Electrical & Computer Engineering
Wireless Communications
Small Scale Fading
Hamid Bahrami
Small-Scale Fading
l Describe the rapid fluctuations of the amplitudes, phases, or multipath delays of a radio signal over a short period of time or travel distance l The large-scale path loss effects might be ignored l Fading is caused by multipath waves
l Rapid changes in signal strength l Random frequency shift l Time dispersion (echoes)
Factors Influencing Small-Scale Fading
l Multipath propagation l Multiple signal paths l Intersymbol interference
l Speed of the mobile l Doppler shift l Random frequency modulation
l Speed of surrounding objects l Surrounding objects move at a greater rate than the MS
l The transmission bandwidth of the signal l Comparing to the bandwidth of the multipath channel
Doppler Shift
l Named after Chris Doppler, an Australian scientist l The apparent change in frequency and wavelength of a
due to the moving receiver or the waving transmitter
A mobile moving at a constant velocity v Along a path between X and Y with the distance d
Signals received from a remote source S
The source is very far away, same
The difference in path lengths θθ coscos tvdl Δ==Δ
The phase change θλπ
λπφ cos22 tvl Δ=Δ=Δ
Doppler Shift
l Doppler shift, the apparent change in frequency
l Moving toward the direction of arrival of the wave à the Doppler shift is positive (increasing frequency)
l Moving away from the direction of arrival of the wave à the Doppler shift is negative (decreasing frequency)
θλ
φπ
cos21 v
tfd =
ΔΔ=
Enhance our understanding…
l Consider a transmitter which radiates a sinusoidal carrier frequency of 1850 MHz. For a vehicle moving 60 mph, compute the received carrier frequency if the mobile is moving (a) directly toward the transmitter, (b) directly away from the transmitter, (C) in a direction which is perpendicular to the direction of arrival of the transmitted signal
Impulse Response Model
l The impulse response model can be used for the small-scale variations of a mobile radio signal ?!
l So far, we know l The impulse response is a wideband channel
characterization and contains all information l The mobile radio channel may be modeled as a linear
filter with a time varying impulse response l We need the proof…
Impulse Response Model
l For a fixed position d, the channel between the Tx and the Rx can be modeled as a linear time invariant system h(t)
l Due the multipath fading, the channel transfer function should be a function of the position of the Rx, h(d, t)
( )∫∞
∞−−=⊗= τττ dtdhxtdhtxtdy ),(),()(),(
The received signal The transmitted signal
For a causal system, ( )∫ ∞−−=
tdtdhxtdy τττ ),(),(
d=vt
( )∫ ∞−−=
tdtvthxtvty τττ ),(),(
Impulse Response Model
l The mobile channel can be modeled as a linear time varying channel, changing with time and distance
( ) ),()(),()(),()( tdhtxtvthtxdtvthxtyt
⊗=⊗=−= ∫ ∞−τττ
Since v is constant over a short time or time interval,
( ) ),()(),()( ττττ thtxdthxty ⊗== ∫∞
∞−
x(t): the transmitted bandpass waveform y(t): the received waveform h(t, τ): the impulse response of the time varying multipath radio channel τ: the channel multipath delay for a fixed value of t
Impulse Response Model
l Discrete model l Equal time delay segments àexcess delay bins l τ0: the first arriving signal l N: the total number of possible multipath components
t1
t2
h tb ,τ( )
τ t2( )
τ t1( )
τ to( )τ 1 τ 4τ 3τ 2 τ N −1τ N − 2
t
delaybin i iΔτ τ τ= −+1
Impulse Response Model
l Discrete model (cont.) l The model may be used to analyze transmitted RF
signals having bandwidths less than 2/ l Excess delay τi: the relative delay of the ith multipath
component as compared to the first arriving component l Maximum excess delay of the channel: N
τΔ
τΔ
Impulse Response Model
l Discrete model (cont.) l If the channel impulse response is assumed to be time
invariant or at least wide sense stationary
l Power delay profile of the channel over a local area
[ ] ( )iN
iiib jah ττδθτ −=∑
−
=
1
0
exp)(
( ) 2),( ττ thkP b=
Parameters of Mobile Multipath Channels
l Review: Power delay profile l A plot of relative received power as a function of excess delay
with respect to a fixed time delay reference l Averaging instantaneous power delay profile measurements over
a local-area
∑−
−
=1
00
22
0 )()(N
kk tatr
Time Dispersion Parameter
l The mean excess delay
( )( )∑
∑∑∑
==
kk
kkk
kk
kkk
P
P
a
a
τ
ττττ 2
2
The rms delay spread
( )( )( )∑
∑∑∑
==
−=
kk
kkk
kk
kkk
P
P
a
a
τ
ττττ
ττστ2
2
22
2
22
The maximum excess delay
dB X within iscomponent multipath aat which delay max. the: arrivalfirst the:0
0max
X
X
ττ
τττ −=
Time Dispersion Parameter
Time Dispersion Parameter
l The rms delay spread and mean excess delay are defined from a single power delay profile, which is averaged over a local area
l Many measurements are made at many local areas in order to determine a statistical range of multipath channel parameters
l In practice, such parameters depend on the choice of noise threshold used to process P() analysis
l …Related to bandwidth
Example: RMS Delay Spread
Environment Freq. MHz RMS Delay Spread Notes
Urban 910 1300 ns avg. New York City
Urban 892 10~25 µs San Francisco
Suburban 910 200~310 ns Typical case
Suburban 910 1960~2110 ns Extreme case
Indoor 1500 10~50 ns Office building
Indoor 850 270 ns max. Office building
Indoor 1900 70~94 ns avg. Office building
Practice: page 201, ex. 5.4 l Compute the RMS delay spread for the following power delay
profile: l If BPSK modulation is used, what is the maximum bit rate that can
be sent through the channel without needing an equalizer?
