Summary

3.4 Summary

investigated which will be addressed in more detail when calculating second order statistics in upcoming chapters.

Narrowband Modeling with Additive 4

Envelope Process

Contents

4.1 The Proposed Model Based on an Additive Envelope Process . . . . . 39 4.2 Results and Discussions . . . . 43 4.3 Special Cases . . . . 51 4.4 Summary . . . . 53

4.1 The Proposed Model Based on an Additive Envelope Process

This chapter proposes and develops a model taking into account the fading process experienced by the different components of the received signal. Here, a narrowband model is introduced using additive complex envelope process with a Doppler PSD generated from a 3D scattering model. The expressions for the pdf of the envelope and phase components are derived first; taking into account appropriate fading process as experienced by the LOS and NLOS components. The mathematical expressions for LCR and AFD are then derived. By comparing with measurement results of LCR and AFD available in literature, it is shown that this model can be used to depict a wide variety of fading situations for flat urban and suburban areas. If LOS component is not included, the model represents the scenario normally encountered in dense urban conditions. With LOS component being present but not experiencing shadow fading, the model represents fading conditions experi- enced in open areas. For such cases, the proposed model give rise to a new distribution which is termed here as extended Nakagami-q or extended Hoyt fading process.

4.1 The Proposed Model Based on an Additive Envelope Process

In the model proposed here, as discussed in Section 3.1.1, only the LOS component gets effected by shadow fading while the local mean power of scattered components is kept constant. This otherwise means that NLOS components of the transmitted signal are not shadow faded before entering into the local cluster of scatterers in the close vicinity of the receiver. Therefore, this leads to an additive process same as equation (3.1) and has the applicability in flat urban and suburban area as explained in Section 3.1.1. Here, X₁ and X₂ are both low pass Gaussian process with zero mean and unequal variances σ²₁ and σ₂² respectively, leading to Nakagami-q or Hoyt distribution of the envelope of scattered components [52, 53]. Y represents the shadow faded LOS component with lognormal distribution given as

p_Y(y) = 1

√2πσ_sy exp

"

−(lny−µ_s)² 2σ_s²

#

, y≥0 (4.1)

where σ_s and µ_s are standard deviation and mean of lny respectively. The mean signal level in a large area (large enough to experience the slow variation of shadow fading) is denoted here asS₀, which is related to µ_s asµ_s = lnS₀.

So, the envelope is given by

R=|R|e = q

(X₁+Y)²+X₂². (4.2)

4.1 The Proposed Model Based on an Additive Envelope Process

4.1.1 First Order Statistics

The probability density function (pdf) of envelopeRcan be derived with the help of conditional probability. Given the condition Y =y, the in-phase component of Re is modified into a non-zero mean Gaussian processX₁⁰ =X₁+y with meany. So, the joint pdf ofX₁⁰ and X₂ is

p_X⁰

1,X2(x⁰₁, x₂) = 1

2πσ₁σ₂ exp Ã

−

"

x⁰₁² 2σ²₁ + x²₂

2σ₂²

#!

. (4.3)

Now, by transformation from cartesian to polar coordinates, we get the joint conditional pdf p_R,Θ(r, θ|Y =y) as [60]

p_R,Θ(r, θ|Y =y) = rexp

³

−

h(rcosθ−y)²

2σ²₁ +^r2_2σ^sin2^2θ 2

i´

2πσ₁σ₂ (4.4)

where r≥0 and 0≤θ≤2π. So, the joint envelope-phase pdfp_R,Θ(r, θ) can be derived as [61]

p_R,Θ(r, θ) = Z _∞

0

p_R,Θ(r, θ|Y =y)p_Y(y)dy. (4.5) In light of the technique given in [60] (pp. 128-131), we deduce the individual pdf of envelope and phase as marginal pdf of equation (4.5), as

p_R(r) = r

√2πσ₁σ₂σ_sexp µ

−r² 4

µ 1 σ₁² + 1

σ²₂

¶¶Z^∞

0

1 yexp

"

−(lny−µ_s)² 2σ_s² − y²

2σ₁²

#

× X∞

k=0

(−1)^k²_kI_k µr²

4 µ 1

σ²₁ − 1 σ₂²

¶¶

I_2k µry

σ²₁

¶

dy (4.6)

where²_k= 2 fork6= 0 (²₀ = 1), I_k(·) is modified Bessel function of first kind with orderk; and p_Θ(θ) = 1

(2π)^3/2σ_s Z _∞

0

1 yexp

"

−(lny−µ_s)² 2σ_s²

# ·σ₁σ₂ G(θ)exp

µ

− y² 2σ₁²

¶

+

√2πσ²₂ycosθ G^3/2(θ) exp

µ

−y²sin²θ 2G(θ)

¶ Q

Ã

−σ₂ycosθ σ₁p

G(θ)

!#

dy (4.7)

whereG(θ) =σ²₂cos²θ+σ₁²sin²θ andQ(·) is the Q-function which is related with error function asQ(x) = 12

· 1−erf

µ

√x 2

¶¸

.

Under certain conditions, if the shadow fading is absent, the joint envelope-phase distribution in equation (4.4) denotes a new distribution. We, hereby, name it as extended Nakagami-q or extended Hoyt distribution and describe its statistics in Section 4.3.2.

