Multipath and Doppler Effects and Models

Transcription

1 Multipath ad Doppler Effects ad Models There are two parts i this lecture. I Part I, we will first itroduce the mutipath propagatio effects ad Doppler frequecy shift/spread effects. I Part II, we will briefly itroduce multipath ad Doppler chael models.

2 Backgroud: Overviews o Wireless Chael Modelig We eed to ask the followig three importat questios while desigig a wireless commuicatio lik:. Fadig & Power Loss: Is the sigal to iterferece plus oise ratio (SINR) large eough for the receiver to detect the trasmitted sigal? 2. Sigal Distortio: Ca the sigal distortio be igored, predicted or removed so that we kow how to recover the trasmitted iformatio at the receiver? 3. Time Variatio: Ca the receiver adapt faster eough to the variatios of the above two features (SINR & sigal distortio)? A complete wireless chael model should provide quatitative measures of SINR, sigal distortio ad time variatio. For SINR (referred i questio ), we eed oly to cosider the time ivariat trasmissio loss at a sigle frequecy (that is the RF carrier frequecy). The frequecy ad time depedet properties of sigal ca be addressed i aswerig questios 2 ad 3. Sigal distortio (referred i questio 2) is caused by the frequecy depedet variatios of the received sigal stregth ad phase. The primary source of frequecy depedet variatios is multipath propagatio. Here, we eed oly to cosider time ivariat situatios ad leave the time varyig features to questio 3. Motios of receivers, trasmitters or wireless eviromets geerate Doppler effects. With Doppler effects, sigal frequecies shift ad spread. These Doppler effects will cause time variatios i the received sigal stregth ad wave shape. This kid of time varyig features is usually radom ad ca be modeled as stochastic processes. I order to addressig these three importat issues, we divide wireless chael modelig ito three parts: Trasmissio loss -sigle frequecy (or arrowbad sigal) -time ivariat eviromet (or short observatio time period) Frequecy depedet chael impulse respose or trasfer fuctio -multiple frequecies (or broadbad sigal) -time ivariat eviromet (or short observatio time period) Time varyig chael impulse respose or trasfer fuctio -multiple frequecies (or broadbad sigal) 2

3 -time varyig eviromet (or log observatio time period) We have discussed trasmissio loss (icludig path loss, short term fadig ad log term fadig) of a sigle frequecy respose i time ivariat eviromets i the previous lecture. Both frequecy depedet ad time varyig features of a chael impulse respose (or trasfer fuctio) will be itroduced i this lecture. 3

4 Part I: Multipath ad Doppler Effects After studyig this ote, studets will be able to. Uderstad multipath chael effects i both time ad frequecy domais 2. Uderstad Doppler effects i both time ad frequecy domais 3. Uderstad multipath ad Doppler effects i both time ad frequecy domais I. Multipath Chael Effects: Time Ivariat Case (No Doppler effects) I wireless commuicatio eviromets, a sigal trasmitted from the trasmitter reaches the receiver through may differet paths as illustrated i Figure. Figure : Multipath propagatio Let s(t) is the trasmitted sigal. The received sigal ca the be writte as a sum of multipath arrivals: L yt ( ) = ast i ( τ i), τ τ2 τ3... τl () i= Here, L is the total umber of multipath arrivals, a i ad τ i are the amplitude ad arrival time of the i th ray, respectively. A. s(t) is a time harmoic (i.e., sigle frequecy or siusoidal) sigal j t Cosider the trasmitted sigal st ( ) = e ω. The, the received sigal is L L j ( ) ( ) ω t τ jωt j = = ( ω) with ( ω) = ωτ = = (2) yt ae H e H ae 4

5 Here, H ( ω ) is defied as the trasfer fuctio of the multipath eviromet. Note that the receiver sigal y(t) remais as a time harmoic sigal with the same agular frequecy ω as the trasmitted sigal s(t). Thus, o distortio i wave shape has occurred durig the trasmissio of s(t) through a time ivariat multipath eviromet. However, the magitude of the sigal has bee modified. The ew magitude is H ( ω ) which is a fuctio of agular frequecy ω. We use the followig matlab code to geerate the Figure 2: ======================================================== clear all; % amplitudes of 7 multipath arrivals a=[ ]; % arrival times of 7 multipath arrivals t=[ ]; i=0; % frequecy idex for omega=0:0.05:00; % agular freuecies multipath_arrival=a.*exp(j*omega*t); i=i+; abs_h(i)=abs(sum(multipath_arrival)); % the i-th trasfer fuctio ed omega=0:0.05:00; plot(omega, abs_h) ylabel('amplitude of trasfer fuctio') xlabel('agular freuecy') title('frequecy depedet multipath fadig') Here, we use 7 multipath arrivals. The amplitudes ad arrival times of these seve multipath arrivals are radomly chose. From Figure 2, the magitude of received sigal fluctuates as agular frequecy chages. For some frequecies, the 7 multipath arrivals iterfere destructively ad yield small H ( ω ). For other frequecies, the 7 multipath arrivals iterfere costructively ad yield large H ( ω ). This pheomeo is called multipath fadig. Figure 2 shows that multipath fadig is frequecy depedet. Other kids of fadig will be discussed i future lectures. 5

6 Figure 2: Multipath fadig is a fuctio of frequecy. Sice the amplitudes ad arrival times of multipath arrivals deped o locatios of trasmitter ad receiver, the received sigal stregth will also deped o the locatios of trasmitter ad receiver. For example, cosider a two ray model where lie of sight (LOS) ad reflected rays are the two multipath arrivals. Let the trasmitter atea height be h t ad the receiver atea height be h r. The horizotal distace betwee the trasmitter ad the receiver is deoted as d. From Figure 3, the travel distace for the LOS ray is r = d + ( h h ) 2 2 LOS t r ad the travel distace for the reflected ray is 2 2 r = d + ( h + h ) ref t r The the trasfer fuctio is b b LOS / ref b LOS jωr c b ref LOS j2 π rlos / λ ref Hd ( ) = e + R e = e + Re r r r r jωr / c j2 π rref / λ LOS ref LOS ref 6

7 where R is the reflectio coefficiet ad the coefficiets b LOS ad b ref are fuctios of atea patters, trasmitted power, etc. For coveiece, we choose b LOS =, b ref = ad R=- i our example. Thus, j2 π r / 2 / ( ) LOS λ j π r Hd e e ref λ = r r LOS ref LOS 0m Reflected d 2m Figure 3: two-ray model We will first plot the magitude of Hd ( ) agaist the distace d usig the followig matlab code. If the frequecy f=ghz, the wave legth is λ = c/ f =0.3 m because the wave speed c=3*0 8 m/sec. Let h t =0m, h r =2m. ============================================================= clear all ht=0;hr=2; c=3e8;f=e9;lambda=c/f; R=-; d=:0.5:0000; d=sqrt(d.^2+(ht-hr)^2); d2=sqrt(d.^2+(ht+hr)^2); a=exp(j*2*pi.*d/lambda)./d; a2=r*exp(j*2*pi.*d2/lambda)./d2; a=abs(a+a2); ld=log0(d);la=log0(a); figure (4) plot(ld,la); xlabel('log0(distace)') ylabel('log0(magitude)') title( 'two ray model') ============================================================== 7

