Chapter 6 Wdeband channels 128
Delay (tme) dsperson A smple case Transmtted mpulse h h a a a 1 1 2 2 3 3 Receved sgnal (channel mpulse response) 1 a 1 2 a 2 a 3 3 129
Delay (tme) dsperson One reflecton/path, many paths Impulse response 0 2 3 4 Delay n excess of drect path 130
Narrowband versus Wdeband Systems -1 Transmtted Sgnal Receved Sgnal In a wde-band system, the shape and duraton of the receved sgnal H ( f ) n the frequency doman or h ( t ) n the tme doman s dfferent from the shape of the transmtted sgnal H s ( f ) frequency doman h s ( t ) tme doman Transmtted Sgnal Receved Sgnal In a narrow-band system, the shape of the receved sgnal s the same as the transmtted sgnal. H c ( f ) = transfer functon frequency doman h c ( t ) = mpulse response tme doman -1 Transmtted Sgnal Receved Sgnal These dagrams show the tme doman and the frequency doman responses of both systems. Transmtted Sgnal Receved Sgnal
Narrow- versus wde-band Channel frequency response H f db B 1 f B 2 132
System functons (1) Tme-varant mpulse response Due to movement, mpulse response changes wth tme Input-output relatonshp: yt Tme-varant transfer functon H(t,f) Perform Fourer transformaton wth respect to Input-output relatonshp Ht,f Y f xt ht,d ht,expj2fd ht, XfHt,fexpj2ftexpj2 f tdfdt becomes Y(f)=X(f)H(f) only n slowly tme-varyng channels 133
Transfer functon, Typcal urban 134
System functons (2) Further equvalent system functons: Snce mpulse response depends on two varables, Fourer transformaton can be done w.r.t. each of them -> four equvalent system descrptons are possble: Impulse response Tme-varant transfer functon Spreadng functon S, ht,expj2tdt Doppler-varant spreadng functon B,f S,expj2fd 135
Wreless Channels Interrelaton Between Determnstc System Functons Fourer Transform wrt t (delay) Tme varant mpulse response Fourer Transform wrt t (tme) Tme varant Transfer functon Doppler varant Impulse response - spreadng functon S(Doppler, delay) Fourer Transform wrt t (tme) Doppler varant transfer functon B(Doppler shft, frequency) Fourer Transform wrt t (delay)
Stochastc system functons ACF - autocorrelaton functon (second-order statstcs) R h t,t,, Eh t,ht, Input-output relatonshp: R yy t,t Rxx t,t R h t,t,, dd Exam physcal propertes of the channel --> smplfy correlaton functon --> WSS (Wde-Sense Statonary) + US (Uncorrelated scatters --> assumptons lead to WSSUS Model 136
The WSSUS model: mathematcs If WSSUS s vald, ACF depends only on two varables (nstead of four) ACF of mpulse response becomes R h t,t t,, P h t, P h t,...delay cross power spectral densty ACF of transfer functon R H t t,f f R H t,f ACF of spreadng functon R s,,, P s, P s,...scatterng functon Wde-Sense Statonary (WSS) assumpton depends only on the dfferences n t - t' where the statstcal propertes of the channel don't change wth tme. Fadng stll a dynamc factor, just the statstcs are statonary whch leads to a quas-statonary envronment over a tme nterval (movement of less than 10 λ). Uncorrelated Scatters (US) assumpton depends on dfferences n frequency. Not truly vald n an ndoor envronment where for example scatters off a wall are correlated. Thus WSSUS assumptons more applcable to the outdoors. Popular model (WSSUS) assumptons but not necessarly vald. 137
Dgtal Representaton WSSUS Model
Condensed parameters Correlaton functons depend on two varables For concse characterzaton of channel, we desre A functon dependng on one varable or A sngle (scalar) parameter Most common condensed parameters Power delay profle Rms delay spread Coherence bandwdth Doppler spread Coherence tme 138
Channel measures Copyrght: Shaker 140
Condensed parameters Power-delay profle One nterestng channel property s the power-delay profle (PDP), whch s the expected value of the receved power at a certan delay: P E ht, 2 t E t denotes expectaton over tme. For our tapped-delay lne we get: N P E exp t t j t 1 N N 2 2 E t t 2 1 1 Average power of tap. 2 141
Condensed parameters Power-delay profle (cont.) We can reduce the PDP nto more compact descrptons of the channel: Total power (tme ntegrated): Pm P d Average mean delay: T m S P d P Average rms delay spread: m 2 P d P m T m For our tapped-delay lne channel: N 2 Pm 2 T m S 1 N 1 2 N 1 P m 2 2 2 2 P m T m 142
Condensed parameters Frequency correlaton A property closely related to the power-delay profle (PDP) s the frequency correlaton of the channel. It s n fact the Fourer transform of the PDP: exp 2 f f P j f d 1 For our tapped delay-lne channel we get: N 2 2 exp 2 f f j f d 1 N 2 2 exp j2f 143
Condensed parameters Coherence bandwdth f f f 0 f 0 2 B C f 144
Channel measures Copyrght: Shaker 145
Condensed parameters The Doppler spectrum Gven the scatterng functon P s (doppler spectrum as functon of delay) we can calculate a total Doppler spectrum of the channel as:, PB PS d For our tapped delay-lne channel, we have: P S 2, P B 2 2 2,max 1 2 2 d N 2 2,max 2 2 2 2,max Doppler spectrum of tap. 146
Condensed parameters The Doppler spectrum (cont.) We can reduce the Doppler spectrum nto more compact descrptons of the channel: Total power (frequency ntegrated): PB, m PB d Average mean Doppler shft: T Bm, S B PB d P Bm, Average rms Doppler spread: 2 P d P Bm, T B, m For our tapped-delay lne channel: N 2 PB, m 2 T 1 Bm, 0 S B N 1 2 2 v,max P Bm, 147
Channel measures Copyrght: Shaker 148
Condensed parameters Coherence tme Gven the tme correlaton of a channel, we can defne the coherence tme T C : t t t 0 t 0 2 T C t 149
Condensed parameters The tme correlaton A property closely related to the Doppler spectrun s the tme correlaton of the channel. It s n fact the nverse Fourer transform of the Doppler spectrum: t exp 2 t P j t d B For our tapped-delay lne channel we get N 2 t t exp j2t d 1 N N 1 1 2 2 2 2,max 2 2 2 2,max 2 J 2 0,max t exp j2t d Sum of tme correlatons for each tap. 150
It s much more complcated than what we have dscussed! Copyrght: Shaker 151
Double drectonal mpulse response TX poston RX poston number of multpath components for these postons N r ht, r TX, r RX,,, h t, r TX, r RX,,, 1 delay drecton-of-departure drecton-of-arrval h t, r TX, r RX,,, a e j 152
Physcal nterpretaton l 153
Drectonal models The double drectonal delay power spectrum s sometmes factorzed w.r.t. DoD, DoA and delay. DDDPS,, APS BS APS MS PDP Often n realty there are groups of scatterers wth smlar DoD and DoA clusters DDDPS,, P c k APS c,bs k APS c,ms k PDP c k k 154
Angular spread Es,,,s,,, P s,,, double drectonal delay power spectrum DDDPS,, P s,,,d angular delay power spectrum ADPS, DDDPS,,G MS d l angular power spectrum APS APDS,d power P APSd 155
Matlab/Smulnk - Communcatons System Toolbox