Rayleigh and Rician Fading

Consider two independent normal random variables X ∼ N(m1, σ2) and

Y ∼ N(m2, σ2). Let us define a complex Gaussian random variable Z via :

Z = X + jY . The envelope and phase of the random variable Z are definedas:

R =√

X2 + Y 2 , Θ = tan−1(

Y

X

)

.

The unique solution of these equations for X and Y in terms of R and Θare given by the polar coordinate representation:

X = R cosΘ , Y = R sinΘ.

The Jacobian for the area transformation is given by:

dxdy = rdrdθ ⇐⇒∣

∣

∣

∣

J

(

r, θ

x, y

)∣

∣

∣

∣

=1

r.

Using the Jacobian method for the joint distribution of R and θ we obtain:

fR,Θ(r, θ) =r

2πσ2exp

(

−r2 + s2

2σ2

)

exp

(

−rm1 cos θ +m2 sin θ

σ2

)

u(r), θ ∈ [−π, π].

Io(x) is the modified zero-th order Bessel function of the first kind definedby

Io(x) =1

2π

∫

2π

0

e−x cos θdθ.

The parameter s is the non-centrality parameter of the faded envelope andis defined as:

s =√

m21+m2

2.

The Rice factor K of the faded envelope is a measure of the power in thefaded envelope that has been produced by the means of X and Y and isdefined as :

K =m21 +m22

2σ2=

s2

2σ2.

Upon integrating this expression over θ ∈ [−π, π] from fR,Θ(r, θ), weobtain the marginal distribution over the envelope:

fR(r) =r

σ2exp

(

−r2 + s2

2σ2

)

Io

(

rs

σ2

)

u(r).

In general the marginal distribution for the phase is dependent on θ and isobtained by integrating out R from the joint distribution:

fΘ(θ) =1

2πexp

(

−s2

2σ2

)

+m̃

2πσexp

(

m̃2 − s2

2σ2

)

Q

(

−m̃

σ

)

, θ ∈ [−π, π]

where the parameter m̃ is defined as :

m̃ = m1cosθ +m2sinθ =√

m21+m2

2cos

(

θ + cos−1(

m1

m2

))

,

and Q(x) denotes the normalized Gaussian tail probability function definedvia:

Q(x) =1√2π

∫ ∞

xexp

(

−y2

2

)

dy.

When K = 0, the marginal for the phase is :

fΘ(θ) =

{

1

2πθ ∈ [−π, π]

0 otherwise.

In other words the phase of the random variable Z is uniformly distributed,i.e., Θ ∼ U([−π, π]) when the rice factor of the envelope is K = 0.

For the specific case when the K = 0, the Rician faded envelope reducesdown to the Rayleigh faded envelope:

fR(r) =r

σ2exp

(

−r2

2σ2

)

u(r)