Chapter 5 Small-Scale Fading and Multipath School of Information Science and Engineering, SDU
Outline Small-Scale Multipath Propagation Impulse Response Model of a Multipath Channel Small-Scale Multipath Measurements Parameters of Mobile Multipath Channels Types of Small-Scale Fading Rayleigh and Ricean Distributions Statistical Models for Multipath Fading Channels
Small Scale Fading Describes rapid fluctuations of the amplitude, phase of multipath delays of a radio signal over short period of time or travel distance Caused by interference between two or more versions of the transmitted signal which arrive at the receiver at slightly different times. 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 Effects of multipath Rapid changes in the signal strength Over small travel distances, or Over small time intervals Random frequency modulation due to varying Doppler shifts on different multiples signals Time dispersion (echoes) caused by multipath propagation delays Multipath occurs because of Reflections Scattering
Multipath At a receiver point Radio waves generated from the same transmitted signal may come from different directions with different propagation delays with (possibly) different amplitudes (random) with (possibly) different phases (random) with different angles of arrival (random). These multipath components combine vectorially at 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 Other Objects in the radio channels may be mobile or stationary If other objects are stationary Motion is only due to mobile Fading is purely a spatial phenomenon (occurs only when the mobile receiver moves) The spatial variations as the mobile moves will be perceived as temporal variations t = d/v Fading may cause disruptions in the communication
Factors Influencing Small Scale Fading Multipath propagation Presence of reflecting objects and scatterers cause 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 Speed of mobile Cause Doppler shift at each multipath component Causes random frequency modulation Speed of surrounding objects Causes time-varying Doppler shift on the multipath components
Factors Influencing Small Scale Fading Transmission bandwidth of the channel 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 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. The change in frequency is called Doppler Shift. It depends on The relative velocity of the receiver with respect to transmitter The frequenct (or wavelenth) of transmission The direction of traveling with respect to the direction of the arriving signal.
Doppler Shift Transmitter is moving The frequency of the signal that is received behind the transmitter will be smaller The frequency of the signal that is received in front of the transmitter will be bigger
Doppler Shift Recever is moving S d = XY Λl = SX SY Λl = vλt cosθ = d cosθ l X θ d Y v A mobile receiver is traveling from point X to point Y The phase changein the receivedsignal: Λl 2πvΛt Φ = 2π = cosθ λ λ Doppler shift (The apparent change in frequency): f 1 2π Φ t d = = v cosθ λ
Doppler Shift The Dopper shift is positive If the mobile is moving toward the direction of arrival of the wave. The Doppler shift is negative If the mobile is moving away from the direction of arrival of the wave.
Impulse Response Model of a Multipath Channel The wireless channel charcteristics can be expressed by impulse response function The channel is time varying channel when the receiver is moving. Lets assume first that time variation due strictly to the receiver motion (t = d/v) 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 The wireless channel characteristics can be expressed by impulse response function The channel is time varying channel when the receiver is moving. Lets assume first that time variation due strictly to the receiver motion (t = d/v) 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 channel impulse response which is dependent on d (hence time-varying d=vt).