8 Figure 4: Mutipath effects as a fuctio of distace betwee source ad receiver. Please see lecture 2 for detailed discussios. Secodly, we plot the magitude of H( f ) agaist the frequecy f for four distaces d=50m, 300m, 800m ad 2000m usig the followig matlab code: ============================================================= clear all ht=0;hr=2; c=3e8;r=-; f0=e8; fi= [::000];fd= ;f= f0+fd*fi; lambda=c./f; da=[50,300,800,2000]; for i=:legth(da) d=da(i); d=sqrt(d.^2+(ht-hr)^2); d2=sqrt(d.^2+(ht+hr)^2); Td=(d2-d)/c a=exp(j*2*pi*d./lambda)/d; a2=r*exp(j*2*pi*d2./lambda)/d2; a(i,:)=abs(a+a2); ed figure (5) 8

9 subplot(2,2,);plot(f,a(,:));title('d=50m');ylabel('magitude') subplot(2,2,2);plot(f,a(2,:));title('d=300m');ylabel('magitude') subplot(2,2,3);plot(f,a(3,:)); title('d=800m');xlabel('frequecy');ylabel('magitude') subplot(2,2,4);plot(f,a(4,:)); title('d=2000m');xlabel('frequecy');ylabel('magitude') ============================================================== Figure 5 Frequecy characteristics of Multipath fadig at four locatios. From Figures 4 ad 5, we coclude that frequecy characteristics of Multipath fadig are locatio depedet. Note that the frequecy separatio of two adjacet deep fades is i each case i Figure 5 is /TD where TD is the travel time differece for the two rays: rref rlos TD = c 9

10 B. s(t) cotais multiple frequecy compoets As show i equatio (2), the trasfer fuctio of a wireless commuicatio chael with multipath arrivals ca be writte as L j H( ω) = ae ωτ = Here, a ad τ are the amplitude ad time-delay of the th ray, respectively. As show i (), for a iput sigal s(t) with multiple frequecies, the output of the chael ca be writte as L yt () = ast ( τ ) = Whe the sigal s(t) cosists of multiple frequecy compoets, jωt st () = S( ω) e dω 2π (3) where S( ω ) is the spectrum of s(t). The spectrum of y(t) ca be writte as L j Y( ω) = H( ω) S( ω) = a S( ω) e ωτ (4) = Cosider the followig 6-ray model as a example. The amplitudes are defied as a : [, 0.3, -0.8, 0.5, -0.4, 0.2]. We will cosider two kids of arrival time distributios: Case : τ : [0, µs, 2µs, 3µs, 4µs, 5µs] Case 2: τ : [0, 0.µs, 0.2µs, 0.3µs, 0.4µs, 0.5µs] The delay separatio betwee the first arrival ad the last arrival is 5µs i Case ad is oly 0.5 µs i Case 2. For the time beig, the delay separatio is called as delay spread. I future lectures, delay spread will be defied i other ways. Cosider the trasmitted sigal is a square pulse with pulse width equal to 5µs,. Time Domai View: We use the followig matlab code to geerate the time domai view of trasmitted sigals ad received sigals for both cases. From Figure 6, we observe that multipath arrivals cause distortio. The larger the delay spread is, the worse the distortio becomes. 0

11 clear all; a=[,0.3,-0.8,0.5,-0.4,0.2];t=[0,,2,3,4,5;0,0.,0.2,0.3,0.4,0.5]; sigal=[0, zeros(,0),oes(,50),zeros(,000)]; % trasmitted sigal for k=:2; %for two cases for i=:6; ray(i,:)=a(i)*[0, zeros(,(00*t(k,i))),oes(,50),zeros(,(000-00*t(k,i)))]; ed y(k,:)=sum(ray(:,:ed)); ed t=((::legth(y(,:)))-)*0^(-2); subplot(2,2,);plot(t,sigal); ylabel('trasmitted sigal s(t)'); title('case & case 2') axis([ ]) subplot(2,2,2);plot(t,y(,:)); ylabel('received sigal y(t)'); title('case : large delay spread') subplot(2,2,4); plot(t,y(2,:)); xlabel('time(us)'); ylabel('received sigal y(t)'); title('case 2: small delay spread')

12 Figure 6: Trasmitted ad received sigals for two cases. 2

13 2. Frequecy Domai View: We use the followig matlab code to geerate the frequecy domai view of trasmitted sigals ad received sigals for both cases. At first, FFT is employed to implemet (3) to fid the iput spectrum. Secodly, (2) is used to compute the chael trasfer fuctios. Fially, (4) is used to compute the output spectrum. clear all; s=[oes(,0),zeros(,90)]; % trasmitted sigal s_f=fft(s); x=s_f([:50]); y=s_f([5:00]); sigal_f=[y,x]; %iput spectrum dt=5/0; % each time iterval is 0.0 micro sec df=/(00*dt); f_s=df*([0:99]-50);% frequecy vector a=[,0.3,-0.8,0.5,-0.4,0.2]; %amplitudes f=f_s; w=2*pi*f; t_=[0,,2,3,4,5]; % arrival times for case for i=:6; h(i,:)=a(i)*exp(-j*w*t_(i)); ed h_=sum(h(:,:ed));%trasfer fuctio y_=h_.*sigal_f;%output spectrum t_2=[0,0.,0.2,0.3,0.4,0.5]; % arrival times for case 2 for i=:6; h2(i,:)=a(i)*exp(-j*w*t_2(i)); ed h_2=sum(h2(:,:ed));%trasfer fuctio y_2=h_2.*sigal_f;%output spectrum figure() subplot(2,3,); plot(f_s,abs(sigal_f)); ylabel('magitude');title('i/p spectrum') subplot(2,3,4); plot(f_s,agle(sigal_f)); ylabel('phase'); xlabel('frequecy(mhz)'); 3

14 subplot(2,3,2); plot(f,abs(h_)); title('chael ') subplot(2,3,5); plot(f,agle(h_)); xlabel('frequecy(mhz)'); subplot(2,3,3); plot(f,abs(h_2)); title('chael 2') subplot(2,3,6); plot(f,agle(h_2)); xlabel('frequecy(mhz)'); figure(2) subplot(2,3,); plot(f_s,abs(sigal_f)); ylabel('magitude');title('i/p spectrum') subplot(2,3,4); plot(f_s,agle(sigal_f)); ylabel('phase'); xlabel('frequecy(mhz)'); subplot(2,3,2); plot(f,abs(y_)); title('o/p spectrum ') subplot(2,3,5); plot(f,agle(y_)); xlabel('frequecy(mhz)'); subplot(2,3,3); plot(f,abs(y_2)); title('o/p spectrum 2') subplot(2,3,6); plot(f,agle(y_2)); xlabel('frequecy(mhz)'); =============================================================== 4

