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Chapter 5-Small Scale Fading and Multipath Chapter 5 Small-Scale Fading and Multipath School of Information Science and Engineering, SDU Outline l Small-Scale Multipath Propagation l Impulse Response Model of a Multipath Channel l Small-Scale Multipath Measurements l Parameters of Mobile Multipath Channels l Types of Small-Scale Fading l Rayleigh and RiceanDistributions l Statistical Models for Multipath Fading Channels Small Scale Fading l Describes rapid fluctuations of the amplitude, phase of multipath delays of a radio signal over short period of time or travel distance l Caused by interference between two or more versions of the transmitted signal which arrive at the receiver at slightly different times. l These waves are called multipath waves and combine at the receiver antenna to give a resultant signal which can vary widely in amplitude and phase. Small Scale Multipath Propagation l Effects of multipath l Rapid changes in the signal strength § Over small travel distances, or § Over small time intervals l Random frequency modulation due to varying Doppler shifts on different multiples signals l Time dispersion (echoes) caused by multipath propagation delays l Multipath occurs because of l Reflections l Scattering Multipath l At a receiver point l Radio waves generated from the same transmitted signal may come l from different directions l with different propagation delays l with (possibly) different amplitudes (random) l with (possibly) different phases (random) l with different angles of arrival (random). l These multipath components combine vectoriallyat the receiver antenna and cause the total signal § to fade § to distort Multipath Components Radio Signals Arriving from different directions to receiver Component 1 Component 2 Component N Receiver may be stationary or mobile. Mobility l Other Objects in the radio channels may be mobile or stationary l If other objects are stationary l Motion is only due to mobile l Fading is purely a spatial phenomenon (occurs only when the mobile receiver moves) l The spatial variations as the mobile moves will be perceived as temporal variations § Dt= Dd/v l Fading may cause disruptions in the communication Factors Influencing Small Scale Fading l Multipath propagation l Presence of reflecting objects and scattererscause multiple versions of the signal to arrive at the receiver § With different amplitudes and time delays § Causes the total signal at receiver to fade or distort l Speed of mobile l Cause Doppler shift at each multipath component l Causes random frequency modulation l Speed of surrounding objects l Causes time-varying Doppler shift on the multipath components Factors Influencing Small Scale Fading l Transmission bandwidth of the channel l The transmitted radio signal bandwidth and bandwidth of the multipath channel affect the received signal properties: § If amplitude fluctuates or not § If the signal is distorted or not Doppler Effect l Whe a transmitter or receiver is moving, the frequency of the received signal changes, i.e. İt is different than the frequency of transmissin. This is called Doppler Effect. l The change in frequency is called Doppler Shift. l It depends on l The relative velocity of the receiver with respect to transmitter l The frequenct (or wavelenth) of transmission l The direction of traveling with respect to the direction of the arriving signal. Doppler Shift – Transmitter is moving The frequency of the signal The frequency of the signal that is received behind the that is received in front of the transmitter will be smaller transmitter will be bigger Doppler Shift –Recever is moving S d = XY Ll = SX - SY = d cosq Ll = vLt cosq The phase change in the received signal: Ll 2pvLt DF = 2p= cosq ll Dl Doppler shift (The apparent change in frequency) : 1 DF v XYq f = = cosq dd 2pDt l v A mobile receiver is traveling from point X to point Y Doppler Shift l The Dopper shift is positive l If the mobile is moving toward the direction of arrival of the wave. l The Doppler shift is negative l If the mobile is moving away from the direction of arrival of the wave. Impulse Response Model of a Multipath Channel l The wireless channel charcteristics can be expressed by impulse response function l The channel is time varying channel when the receiver is moving. l Lets assume first that time variation due strictly to the receiver motion (t = d/v) l Since at any distance d = vt, the received power will be combination of different incoming signals, the channel charactesitics or the