Moelg of he lear me-vara chael Sve-Gusav Häggma
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1. Characerzao of he lear me-vara chael 3 The rasmsso chael (rao pah) of a rao commucao sysem s mos cases a mulpah chael. Whe chages ae place he propagao evrome e.g. he rao saos are moble, reflecors a scaerers are movg, or he meum self (roposphere, oosphere) s chagg, he he chael respose wll also chage as a fuco of me, he rao pah s fag. If he chages are slow so ha he chael s almos cosa uer he urao of a sgle aa symbol or eve uer he urao of a frame coag hures of symbols, he chael ca be characerze as quasvara, a he chael respose o oe symbol or oe frame ca be calculae usg he formulas of me vara sysems. I he rasmsso of a log message he chael s, however, graually chagg a he chages ca be represee e.g. wh he jo esy fuco of he mulpah chael ap pah amplues a elays a her correlao fucos. Whe he chael s sgfcaly chagg urg he rasmsso of a aa frame ew chael represeaos mus be use. The feaures of hese mus also be ow, so ca be ece whe he chael ca be moele as quas-vara. The mpulse respose h() or he rasfer fuco H(f), whch s he Fourer rasform of he mpulse respose, fully escrbes he lear mevara sysem. I he characerzao of lear me-vara sysems several ew sysem fucos are rouce. Ther physcal erpreao s o always easy, a her cocepo s raher laborous. Hereafer we wll use he abbrevaos LTV-chael for he Lear Tme-Vara chael a LTI-chael for he Lear Tme-Ivara chael. Bello has evelope he heory of LTV-sysem alreay 1963 [1] for characerzao of he roposcaer chael. Goo escrpos are also clue refereces [2] a [3]. I Fg. 1 he approach use here s presee. 1 [ 1] P.A.Bello: Characerzao of raomly me-vara lear chaels. IEEE Tras., Vol CS- 11, No. 4, Dec. 1963, pp. 36-393. [2] R.Seele (eor): Moble rao commucaos. Loo 1992, Peech Press, 779p.
LTV-chael approach 4 Trasmsso chaels LTI-chaels LTV-chaels Raom LTV-chaels Deermsc LTV-chaels Sysem fucos of he eermsc LTV-chael Aeuao srbuos of he fla fag chael 4-mesoal auocorrelao fucos of he raom LTVchael 2-mesoal auocorrelao fucos of he WSSUSchael 1-mesoal specral fucos Scalar parameer represeaos Fg. 1 [3] D.Parsos: The moble rao propagao chael. Loo 1992, Peech Press, 316p.
1.1 Calculao of LTV-chael sgal respose. 5 The calculao of he sgal respose s frs presee for he scree LTV-mulpah chael. The resul s he geeralze for a arbrary LTVchael. The sgal aalyss s performe for complex low-pass sgals. Fg. 2 shows he behavor of he LTV-mulpah chael a ealze suao. The ealsao meas e.g. ha he sgal passes usore over all pahs. The sgal urao s so shor ha he sgals over ffere pah o o overlap a he recever. The pu sgal s a arrow pulse, whch s rasme a he me sas 1, 2, 3, a 4. The chael has of course some mmum elay, before he respose sars. O he frs me sa 1 he chael s a hree-pah chael a he pulse s aeuae vually o each pah bu he pulse shape remas uchage. The me-vara aure of he chael ca be see from he varyg umber, aeuao, a elay of he pahs a ffere me sas. From hs ca be coclue ha he mulpah chael respose epes boh o he me of arrval of he pu pulse (me) a of he me pas sce ha (elay). The mpulse respose s a fuco of boh me a elay, whle he me-vara chael s oly a fuco of elay. The pu a oupu sgals are real he fgure bu geerally he oupu sgal s complex whe low-pass represeao of he sysem s use. Also he pu sgal ca be complex. If he elay fferece bewee he pahs s less ha he sgal urao, he pah resposes wll overlap a he respose wll be very complex a he mulpah srucure cao be recly observe. Complex low-pass sysem represeao of he sgals s use. The physcal ba-pass pu a oupu sgals are hus: L NM = R ST U VW 1L = 2NM + R s () = Re ST ze j 2π f cu () VW 1 ze j 2π () f c = + z () e j 2π f c 2 r we j 2π f c we j 2π f c () Re () () w() e j 2π f c O QP O QP (1) (2) where fc s he carrer frequecy use.
