ELG4179: Wireless Communication Fundamentals S.Loyka. Frequency-Selective and Time-Varying Channels

Transcription

1 Frequeny-Seletve and Tme-Varyng Channels Ampltude flutuatons are not the only effet. Wreless hannel an be frequeny seletve (.e. not flat) and tmevaryng. Frequeny flat/frequeny-seletve hannels Frequeny response of the hannel: Channel as a lnear flter FT ( ) ( ) h τ H f (5.) H x( f ) ( f ) H ( f ) x( f ) ( ) ( ) f f H f - hannel frequeny response X f - sgnal s spetrum f - hannel oherene bandwdth f - sgnal bandwdth f f f f a) f < f - frequeny flat; b) f > f--frequeny seletve Dstortonless transmsson: j2 f H ( f ) = a e π τ (5.2) Leture 5 -Ot-7 (9)

2 Impulse Response of a Wreless Channel The ause of frequeny seletve hannel: delay spread. Consder mpulse response of the hannel. Gven the nput sgnal ( ) A - omplex ampltude, s t, the sgnal ( ) A x t at the hannel output s N (5.3) = ( ) = A s( t τ ) x t = j ae ϕ τ - delay of -th multpath, there are N delayed omponents, LOS always arrves frst. The mpulse response s One mpulse at Tx -> many mpulses at Rx (why?) h N (5.4) = ( τ ) = A δ( τ τ ) Leture 5 -Ot-7 2(9)

3 P.M. Shankar, Introduton to Wreless Systems, Wley, Input-output relatonshp ( ) ( ) ( ) ( ) ( ) x t = h τ s t τ dτ = s τ h t τ dτ 0 t (5.5) Wreless hannel an be modeled as a lnear system (may be tmevaryng). Delay spread s a key to FS hannels. Average delay and meansquare delay are: τ = Pτ where P - power of the -th omponent. P 2 τ = P 2 τ P (5.6) Leture 5 -Ot-7 3(9)

4 Delay spread (RMS) s ELG479: Wreless Communaton Fundamentals S.Loyka 2 ( ) ( ) 2 2 τ = τ τ = τ τ (5.7).e. the standard devaton of the delay. τ haraterzes tmespreadng of the pulse n the hannel. Realst example: T.S. Rappaport, Wreless Communatons, Prente Hall, 2002 Leture 5 -Ot-7 4(9)

5 T.S. Rappaport, Wreless Communatons, Prente Hall, 2002 Leture 5 -Ot-7 5(9)

6 Frequeny-Seletve Propertes Compare delay spread τ and symbol duraton T : Frequeny-seletve : T τ, Frequeny-flat: T >> τ (5.8) Coherene bandwdth f of the hannel: ~ / τ or f = / τ (5.9) f where s a onstant, usually ; e.g. = 0.2 for 0.5 orrelaton. The same an be expressed usng sgnal (RF) bandwdth f 2 / T : s frequeny-seletve: fs > f frequeny-flat: f f T 0 τ (5.0) s Error floor effet (to be dsussed later). Leture 5 -Ot-7 6(9)

7 Example: two-ray model ( ) ( ) ( ) ( ) h τ = δ τ + aδ τ τ H f = + ae 2 2 ( ) = ( + osθ ) + ( sn θ) H f a a 2 = + a + 2aosθ where θ = ω τ = 2πf τ jω τ (5.) Tap-delay model: s( t ) τ a x( t ) 2 Magntude Channel Frequeny Response normalzed frequeny f τ a= a=0.5 a=0. H ( f ) = 2 + a + 2a os 2πf τ Leture 5 -Ot-7 7(9)

8 Consder spef ases: () Frequeny-flat hannel: 2πf τ << θ f <<, so that 2 π τ H ( f ) + a - frequeny-ndependent (flat). Another rteron (less strt): 2 πf τ < π / 2 f < 4 τ (2) Frequeny-seletve: f, or 2 π τ f 4 τ Q.: Usng (5.7), fnd the delay spread (RMS) for the two-ray model. Leture 5 -Ot-7 8(9)

9 Doppler Spread and Tme-Varyng Channels Consder movng MS: v v BS MS Doppler effet: frequeny shft by v fd = f0 fms = f0 ± fd (5.2) Consder movng MS at angle: BS θ v MS v v fd = f0 os θ, fms = f 0 os + θ (5.3) Multpath hannel: -th path: f d v = f0 osθ (5.4) Leture 5 -Ot-7 9(9)