15 The iput spectrum ad chael fuctios for both cases are show i Figure 7. The magitude fuctios are o the top row ad the phase fuctios are o the bottom row. From the left colum, oe ca see the iput spectrum lies primarily i [-200kHz 200kHz]. For chael 2 (right colum), the magitude of trasfer fuctio is basically flat ad the phase of the trasfer fuctio is basically liear i this iterval. Therefore, chael 2 will cause little distortio ad this kid of chael is called to have flat fadig. For chael (ceter colum), the magitude of trasfer fuctio is ot flat ad the phase of the trasfer fuctio is ot liear i this iterval. Therefore, chael will cause more distortio ad this kid of chael is called to have frequecy selective fadig. Figure 7: Iput spectrum (left colum), trasfer fuctio of case (ceter colum), ad trasfer fuctio of case 2 (right colum). 5

16 The iput spectrum ad output spectra for both cases are show i Figure 8. The magitude fuctios are o the top row ad the phase fuctios are o the bottom row. The iput spectrum, show i the left colum, is the same as that i Figure 7. As metioed before, it lies primarily i [-200kHz 200kHz]. For chael 2, the output spectrum (right colum) is very similar to the iput spectrum. Therefore, chael 2 will cause little distortio ad this kid of chael is called to have flat fadig. For chael, the output spectrum (right colum) is ot similar to the iput spectrum. Therefore, chael will cause more distortio ad this kid of chael is called to have frequecy selective fadig. Figure 8: Iput spectrum (left colum), output spectrum of case (ceter colum), ad output spectrum of case 2 (right colum). From Figures 7, the variatio rate (with respect to frequecy) of the trasfer fuctio is proportioal to the delay spread. A larger delay spread causes a faster variatio rate i the trasfer fuctio. This is illustrated by Figure 9 where the absolute value of the trasfer fuctios for four delay spreads are plotted agaist the frequecy. The code of geeratig this figure is also show below. For the case with delay spread=0.2 µ sec, a cycle of variatio (from a local peak to the ext local peak) is of the order of 5 MHz. Similarly, for the case with delay spread= µ sec, 5 µ sec, or 0 µ sec, a cycle of 6

17 variatio (from a local peak to the ext local peak) is of the order of MHz, 0.2 MHz, or 0. MHz, respectively. Figure 9 Absolute values of the trasfer fuctios of four delay spreads. clear all; N=20 %umber of rays a=rad(,n); % amplitudes of N multipath arrivals tt=rad(,n); f=880:0.005:900; delay_spread=0.2; t=tt*delay_spread; % arrival times of N multipath arrivals, micro sec i=0; % frequecy idex for fi=880:0.005:900; % agular freuecies multipath_arrival=a.*exp(j*2*pi*fi*t); i=i+; abs_h(i)=abs(sum(multipath_arrival)); % the i-th trasfer fuctio ed subplot(2,2,) plot(f, abs_h) 7

18 ylabel('delay_spread=0.2 micro sec') xlabel('freuecy, MHz') delay_spread=; t=tt*delay_spread; % arrival times of N multipath arrivals, micro sec i=0; % frequecy idex for fi=880:0.005:900; % agular freuecies multipath_arrival=a.*exp(j*2*pi*fi*t); i=i+; abs_h(i)=abs(sum(multipath_arrival)); % the i-th trasfer fuctio ed subplot(2,2,2) plot(f, abs_h) ylabel('delay_spread= micro sec') xlabel('freuecy, MHz') delay_spread=5; t=tt*delay_spread; % arrival times of N multipath arrivals, micro sec i=0; % frequecy idex for fi=880:0.005:900; % agular freuecies multipath_arrival=a.*exp(j*2*pi*fi*t); i=i+; abs_h(i)=abs(sum(multipath_arrival)); % the i-th trasfer fuctio ed subplot(2,2,3) plot(f, abs_h) ylabel('delay_spread=5 micro sec') xlabel('freuecy, MHz') delay_spread=0; t=tt*delay_spread; % arrival times of N multipath arrivals, micro sec i=0; % frequecy idex for fi=880:0.005:900; % agular freuecies multipath_arrival=a.*exp(j*2*pi*fi*t); i=i+; abs_h(i)=abs(sum(multipath_arrival)); % the i-th trasfer fuctio ed subplot(2,2,4) plot(f, abs_h) ylabel('delay_spread=0 micro sec') xlabel('freuecy, MHz') ======================================================= 8

19 II. Doppler Effect & Multipath Effects A. Doppler Frequecy Shift (sigle path) Deote the speed ad the frequecy of a radio wave as c ad f, respectively. Defie a wavefrot as a surface with a costat phase. For example, cosider a spherical wave j2 π f ( r/ c t e ) / r emitted by a statioary source. The phase term is 2 π f ( r/ c t) ad a wavefrot (a costat phase surface) ca be defied as 2 π f ( r/ c t) = a. Here, a is a costat. For ay t, the wavefrot is a sphere located at r c( t a/2π f ) = + (see Figure 0). Note that the wavefrot travels with the speed c. Let two adjacet wavefrots be defied ja j( a 2 π ) as 2 π f ( r/ c t) = a ad 2 π f( r'/ c t) = a 2π. Sice e = e, the wavelegth ca the be defied as the distace betwee these two wavefrots (for ay time t): λ = r' r = c/ f. Figure 0 Wavefrots of a statioary source. Movig Source (sigle path) Doppler frequecy shift due to a movig source ca be explaied by the followig sceario. Whe a ambulace, police car, or fire truck races by you, the soud pitch emitted by the vehicle chages as the vehicle races towards, the away from you. This chage i pitch results from a frequecy shift of the soud waves (see Figure ). As the vehicle approaches, the soud waves from its sire are compressed towards the listeer. The itervals betwee waves dimiish, which traslates ito a icrease i frequecy or pitch. As the vehicle recedes, the soud waves are stretched relative to the listeer, causig the sire's pitch to decrease. By aalogy, the radio wave radiatio emitted by a movig object also exhibits the Doppler frequecy shift. The radiatio emitted by a object movig toward a observer is 9

20 squeezed ad its frequecy appears to icrease. I cotrast, the radiatio emitted by a object movig away is stretched ad its frequecy appears to decrease. Figure : Doppler frequecy shift of a movig source Now cosider the radiatio source is movig with a speed v toward the receiver. Deote the locatio of the radiatio source at t=0 as r=0. The wavefrot radiated at t=0 is a sphere cetered at r=0. The ext wavefrot will be radiated at t = / f (i.e., oe cycle after the previous wavefrot is radiated). As show i Figure, sice the radiatio source is movig with a speed v, the source locatio at t = / f is r=v/f. I other words, this secod wavefrot will be a sphere cetered at r=v/f. Therefore, the wavelegth observed at the receiver is compressed by a amout of v/f ad the equivalet wavelegth is λv = λ v/ f = c/ f v/ f. Thus, the equivalet frequecy observed at the observed becomes c v fv = c/ λv = = f f( + ) c/ f v/ f v c c The Doppler shift frequecy is v fv f f( ) c 2. Movig Observer (sigle path) Whe the radiatio source is statioary, wavefrots at t=0 are previously show i Figure 0. The distace betwee two adjacet wavefrots is λ. These wavefrots propagate outward (away from the source) with a speed of c. If the observer is movig toward the source with a speed v, the relative speed betwee the wavefrots ad the observer is v+c. Thus, the time for the observer to pass through a wavelegth is 20