Multipath Channel Model Building 2 nd MC Multipath Channel Base Station 1 st MC Mobile 2 Building Building Building 1 st MC 2 nd MC 4 th MC 3 rd MC (Multipath Component) Mobile 1 Multipath Channel
Impulse Response Model of a Multipath Channel x(t) Wireless Multipath Channel h(d,t) y(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) h( d, t) = For a causal system, h(d,t) = 0 for y( d, t) = t x( τ ) h( d, t τ ) dτ x( τ ) h( d, t τ ) dτ t < 0; hence
Impulse Response Model d = vt assume v is constant over time t yvt (,) t = x( τ) hvt (, t τ) dτ t yt () = x( τ) hvt (, t τ) dτ = xt () hvt (,) t = xt () hdt (,) We assume v is constant over short time. x(t): transmitted waveform y(t): received waveform h(t,τ): 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. τ is the multipath delay of the channel for a fixed value of t. y( t) = x( τ ) h( t, τ ) dτ = x( t) h( t, τ )
...Continue with Multipath Channel Impulse Response Model
Impulse Response Model x( t) = Re x(t) { } j ωc c( t) e t { j ω t } c h ( t, τ e h( t, τ ) = Re ) Bandpass Channel Impulse Response Model b y(t) y( t) y( t) = = Re { jωct r( t) e } x( t) h( t, τ ) c(t) 1 2 h b ( t, τ ) r(t) 1 r( t) = c( t) hb ( t, τ ) 2 Baseband Equivalent Channel Impulse Response Model
Impulse Response Model { } { } t f j c c t f j b c c e t r t y f e t c t x t h t c t r π π π ω τ 2 2 ) ( Re ) ( 2 ) ( Re ) ( ), ( 2 1 ) ( ) ( = = = = c(t) is the complex envelope representation of the transmitted signal r(t) is the complex envelope representation of the received signal h b (t,τ) is the complex baseband impulse response
Discrete-time Impulse Response Model of Multipath Channel Amplitude of Multipath Component Excess Delay Bin There are N multipath components (0..N-1) τ o = 0 τ 1 = τ τ i = (i) τ τ N-1 = (N-1) τ τ 2 τ τ 0 τ N-1 τ i τ (excess delay) Excess delay: relative delay of the ith multipath componentas compared to the first arriving component τ i : Excesss delay of i th multipath component, NDt: Maximum excess delay
Multipath Components arriving to a Receiver Ignore the fact that multipath components arrive with different angles, and assume that they arriving with the same angle in 3D. 1 2 N-2 N-1 N th Component... τ 0 =0 τ 1 τ Ν 3 τ Ν 2 τ Ν 1 Each component will have different Amplitude (a i ) and Phase (θ i ) τ (relative delay of multipath Comnponent)
Baseband impulse response of the Channel h a τ b i i ( t, τ ( t) 2πf ( t, τ c : τ ) ) : excess delay of the ith multipath component i free space propagatio n of δ ( ) : unit N = 1 i= 0 the real amplitude ( t) + φ impulse a i i ( t, τ ( t, τ ) : function. ) e j(2πf τ of the ith multipath component Phase c ( t ) + φ term that the ith component. i i ( t, τ )) δ ( τ at represents τ time phase Simply represent i t. ( t)) at it time shift with t. due : to θ ( t, T i )
Discrete-Time Impulse Response Model for a Multipath Channel h b (t,τ) t t 3 τ(t 3 ) t 2 τ(t 2 ) t 1 τ(t 1 ) t 0 τ τ(t o τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ N-2 τ 0 ) N-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: h b ( τ ) = N 1 i= 0 a e i jθ i δ ( τ τ i ) When measuring or predicting h b (τ), a probing pulse p(t) which approximates the unit impulse function is used at the transmitter. That is: p( t) δ ( t τ ) This is called sounding the channel to determine impulse response.
Complex Baseband Impulse Response Baseband impulse response h b (τ) is a complex number and therefore has a magnitude (amplitude) a i and a phase θ i. a i θ i h b (τ) h b (τ) = a i e jθι h b (τ) = a i (cosθ i +jsinθ i ) h b (τ) = a i you can think of it also as a vector that starts at origin.
Amplitudes and Phases of Multipath Components f c 1 st Arriving Multipath Component (Say 0 th Component) θ 0 =0 2a 0 (phase) Two components emerge from the same source at the same time. They belong to the same transmitter signal. But they travel different paths. They arrive at the same receiver with time difference equal to t i. f c 2a i θ i is expressed in radians i th Multipath Component 0 θ i =2πf c τ i τ i τ
Components arriving at the same time What happens if two or more multipath components are with the same access delay bin (arrive at the same time)? Then the received signal is the vectorial addition of two multipath signals. S2 R a 3 θ 3 a 1 S1 Example: Lets assume two signals S1 and S2 arrive at the same time at the receiver: a 2 θ 2 θ 1 S R = 1 = = a e 1 a S 1 1 jθ e 1 + jθ + 1 S 2 a 2 e S 2 jθ 2 = = a a 2 3 e e jθ 2 jθ 3 R is the combined receiver signal.