0 dB 0 dB
0 1 us
1.0≤sTτσ
Coherence Bandwidth
l A defined relation derived from the rms delay spread l A statistical measure of the range of frequencies over
which the channel can be considered “flat” l The range of frequencies over which two frequency
components have a strong potential for amplitude correlation
τσ501≈cB
τσ51≈cB
As the bandwidth over which the frequency correlation function is above 0.9
As the bandwidth over which the frequency correlation function is above 0.5
Doppler Spread and Coherence Time
l Describe the time varying nature of the channel in a small-scale region
l Doppler spread BD l A measure of the spectral broadening caused by the time rate of
change of mobile radio channel l As the range of frequencies over [fc-fd, fc+fd] which the received
Doppler spectrum is essentially non-zero l Slow fading channel: if the baseband signal bandwidth is much
greater than BD, the effect of Doppler spread are negligible at the receiver
Doppler Spread and Coherence Time
l Coherence time TC l The time domain dual of Doppler spread l Characterize the time varying nature of the frequency
dispersiveness of the channel
mmc
mc
mc
ffT
fT
fT
423.0169
169
1
2 ==
≈
≈
π
π
The maximum Doppler shift fm=v/
The time over which the time correlation function is above 0.5
Practical
Doppler Spread and Coherence Time
l Coherence time is actually a statistical measure of the time duration over which the channel impulse response is essentially invariant
l Coherence time is the time duration over which two received signals have a strong potential for amplitude correlation
l Two signals arriving with a time separation greater than TC are affected differently by the channel
l If the reciprocal bandwidth of the baseband signal is greater than the coherence time of the channel, then the channel will change during the transmission of the message, thus causing distortion at the receiver.
Practice: page 202, ex. 5.5
l Calculate the mean excess delay, rms delay spread, and the max. excess delay (10dB) for the multipath of the channel. Calculate the coherence bandwidth.
-20 dB
-10 dB
0 dB
Pr()
0 1 2 5 (us)
Types of Small-Scale Fading
l Depending on the relation between the signal parameters: bandwidth, symbol period
l Depending on the relation of the channel parameters: delay spread, Doppler spread
l Types l Time dispersion fading l Frequency selective fading l Frequency dispersion fading l Time selective fading
Multipath Time Delay Spread
l Flat fading l A constant gain and linear phase response over a bandwidth l A bandwidth is greater than the BW of the transmitted signal l Most common type l The spectral characteristics of the transmitted signal are preserved at
the receiver, even though the strength of the received signal changes with time
l Narrowband channels or amplitude varying channels
ττ σσ 10,>><<
s
cs
TBB
Multipath Time Delay Spread
l Frequency selective fading l A constant gain and linear phase response over a bandwidth l A bandwidth is smaller than the BW of the transmitted signal l More difficult to model l The received signal includes multiple versions of the transmitter
waveform which are faded and delayed l Wideband Channel or Intersymbol Interference (ISI)
ττ σσ 10<>
s
cs
TBB
Flat Fading
Frequency Selective Fading
Due to Doppler Shift
l Fast fading l The channel impulse response changes rapidly within the
symbol duration l Resulting in frequency dispersion à signal distortion l Only dealing with the rate of the change of the channel due to
the motion. It does not imply flat/frequency selective fading l In practice, fast fading only occurs for very low data rates
Ds
cs
BBTT
<>
Due to Doppler Shift
l Slow fading l The channel impulse response changes slower than the
transmitted baseband signal l The channel may be assumed to be static over one or several
reciprocal bandwidth intervals à the Doppler spread of the channel is much less than the bandwidth of the baseband
Ds
cs
BBTT
>><<
Types of Small-scale Fading
Based on mulitpath time delay spread
Flat fading BW of signal < BW of channel Delay spread < Symbol period
Frequency selective fading BW of signal > BW of channel Delay spread > Symbol period
Based on Doppler spread
Fast fading High Doppler spread
Coherent time < Symbol period Channel variation faster than
baseband signal variations
Slow fading Low Doppler spread
Coherent time > Symbol period Channel variation slower than
baseband signal variations
Types of Small-scale Fading Sym
bol Period of Transm
itting symbol
Transmitted B
aseband Signal B
andwidth
Transmitted Symbol Period
Transmitted Baseband Signal Bandwdith
Flat Slow Fading Flat Fast Fading
Frequency Selective Slow Fading
Frequency Selective Fast Fading
TS
BS
BS
TS TC
BC
Bd
Frequency Selective Fast Fading
Frequency Selective Slow Fading
Flat Slow Fading Flat Fast Fading
Practice: page 253, 5.30
l For each of the three scenarios below, decide if the received signal is best described as undergoing fast fading, frequency selective fading or flat fading. l A binary modulation has a data rate of 500 kbps, fc=1 GHz and a
typical urban radio channel is used to provide communications to cars moving on a highway.