21 t v λ c/ f = = c+ v c+ v ad the equivalet frequecy is c+ v v fv = = = f( + ) t c/ f c v Thus, the Doppler shift frequecy is also v fv f f( ) c 3. Movig Source ad Receiver (sigle path) As the distace betwee the source ad the receiver icreases, the spherical wavefrots become plae wavefrots at the observer. Let θ o be the agle betwee the propagatio directio of the radio wave ad the movig directio of the observer (see Figure 2). Similarly, let θ s be the agle betwee the propagatio directio of the radio wave ad the movig directio of the source (also see Figure 2). observer movig directio θ s source movig directio θ o wave propagatio directio wavefrot Figure 2: Icidet plae wave o a movig receiver Let the speed of the observer is v o ad the speed of the source is v s. As discussed i the movig source case, the equivalet wavelegth is λ = λ v cos θ / f = c/ f v cos θ / f v s s s s Sice the equivalet speed of the observer with respect to a wavefrot is takes λv t = c + v cosθ o o c+ v cosθ, it o o 2

22 for the observer to pass from oe wavefrot to the ext wavefrot. If the speeds v s ad v o are much smaller tha c, the Doppler shifted frequecy is approximated as c+ vocosθo c+ vocosθo vocos θo / c fs+ o= = = = f t λv c/ f vscos θs / f vscos θs / c f (+ v cos θ / c)(+ v cos θ / c) f(+ v cos θ / c+ v cos θ / c) o o s s o o s s Thus, the Doppler frequecy shift iduced by a movig source ad/or a movig receiver is vs cosθs + vocosθo f = fv f f( ) (5) c For a sigal s(t), its spectrum jωt S( ω) = s( t) e dt (6) Havig a frequecy shift of f, the ew sigal becomes jωt s f () t = S( ω 2 π f) e dω 2π + (7) If s(t) is a sigle frequecy sigal (say st () j2 ft = e π ), s t e π + j2 ( f f ) t f () =. B. Doppler Frequecy Shifts of a Time Harmoic Sigal (Multipath) As show i Figure, multipath sigals emit from the source at differet agles ad arrive at the observer at differet agles. Thus, Doppler shifts of differet arrivals are usually differet from oe aother. For coveiece ad without loss of geerality, let the relative speed betwee the source ad the receiver is v to accout for both v s ad v o i (5). Sice the rage of cos is withi - ad +, the maximum Doppler frequecy shift is v ± fd = f (8) c Whe there is o Doppler effect, as show i (2), the received sigal is j t j yt ( ) H( ) e ω ω with H( ω) ae ωτ = = N = Here, N is the total umber of multipath arrivals, a ad τ are the amplitude ad arrival time of the th ray, respectively. Whe there is Doppler effect, let ω = 2π f, f fd, be the Doppler agular frequecy shift of the th ray. The received sigal becomes N j t j j t yt ( ) H(, te ) ω ω where H( ω, t) ae ωτ + = = ω (9) = is a time varyig trasfer fuctio which is o loger a time harmoic sigal. 22

23 Usig the followig matlab program, we geerate Figure 3. clear all; N=20 ;% umber of multipath arrivals a=rad(,n); %amplitude tau=rad(,n); %arrival time f_d= shift=rad(,n)*2*f_d-f_d; %Doppler shifts f=0; % the frequecy of the trasmitted time harmoic sigal f_shift=f+shift; t=[22:0.0:25]; %No Doppler shift s_t=exp(j*2*pi*f*t); %trasmitted sigal y_t=sum(a.*exp(-j*2*pi*f*tau))*exp(j*2*pi*f*t); %received sigal =; y_d_t=sum(a()*exp(-j*2*pi*f*tau()))*exp(j*2*pi*(f_shift())*t); for =2:N y_d_t=y_d_t+a()*exp(-j*2*pi*f*tau())*exp(j*2*pi*(f_shift())*t); %received sigal ed figure () subplot(2,2,3) stem(f_shift,a) xlabel('frequecy, Hz') ylabel('ray amplitude') title('doppler Shifts') subplot(2,2,2) plot(t,y_t,'r') title('o Doppler shift') ylabel('received sigal') xlabel('time, sec') subplot(2,2,) stem(tau,a) xlabel('time, Sec') ylabel('ray amplitude') title('time Delay') subplot(2,2,4) plot(t,y_d_t) title('with Doppler shifts') ylabel('received sigal') xlabel('time, sec') 23

24 Here, a time harmoic sigal with f=0hz is trasmitted. There are 20 multipath arrivals (rays). The radomly geerated amplitudes ad arrival times of the 20 rays are show i the upper left graph. Whe there are o Doppler shifts, a portio of the received sigal is show i the upper right graph. Oe ca see that the received sigal is still a time harmoic sigal with f=0hz. There is o distortio. Whe there are Doppler frequecy shifts, the radomly geerated amplitudes ad Doppler shifted frequecies of the 20 rays are show i the lower left graph. A portio of the received sigal is show i the lower right graph. The sigal is distorted ad varies as time icreases. Figure 3 Multipath effect (without Doppler shifts) o a time harmoic sigal ad combied multipath ad Doppler effects o a time harmoic sigal. The variatio rate (with respect to time) of the evelope of the time varyig sigal is proportioal to the Doppler spread. A larger Doppler spread causes a faster variatio i the evelope of the time varyig sigal. This is illustrated by Figure 4 where the evelope of the time varyig sigal for four Doppler spreads is plotted agaist the time. The code of geeratig this figure is also show below. For the case with Doppler spread=0.0hz, a cycle of variatio (from a local peak to the ext local peak) is of the order of 00 sec. Similarly, for the case with delay spread=0.05 Hz, 0. Hz, or 0.5 Hz, a 24

25 cycle of variatio (from a local peak to the ext local peak) is of the order of 20 sec, 0 sec, or 2 sec, respectively. Figure 4: The received sigal of a time varyig chael where the trasmitted sigal is a time-harmoic sigal with frequecy f=0hz. clear all; N=20 ;% umber of multipath arrivals a=rad(,n); %amplitude tau=rad(,n); %arrival time f_d=0.0 shift=rad(,n)*2*f_d-f_d; %Doppler shifts f=0; % the frequecy of the trasmitted time harmoic sigal f_shift=f+shift; t=[0:0.0:50]; =; y_d_t=sum(a()*exp(-j*2*pi*f*tau()))*exp(j*2*pi*(f_shift())*t); for =2:N 25