Components arriving at the same time The amplitude and phase of the combined signal (R) depends on the amplitudes and phases of the two components. Depending on the values of the phases of the components, the combined affect may weaken or strengthen the amplitude of the combined signal. It is possible that the two signals may totally cancel each other depending on their relative phases on their amplitudes.
Example 1 Addition of Two Signals MC: Multipath Component 1 st MC 2 st MC 3 2 cos(x+pi/16) cos(x+pi) cos(x+pi/16)+cos(x+pi) Combined Signal a 1 /a 2 =1 θ 1 =π/16 θ 2 =π 1 0-1 -2-3 -10-5 0 5 10
Example 2 Addition of Two Signals 1 st MC 2 st MC 3 2 cos(x+pi/16) 3*cos(x+pi) cos(x+pi/16)+3*cos(x+pi) Combined Signal a 1 /a 2 =1/3 θ 1 =π/16 θ 2 =π 1 0-1 -2-3 -10-5 0 5 10
Power Delay Profile For small-scale fading, the power delay profile of the channel is found by taking the spatial average of 2 over a local area (small-scale area). h b ( t; τ ) If p(t) has a time duration much smaller than the impulse response of the multipath channel, the received power delay profile in a local area is given by: P( τ ) k h ( t; τ b ) 2 The bar represents the average over the local area of h b ( t; τ ) 2 Gain k relates the transmitter power in the probing pulse p(t) to the total received power in a multipath delay profile.
Example power delay profile Taken from Dimitrios Mavrakis Homepage:http://www.ee.surrey.ac.uk/Personal/D.Mavrakis/
Relationship between Bandwidth and Receiver Power What happens when two different signals with different bandwidths are sent through the channel? What is the receiver power characteristics for both signals? We mean the bandwith of the baseband signal The bandwidth of the baseband is signal is inversely related with its symbol rate. One symbol
Bandwidth of Baseband Signals Highbandwidth (Wideband) Signal Lowbandwidth (Narrowband) Signal Continuous Wave (CW) Signal t
A pulsed probing signal (wideband) T bb p(t) Transmitter x(t): transmitted signal T REP T REP x( t) >> τ ( τ : maximum measured excess delay) max = Re{ p( t) e max j 2πf c t } = p( t) cos(2πf c t) x(t) Multipath y(t) p(t) Multipath Wireless Channel Wireless Channel r(t) Bandpass signals Baseband signals
Received Power of Wideband Sİgnals p(t) Multipath Wireless Channel r(t) The output r(t) will approximate the channel impulse response since p(t) approximates unit impulses. N 1 1 jθi r( t) = aie p( t τ 2 i= 0 Assume the multipath components have random amplitudes and phases at time t. N 1 2 N 1 jθ i 2 Ea θ [ PWB ] = Ea, θ aie = ai = i= 0 i= 0 i ) E[ P, WB ]
Received Power of Wideband Sİgnals This shows that if all the multipath components of a transmitted signal is resolved at the receiver then: The average small scale received power is simply the sum of received powers in each multipath component. In practice, the amplitudes of individual multipath components do not fluctuate widely in a local area (for distance in the order of wavelength or fraction of wavelength). This means the average received power of a wideband signal do not fluctuate significantly when the receiver is moving in a local area.
