIMO systems. As previously discussed, correlation between MIMO channelsmay have a negative effect on the ability to separate the multiple data streamsat the MIMO receiver. A high-performance channel emulator must providecorrelation characteristics when compared with a traditional SISO channelemulator. It is also important to validate the channel emulators performance bymeasuring the correlation matrix produced in comparison with the instrumentsconfiguration. As a measurement example, the PXB is configured with athree-path 2x2 MIMO channel using cross-polarized (0/90) MS antennas andslant-polarized (45/45) BS antennas similar to the conditions in Equation 41,above. Table 4 shows the correlation matrix for the configured system. Themeasured correlation matrices for paths 1 through 3 are shown in Table 5.

Table 4. Desired correlation matrix for the 2x2 MIMO channel

1 0 0.7264 0

0 1 0 0.7264

0.7264 0 1 00 0.7264 0 1

Table 5. Measured correlation matrix as a function of path

Measured correlation matrix for Path 1 is:

1 0.005 0.693 0.025

0.005 1 0.004 0.732

0.693 0.004 1 0.023

0.025 0.732 0.023 1

Measured correlation matrix for Path 2 is:1 0.014 0.704 0.008

0.014 1 0.009 0.742

0.704 0.009 1 0.005

0.008 0.742 0.005 1

Measured correlation matrix for Path 3 is:

1 0.026 0.687 0.007

0.026 1 0.004 0.728

0.687 0.004 1 0.014

0.007 0.728 0.014 1

The measured results are very consistent with the correlation coefficientsdesired for this configuration. As a result, the PXB does an excellent job ofcreating MIMO channels with a desired amount of spatial correlation thatsimulates real-world test conditions.


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Per-path correlation versus per-channel correlation

When standardizing test conditions for emulating MIMO channels, the correla-tion properties can be per-path or per-channel. Per-path correlation meanseach tap uses a different correlation matrix, while per-channel correlationmeans all the taps use same correlation matrix. As shown in Equation 34, thespatial correlation coefficient between two antenna elements is a function ofantenna spacing, the PAS, and the radiation pattern of the antenna elements.The PAS is a function of path AoA/AoD, and path AS. In real-world conditionsit is not universal that all the paths have the same AoA/AoD and AS values,therefore different paths could have different correlation coefficients. The useof per-path correlation may improve the channel emulation performance. Inorder to emulate this real-world scenario, the MIMO channel models used forMobile WiMAX and the WLAN 802.11n standards use the per-path correlationbased on different AoA/AoD for each path. While per-path spatial correlationcan closely model a real wireless channel, it has a high level of computationalcomplexity. Fortunately, the PXB has pre-defined channel models to automati-cally configure the instruments path correlations.

Some wireless standards such as LTE, in an effort to reduce the modelscomplexity, have recommended the per-channel correlation model withoutconsidering the path AoA/AoD information. When testing a MIMO systemagainst these specifications, the PXB can also provide per-channel correlationsusing pre-defined models, or, as shown in Figure 32, provide a simple tableentry for custom per-channel emulation models.

When emulating wireless channels it is important to understand the testrequirements for spatial correlations. Fortunately the PXB has the flexibility tosupport both per-path correlation and per-channel correlation, either individuallyor at the same time.

Figure 32. PXB per-channel correlation setup screen.


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Theoretical MIMO channel capacity

In order to give a more intuitive impression of channel capacity loss caused byfading correlation, Figure 33 compares the capacity as a function of SNR for a2x2 MIMO system with different correlation coefficients at the transmitter ()and the receiver (). Compared with the completely independent MIMO channel(= = 0), the figure shows that there is a little capacity loss for the lowcorrelated channels (= = 0.3). For highly correlated channels (= = 0.95)at high SNR, the capacity decreases by 3.9 bps/Hz as compared to the idealuncorrelated case. For completely correlated channels (= = 1.0), the capacitydecreases by 4.4 bps/Hz at high SNR. Note that even when the correlationcoefficients are 1, there is still an increase in capacity relative to SISO, thoughthe improvements are small, by increasing the number of antenna pairs. Thelargest improvements are achieved when the channels are independent. In thiscase, the MIMO capacity is improved by, approximately, the SISO capacitymultiplied by min(Nt, Nr)

1. Appendix A provides additional details for thetheoretical derivation of the MIMO channel capacity.

Figure 33.Ergodic capacity for a 2x2 MIMO system with different transmit and receive correlationcoefficients.

1. C. Chuah, D. Tse, J. Kahn and R. Valenzuela, Capacity scaling in MIMO wireless systems under correlatedfading, IEEE Trans. Inf. Theory, 48(3), 637-650, March 2002.


