Rec. ITU-R P.47-3 RECOMMEDATIO ITU-R P.47-3 Multpath propagaton and parameterzaton of ts characterstcs (Queston ITU-R 3/3) (999-3-5-7) Scope Recommendaton ITU-R P.47 descrbes the nature of multpath propagaton and defnes the approprate parameters for the statstcal descrpton of multpath effects, and provdes examples of correlaton effects among multple propagaton paths and ther computaton. The ITU Radocommuncaton Assembly, consderng a) the necessty of estmatng the effects of multpath on servces employng dgtal systems; b) that t s desrable to standardze the termnology and expressons used to characterze multpath, recommends that, to descrbe the concepts of multpath n a consstent manner, the terms and defntons gven n Annex should be employed. Annex Introducton In rado systems wth low antenna heghts, there are often multple ndrect paths between the transmtter and recever due to reflectons from surroundng objects, n addton to the drect path when there s lne-of-sght. Such multpath propagaton s partcularly sgnfcant n urban envronments, where the sdes of buldngs and paved road surfaces provde strong reflectons. As a result, the receved sgnal conssts of the summaton of several components havng varous ampltudes, phase angles and drectons of arrval. The resultng spatal varablty of sgnal strength can be vewed as havng two regmes: a) rapd fadng whch vares over dstances of the order of a wavelength due prmarly to changes n phase angles of dfferent sgnal components; b) slow fadng whch vares over larger dstances due prmarly to changes n shadowng loss by surroundng objects. In addton, the varous sgnal components can be Doppler shfted by dfferent amounts due to the movement of the moble or of reflectng objects such as vehcles. The multpath moble channel can be characterzed n terms of ts mpulse response whch vares at a rate dependent on the speed of the moble and/or the scatterers. Therefore, a recever has to be
Rec. ITU-R P.47-3 able to cope wth the sgnal dstorton arsng from echoes n the channel as well as the rapd changes n the nature of ths dstorton. Such characterstcs of the moble rado channel are descrbed by the power delay profles and the Doppler spectra whch are obtaned from wdeband channel soundng measurements. Sgnals transmtted to and from movng vehcles n urban or forested envronments exhbt extreme varatons n ampltude due to multple scatterng. Fades of 3 db or more below the mean level are common. The nstantaneous feld strength when measured over dstances of a few tens of wavelengths s approxmately Raylegh-dstrbuted. The mean values of these small sector dstrbutons vary wdely from area to area, dependng on the heght, densty and dstrbuton of hlls, trees, buldngs and other structures. Multpath propagaton characterstcs are a major factor n controllng the qualty of dgtal moble communcatons. Physcally, multpath propagaton characterstcs mply multpath number, ampltude, path-length dfference (delay), and arrval angle. These can be characterzed from the power delay profle. Alternatvely, the Fourer transform of the complex mpulse response results n the complex transfer functon whose ampltude versus frequency characterstcs gve the frequency selectvty of the multpath and s related to the correlaton bandwdth. Defntons of small sector (or small-scale) channel parameters are gven n and 3. Statstcs of small-scale parameters are subsequently used to produce cumulatve dstrbuton functons (CDF). Medum-scale CDF covers a partcular route of measurement, whch s of the order of tens to hundreds of metres. The combned data set from a number of medum-scale routes s consdered to be large-scale or global characterzaton whch s representatve of the surveyed envronment e.g. hlly terran, urban, suburban, ndoor large rooms, corrdors, etc. A tme-varyng lnear channel can be characterzed by a lnear transversal flter. The output of ths flter contans a sum of delayed, attenuated and Doppler shfted versons of the nput sgnal. The channel s then represented by the delay-doppler-spread functon, sometmes referred to as the scatterng functon. Ths functon represents the multpath phenomenon n the three dmensons of excess delay, Doppler frequency and power densty. Ths formulaton s partcularly sutable for realzng a hardware smulator n the form of a dynamc transversal flter. Multpath parameters. Defntons The approprate parameters for the statstcal descrpton of multpath effects are gven below. The average delay s the power weghted-average of the excess delays measured and s gven by the frst moment of the power delay profle (the square of the ampltude of the mpulse response). The r.m.s. delay spread s the power weghted standard devaton of the excess delays and s gven by the second moment of the power delay profle. It provdes a measure of the varablty of the mean delay. The delay wndow s the length of the mddle porton of the power delay profle contanng a certan percentage (typcally 9%) of the total energy found n that mpulse response. The delay nterval s defned as the length of the mpulse response between two values of excess delay whch mark the frst tme the ampltude of the mpulse response exceeds a gven threshold, and the last tme t falls below t. The threshold used depends on the dynamc range of the measurng equpment: a typcal value s db below the peak level of the delay profle. The correlaton bandwdth s defned as the band of frequences for whch the autocorrelaton functon of the transfer functon s above a gven threshold; a typcal threshold value s.5.