Received Power of Narrowband Sİgnals A CW Signal Transmitter x(t): transmitted signal c(t) Assume now A CW signal transmitted into the same channel. Let comlex envelope will be: c( t) = 2 The instantaneous complex envelope of the received signal will be: r( t) = N 1 i= 0 a e i jθ i ( t, τ ) The instantaneous power will be: r N 1 2 ( ) t = i= 0 a e i jθ i ( t, τ ) 2
Received Power of Narrowband Sİgnals Over a local area (over small distance wavelengths), the amplitude a multipath component may not change signicantly, but the phase may change a lot. For example: -if receiver moves λ meters then phase change is 2π. In this case the component may add up posively to the total sum Σ. -if receiver moves λ/4 meters then phase change is π/2 (90 degrees). In this case the component may add up negatively to the total sum Σ, hence the instantaneous receiver power. Therefore for a CW (continues wave, narrowband) signal, the small movements may cause large fluctuations on the instantenous receiver power, which typifies small scale fading for CW signals.
Wideband versus Narrowband Baseband Signals However, the average received power for a CW signal over a local area is equivalent to the average received power for a wideband signal on the local area. This occurs because the phases of multipath components at different locations over the small-scale region are independently distributed (IID uniform) over [0,2π]. In summary: 1. Received power for CW signals undergoes rapid fades over small distances 2. Received power for wideband signals changes very little of small distances. 3. However, the local area average of both signals are nearly identical.
Small-Scale Multipath Measurements Several Methods Direct RF Pulse System Spread Spectrum Sliding Correlator Channel Sounding Frequency Domain Channel Sounding These techniques are also called channel sounding techniques
Direct RF Pulse System Tx f c Pulse Generator RF Link Rx BPF Detector Digital Oscilloscope
Parameters of Mobile Multipath Channels Time Dispersion Parameters Grossly quantifies the multipath channel Determined from Power Delay Profile Parameters include Mean Access Delay RMS Delay Spread Excess Delay Spread (X db) Coherence Bandwidth Doppler Spread and Coherence Time
Measuring PDPs Power Delay Profiles Are measured by channel sounding techniques Plots of relative received power as a function of excess delay They are found by averaging intantenous power delay measurements over a local area Local area: no greater than 6m outdoor Local area: no greater than 2m indoor Samples taken at λ/4 meters approximately For 450MHz 6 GHz frequency range.
Timer Dispersion Parameters τ ( ) = = = k k k k k k k k k k P P a a ) ( ) )( ( 2 2 2 2 2 2 2 τ τ τ τ τ τ τ σ τ Determined from a power delay profile. Mean excess delay( ): Rms delay spread (s t ): = = k k k k k k k k k k P P a a ) ( ) )( ( 2 2 τ τ τ τ τ
Timer Dispersion Parameters Maximum Excess Delay (X db): Defined as the time delay value after which the multipath energy falls to X db below the maximum multipath energy (not necesarily belonging to the first arriving component). It is also called excess delay spread.
RMS Delay Spread
PDP Outdoor
PDP Indoor
Noise Threshold The values of time dispersion parameters also depend on the noise threshold (the level of power below which the signal is considered as noise). If noise threshold is set too low, then the noise will be processed as multipath and thus causing the parameters to be higher.
Coherence Bandwidth (B C ) Range of frequencies over which the channel can be considered flat (i.e. channel passes all spectral components with equal gain and linear phase). It is a definition that depends on RMS Delay Spread. Two sinusoids with frequency separation greater than B c are affected quite differently by the channel. f 1 f 2 Receiver Multipath Channel Frequency Separation: f 1 -f 2
Coherence Bandwidth Frequency correlation between two sinusoids: 0 <= C r1, r2 <= 1. If we define Coherence Bandwidth (B C ) as the range of frequencies over which the frequency correlation is above 0.9, then B C = 1 50σ σ is rms delay spread. If we define Coherence Bandwidth as the range of frequencies over which the frequency correlation is above 0.5, then B C = 1 5σ This is called 50% coherence bandwidth.