6 The me-vara chael pulse respose a four ffere me sas z(- ) 1 w() z(- 2 ) w() z(- ) 3 w() z(- ) 4 w() M() 1 α = j f 2π cτ() w () = () e z τ () c h Fg. 2
7 From Fg. 2 ca be see ha he physcal oupu sgal of he LTVmulpah chael ca be expresse as M () 1 α = r () = () s τ () (3) where - α() s he ga of he :h propagao pah as fuco of me, - τ() s he propagao elay of he :h propagao pah as fuco of me, - M() s he umber of propagao pahs as fuco of me. Isero of Eq. (1) Eq. (3) gves: M () 1 r () = α ()Re z τ () e = R S M () 1 2π c τ () = Re α() z τ() e T = RM () 1 j2πf j Re () e cτ() 2π = α z τ () e S T = R S T j2πf c τ () j f U V W U V W fcu V W (4) I he las form of Eq. (4) he complex low-pass oupu sgal ca be recogze: M () 1 α = j f 2π w e cτ () () () = z τ () (5) Eq. (5) ca also be wre as: L N M M () 1 w () = h () δλ τ () P z( λ) = O Q (6) where eoes covoluo a j2πf h () () e cτ () = α (7)
From hs follows he mpulse respose of he LTV-mulpah chael: 8 M () h( λ, ) = h ( ) δ λ τ ( ) = (8) where h(λ,) s he respose a me o a mpulse arrve λ secos earler. Eqs. (5) - (8) are val for a scree mulpah chael. I some rao chaels e.g. he roposcaer chael a me-couous mpulse respose s a more aequae moel. The he sgal respose s calculae wh a geeralze covoluo egral: z w () = h( λ,)( z λ ) λ (9) The respose of he mulpah chael o a sewave z e j 2π ()= f z (1) s obae by serg he pu sgal Eq. (1) Eq. (5), whch gves M () 1 2π τ 2π τ w () = α () e e = M () 1 = = α () e j f j f c () z () j2 fc+ fz j2 fz π τ () π e (11) A smple example wll show ha he pah elay varao a o he amplue varao wll maly eerme he yamc characerscs of he LTV-chael. Example 1. I Fg. 3 s assume ha he refleco coeffce s -1. The geomercal legh of he propagao pahs s calculae usg he rgh-agle ragles a he mage prcple. The pah amplue gas a elays follow recly from he legh of he propagao pahs. The logarhmc amplue of he receve sgal as a fuco of me s calculae assumg he carrer frequecy 2 GHz a he spee of he recever 15 m/s. The geomercal values are gve Fg. 3.
The graph Fg. 3 shows he logarmc sgal amplu as fuco of me assumg frs ha oly he elays epe o me, a he ha boh elays a gas are chagg. I hs smple suao he me epeece s very regular, whch s early equal uer boh assumpos. From hs ca be coclue ha he behavor s eerme almos solely by he elay chages a he chages pah gas oly have mor effecs. Ths ca be geeralse o almos all rao chaels. The reaso for hs behavour s of course ha elay chages are mulple by he carrer frequecy he expoeal fuco Eq. (11). 9 Example 2. The moble chael ca a small area be moele wh he mpulse respose M 1 j2 = πν h( λ, ) = h e δ λ τ (12) Ths s a scree M-pah chael where he complex pah gas h a he elays τ o o chage wh me. As a cosequece of he cosa spee each pah has s ow cosa Doppler-shf ν = v α c f cos (13) c where v o spee of he moble sao, c s he spee of he rao wave, fc s he carrer frequecy, a α s he agle bewee he :h propagao pah a he moble sao velocy vecor. I realy h, τ, α, a M are fucos of me bu he small rego where he moble sao moves uer oe rasmsso frame hey are approxmaely cosa. The raomess of hs chael appears so ha he moel parameers afer some few frames have obae ew values.