Rec. ITU-R P.47-3 3 The number of multpath or sgnal components s the number of peaks n a power delay profle whose ampltude are wthn A db of the hghest peak and above the nose floor.. Dscusson The approprate parameters for the statstcal descrpton of multpath effects can be computed ether from nstantaneous power delay profles or from average power delay profles whch represent ether tme averages obtaned when the recever s statonary and represents the movement n the envronment or spatal averages obtaned when the recever s n moton. Examples of these are shown n Fg. whch were obtaned wth a van where the mddle profles were obtaned wth the van statonary and the other two profles were obtaned whle the van was movng. Both types of averages should be computed over a number of mpulse responses wthn the coherent ntegraton tme of the channel defned as the tme duraton (or spatal nterval) over whch the multpath components have not moved by + half a tme delay bn (or range bn). FIGURE Uplnk Downlnk (db) (db) 4 4 5 5 5 5 (db) (db) 4 5 5 4 5 5 (db) (db) 4 5 5 4 5 5 Power delay profles for the frequency dvson duplex bands of UMTS wth tme averagng (mddle profle) and spatal averagng (top and bottom profles). The profles are normalzed to dsplay relatve power densty as a functon of τ. Horzontal lne shows the db delay nterval. The total energy, P m, of the mpulse response s: t3 P = t) d t () m t
4 Rec. ITU-R P.47-3 where: t) : power densty of the mpulse response t : delay wth respect to a tme reference t : nstant when t) exceeds the cut-off level for the frst tme t 3 : nstant when t) exceeds the cut-off level for the last tme. The average delay, T D, s gven by the frst moment of the power delay profle: where: T D = τ e τ τ τ) e τ) d τ d τ τ a (a) τ : excess tme delay varable and s equal to t t τ a : τ e = t 3 t. In dscrete form equaton (a) becomes: arrval tme of the frst receved multpath component (frst peak n the profle) T D = = = τ τ ) τ ) where = and are the ndces of the frst and the last samples of the delay profle above the threshold level, respectvely, and M s the ndex of the frst receved multpath component (frst peak n the profle). The delays may be determned from the followng relatonshp: r τ M t ( µ s) = 3. 3 km where r s the sum of the dstances from the transmtter to the multpath reflector, and from the reflector to the recever, or s the total dstance from the transmtter to recever for t LOS. The root mean square (r.m.s.) delay spread, S, s defned by the square root of the second central moment: (b) S = τ e ( τ T D τ e τ a ) τ) d τ τ) d τ (3)
Rec. ITU-R P.47-3 5 or, n dscrete form: S = = ( τ T D = τ M τ ) ) τ ) The delay wndow, W q, s the length of the mddle porton of the power delay profle contanng a certan percentage, q, of the total power: W q = (t t ) (5) whereby the boundares t and t are defned by: t t q t) d t = t t 3 q t) d t = P q and the energy outsde of the wndow s splt nto two equal parts P m. The delay nterval, I th, s defned as the tme dfference between the nstant t 4 when the ampltude of the power delay profle frst exceeds a gven threshold P th, and the nstant t 5 when t falls below that threshold for the last tme: I th = (t 5 t 4 ) (7) The Fourer transform of the power densty of the mpulse response provdes the autocorrelaton C( f ) of the transfer functon: τ e m C( f ) = τ) exp ( j π f τ) d τ (8) For a Rcan channel, equaton (8) underestmates the correlaton bandwdth. For such channels t s more accurate to estmate the correlaton bandwdth from the spaced frequency correlaton functon, whch s obtaned from the tme varant complex