26 y_d_t=y_d_t+a()*exp(-j*2*pi*f*tau())*exp(j*2*pi*(f_shift())*t); %received sigal ed subplot(2,2,) plot(t,y_d_t) xlabel('time, sec') ylabel('f_d=0.0') N=20 ;% umber of multipath arrivals a=rad(,n); %amplitude tau=rad(,n); %arrival time f_d=0.05 shift=rad(,n)*2*f_d-f_d; %Doppler shifts f=0; % the frequecy of the trasmitted time harmoic sigal f_shift=f+shift; t=[0:0.0:50]; =; y_d_t=sum(a()*exp(-j*2*pi*f*tau()))*exp(j*2*pi*(f_shift())*t); for =2:N y_d_t=y_d_t+a()*exp(-j*2*pi*f*tau())*exp(j*2*pi*(f_shift())*t); %received sigal ed subplot(2,2,2) plot(t,y_d_t) xlabel('time, sec') ylabel('f_d=0.05') N=20 ;% umber of multipath arrivals a=rad(,n); %amplitude tau=rad(,n); %arrival time f_d=0. shift=rad(,n)*2*f_d-f_d; %Doppler shifts f=0; % the frequecy of the trasmitted time harmoic sigal f_shift=f+shift; t=[0:0.0:50]; =; y_d_t=sum(a()*exp(-j*2*pi*f*tau()))*exp(j*2*pi*(f_shift())*t); for =2:N y_d_t=y_d_t+a()*exp(-j*2*pi*f*tau())*exp(j*2*pi*(f_shift())*t); %received sigal 26

27 ed subplot(2,2,3) plot(t,y_d_t) xlabel('time, sec') ylabel('f_d=0.') N=20 ;% umber of multipath arrivals a=rad(,n); %amplitude tau=rad(,n); %arrival time f_d=0.5 shift=rad(,n)*2*f_d-f_d; %Doppler shifts f=0; % the frequecy of the trasmitted time harmoic sigal f_shift=f+shift; t=[0:0.0:50]; =; y_d_t=sum(a()*exp(-j*2*pi*f*tau()))*exp(j*2*pi*(f_shift())*t); for =2:N y_d_t=y_d_t+a()*exp(-j*2*pi*f*tau())*exp(j*2*pi*(f_shift())*t); %received sigal ed subplot(2,2,4) plot(t,y_d_t) xlabel('time, sec') ylabel('f_d=0.5') ============================================================== 27

28 C. Doppler ad Multipath Effects of Sigals with Multiple Frequecies I a time varyig (i.e., with Doppler effect) multipath eviromet, as show i (9), the j t received sigal (due to a trasmitted time harmoic sigal e ω ) is j t j j t yt ( ) H(, te ) ω ω where H( ω, t) ae ωτ + ω = = N = is the time varyig spectrum. Cosider a sigal with multiple frequecy compoets jωt st () = S( ω) e dω 2π The time varyig spectrum of the received sigal is S( ω) H( ω, t) ad the time-domai received sigal is N jωt jω( t τ) + jωt yt () = S( ω) H( ω,) te dω a S( ω) e dω 2π = 2π = N N jω( t τ) jωt jωt = a S( ω) e dω e as( t τ) e 2π = = I the first example, cosider 2 cases with differet Doppler spreads. I each case, there are 6 rays ad the ray amplitudes of these 6 rays are a : [, 0.3, -0.8, 0.5, -0.4, 0.2] The Doppler shifts are Case : ω : [0, 2Hz, 0Hz, 6Hz, 8Hz, 4Hz] Case 2: ω : [0, 20Hz, 00Hz, 60Hz, 80Hz, 40Hz] Here, we artificially make all travel time delays zero ( τ =0). Cosider s(t) is betwee t 0 to t 0 +5µs ad is zero elsewhere. Plot y(t) for two observatio times t 0 =0sec ad 20msec i Figure 5. We observe that, at first, there is o distortio at the receiver because all travel time delays are zero ( τ =0). Secodly, Doppler spread causes the sigal to vary with time. At differet observatio times, the received sigals are differet. Thirdly, a larger Doppler spread causes a faster time variatio rate. The code to geerate Figure 5 is listed below. ========================================================= clear; t=(0:0.0:0);a=[,0.3,-0.8,0.5,-0.4,0.2];w=[0,2,0,6,8,4]; for i=:6; s(i,:)=a(i)*exp(j*0*w(i))*[oes(,50),zeros(,500)]; s2(i,:)=a(i)*exp(j*0.02*w(i))*[oes(,50),zeros(,500)]; ed y(,:)=sum(s);y2(,:)=sum(s2);w=[0,20,00,60,80,40]; for k=:6; 28

29 s3(k,:)=a(k)*exp(j*0*w(k))*[oes(,50),zeros(,500)]; s4(k,:)=a(k)*exp(j*0.02*w(k))*[oes(,50),zeros(,500)]; ed y3(,:)=sum(s3);y4(,:)=sum(s4); subplot(2,,); plot(t,abs(y));ylabel('y(t)'); xlabel('time(us)'); title('case : small Doppler spread') ylim([ ]);hold o; plot(t,abs(y2),'r');hold off leged('t_0=0msec','t_0=20msec') subplot(2,,2); plot(t,abs(y3));xlabel('time(us)');ylabel('y(t)'); title('case 2: large Doppler spread') ylim([ ]);hold o; plot(t,abs(y4),'r');hold off leged('t_0=0msec','t_0=20msec') ============================================================ Figure 5: Doppler effects o a sigal with multiple frequecy compoets. 29

