2.3 Radio Channel
2.3.2 Stochastic Channel Modeling
In this section, the stochastic modeling of both long-term and short-term fading is intro-duced. The most often used stochastic model for long-term fading is exponential path-loss plus log-normal shadowing [Rap02]. LetXσ denote the log-normal shadowing, which is a zero-mean Gaussian distributed random variable in unit of dB with standard deviation σ(also in unit of dB). Further, by lettingdbe the distance between the transmitter and the receiver, the attenuation due to the long-term fading in unit of dB is expressed as
loss(d)[dB] =A+ 10γlog10 d
d0
+Xσ, (2.1)
whered0,Aandγare constant real values. Since the attenuation of a signal is proportional to the square of the propagation distance in free space, the value of γ, known as path-loss exponent, is generally greater than 2. Usually, the values ofA, γ andσ are derived from field measurements [Rap02]. According to [IST05c], the model of the attenuation due to the long-term fading for the considered urban macro scenario is obtained from field measurements as
loss(d)[dB] = 37.49 + 35.74 log10 d
d0
+Xσ with σ = 8 dB. (2.2)
2.3 Radio Channel
The presence of reflectors and scatters results in multiple versions of the transmit signal that arrive at the receiver, displaced with respect to one another in time and spatial orienta-tion. In the geometric or ray-based model based on stochastic modeling of scatterers, the receive signal is assumed to consist ofN time-delayed multi-path replicas of the transmit signal [IST05c]. As shown in Figure 2.2, each of the N paths represents a cluster of M sub-paths. Sub-paths within each path are assumed to have different initial phases but iden-tical delay, because the delay difference among them is too small to be resolvable within the transmission signal bandwidth. Path powers, path delays, and angle properties at both
x x x x xx x x
path n sub-path m
Array at Tx
Array at Rx Array broadside
at Rx Array Broadside
at Tx
θv n
ϕm,
distance d
n
φm,
AoD ,
δn
AoA ,
δn v
1
Tx: Transmitter Rx: Receiver
Figure 2.2: Geometric model of multi-path propagation [IST05c].
sides of the link are modeled as random variables defined through individual probability density functions and cross-correlations [IST05c].
To mathematically describe the multi-path propagation, the following notations are in-troduced. Pn and τn denote the power and the delay of the n-th path, respectively. φm,n
andϕm,nrepresent the angle of departure (AoD) and the angle of arrival (AoA) of them-th sub-path in then-th path with respect to the array broadside of the transmitter and receiver, respectively. Further,v is the velocity of the relative motion between the mobile terminals and the surrounding, and its direction with respect to the array broadside of the receiver is represented byθv. Note that all defined angles that are measured in a clockwise direction are assumed to be negative in value.
Doppler shift, also referred to as Doppler frequency, is the difference between the ob-served carrier frequency and the emitted one. It depends on the velocity of the relative motionv, the speed of light c, the carrier frequencyfc, and the angle between the direc-tions of the signal propagation and the relative motion. Since the AoAs of sub-paths differ from each other, different Doppler frequency is observed on each sub-path. The Doppler
frequency for them-th sub-path of then-th path is calculated as fD,m,n = vfc
c cos(ϕm,n−θv) (2.3) [Rap02].For a given velocity, the maximum Doppler frequency
fD,max= vfc
c (2.4)
is observed when the direction of a certain sub-pathϕm,n coincides with the direction of the relative motionθv [Rap02].
On each sub-path, by taking the signal transmitted/received at the first antenna element as reference, the signal transmitted/received at each of the other antenna elements experiences a phase shift. For the sake of simplicity, the sub-path and path indices are omitted when presenting the calculation of the phase shift in the following. The phase shift experienced at thei-th antenna element with reference to the first antenna element is given by
ai =ej2πfcτi, (2.5)
whereτi is the time for the signal wave front to pass from the first antenna element to the i-th antenna element [Hay96]. As shown in Figure 2.3, for a given antenna configuration, τi only depends on the direction of the incoming wave front, as long as the distance to the source is far enough to make the wave front planar. Thus, the phase shift of the signal on thes-th antenna element with respect to the reference at the transmitter can be formulated as a function of its AoDφand its distance from the reference antenna elementds, i.e.
a(tx)s (φ, ds) =ej2πfcc dssin(φ), (2.6) and the phase shift of the signal on theu-th antenna element with respect to the reference at the receiver can be formulated as a function of its AoAϕand its distance from the reference antenna elementdu, i.e.
a(rx)u (ϕ, du) =ej2πfccdusin(ϕ), (2.7)
By letting ψm,n be the initial phase for the m-th sub-path of the n-th path, and GTx
and GRx represent the antenna gain of the transmitter and the receiver, respectively, the amplitude of the time-variant channel impulse response (CIR) gu,s,n(t) on the n-th path between each antenna pair(u, s)is given by
gu,s,n(t) =√
PnPM
m=1 ejψm,n ·ej2πfD,m,nt. . .
·p
GTx(φm,n)a(tx)s (φm,n, ds). . .
·p
GRx(ϕm,n)a(rx)u (ϕm,n, du). . .
(2.8)
2.3 Radio Channel
Antenna array broadside 1
Wave front
s sin d 1 1
ds
s
1
Antenna array broadside
1 Wave front
u sin d 1 2
du
u
2
Transmit side Receive side
Figure 2.3: Spatial delay incurred when a plane wave impinges on a linear array.
based on (2.3), (2.6) and (2.7).
By denoting withδ(·)the Kronecker delta function, the CIR between each antenna pair (u, s)at timetis obtained as the superposition of allN paths according to
gu,s(τ, t) =
N
X
n=1
gu,s,n(t)δ(τ −τn). (2.9)