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Conguring the channel emulator to achievethe desired correlation

Under ideal conditions, MIMO systems provide dramatic channel capacity gainthrough increased spatial diversity. As previously shown however, the capacitygain is reduced if the fading characteristic among various channels is correlated.

Many wireless standards, such as WiMAX and LTE, recommend test scenariosthat use correlated channel matrices. One approach that is widely accepted fordefining the correlation properties of a MIMO channel relies on the -parameter.In this case, provides an indication of the correlation as it relates to capacity.The capacity, operating under a specified SNR, is defined as a linear interpola-tion of the capacity for a completely correlated MIMO channel to that of anuncorrelated MIMO channel. Using the -parameter, the resulting capacity isdefined as

Equation 47

where C0is the channel capacity without correlation and C1 is the channelcapacity when the channels are completely correlated. With this approach, it issimple to specify the expected correlation degree through the -parameter. Forexample, in LTE channel model for 2x2 antenna configurations, the medium andhigh correlation matrices are defined using values of 0.5 and 0.9,respectively. For a system operating with a target value for and under a speci-fied SNR, the correlation matrix can be tuned to achieve the desired correlation.

( ) 0 11C C C = +


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The correlation matrix that can guarantee the expected capacity, C, is notunique, and different correlation matrices can be chosen to satisfy this capacityrequirement. One very flexible method to achieve the desired correlation matrixis to adjust the antenna configuration, such as element spacing and/or polar-ization. As an example, the BS antenna spacing is adjusted using a 2x2 MIMOconfiguration with vertically polarized antennas, as shown in Figure 30(a), until

the desired correlation is achieved. The antenna parameters for the MS arefixed with values shown in Table 6. The receiver correlation coefficient, , iscalculated using Equation 29.

For this example, the BS uses a 3-sector antenna configuration with AS=2degrees and AoD = 50 degrees. The BS correlation coefficient, , changeswhen adjusting the spacing between the BS antenna elements. Under thisconfiguration, the combined spatial-polarization correlation matrix can be calcu-lated using Equation 45. If the calculated channel capacity is lower than thedesired channel capacity, the antenna spacing is increased to reduce thecorrelation and thus increase the channel capacity. By iteratively adjusting theantenna spacing, the desired can be achieved. Table 7 shows correlationindex as a function of BS antenna spacing for this 2x2 MIMO example.

Table 7. Relationship between BS antenna spacing, correlationcoefficient and channel capacity under specified SNR values

Antennaspacing d

Correlationcoefficient

SNR = 10 dB

SNR = 20 dB

0 1.0000 0.9060 0.9445

0.5 0.7390 + 0.6700i 0.9004 0.9270

1.0 0.0969 - 0.9854i 0.8921 0.8806

1.5 0.5827 + 0.7857i 0.8543 0.8189

2.0 0.9433 - 0.1881i 0.8252 0.7598

3.0 0.2687 + 0.8779i 0.7591 0.6542

4.0 0.7955 + 0.3350i 0.6958 0.5636

5.0 0.3854 - 0.7028i 0.6246 0.4951

6.0 0.6061 - 0.4196i 0.5704 0.4389

7.0 0.4388 + 0.5106i 0.5232 0.3971

Table 6. MS (Receiver) antenna configuration

Antenna spacingin wavelength

Antennatype

AS(degrees)

AoA(degrees)

Correlationcoefficient ()

MS 0.5 Omni 35 67.5 0.6905 + 0.3419i


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Applying SNR to MIMO channels

A convenient place for setting the channels Signal to Noise Ratio (SNR) istypically at the receiver. The signal power can be accurately measured with apower meter and the channel emulator can generate the required noise accordingto the desired SNR. This technique is valid for SISO systems and MIMO systemsthat have uncorrelated channels. When the MIMO channels are correlated, analternate approach for measuring the signal power and generating noise isrequired.

SNR for SISO and uncorrelated MIMO channels

For SISO systems, the received signal, Y, is defined as

Equation 48

where X is the transmitted data, H is the channel coefficient and N is the noise.For a specified SNR, the signal power, S, is first measured at the output of thechannel emulator in the absence of noise. The covariance of the noise, 2,being a random Gaussian process, can be calculated and added by the channel

emulator to simulate the effect of applying SNR to the SISO channel. As shownin Appendix B, this technique is also valid for uncorrelated MIMO channels. Inthis case, the signal at the receiver can also be defined using Equation 48where X is now a vector of Mttransmitted signals, H is the channel coefficientmatrix with Mrrows and Mtcolumns, and Yis a vector of Mrreceived signals.In the MIMO case, Nis an Mrrow of random Gaussian processes. It is alsoshown in Appendix B that the signal power can be measured at either thereceiver or the transmitter for the uncorrelated MIMO system.