30 Cotiued from the previous example, we will cosider o-zero travel time delays. Cosider 4 cases with differet combiatios of delay spreads ad Doppler spreads. I each case, there are 6 rays ad the ray amplitudes of these 6 rays are a : [, 0.3, -0.8, 0.5, -0.4, 0.2] The Doppler shifts are Cases ad 3 are with small Doppler spread: ω : [0, 2Hz, 0Hz, 6Hz, 8Hz, 4Hz] Cases 2 ad 4 are with large Doppler spread: ω : [0, 20Hz, 00Hz, 60Hz, 80Hz, 40Hz] The travel time delays are Cases ad 2 are with large delay spread: τ : [0, µs, 2µs, 3µs, 4µs, 5µs]; Cases 3 ad 4 are with small delay spread: τ : [0, 0.µs, 0.2µs, 0.3µs, 0.4µs, 0.5µs]; We will discuss the frequecy-domai view first. We use the followig matlab code to compute ad plot i Figure 6 the trasfer fuctio (amplitude ad phase) for frequecies from MHz to +MHz for two the observatio times at t 0 =0sec ad 20msec. clear; a=[,0.3,-0.8,0.5,-0.4,0.2]; t=[0,,2,3,4,5;0,,2,3,4,5;0,0.,0.2,0.3,0.4,0.5;0,0.,0.2,0.3,0.4,0.5]; w=[0,2,0,6,8,4;0,20,00,60,80,40;0,2,0,6,8,4;0,20,00,60,80,40]; f=-:0.0:; w=2*pi*f; for k=:4; for i=:6; h(i,:)=a(i)*exp(-j*w*t(k,i))*exp(j*w(k,i)*0); h2(i,:)=a(i)*exp(-j*w*t(k,i))*exp(j*w(k,i)*0.02); ed h_(k,:)=sum(h(:,:ed)); h_2(k,:)=sum(h2(:,:ed)); ed subplot(4,2,);plot(f,abs(h_(,:)));ylabel('case ');ylim([0 4]); hold o plot(f,abs(h_2(,:)),'r');title('amplitude'); %leged('t_0=0msec','t_0=20msec') hold off subplot(4,2,2);plot(f,agle(h_(,:))); ylim([-4 4]); hold o plot(f,agle(h_2(,:)),'r'); title('phase'); % hold off subplot(4,2,3);plot(f,abs(h_(2,:)));ylabel('case 2');ylim([0 4]); hold o plot(f,abs(h_2(2,:)),'r'); hold off subplot(4,2,4);plot(f,agle(h_(2,:)));ylim([-4 4]); hold o plot(f,agle(h_2(2,:)),'r'); hold off 30

31 subplot(4,2,5);plot(f,abs(h_(3,:)));;ylabel('case 3');ylim([0 4]); hold o plot(f,abs(h_2(3,:)),'r'); hold off subplot(4,2,6);plot(f,agle(h_(3,:))); ylim([-4 4]); hold o plot(f,agle(h_2(3,:)),'r'); hold off subplot(4,2,7);plot(f,abs(h_(4,:)));xlabel('frequecy(mhz)'); ylabel('case 4');ylim([0 4]); hold o plot(f,abs(h_2(4,:)),'r'); hold off subplot(4,2,8);plot(f,agle(h_(4,:)));xlabel('frequecy(mhz)'); ylim([-4 4]); hold o plot(f,agle(h_2(4,:)),'r'); hold off; ================================================================ Figure 6: The larger delay spread causes faster variatio rate with respective to frequecy (cases & 2). The larger Doppler spread causes faster variatio rate with respective to the observatio time (cases 2 &4). Leged: red (blue) lie represets the observatio made at t 0 =20msec (0msec). 3

32 Now we will discuss the time-domai view. Cosider s(t) is betwee t 0 to t 0 +5µs ad is zero elsewhere. We use the followig matlab code to compute ad plot i Figure 7 the received sigal y(t) for the two observatio times at t 0 =0sec ad 20msec. clear all; a=[,0.3,-0.8,0.5,-0.4,0.2]; t=[0,,2,3,4,5;0,,2,3,4,5;0,0.,0.2,0.3,0.4,0.5;0,0.,0.2,0.3,0.4,0.5]; w=[0,2,0,6,8,4;0,20,00,60,80,40;0,2,0,6,8,4;0,20,00,60,80,40]; for k=:4; for i=:6; s(i,:)=a(i)*exp(j*0*w(k,i))*[zeros(,(00*t(k,i))),oes(,50),zeros(,(000-00*t(k,i)))]; s2(i,:)=a(i)*exp(j*0.02*w(k,i))*[zeros(,(00*t(k,i))),oes(,50),zeros(,(000-00*t(k,i)))]; ed y(k,:)=sum(s(:,:ed)); y2(k,:)=sum(s2(:,:ed)); ed t=((::legth(y(,:)))-)*0^(-2); subplot(4,2,);plot(t,real(y(,:)));ylabel('case ');ylim([-2 2]); hold o plot(t,real(y2(,:)),'r');title('real part') hold off subplot(4,2,2);plot(t,imag(y(,:)));ylim([-2 2]); hold o plot(t,imag(y2(,:)),'r');title('imagiary part') hold off subplot(4,2,3);plot(t,real(y(2,:)));ylabel('case 2');ylim([-2 2]); hold o plot(t,real(y2(2,:)),'r'); hold off subplot(4,2,4);plot(t,imag(y(2,:)));ylim([-2 2]); hold o plot(t,imag(y2(2,:)),'r'); hold off subplot(4,2,5);plot(t,real(y(3,:)));;ylabel('case 3');ylim([-2 2]); hold o plot(t,real(y2(3,:)),'r'); hold off subplot(4,2,6);plot(t,imag(y(3,:)));ylim([-2 2]); 32

33 hold o plot(t,imag(y2(3,:)),'r'); hold off subplot(4,2,7);plot(t,real(y(4,:)));xlabel('time(us)');ylabel('case 4'); hold o plot(t,real(y2(4,:)),'r');ylim([-2 2]); hold off subplot(4,2,8);plot(t,imag(y(4,:)));xlabel('time(us)');ylim([-2 2]); hold o plot(t,imag(y2(4,:)),'r'); Figure 7: The larger delay spread causes more distortio (cases & 2). The larger Doppler spread causes faster variatio rate with respective to the observatio time (cases 2 &4). Leged: red (blue) lie represets the observatio made at t 0 =20msec (0msec). Results i Figures. 6 & 7 will be re-plotted i below with five observatios: t 0 =0sec, 0msec, 20msec, 30msec, 40msec, ad 50msec. 33

34 Case : a : [, 0.3, -0.8, 0.5, -0.4, 0.2]; t : [0, µs, 2µs, 3µs, 4µs, 5µs]; ω : [0, 2Hz, 4Hz, 6Hz, 8Hz, 0Hz] Part a: H ( N j( ωt ωt ) ω, t) = ae Part b: = y( t) = N = a s( t t ) e j( ω t) 2.5 H(w).5 real(y(t)) π MHz phase(h(w)) MHz imag(y(t)).5 x sec x 0-5 Doppler spread is ot large (with respect to the time iterval betwee observatio times, i.e., t 0 ): Frequecy Domai View: slow time selective fadig Time Domai View: Time variatio is slow Delay spread is large (with respect to symbol duratio): Frequecy Domai View: Frequecy selective fadig is severe Time Domai View: Distortio is severe. 34

35 Case 2: a : [, 0.3, -0.8, 0.5, -0.4, 0.2]; t : [0, µs, 2µs, 3µs, 4µs, 5µs]; ω : [0, 20Hz, 40Hz, 60Hz, 80Hz, 00Hz] Part a: H ( N j( ωt ωt ) ω, t) = ae Part b: = y( t) = N = a s( t t ) e j( ω t) 2.5 H(w) 2 real(y(t)) π MHz phase(h(w)) MHz imag(y(t)).5 x sec x 0-5 Doppler spread is large (with respect to the time iterval betwee observatio times, i.e., t 0 ): Frequecy Domai View: fast time selective fadig Time Domai View: Time variatio is fast Delay spread is large (with respect to symbol duratio): Frequecy Domai View: Frequecy selective fadig is severe Time Domai View: Distortio is severe. 35