Coherence Bandwidth Example: For a multipath channel, σ is given as 1.37µs. The 50% coherence bandwidth is given as: 1/5σ = 146kHz. This means that, for a good transmission from a transmitter to a receiver, the range of transmission frequency (channel bandwidth) should not exceed 146kHz, so that all frequencies in this band experience the same channel characteristics. Equalizers are needed in order to use transmission frequencies that are separated larger than this value. This coherence bandwidth is enough for an AMPS channel (30kHz band needed for a channel), but is not enough for a GSM channel (200kHz needed per channel).
Coherence Time Delay spread and Coherence bandwidth describe the time dispersive nature of the channel in a local area. They don t offer information about the time varying nature of the channel caused by relative motion of transmitter and receiver. Doppler Spread and Coherence time are parameters which describe the time varying nature of the channel in a small-scale region.
Doppler Spread Measure of spectral broadening caused by motion We know how to compute Doppler shift: f d Doppler spread, B D, is defined as the maximum Doppler shift: f m = v/λ If the baseband signal bandwidth is much greater than B D then effect of Doppler spread is negligible at the receiver.
Coherence Time Coherence time is the time duration over which the channel impulse response is essentially invariant. If the symbol period of the baseband signal (reciprocal of the baseband signal bandwidth) is greater the coherence time, than the signal will distort, since channel will change during the transmission of the signal. T S T C Coherence time (T C ) is defined as: 1 TC f m f 1 f 2 t=t 2 -t 1 t 1 t 2
Coherence Time Coherence time is also defined as: T C 9 2 = 16πf m 0.423 f m Coherence time definition implies that two signals arriving with a time separation greater than T C are affected differently by the channel.
Types of Small-scale Fading Small-scale Fading (Based on Multipath Tİme Delay Spread) Flat Fading 1. BW Signal < BW of Channel 2. Delay Spread < Symbol Period Frequency Selective Fading 1. BW Signal > Bw of Channel 2. Delay Spread > Symbol Period Small-scale Fading (Based on Doppler Spread) Fast Fading 1. High Doppler Spread 2. Coherence Time < Symbol Period 3. Channel variations faster than baseband signal variations Slow Fading 1. Low Doppler Spread 2. Coherence Time > Symbol Period 3. Channel variations smaller than baseband signal variations
Flat Fading Occurs when the amplitude of the received signal changes with time For example according to Rayleigh Distribution Occurs when symbol period of the transmitted signal is much larger than the Delay Spread of the channel Bandwidth of the applied signal is narrow. May cause deep fades. Increase the transmit power to combat this situation.
Flat Fading s(t) h(t,τ) r(t) τ << T S 0 T S 0 τ 0 T S +τ Occurs when: B S << B C and T S >> σ τ B C : Coherence bandwidth B S : Signal bandwidth T S : Symbol period σ τ : Delay Spread
Frequency Selective Fading Occurs when channel multipath delay spread is greater than the symbol period. Symbols face time dispersion Channel induces Intersymbol Interference (ISI) Bandwidth of the signal s(t) is wider than the channel impulse response.