impulse response funcion depends on the distance d between trandmitter and receiver. Multipath Channel Modeling Impulse Response Model of a Multipath Wireless Channel Impulse Response Model of a Multipath Channel l The wireless channel characteristics can be expressed by impulse response function l The channel is time varying channel when the receiver is moving. l Lets assume first that time variation due strictly to the receiver motion (t = d/v) l Since at any distance d = vt, the received power will ve combination of different incoming signals, the channel charactesitics or the impulse response funcion depends on the distance d between trandmitter and receiver Impulse Response Model of a Multipath Channel d = vt v d A receiver is moving along the ground at some constant velocity v. The multipath components that are received at the receiver will have different propagation delays depending on d: distance between transmitter and receiver. Hence the channel impulse response depends on d. Lets x(t) represents the transmitter signal y(d,t) represents the received signal at position d. h(d,t) represents the channelimpulse response which is dependent on d (hence time-varying d=vt). Multipath Channel Model Building Multipath Channel 2nd MC Base 1st MC Mobile 2 Station B u il di ng B u ild 1st MC in 4th MC g Multipath Channel 2nd MC B Mobile 1 uild 3rd MC ing (Multipath Component) Impulse Response Model of a Multipath Channel Wireless Multipath Channel x(t) y(t) h(d,t) The channel is linear time-varying channel, where the channel characteristics changes with distance (hence time, t = d/v) ¥ y(d,t) = x(t) Ä h(d,t) = ò x(t )h(d,t -t )dt -¥ For a causal system, h(d,t) = 0 for t < 0; hence t y(d,t) = ò x(t )h(d,t -t )dt -¥ Impulse Response Model d= vt assume v is constant over time t y(vt,t)=-òx(t)h(vt,)tdtt -¥ t y(t)=òx(t)h(vt,t-tt)d=x(t)Äh(vt,t)=Äx(t)h(dt,) -¥ We assume v is constant over short time. x(t): transmitted waveform y(t): received waveform h(t,t): impulse response of the channel. Depends on d (and therefore t=d/v) and also to the multiple delay for the channel for a fixed value of t. t is the multipath delay of the channel for a fixed value of t. ¥ y(t) = ò x(t )h(t,t )dt = x(t) Ä h(t,t ) -¥ ...Continue with Multipath Channel Impulse Response Model Impulse Response Model x(t) y(t) jwct h(t,t ) = Re{hb (t,t )e } jw t x(t) = Re{c(t)e jwct } y(t) = Re{r(t)e c } Bandpass Channel Impulse Response Model y(t) = x(t) Ä h(t,t ) c(t) 1 r(t) hb (t,t ) 2 1 r(t) = c(t) Ä h (t,t ) 2 b Baseband Equivalent Channel Impulse Response Model Impulse Response Model 1 r(t) = c(t) Ä h(t,t) 2 b j2pfct x(t) = Re{}c(t)e wc= 2pfc y(t) = Re{}r(t)ej2pfct c(t) is the complex envelope representation of the transmittedsignal r(t) is the complex envelope representation of the received signal hb(t,t) is the complex baseband impulse response Discrete-time Impulse Response Model of Multipath Channel Amplitude of Multipath Component There are N multipath components (0..N-1) to= 0 t1= Dt Excess Delay ti= (i)Dt Bin tN-1= (N-1)Dt t (excess delay) Dt tN-1 t0 t2 ti Excess delay: relative delay of the ithmultipath componentascompared to the first arriving component th ti : Excesssdelay of i multipath component, NDt: Maximum excess delay Multipath Components arriving to a Receiver Ignore the fact that multipath components arrive with differentangles, and assume that they arriving with the same angle in 3D. 1 2NN-2 N-1 th Component ....... t t0=0 t1 tN-3 tN-2 tN-1 (relative delay of multipath Comnponent) Each component will have different Amplitude (ai) and Phase (θi) Baseband impulse response of the Channel N -1 j(2pfct i (t)+fi (t,t )) hb (t,t ) = å ai (t,t )e d (t -t i (t)) i=0 ai (t,t ) : the real amplitude of the ith multipath component at time t. t i (t) : excess delay of the ith multipath component at time t. 2pfct i (t) +fi (t,t ) : Phase term that represents phase shift due to free space propagatio n of the ith component. Simply represent it with : q (t,T ) i d (·) : unit impulse function. Discrete-Time Impulse Response Model for aMultipath Channel hb(t,t) t t3 t(t3) t2 t(t2) t1 t(t1) t0 t t(t0) o t1 t2 t3 t4 t5 t6 tN-2 tN-1 Time-Invariance Assumption If the channel impulse response is assumed to be time-invariant over small-scale time or distance interval, then the channel impulse response can be simplified as: N -1 jqi hb (t ) = å aie d (t -t i ) i=0 When measuring or predicting hb(t), a probing pulse p(t) which approximates the unit impulse function is used at the transmitter.
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