36 Case 3: a : [, 0.3, -0.8, 0.5, -0.4, 0.2]; t : [0, 0.µs, 0.2µs, 0.3µs, 0.4µs, 0.5µs]; ω : [0, 2Hz, 4Hz, 6Hz, 8Hz, 0Hz] Part a: H ( N j( ωt ωt ) ω, t) = ae Part b: = y( t) = N = a s( t t ) e j( ω t).6 H(w).5 real(y(t)) π MHz phase(h(w)) MHz imag(y(t)).5 x sec x 0-5 Doppler spread is ot large (with respect to the time iterval betwee observatio times, i.e., t 0 ): Frequecy Domai View: slow time selective fadig Time Domai View: Time variatio is slow Delay spread is small (with respect to symbol duratio): Frequecy Domai View: Frequecy selective fadig is ot severe Time Domai View: Distortio is ot severe. 36

37 Case 4: a : [, 0.3, -0.8, 0.5, -0.4, 0.2]; t : [0, 0.µs, 0.2µs, 0.3µs, 0.4µs, 0.5µs]; ω : [0, 20Hz, 40Hz, 60Hz, 80Hz, 00Hz] Part a: H ( N j( ωt ωt ) ω, t) = ae Part b: = y( t) = N = a s( t t ) e j( ω t) 2.5 H(w) 2 real(y(t)) π MHz phase(h(w)) MHz imag(y(t)).5 x sec x 0-5 Doppler spread is large (with respect to the time iterval betwee observatio times, i.e., t 0 ): Frequecy Domai View: fast time selective fadig Time Domai View: Time variatio is fast Delay spread is ot large (with respect to symbol duratio): Frequecy Domai View: Frequecy selective fadig is ot severe Time Domai View: Distortio is ot severe 37

38 Part II. Multipath & Doppler Chael Models After studyig this ote, studets will be able to. Uderstad multipath chael modelig i both time ad frequecy domais 2. Uderstad Time varyig chael modelig i both time ad frequecy domais 3. Uderstad key chael parameters: delay spread & coheret badwidth; Doppler spread & coheret time 4. Uderstad the defiitio of arrowbad ad broadbad sigals 5. Uderstad chagig rate of chael characteristics 38

39 I. Multipath Chael Modelig (without Doppler Effects):: Time Ivariat Case I wireless commuicatio eviromets, a sigal trasmitted from the trasmitter reaches the receiver through may differet paths as illustrated i Figure. Figure : Multipath propagatio At a certai observatio time t, the chael impulse respose (see Figure 2) ca be expressed as a sum of rays (multipath arrivals) L h( τ ) = aiδτ ( τi), τ τ2 τ3... τl (a) i= Here, L is the total umber of rays, a i ad τ i are the amplitude ad arrival time of the i th ray, respectively. Note that a i ad τ i are sometimes modeled as radom variables ad h(t) is modeled as a radom process. h(τ) Chael Impulse Respose τ (sec~µsec) Delay Spread T D Figure 2 Discrete ad cotiuous models for the impulse respose of a time ivariat wireless chael Theoretically speakig, there always are ifiite rays propagatig from the source to the receiver. However, i practice, there will be oly a fiite umber of relevat rays. The 39

40 spa of arrival times of these relevat rays is characterized by a parameter called time delay spread T D. There are several ways to defie T D. Here, for coveiece, we will defie T D as the maximum separatio of arrival times of relevat rays. Note that T D is usually i the order of 0µsec for rural outdoor eviromets, µsec for urba or suburba outdoor eviromets, ad of 00 sec for large idoor eviromets. Whe the umber of rays is large ad/or the badwidth of the trasmitted sigal is arrow, it may be more coveiet to use a cotiuous time model for the chael impulse respose (see the evelope of the pulse i Figure 2): 0, 0 τ T h( τ ) D = 0, elsewhere The chael frequecy respose (trasfer fuctio) is represeted by the Fourier trasform of the impulse chael respose i (a) or (b): L jωτi ae discrete model i jωτ i= H( ω) = h( τ) e dτ = T D jωτ h( τ) e dτ Cotiuous model 0 A typical curve for the magitude of the trasfer fuctio is show i Figure 3: (b) (2) H(f) Chael Trasfer Fuctio f(ghz~mhz) Figure 3 magitude of the trasfer fuctio of a time ivariat wireless chael Figure 3 shows small values of H ( ω) at certai frequecies. The spacig amog these frequecies depeds o T D. This depedecy will be explored i the ext sectio i more detailed. Whe the trasmitted sigal has a fixed power spectral desity, the frequecy compoets of the received sigal at or ear these frequecies will be much weaker tha other frequecy compoets. This pheomeo of radom fluctuatio of sigal stregth is called fadig. If h( τ ) is a radom process, H ( ω) is also a radom process. Thus fadig is also a radom pheomeo. It should be characterized by some statistical models. We would like to defie the coheret badwidth as 40

41 Bc = (3) TD The average spacig of two adjacet deep fades i Figure 3 is the order of Bc =. T (Note the defiitio of T D ad B c are ot uique. We are just itroducig the cocepts here. Rigorous defiitios will be provided i lecture 9.) D 4

42 II. Broadbad ad Narrowbad Sigals A broadbad (or arrowbad) sigal is a sigal whose badwidth B is broader (or arrower) tha a certai referece frequecy. There are several ways to choose the referece. A covetioal defiitio i cellular commuicatios is referrig to a existig system. For example, the secod geeratio IS-95 CDMA system (where sigal badwidth of a sigle chael is.28mhz) is a broadbad system compared to the first geeratio AMPS system (where sigal badwidth of a sigle chael is 30KHz ). However, IS-95 is cosidered as a arrowbad system if the referece is istead the third geeratio WCDMA system (where sigal badwidth of a sigle chael is 5 MHz). I radio egieerig, the referece frequecy is % of the carrier frequecy f c. A sigal is broadbad (or arrowbad) if its badwidth B is larger (or smaller) tha 0.0f c. If the carrier frequecy f c =2GHz, the referece frequecy is 20MHz. Thus, all three systems (AMPS, IS-95 ad WCDMA) are arrowbad. Here, we will use a chael characteristic called coheret badwidth B c as the frequecy referece. Cosider a two-ray discrete model with a = a2 = ad τ = τ 2 = T D /2. jωτ jωτ 2 From (2), H( ω) = ae + a e = 2si( ωt D /2). As show i Figure 4, deep fades 2 occurs at frequecies with miimum frequecy spacig = T D. Note that ay two rays arrivig betwee τ ad τ 2 give rise to a iterferece patter similar to that i Figure 4, but with a frequecy spacig > T D Bc T D. We defie the coheret badwidth as = (3) The physical meaig of coheret badwidth ca be explaied by the followig example. 2 Cosider two time harmoic sigals j f t e π 2 2 ad j f t e π. Their chael frequecy resposes H(2 π f) ad H(2 π f2) are cosidered to be similar to each other if f f2 << Bc. But the chael frequecy resposes H(2 π f) ad H(2 π f2) are cosidered to be ot similar to each other if f f2 > Bc. Cosider a square pulse with a pulse width=t. The magitude of its Fourier trasform is T jωτ T e jωτ 0 2 T e dτ = = si( ω ) jω ω