Frequency Selective Fading s(t) r(t) h(t,τ) τ >> T S 0 T S 0 τ 0 T S +τ T S Causes distortion of the received baseband signal Causes Inter-Symbol Interference (ISI) Occurs when: B S >B C and T S < σ τ As a rule of thumb: T S < σ τ
Fast Fading Due to Doppler Spread Rate of change of the channel characteristics is larger than the Rate of change of the transmitted signal The channel changes during a symbol period. The channel changes because of receiver motion. Coherence time of the channel is smaller than the symbol period of the transmitter signal Occurs when: B S < B D and T S >T C B S : Bandwidth of the signal B D : Doppler Spread T S : Symbol Period T C : Coherence Bandwidth
Slow Fading Due to Doppler Spread Rate of change of the channel characteristics is much smaller than the Rate of change of the transmitted signal Occurs when: B S >> B D and T S << T C B S : Bandwidth of the signal B D : Doppler Spread T S : Symbol Period T C : Coherence Bandwidth
Different Types of Fading T S Flat Slow Fading Flat Fast Fading Symbol Period of Transmitting Signal σ τ Frequency Selective Slow Fading Frequency Selective Fast Fading T C Transmitted Symbol Period T S With Respect To SYMBOL PERIOD
Different Types of Fading B S Transmitted Baseband Signal Bandwidth B C Frequency Selective Fast Fading Frequency Selective Slow Fading Flat Fast Fading Flat Slow Fading B D Transmitted Baseband Signal Bandwidth B S With Respect To BASEBAND SIGNAL BANDWIDTH
Fading Distributions Describes how the received signal amplitude changes with time. Remember that the received signal is combination of multiple signals arriving from different directions, phases and amplitudes. With the received signal we mean the baseband signal, namely the envelope of the received signal (i.e. r(t)). Its is a statistical characterization of the multipath fading. Two distributions Rayleigh Fading Ricean Fading
Rayleigh and Ricean Distributions Describes the received signal envelope distribution for channels, where all the components are non-los: i.e. there is no line-of sight (LOS) component. Describes the received signal envelope distribution for channels where one of the multipath components is LOS component. i.e. there is one LOS component.
Rayleigh Fading
Rayleigh Rayleigh distribution has the probability density function (PDF) given by: p( r) = r σ 0 2 e r 2σ 2 2 ( 0 ( r < r ) 0) σ 2 is the time average power of the received signal before envelope detection. σ is the rms value of the received voltage signal before envelope detection Remember: 2 P (average power) V rms (see end of slides 5)
Rayleigh The probability that the envelope of the received signal does not exceed a specified value of R is given by the CDF: r r r P mean median rms R R = = σ ( R) Pr ( r R) p( r) dr = 1 e 0 = = E[ r] = = 1.177σ 2σ 0 rp( r) dr = σ π 2 found by solving 2 2 2 = 1.2533σ 1 2 = r median 0 p( r) dr
Rayleigh PDF 0.7 0.6 0.5 0.6065/σ mean = 1.2533σ median = 1.177σ variance = 0.4292σ 2 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 σ 2σ 3σ 4σ 5σ
Ricean Distribution When there is a stationary (non-fading) LOS signal present, then the envelope distribution is Ricean. The Ricean distribution degenerates to Rayleigh when the dominant component fades away.
Level Crossing Rate (LCR) Threshold (R) LCR is defined as the expected rate at which the Rayleigh fading envelope, normalized to the local rms signal level, crosses a specified threshold level R in a positive going direction. It is given by: N R = 2π f m ρe ρ 2 ρ N R : where = R / r rms (specfied envelope crossings per second value normalized to rms)
Average Fade Duration Defined as the average period of time for which the received signal is below a specified level R. For Rayleigh distributed fading signal, it is given by: ( ) rms m R R r R f e e N R r N = = = = ρ π ρ τ τ ρ ρ, 2 1 1 1 ] Pr[ 1 2 2
Statistical Models for Multipath Fading Channels Clarke's Model for Flat Fading Two-ray Rayleigh Fading Model Saleh and Valenzuela Indoor Statistical Model SIRCIM and SMRCIM Indoor and Outdoor Statistical Models
Fading Model Gilbert-Elliot Model Signal Amplitude Fade Period Threshold Time t Good (Non-fade) Bad (Fade)
Gilbert-Elliot Model 1/AFD Good (Non-fade) 1/ANFD Bad (Fade) The channel is modeled as a Two-State Markov Chain. Each state duration is memory-less and exponentially distributed. The rate going from Good to Bad state is: 1/AFD (AFD: Avg Fade Duration) The rate going from Bad to Good state is: 1/ANFD (ANFD: Avg Non-Fade Duration)