SNR for correlated MIMO channels

When the MIMO channels are correlated, the measured signal power at thereceiver side can be dependent on the correlation of the channels. This correla-

tion dependency prevents a channel emulator from accurately configuring theMIMO system for a desired SNR, using power measurements at the receiver. Toovercome this difficulty, the channel emulator can use measurements of thesignal power at the transmitter to appropriately set the required SNR. Thefollowing derivation shows a simplified example using a 2x1 MISO system todemonstrate an appropriate measurement technique for configuring the SNR ina channel emulator when the channels are correlated.

Y HX N = +


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The MISO pre-coding matrix is defined as

Equation 49

The signal transmitted from antenna 1 is2

X, the signal transmitted from

antenna 2 is2

Xe j, and the transmitted signal power from each antenna is S.

The channel between transmit antenna 1 and the receive antenna is H1. Thechannel between transmit antenna 2 and the receive antenna is H2. UsingEquation 48, the received signal becomes

Equation 50

If H1is independent with H2 , the received signal power is

Equation 51

where H1and H2represent average channel gains of H1and H2respectively.When H1 =H2 = H, then E(YY*) = 2HS.

If H1is completely correlated with H2, meaning that H1 =H2 = H, then thereceived signal becomes

Equation 52

and the received signal power is

Equation 53

je

1

2

1

2

j

1 H2

XeH

2

XY

+=

S)HH()YY(E 21* +=

2

XH)e1(Y j+=

SH))cos(1(2)YY(E * +=


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When = /4 , the received signal power becomes 2(1 + 2/2)HS, which isdifferent from the case with independent channel conditions. Therefore if themeasured signal power at receiver is used to calculate the noise power requiredfor a specific SNR value, then the added noise power will vary according to thefading correlation property. Continually adjusting the noise power as a functionof correlation property introduces unnecessary complexity into the measure-

ment and may result in reduced accuracy when configuring the measurementSNR. To overcome this difficulty, the PXB defines the SNR relative to thetransmitted signal power and uses the following SNR definition

Equation 54

where S1and S2are the signal powers from each transmitter. With thisdefinition, the PXB measures the signal power at the transmitters prior tofading and then adds the appropriate noise power to achieve the desired SNR.In this technique the noise contribution can be determined without consideringthe fading correlation property of the channel. Additional information regardingSNR in correlated MIMO channels is provided in Appendix B.

22211 HSHSSNR

+=


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Conguring SNR using the PXB

The complexity of controlling and calibrating the signal power and noise powerin a MIMO test are eliminated when using the PXB. The PXB uses automatedsignal routing and power calibration to precisely control the SNR in eachchannel of the MIMO setup. Figure 34 shows the PXB menu for entering therequired SNR over the desired integration and noise bandwidths. The integrationbandwidth (BW) is typically set to the channel bandwidth of the MIMO receiver.The noise power is usually spread over a wide bandwidth but is calibrated overthe integration BW when calculating SNR, see Figure 35.

Figure 34. AWGN settings using the PXB.

Figure 35. Signal and noise bandwidth definitions.

IntegrationBW

NoiseBW

SNR (dB)


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Configuring custom correlation matrices targeted for a specific application canbe a lengthy process. Fortunately, the PXB provides a set of pre-defined MIMOchannel models based on the specifications of several wireless standardsincluding Mobile WiMAX and LTE. The instrument is configured using a simplemenu structure for selecting the channel model. In addition, custom correlationmatrices and path definitions may be created and saved using the instruments

interface as shown in Figure 36.

Figure 36. PXB fading model selection menu.

For signal generation, the PXB has up to 4 built-in baseband generatorsallowing the creation of standard-compliant signals with support for up to4x2 MIMO in one instrument. The baseband generators can also play backwaveforms created by the Keysight Signal Studio1software tool which providesan extensive library of standard-compliant and specialized waveforms.

ConguringStandard-CompliantMIMO Channelsusing the PXB

1. For more information about Keysight Signal Studio, visit www.keysight.com/find/signalstudio.
http://www.keysight.com/find/signalstudiohttp://www.keysight.com/find/signalstudiohttp://www.keysight.com/find/signalstudio


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The PXB can be configured to fade RF signals with the addition of MXA signalanalyzers, for downconverting and digitizing the RF signal for real-time fadinginside the PXB. For example, Figure 37 shows the PXB configuration blockdiagram for a 2x2 MIMO system with RF fading using two MXA signal analyzers.For this configuration, the PXB connects the faded signals to MXG signal gen-erators for upconversion back to RF.

Figure 37. PXB configuration block diagram for RF-to-RF fading of a 2x2 MIMO signal.