43 The first zeros occurs at f =±. Thus, we cosider T T sigal with data rate equal to. We defie a sigal is T as the badwidth B of a digital Broadbad: sigal badwidth B or > chael coheret badwidth B c Narrowbad: sigal badwidth B << chael coheret badwidth B c (4) The trasfer fuctio of a two ray chael model, the spectrum of a broadbad sigal (Gree) ad the spectrum of a arrowbad sigal (Red) are show i Figure 4. H(f) B = sigal badwidth /T B f T D Figure 4: Trasfer fuctio of a 2-ray chael (Blue), Spectrum of a broadbad sigal (Gree) ad Spectrum of a arrowbad sigal (Red). Sice T=/B ad T D =/B c, Equatio (4) ca also be writte as Broadbad: sigal bit(chip) duratio T or < chael delay spread T D Narrowbad: sigal bit duratio T >> chael delay spread T D (5) The physical meaig of Equatio (5) is show i Figure 5: Chael Output Chael Iput T D /T small 0 T 2T 0 T 2T T D /T large 0 T 2T Figure 5 Multipath causes sigal distortio. Cosider a 2-ray model where the chael output is the sum of two rays. Whe T>T D (arrowbad), the distortio is small ad whe T T D or T< T D (broadbad), the distortio is large. 43

44 III. Multipath Chael Modelig (with Doppler Effects): Time Varyig Case If the receiver, trasmitter ad eviromet are time ivariat, the chael respose i Equatio (a) or (b) is also time ivariat. This meas that at a differet observatio time t 2, the chael impulse respose remais uchaged. However, if the receiver, trasmitter or eviromet is movig, the chael respose i Equatio (a) depeds o the observatio time ad is rewritte as Lt () ht (, τ) = ai() tδ( τ τi()) t (6) i= Here, the umber of rays L, the ray amplitudes a i ad arrival times τ i becomes fuctios of observatio time (see also Figure 6). There are two time scales i (6). The large time scale t represets the observatio time ad the small time scale τ represets the multipath arrival time. The Fourier trasform of (6) with respect to the small time scale variable τ is Htω (, ) which is illustrated i Figure 7. If the time variatio is caused by Doppler effects, Equatio (6) ca be writte as L j it ht (, τ) aiδ( τ τi) e ω (7) i= where ω i is the Doppler frequecy shift of the i th ray. The Fourier spectrum with respect to the small time scale τ of Equatio (7) ca be writte as L j( ωτ ω t) ae i (8) i= i i Ht (, ω) The Doppler frequecy shifts ω i will cause the spectrum of the receivig sigal to spread. j t For example, cosider a time-harmoic source sigal e ω which has oly a lie spectrum atω = 2π f. However, the received sigal is L L j( ωτ ω t) jωt jωτ j( ω+ ω ) t i i i= i= (9) jωt i i i i Ht (, ω) e ae e = ae e which has L lie spectra at ω + ωi for i=,2,,l. Whe L is large, these lie spectra distributes over a iterval [ ω ωd ω + ωd ] with ωd = 2π fd (see Figure 8). Here f d is the maximum Doppler spread: v f = d f tc C f (0) d where f is the carrier frequecy, v is the maximum relative velocity betwee trasmitter ad receiver, C is the speed of light (3*0 8 m/sec). For example, if a vehicle speed is 30 m/sec ad the carrier frequecy is 2 GHz, the maximum Doppler spread is 200 Hz ad coheret time is 5 msec. 44

45 t = t 4 t = t 3 t = t 2 t = t Arrival time τ i Figure 6 Discrete impulse resposes of a time-varyig wireless chael t (msec~sec) H(t,f) t=t 4 t=t 3 t=t 2 H(t,f) t=t f f(ghz~mhz) Figure 7 Fourier spectrum of a time-varyig wireless chael f-f f f+f D D Figure 8 Doppler spread i betwee [ f fd f fd + ] 45

46 IV. Slow ad Fast Fadig (Varyig) Chael The time varyig rate of the chael respose is usually characterized by the coheret time t c. If the separatio betwee two observatio times is much less tha t c, we claim that the two impulse resposes (or two trasfer fuctios) are similar to each other: ht (, τ) ht ( 2, τ) or Ht (, ω) Ht ( 2, ω) if t t2 << tc () Otherwise, the two chael impulse resposes (or two trasfer fuctios) are sigificatly differet. Thus, if the bit duratio is smaller tha the coheret time, we claim that the chael respose remais uchaged durig a data bit period. Usually the coheret time is defied as the iverse of the maximum Doppler spread: tc = (2) f d Derivatio of Equatio (2) for geeral radom processes requires advaced statistical aalysis. We will use the worst case (L=2, a = a2 =, τ = τ 2 = 0, ad ω = ω = π f ) i (8) to demostrate () ad (2): 2 2 d L j( ωτi ωi t) j2π fd t j2π fd t (, ω) i = = 2 si(2 π d ) i= H t ae e e f t Therefore, Ht ( + t, ω) = 2 si(2 π fd ( t+ t)) = 2 si(2 π fdt)cos(2 π fd t) + cos(2 π fdt)si(2 π fd t) 2 si(2 π fd( t)) = H( t, ω) if t << = tc f d For example, if a vehicle speed is 30 m/sec ad the carrier frequecy is 2 GHz, the maximum Doppler spread is 200 Hz ad the coheret time is 5 msec. If the data rate is 2kbps, the bit duratio is 0.5 msec ad therefore the chael respose ca be cosidered to be a costat durig a bit period. Cosider a umerical example where a sigle frequecy sigal is trasmitted through a time varyig chael. The evelopes of the received sigals for f d =0.0, 0.05, 0. & 0.5 khz are show i Figure 9. Note that if there were o Doppler spread (i.e., f d =0), the evelope of the received sigal would have remaied costat. Now, there is Doppler spread, the variatio rate (/ t c ) of the evelope is proportioal to f d : = tc fd I aalyzig the umerical results, we ca defie the coheret time as the average time differece betwee two adjacet peaks of the evelope. Comparig theoretical result ad umerical result, i the upper left graph, the coheret time is aroud 00 msec. I the upper right graph, the coheret time is aroud 20 msec. I the lower left graph, the 46

47 coheret time is aroud 0 msec. I the upper right graph, the coheret time is aroud 2 msec. We gai a good isight what does the coheret time mea. Figure 9: The received sigal of a time varyig chael where the trasmitted sigal is a time-harmoic sigal with frequecy f=0khz 47