1 x v y ()= c + h+ 2 2 ()= c + h+ c2 h r x v y o 2 2 r x v y 1 c α o () = τ r () () = r o () α 1 () = r 1 () τ c () = 1 r 1 () 1 B 5 5,4 B 5,2-5 5, 1,8 1,9 /s 2-1 -15-16,5 B -17, -2,2,4,6,8 1, 1,2 1,4 /s 1,6 1,8 2, Fg. 3-17,5 1,945 1,95 /s 1,955
3.2.2 The sysem fucos of a LTV-chael 11 The basc sysem fuco of a eermsc LTV-chael a of a sample fuco of a raom LTV-chael s me vara mpulse respose h(λ,) escrbe above, also ow as he chael pu elay sprea fuco. Oher frequely use sysem fucos are: he me-vara chael rasfer fuco: p a f H( f, ) = λ h( λ, ) = zh( λ, )exp j2 πfλ λ (14) whch s obae by Fourer-rasformg he mpulse respose wh respec o he elay varable λ. I escrbes he complex evelope of he oupu sgal whe he pu sgal s ej2πf. he chael oupu Doppler sprea fuco: p a f D( f, ν) = H( f, ) = zh( f, )exp j2 πν (15) whch s obae by Fourer-rasformg he rasfer fuco wh respec o he me varable, a whch escrbes he chael frequecy respose o he frequecy f + ν, whe he pu sgal s ej2πf. he elay-doppler sprea fuco p a f S( λν, ) = h( λ, ) = zh( λ, )exp j2 πν (16) s obae by Fourer-rasformg h(λ,) wh respec o he me varable or by ag he verse Fourer-rasform of he Doppler-sprea fuco wh respec o frequecy f. I gves he complex ga of he chael o he elay erval τ + τ a he Doppler-shf erval ν + ν. The Fourer-rasform relaos of hese four sysem fucos are show Fg. 4. I ao four oher ual sysem fucos ca be efe.
REPRESENTATION OF THE LTV-CHANNEL 12 SYSTEM FUNCTIONS OF A DETERMINISTIC CHANNEL z() h(λ,) w() λ a f λ a f a ν f aλν f λ w () = zh( λ,)( z λ) λ j2πf = H( f, ) Z( f) e f u j = z u 2πνλ S( λν, ) z( λ) e νλ j f = z u 2π D( f νν, ) Z( f ν) e νf Fg. 4
Example 2 coues The saaeous sysem fucos of hs chael moel are: he me-vara mpulse respose (moel efo): 13 M 1 j2 = πν h( λ, ) = h e δ λ τ he me-vara rasfer fuco: (17) M 1 p j2 j2πfτ = πν H( f, ) = h(, ) = h e λ λ e (18) he oupu Doppler-sprea fuco: M 1 p = D( f, ν) = H( f, ) = h δ ν ν e j2πτ f (19) he elay-doppler-sprea fuco: M 1 p = S( λν, ) = h( λ, ) = h δν ν δλ τ (2) I Fg. 5 he mpulse respose, amplue frequecy respose, a he elay- Doppler-sprea fucos are show for a 2-pah chael wh he parameer values: h1 = 1, h2 = 9, τ = 1µ s, τ = 5µ s 1 2 ν = 1Hz, ν = 5Hz 1 2 From he fgure or from Eqs. (17) - (2) appears ha he chael mpulse respose epeece of me ca be see oly from he phase behavour of he complex pah ga, whch s a lear fuco of me (I he fgure raw moulo 2π). I he rasfer fuco ca be see as a glg of he rasmsso mmum hrough he sgal bawh. he elay-doppler-sprea fucos coas mpulses whch ell he elay a Doppler-shf of each pah.
14 h(λ,) 1,5 λ/µs 5,1,2 /s Arg{h(λ,)} 2 ra -2 ra λ/µs 5,1,2 /s H(f,) 1 f/hz 5 1 2 3 4 /ms 25 1 S(λ,ν) -1 ν/hz 1 λ/µs 5 Fg. 5