transfer functon by computng the correlaton coeffcent for dfferent frequency spacngs. The correlaton bandwdth B x s defned as the frequency for whch C( f ) equals x% of C( f = ). Delay wndows for 5%, 75% and 9% power, delay ntervals for thresholds of 9, and 5 db below the peak and correlaton bandwdth for 5% and 9% of correlaton are recommended for analyss of data. It s worth notng that the effects of nose and spurous sgnals n the system (from RF to data processng) can be very sgnfcant. Therefore, t s mportant to determne the nose and/or spurous threshold of the systems accurately and to allow a safety margn on top of that. A safety margn of 3 db s recommended, and n order to ensure the ntegrty of results, t s recommended that a mnmum peak-to-spurous rato of, for example, 5 db (excludng the 3 db safety margn) s used as an acceptance crteron before an mpulse response s ncluded n the statstcs. An example of the use of some of these terms s gven n Fg.. (4) (6)
6 Rec. ITU-R P.47-3 55 FIGURE t t 6 Power densty (dbm) 65 7 75 t 4 t 5 8 t t 3.5.5 Power delay profle llustratng the followng parameters: the delay wndow, W 9, contanng 9% of the receved power s marked between the two vertcal dashed lnes (t and t ), the delay nterval, I 5, contanng the sgnal above the level 5 db below the peak, les between t 4 and t 5. t and t 3 ndcate the start and the end of the profle above the nose floor. FIGURE 3 Power delay profle ndcatng multpath components above threshold level
Rec. ITU-R P.47-3 7 3 Parameters of drecton of arrval 3. Defntons The defntons of the approprate parameters for the statstcal descrpton of multpath effects are gven below: The average arrval angle s the power-weghted average of the measured drectons of arrval and s gven by the frst moment of the power azmuth spectrum. (It can also be called the power angular profle.) The power angular profle s the angular power characterstc wthn the azmuth/horzontal plane. The r.m.s. angular spread s the power-weghted standard devaton of the drecton of arrval and s gven by the second moment of the power angular profle. It provdes a measure of the varablty of the mean arrval angle. The angular wndow s the wdth of the mddle porton of the power angular profle contanng the defned certan percentage of the total energy found n that power angular profle measurement. The angle nterval (or angular spacng) s defned as the wdth of the mpulse response (or wdth of the angular profle) between two values of drecton of arrval. It marks the frst angle at whch the ampltude of the angular profle exceeds a gven threshold, and the last angle at whch t falls below that threshold. The threshold used depends on the dynamc range of the measurng equpment: a typcal value s db below the peak level of the angular profle. 3. Dscusson The approprate parameters for the statstcal descrpton of multpath effects can be computed from ether nstantaneous power angular profles or short-term power angular profles or long-term power angular profles. Ths quantty represents ether tme averages obtaned when the recever s statonary and represents the movement n the envronment, or spatal averages obtaned when the recever s n moton. As shown n Fg. 4, short-term power angular profles are obtaned by spatally averagng the nstantaneous power angular profles over several tens of wavelengths n order to suppress the varaton of rapd fadng; long-term power angular profles are obtaned by spatally averagng the short-term power angular profles at approxmately the same dstance from the base staton (BS) n order to suppress the varaton due to shadowng.