l A binary modulation has a data rate of 5 kbps, fc=1GHz and a typical urban radio channel is used to provide communications to cars moving on a highway.
l A binary modulation has data rate of 10bps, fc=1GHz and a typical urban radio channel is used to provide communications to cars moving on a highway.
Rayleigh Distribution
l To describe the statistical time varying nature of l The received envelope of a flat fading signal l The envelope of an individual multipath component l The envelope of the sum of two quadrature Gaussian
noise signals l Probability Density Function (PDF)
⎪⎩
⎪⎨⎧
>
∞≤≤⎟⎟⎠
⎞⎜⎜⎝
⎛−=
r
rrrrP
00
02
exp)( 2
2
2 σσ
: the rms value of the received voltage signal before envelope detection 2: the time average power of the received signal before envelope detection, variance
σ2
Rayleigh Distribution l Cumulative Density Function (CDF): the probability of the
received signal does not exceed a specified value R
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−−==≤= ∫ 2
2
0 2exp1)(Pr)(
σRdrrPRrRP
R
l The mean value and variance of the Rayleigh distribution
2
0
2222
0
4292.02
)(][][
2533.12
)(][
σπσσ
σπσ
=−=−=
====
∫
∫∞
∞
drrPrrErE
drrrPrEr
r
mean
l The median value of r
σ177.1)(21
0=⇒= ∫ median
rrdrrPmedian
Rayleigh Distribution
PDF CDF
l Without LOS - there are many objects in the environment that scatter the radio signal before it arrives at the receiver
l The urban environment l Build-up city center
l Small-scale fading effect
Rayleigh Distribution
Ricean (Rice) Distribution
l The small-scale fading envelope distribution l When there is a dominant stationary (nonfading) signal
component present (LOS) l As the dominant signal becomes weaker à Rayleigh
⎪⎩
⎪⎨⎧
<
≥∞≤≤⎟⎠⎞⎜
⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛ +−=ro
ArArIArrrP
0
0,02
exp)( 202
22
2 σσσ
A: the peak amplitude of the dominant signal I(.): Bessel function of the first kind and zero-order
Ricean Distribution
l Ricean factor K – the ratio between the deterministic signal power and the variance of the multipath l Aà0, Kà- dB, the dominant path decreases à
Rayleigh l K>>1: less severe fading l K<<1: severe fading
2
2
2
2
2log10)(
2
σ
σAdBK
AK
=
=
Ricean Distribution
Nakagami Distribution l An empirical model that fits measured data in many mobile
environment better than Rayleigh or Ricean model
( )( )
22 2 1mm mmf x exp xx mα⎛ ⎞ ⎛ ⎞−= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠Γ Ω Ω
[ ] [ ]( )[ ]
222 1
2E xE x ; mVar x
Ω = = ≥
12112
, one sided Gaussian, Rayleigh pdf
m, approximates Ricean, no fading
>
⎧⎪⎪= ⎨⎪⎪→∞⎩
Nakagami Distribution l Power distribution
( )( )2
1m mx mmf exp xxmα
− ⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟ ⎝ ⎠Γ Ω⎝ ⎠Ω
( ) [ ]
( )
( )
( )
2
2
2
0
10
1
1
x
mx m
F Px x
f dyy
mm y exp y dym
mG m, xm
α
α
α
−
= ≤
= ∫
⎛ ⎞⎛ ⎞= −⎜ ⎟∫⎜ ⎟ ⎝ ⎠Γ Ω⎝ ⎠Ω⎛ ⎞= ⎜ ⎟Γ ⎝ ⎠Ω ( ) ( ) ( ) 1
0x a yG y e dya,x a,xa − −= Γ −Γ = ∫
Practice: page 251, 5.19
l Show that the magnitude (envelope) of the sum of two independent identically distributed complex (quadrature) Gaussian sources is Rayleigh distributed. Assume that the Gaussian sources are zero mean and have unit variance.