4.1 The Proposed Model Based on an Additive Envelope Process

4.1.2 Second Order Statistics 4.1.2.1 LCR and AFD

Level Crossing Rate (LCR) is the rate at which envelope of received signal crosses a specified threshold. In analytical representation, it is expressed as

N_R(r) = Z∞

0

˙

rp_RR˙(r,r)d˙ r˙ (4.8)

where R and ˙R are received signal envelope and its time derivative, respectively, and p_R,R˙(r,r) is˙ the joint pdf of R and ˙R. p_R,R˙(r,r) can be derived with the use of conditional probability in a˙ similar way as in Section 4.1.1.

p_R,R˙(r,r) =˙ Z _∞

−∞

Z _∞

0

p_R,R˙(r,r|Y˙ =y,Y˙ = ˙y)p_Y,Y˙(y,y)dyd˙ y˙ (4.9) where r≥0, −∞<r <˙ ∞, 0≤θ≤2π and −∞<θ <˙ ∞ and p_Y,Y˙(y,y) is the joint pdf of˙ Y and Y˙ given as [62]

p_YY˙(y,y) =˙ 1

2πσ_sσ_s⁰y²exp



−(lny−µ_s)² 2σ²_s −

³y˙/y

´₂

2σ²_s0



 (4.10)

where y ≥ 0,−∞ <y <˙ ∞ and σ_s⁰ is the standard deviation of time derivative of the normal or Gaussian variate associated with ˙Y.

Given the conditionY =y, the in-phase component of Re is a Gaussian process X₁⁰ =X₁+y, as shown in Section 4.1.1. So, its derivative, ˙X₁⁰ = ˙X₁+ ˙y, is also a Gaussian process with non-zero mean ˙ywhile ˙X₂is zero mean Gaussian distributed. The joint pdfp_X0

1,X2,X˙₁⁰,X˙2(x⁰₁, x₂,x˙⁰₁,x˙₂) comes out to be a four variable joint Gaussian distribution as

p_X0

1,X2,X˙⁰₁,X˙2(x⁰₁, x₂,x˙⁰₁,x˙₂) = 1 (2π)²σ₁σ₂√

β₁β₂exp Ã

−1 2

"

(x⁰₁−y)² σ²₁ +x²₂

σ₂² +( ˙x⁰₁−y)˙ ² β₁ +x˙²₂

β₂

#!

(4.11) where β₁ and β₂ are variances of ˙X₁⁰ and ˙X₂ respectively and can be related with σ₁ and σ₂ by method shown in Section 4.1.2.2.

Now, by transformation from cartesian to polar coordinates, we get the joint conditional pdf as

p_{R,R,Θ,˙ Θ˙}(r,r, θ,˙ θ|Y˙ =y,Y˙ = ˙y) = r² (2π)²σ₁σ₂√

β₁β₂ exp

"

−(rcosθ−y)²

2σ₁² −r²sin²θ 2σ²₂

4.1 The Proposed Model Based on an Additive Envelope Process

− ( ˙rcosθ−rθ˙sinθ−y)˙ ²

2β₁ −( ˙rsinθ+rθ˙cosθ)² 2β₂

#

. (4.12)

Equation (4.12) is integrated with respect toθ and ˙θto derive p_R,R˙(r,r|Y˙ =y,Y˙ = ˙y).

p_R,R˙(r,r|Y˙ =y,Y˙ = ˙y) = r (2π)^3/2σ₁σ₂

Z2π

0

exp Ã

−(rcosθ−y)²

2σ²₁ −r²sin²θ 2σ₂²

!

× 1

pβ₂+ (β₁−β₂) cos²θexp Ã

− ( ˙r−y˙cosθ)² 2 [β₂+ (β₁−β₂) cos²θ]

!

dθ.(4.13)

Then using equation (4.9), expression for p_R,R˙(r,r) is derived as˙

p_R,R˙(r,r) =˙ r (2π)²σ₁σ₂σ_s

Z∞

0

1 yexp

"

−(lny−µ_s)² 2σ_s²

#Z_2π

0

exp Ã

−(rcosθ−y)²

2σ₁² −r²sin²θ 2σ₂²

!

× 1

q β₂+¡

β₁−β₂+σ_s²0y²¢ cos²θ

× exp



− r˙²

2 hq

β₂+¡

β₁−β₂+σ_s²0y²¢ cos²θ

i



dθdy. (4.14)

Finally, using equation (4.8), expression for LCR is derived.

N_R(r) = r (2π)²σ₁σ₂σ_s

Z∞

0

1 yexp

"

−(lny−µ_s)² 2σ_s²

#Z_2π

0

q β₂+¡

β₁−β₂+σ²_s0y²¢ cos²θ

×exp Ã

−(rcosθ−y)²

2σ₁² −r²sin²θ 2σ²₂

!

dθdy. (4.15)

Using the knowledge of computed LCR, average fade duration (AFD), which is defined as the average period of time for which the received signal is below a specified threshold level R, can be computed as

T_R(R) = Z _R

0

p_R(r)

N_R(R) . (4.16)

In other words, it is the ratio of cumulative distribution function to LCR at a particular signal level.