10 Tme-varyng frequeny response j If 0t e ω s transmtted, the Rx sgnal (at MS) s jω0t j( ϕ + 2πft) jω x t e a e e 0τ = (5.5) ( ) N = v where f = f0 os θ. The frequeny response s N j( 0 ) 2 ( 0, j ft H f t) ae ϕ ω τ π = e h( τ, t) (5.6) =.e., a funton of tme! T.S. Rappaport, Wreless Communatons, Prente Hall, 2002 Leture 5 -Ot-7 0(9)

11 Tme-varyng mpulse response If f = 0, the hannel s fxed (not tme-varyng). If f s large, the hannel s fast-fadng (varyng). If f s small, the hannel s slow-fadng. How large (small) s large (small)? Compare f d wth T.S. Rappaport, Wreless Communatons, Prente Hall, 2002 T s : 2π fdts > fast fadng 2 π fdts slow fadng (5.7) where T s s the symbol/blok duraton. θ θ + θ? Q: how to dede fast/slow when all [ ] 0 Leture 5 -Ot-7 (9)

12 Coherene tme of the hannel Introdue a oherene tme of the hannel*: T = (5.8) 2 π f Ths s the tme when the hannel approxmately does not hange, an be onsdered fxed (stat). d The hannel s onsdered stat for t > T. t < T and tme-varyng for Fast/slow fadng an be expressed as Ts > T Ts << T fast fadng slow fadng (5.9) Note: the error floor effet exsts for both ases. *) another defnton: π 2π fdt = T = 2 4 f d Leture 5 -Ot-7 2(9)

13 Consder τ = 0 :.e. (, ) ELG479: Wreless Communaton Fundamentals S.Loyka Example: two-ray model ( ) j ω τ j d t H f = + ae e ω (5.20) ( ) H f = + e ( ) jω H f = + a + 2a osω t H f t. It s the same as before f θ = ω dt = 2π fdt. Reall that ω = ω 0 v / d d t 2 d (5.2) 2 Magntude Channel Frequeny Response normalzed tmef d t a= a=0.5 a=0. Leture 5 -Ot-7 3(9)

14 Doppler Spetrum Consder many multpath omponents wth unform θ v What s the pdf of fd = f0 os θ? ( ) ( ) θ ( ) / 2, [ 0,2 ] ρ θ dθ = ρ f df ρ ( f ) = θ ρ θ = π θ π (5.22) f d d f d f a f d d,max where a s a normalzaton onstant, and f,max f0 v / 2 (5.23) d = s the maxmum Doppler frequeny. It an be shown that Doppler power spetrum PSD( f d ) s the same as ρ f ( f d ) (provded that unform angular dstrbuton holds). ( ) PSD f d f = f + f d 0 f d,max f d,max f d Leture 5 -Ot-7 4(9)

15 Note that a sngle-tone Tx sgnal results n spread-out spetrum at Rx! ( fd = 2 fd,max ) Moble wreless hannel s a funton of spae and tme! Random hannel: oherene tme s defned as a tme nterval for whh envelope orrelaton 0.5. Coherene bandwdth: frequeny nterval for whh envelope orrelaton 0.5. Note: other orrelaton level an be used. Leture 5 -Ot-7 5(9)

16 T.S. Rappaport, Wreless Communatons, Prente Hall, 2002 ELG479: Wreless Communaton Fundamentals S.Loyka Leture 5 -Ot-7 6(9)

17 Dfferent Forms of Fadng?? P.M. Shankar, Introduton to Wreless Systems, Wley, Leture 5 -Ot-7 7(9)

18 Overvew of System-Level Propagaton Effets ~ 's λ < λ temporal varatons multpath P.M. Shankar, Introduton to Wreless Systems, Wley, (modfed) Leture 5 -Ot-7 8(9)

19 Summary Impulse and frequeny responses of a wreless hannel. Delay spread and frequeny seletve hannels Tap-delay model. Power delay profle. Doppler spread and tme-varyng hannels. Envelope orrelaton. Coherene bandwdth and oherene tme of the hannel. Classfaton of fadng and propagaton effets Readng: o Rappaport, Ch. 5 (exept 5.8). o Other books (see the referene lst). Note: Do not forget to do end-of-hapter problems. Remember the learnng effeny pyramd! Leture 5 -Ot-7 9(9)