Practical Models
l Clark model l Saleh and Valenzuela indoor statistical model l Two-ray Rayleigh fading model l SIRCIM and SMRCIM indoor and outdoor
statistical models l And more…
Level crossing rate by Rice
l Level crossing rate (LCR): the expected rate at which the Rayleigh fading envelope, normalized to the local rms signal level, crosses a specified level in a positive-going direction
2
2),(0
ρρπ −∞== ∫ efrdrRprN mR
The time derivative of r(t)
The joint density function of r and at r=R
The value of specified level R, normalized to the local rms amplitude
R/Rrms
The Average Duration
l The Average Duration: the average period of time for which the received signal is below a specified level R.
[ ]
[ ] [ ] ( )
signal fading theof intervaln observatio the:T fade theofduration the:
exp1Pr1Pr
Pr1
2
i
ii
R
RrT
Rr
RrN
τ
ρτ
τ
−−=≤=≤
≤=
∑
πρτ
ρ
21
2
mfe −=
Practice: page 224, ex. 5.7
l For a Rayleigh fading signal, compute the positive-going level crossing rate for ρ=1, when the maximum Doppler frequency is 20 Hz. What is the maximum velocity of the mobile for this Doppler frequency if the carrier frequency is 900 MHz?
Practice: page 225, ex. 5.8
l Find the average fade duration for threshold levels ρ=0.01, 0.1 and 1, when the Doppler frequency is 200 Hz.
l Find the average fade duration for a threshold levels of ρ=0.707 when the Doppler frequency is 20 Hz. For a binary digital modulation with bit duration of 50 bps, is the Rayleigh fading slow or fast? What is the average number of bit errors per second for the given data rate. Assume that a bit error occurs whenever any portion of a bit encounters a fade for which ρ<0.1.
The average duration of a signal fade helps determine the most likely number of signal bits that may be lost during a fade.
Summary: Small Scale Fading
l Parameters of mobile multipath channels l Time dispersion parameters and coherence bandwidth l Doppler spread and coherence time
( )( )∑
∑∑∑
==
kk
kkk
kk
kkk
P
P
a
a
τ
ττττ 2
2
( )( )( )∑
∑∑∑
==
−=
kk
kkk
kk
kkk
P
P
a
a
τ
ττττ
ττστ2
2
22
2
22
τσ501≈cB
τσ51≈cB
mmc
mc
mc
ffT
fT
fT
423.0169
1691
2 ==
≈≈
π
π
θλ
φπ
cos21 v
tfd =
ΔΔ=
Summary: Small Scale Fading
l Types of small-scale fading l Flat fading vs. frequency selective fading l Fast fading vs. slow fading
Flat fading BW of signal < BW of channel Delay spread < Symbol period
Frequency selective fading BW of signal > BW of channel Delay spread > Symbol period
Fast fading High Doppler spread
Coherent time < Symbol period Channel variation faster than
baseband signal variations
Slow fading Low Doppler spread
Coherent time > Symbol period Channel variation slower than
baseband signal variations
Summary: Small Scale Fading
l Rayleigh fading distribution
⎪⎩
⎪⎨⎧
>
∞≤≤⎟⎟⎠
⎞⎜⎜⎝
⎛−=
r
rrrrP
00
02
exp)( 2
2
2 σσ
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛−−==≤= ∫ 2
2
0 2exp1)(Pr)(
σRdrrPRrRP
R
l Level crossing rate and average duration
Practice l Page 250, example 5.12: The fading characteristics of a CW
carrier in an urban area are to be measured. The following assumptions are made (1) The mobile receiver uses a simple vertical monopole. (2) Large-scale fading due to path loss is ignored. (3) The mobile has no line-of-sight path to the base station. (4) The pdf of the received signal follows a Rayleigh distribution. l Derive the ratio of the desired signal level to the rms signal level that
maximizes the level crossing rate. Express your answer in dB. l Assuming the maximum velocity of the mobile is 50 km/hr, and the
carrier frequency is 900 MHz, determine the maximum number of times the signal envelope will fade below the level found in (a) during a 1 minutes test.
l How long on average will each fade in (b) last?
Practice: page 248, 5.2
l Describe all the physical circumstances that relate to a stationary transmitter and a moving receiver such that the Doppler shift at the receiver is equal to (a) 0 Hz, (b) fdmax , (c) -fdmax and (d) fdmax/2