8 Rec. ITU-R P.47-3 FIGURE 4 Defnton of power angular profles 3.. Total energy Let the receved power n the drecton θ be θ). The total energy, P, of the angular profle s defned as the power beyond the threshold level L whch separates the sgnal from nose, as shown n Fg. 5: where: θ P = θ)dθ (9a) θ 3 θ: measured from the drecton of the prncpal sgnal (assumed to be statonary wthn the duraton of the measurement) (rad) θ): power of the angular profle above the threshold level L ; below L, θ) = L : θ : θ 3 : level wth some margn (3 db recommended) over the nose floor arrval angle when θ) exceeds the threshold level L for the frst tme n ( π, π) θ max In dscrete form equaton (9a) becomes: arrval angle when θ) exceeds the threshold level L for the last tme n θ max ( π, π). = P θ ) = P ( (9b) where = and are the ndces of the frst and the last samples of the power angular profle above the threshold level, respectvely.
Rec. ITU-R P.47-3 9 FIGURE 5 Total energy 3.. Average arrval angle The average arrval angle, T A, s gven by the frst moment of the power angular profle: In dscrete form equaton (a) becomes: θ 3 T A = θ θ)dθ (a) P T θ = A = = θ θ ) θ ) (b) where = and are the ndces of the frst and the last samples of the power angular profle above the threshold level, respectvely. 3..3 r.m.s. angular spread The r.m.s. angular spread S A of the drecton of arrval s defned as follows: In dscrete form: equaton (a) becomes: θ3 ( θ TA) θ S A = θ)dθ (a) P S A = = ( θ T ) = A θ ) θ ) (b) where = and are the ndces of the frst and the last samples of the power angular profle above the threshold level, respectvely.
Rec. ITU-R P.47-3 3..4 Angular wndow The angular wndow, θ w, s the wdth of the mddle porton of the power angular profle contanng a percentage q, of the total power as shown n Fg. 6: whereby the boundares θ w and θ w are defned by: θ θ w θ = θ θ () w q θ)dθ = w w w θ 3 θ q θ)dθ = P q and the energy outsde of the wndow s splt nto two equal parts P. (3) FIGURE 6 Angular wndow 3..5 Angle nterval (angular spacng) The angle nterval A th, s defned as the angle dfference between the angle θ 4 when the ampltude of the power angular profle frst exceeds a gven threshold L th, and the angle θ 5 when t falls below that threshold for the last tme as shown n Fg. 7: A = θ (4) th 5 θ 4 FIGURE 7 Angle nterval
Rec. ITU-R P.47-3 3..6 Spatal correlaton dstance In partcular for multple-output multple-nput (MIMO) channels, the spatal correlaton coeffcent for dfferent spacng d s obtaned from the angle varant complex transfer functon of the power angular profle. The spatal correlaton coeffcent R(d), s defned as follows: where: θ3 θ θ) exp( jπd sn θ / λ)dθ R ( d) = (5) θ3 θ θ)dθ d : dstance for dfferent spacng λ: wavelength. As shown n Fg. 8, the spatal correlaton dstance d c s defned as the frst cut off dstance at whch R(d) equals x% of R(d = ). R( d c ) / R() = x / (6) FIGURE 8 Spatal correlaton dstance 3..7 Recommended parameters Angular wndows for 5%, 75% and 9% power, angle ntervals for thresholds of 9, and 5 db below the peak, and correlaton dstances for 5% and 9% of correlaton are recommended to permt a detaled analyss of data. Furthermore, t s worth notng that the effects of nose and spurous sgnals n the system (from RF to data processng) can be very sgnfcant. Therefore, t s mportant to determne the nose and/or spurous threshold of the systems accurately and to provde a safety margn on top of that. A safety margn of 3 db s recommended, and n order to ensure the ntegrty of results, t s recommended that a mnmum peak-to-spurous rato of, for example, 5 db (excludng the 3 db safety margn) be used as an acceptance crteron lmtng the angular profles ncluded n the statstcs. Fgure 9 shows an example of the effect of settng the magntude of mnmum peak-to-l th rato ( L). In ths Fgure, the power angular profle s assumed to be a Laplace dstrbuton (double exponental dstrbuton) wth angular spread of 4 ; angular spread and angular nterval are calculated as functons of the peak power-to-l th rato. Ths fgure shows