The PXB hardware can be configured with up to 12 DSP blocks. Each DSPblock can be configured as a baseband generator with 512 Msamples ofplayback memory or as a real-time fader with up to 24 paths. The flexibility inconfiguring the channel hardware allows up to 4 baseband generators to be usedwith up to 8 faders. The DSP blocks deliver modulation and fading bandwidthsup to 120 MHz. The PXB can also sum up to 4 baseband generators for inter-ference and mixed-modulation testing. Power and noise calibration is quicklyand accurately performed by the PXB eliminating the need for complicated andlengthy system calibrations. The PXB supports analog and digital I/Q outputconnections to numerous N5102A digital signal interface modules and KeysightRF signal generators. It also supports RF inputs from MXA signal analyzers.

The flexibility of the PXB to support RF, analog and digital interfaces for signalgeneration and real-time fading of MIMO signals delivers an advanced testcapability when developing and validating components and systems for currentand emerging wireless systems.


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Keysight Application Note, Concepts of High Speed Downlink Packet Access:Bringing Increased Throughput and Efficiency to W-CDMA,Literature number 5989-2365EN, January 18, 2007

Keysight Application Note 1509, MIMO Wireless LAN PHY Layer [RF] Operation& Measurement, Literature number 5989-3443EN, September 16, 2005

Keysight Application Note 1578, IEEE 802.16e WiMAX OFDMA SignalMeasurements and Troubleshooting, Literature number 5989-2382EN,June 6, 2006

Related Literature


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The wireless channel can be modeled as a linear time-varying system by usingh(tk, t)to define the channel impulse response at time tand delaytk, wherek = 0, , L 1and L is the multi-path number. Denoting the impulse responsebetween the jth transmit antenna and the ith receive antenna by hi, j(tk, t), theMIMO channel with Nt transmit antennas and Nr receiver antennas is given bythe Nrx Ntmatrix h(tk, t)as

(A-1)

Further, given that the signal sj(t)is launched from the jth transmit antenna,the signal, yj(t), received at the ith receive antenna, is given by

(A-2)

For the ideal MIMO channel, each hi, j(drop the tk, tfor convenience) has thesame property as a SISO wireless channel where all the channels are indepen-dent and uncorrelated. In a realistic MIMO channel, there exists some degreeof correlation between the channels and as a result, directly affects the diversitygains achievable by the MIMO system.

Defining the ideal channel matrix as Hwand the realistic channel matrix as H, itis known that Hcan deviate significantly from Hwas the result of the spatialcorrelation characteristics previously discussed. Correlation in the MIMOchannel implies that the elements of Hare correlated and may be modeled by

(A-3)

Appendix A:Theoretical Model forMIMO Channel Capacity

( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

=

t,ht,ht,h

t,ht,ht,h

t,ht,ht,h

t,H

kN,Nk2,Nk1,N

kN,2k2,2k1,2

kN,1k2,1k1,1

k

trrr

t

t

( ) ( ) ( )=

=

=tN

1j

1L

0k

kjkj,ii tst,hty

( ) ( )w21 HvecRHvec =


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Where Ris the correlation matrix previously defined in the section titledCombined spatial and antenna polarization correlation. If R = INtNr, thenH=Hw. Although the model described above is capable of capturing any corre-lation effects between the elements of H, a simpler and less generalized modelis often adequate and is given by

(A-4)

Where Rrand Rtare positive definite Hermitian matrices that specify thereceive and transmit correlations, respectively. Note that when describing theBS as the transmitter and the MS as the receiver, Rt = RBSand Rr = RMS,respectively. It can be shown that R, Rt,and Rrare related by

(A-5)

Where Wdenotes Kronecker product.

Channel capacity is defined as the maximum error-free data rate that a channel

can support. In an independent, identically distributed (i.i.d.) Rayleigh fadingenvironment, the capacity of a MIMO system with Nttransmit antennas and Nrreceive antennas is defined as

(A-6)

Where idenotes the eigenvalues of Hw Hw,may be interpreted as the aver-age SNR at each of the receive antennas. The channel capacity scales almostlinearly with min(Nt,Nr)for high SNR. This linear growth is due to the fact thatin a richly scattered MIMO channel, path gains between different transmit/receive antenna pairs tend to fade independently, which makes it likely that

multiple parallel channels will be formed, allowing several independent datastreams to be transmitted simultaneously. In a practical MIMO channel, thecapacity potential offered by multiple antennas suffers from correlationbetween local antenna elements. With the correlated channel model describedin Equation (A-4) above, the capacity of the MIMO channel in the presence offading correlation is

(A-7)

This capacity equation assumes that the transmitter does not have anyknowledge of the channel characteristics.

21tw21r RHRH =

rt RRR W=

MIMO Channel Modelling

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