Rec. ITU-R P.47-3 that these parameters undergo sgnfcant changes even for essentally dentcal values. Therefore the value used as L n the statstcal evaluaton should be specfed. FIGURE 9 Example of effect for mnmum peak-to-l th rato ( L) Annex Introducton Ths Annex llustrates some results of computng the correlaton coeffcents from a power angular profle and the effect of the correlaton coeffcents on MIMO capacty. Computng the spatal correlaton coeffcents The defnton n equaton (5) of Annex has been used to compute the spatal correlaton. Ths Annex brefly ntroduces a result and llustrates how the correlaton s affected by antenna spacng. Fgure shows an deal truncated Laplacan power-azmuth spectrum (PAS) such as: PAS L c QL, k, k ( ϕ) = exp { ε[ ϕ ( ϕ, k ϕk )] ε[ ϕ ( ϕ, k + ϕk )]} k= σ L, k ϕ ϕ σ L, k where ε(ϕ) s the step functon and c the number of clusters, ϕ,k s the mean angle of ncdence of k-th cluster, σ L,K s the angular spread. PAS s defned over[ϕ ϕ,ϕ + ϕ]. The power normalzaton condton s assumed as: c k Q L, k exp = = σ k L, k (7) ϕ (8)
Rec. ITU-R P.47-3 3 FIGURE Ideal truncated Laplacan power-azmuth spectrum (PAS) ormalzed Laplacan PAS for a two-cluster case. AS = 3, ϕ [ 9, + 9 ]. Addtonally, the +9º cluster has half the power of the 9º case. o o Then the envelope correlaton coeffcent s gven by: ρ ( D) = R ( D) jr ( D) (9) e XX + where: D = πd/λ d: antenna spacng λ: wavelength, and the cross-correlaton functons R XX (D) and R XY (D) are defned n equaton (6). Fgure llustrates the resultng spatal correlaton. XY
4 Rec. ITU-R P.47-3 FIGURE Resultng spatal correlaton Envelope correlaton coeffcent versus the normalzed dstance =d/λ for the two-cluster case shown n Fg.. 3 Effect of the correlaton coeffcents on MIMO capacty For Raylegh fadng channels, the ergodc MIMO capacty wthout channel knowledge at the transmtter s: / H ( ) = + P H H RR log In HwRT Hw R R R nt σ P / H C = log det I + n RR HwRT H R w nt σ det () where: n R and n T : number of recever and transmtter antennas, respectvely P: total transmt power I n R : n R n R dentty matrx ( ) H and det( ): Hermtan and determnant operaton, respectvely H w : matrx whose elements are ndependent dentcally-dstrbuted complex Gaussan Random varables wth zero mean and unt varance ( ) ½ : Hermtan square root of a matrx. The matrces R R and R T determne the spatal correlatons between the recevers and the / transmtters, respectvely, where the channel matrx H s defned by H = R H / R / R, / T and R are postve defnte Hermtan matrces, and fnally they are assumed to be normalzed such that [R R ] j,j for j =,K,n R and [R T ], for =,K,n T. R w T, R
Rec. ITU-R P.47-3 5 By assumng that R R and R T have full rank and n R = n T = n, then at hgh S/ the capacty can be approxmated as: C P ( R ) log det( R ) H log det HwH w + log det R + T () n Tσ. From the arthmetc mean- If we denote the egenvalues of R R by λ, =,K,n, then = λ = geometrc mean nequalty: n Snce det ( R R ) = λ n = λ n n (), t follows that log det(r R ), and s zero only f all egenvalues of R R are = equal,.e. R R = I n. Thus, the correlaton determnes the MIMO capacty and the loss n the ergodc capacty at hgh S/ s gven by (log det(r R ) + log det(r T )) bt/s/hz. Fgure llustrates the effect of spatal correlatons on the ergodc capacty of a MIMO channel wth n R = n T =. In the fgure R T = I, s assumed. The recever correlaton matrx s chosen accordng to: ρ R = R R *R ρ (3) where ρ R denotes the spatal correlaton between the receve antennas. FIGURE Ergodc capacty wth low and hgh receve correlaton