Nakagami-m Fading Channel

Fading in wireless communication has been a cause of severe degradation in the strength of the
signal. When the transmitted signal undergoes reflection or diffraction such that it follows two
or more than two paths to reach the destination antenna, it is said that a multipath fading has
occurred. This multipath fading results in an ambiguous signal reception at the receiver as the
same transmitted signal arrives at different times and phases to the receiver end. It is how the
environment plays with the signal that defines the degree of fading. So, it is difficult to predict
in a concrete manner, the rules of fading. But then, there have been several statistical models
that could help realize and analyze the fading of the received signal.
For a small geographical region, a Rayleigh or Rician model can be helpful in determining the
fading envelop of the received signal whereas a log-normal fading channel can be used for large
geographical regions. The Nakagami-m fading channel which includes Rayleigh as well as closely
approximates the Rician fading channel, is a probability distribution related to a gamma
distribution. In case of urban cities, where the infrastructures are closely spaced, the fading
14
observed can be modeled better with Nakagami-m fading channel. The Nakagami-m fading
channel is thus more preferable as it enables to model a wider class of fading channel
conditions and to fit well with empirical data [19]. The Nakagami-m distribution is given by
􀝌􀰊􁈺􀟛􁈻 􀵌 􀍴 􁉀􀯠
􀰆 􁉁
􀯠
􀀃 􀰊􀰮􀳘􀰷􀰭
􀯰􁈺􀯠􁈻 􀂇􀂚􀂒􀀃􁈺􀵆 􀯠􀰊􀰮
􀰆 􁈻 (2.1)
where 􀟛 is the channel amplitude, m is the fading severity parameter which varies from 0.5 to
􀗟, 􀟗 = E[􀟛􀬶] which controls the spread. For m = 1, (2.1) gives Rayleigh distribution and for
m = 0.5, we obtain a one sided Gaussian distribution [20]. Different values of m determine the
changing severity of the fading compared to Rayleigh fading.

complex-valued random variable

We call a complex-valued random variable z=x+iy a (circular symmetric) complex Gaussian variable, or it follows complex Gaussian distribution, if its real and imaginary parts, x and y, are jointly Gaussian (i.e. (x,y) follows a two-dimensional Gaussian distribution), uncorrelated (therefore alsoindependent in this case), and they have the same variance of σ². Denoting the mean of x and y by m_x and m_y respectively, we call m_z=m_x+im_y=E[z] the mean of z, and σ²=E[(x-m_x)²]=E[(y-m_y)²]=(1/2)E[|z-m_z|²] is called z's variance per real dimension.

From above we know that the probability density function of the 2-D random variable (x,y) is

p(x,y)=(2πσ²)^-1 exp(-((x-m_x)²+(y-m_y)²)/(2σ²)).

Noting that z=x+iy, the probability density function can also be represented in terms of z:

p(z)=(2πσ²)^-1 exp(-|z-m_z|²/(2σ²))

This is preferred representation of the probability density function of complex-valued random variables. Just remember to separate the real and imaginary parts, x and y, when calculating expectations involving such random variables, rather than integrating over the complex z directly. This applies to the multi-dimensional cases below as well.

The above definition can be easily extended to multi-dimensional cases, by using a 2d-dimensional real Gaussian variable to represent the real and imaginary parts of the d-dimensional complex Gaussian variable. The probability density function of a d-dimensional complex Gaussian variablez=x+iy (a column vector) is

p(z)=(2π)^-d (det(Σ))^-1 exp(-(z-m_z)^HΣ^-1(z-m_z)/2)

Here, m_z=E[z] is the mean of the random vector z, Σ=(1/2)E[(z-m_z)(z-m_z)^H] is its covariance matrix multipled by 1/2 (assuming it is not singular), det(Σ) is Σ's determinant, and z^H means z's Hermitian transposition, or z^T with its elements conjugated. Note that it looks very similar to the probability density function of a d-dimensional real Gaussian variable (see here):

p(x)=(2π)^-d/2 (det(Σ))^-1/2 exp(-(x-m_x)^TΣ^-1(x-m_x)/2)

Above, we have mandated that the real and imaginary parts of a complex Gaussian variable z to be uncorrelated with the same variance, in which case we call z to be circular symmetric, and the above formula for p(z) applies. If z is a d-dimensional complex Gaussian variable, circular symmetricity is more complicated, but it basically means that p(z) can be represented as above. Circular symmetricity gives complex Gaussian variables some desirable properties, as follows:

First, suppose z is a zero-mean d-dimensional complex Gaussian variable (i.e. m_z=0), then its distribution is rotationally invariant, or e^iθz has the same probability density function as z for any real θ (try proving it by yourself). This is probably why it is called "circular symmetric".

Second, let A=E[z₁z₂...z_n z_n+1^*...z_n+m^*], where the unique ones of z₁, ..., z_n+m (we allow for duplicates among them) are jointly Gaussian zero-mean complex random variables (i.e., they form a zero-mean multi-dimensional complex Gaussian variable z), and * denotes complex conjugation, then A=0 if n≠m. To prove this, let w_k=e^iθz_k, k=1,...,n+m, &theta∈R, and define B in the same way as A only with z_k replaced by w_k. Now w=e^iθzshould contain the unique ones of w_k, and w and z should have the same distribution due to circular symmetricity, therefore A=B. However, using w_k=e^iθz_k, we obtain B=e^i(n-m)θA. Since the choice of θ is arbitrary, A must be zero if n≠m. In particular, if z is a zero-mean complex Gaussian variable, then E[z²]=0 (although E[|z|²]=2σ²).

By taking the modulus of a complex Gaussian variable, we get two important distributions: Rayleigh distribution and Rice distribution.

Let z=x+iy be a complex Gaussian variable, a=|E[z]| and σ²=(1/2) E[|z|²], the probability distribution function of r=|z| can be easily obtained by doing variable substitution x=r cosθ, y=r sinθ in p(x,y), then integrating over [0,2π] on θ. When a=0 (or z has zero mean), r=|z| is said to followRayleigh distribution, whose probability density function is

p(r)=(r/σ²) exp(-r²/(2σ²)), r≥0.

Here, p(r) is almost linear with r for small r, and decreases rapidly when r becomes large.

When a is not zero, r=|z| is said to follow Rice distribution, whose probability density function is

p(r)=(r/σ²) exp(-(r²+a²)/(2σ²)) I₀(a r/σ²), r≥0

where I₀(.) is the zeroth-order modified Bessel function, one of the definition equations of which being

I₀(x)=(1/2π) ∫₀^2π exp(x cosθ) dθ.

Here, p(r) is large for r near a, and decreases rapidly for larger and smaller r.

Complex Gaussian variables are often used in engineering. For example, in communication theory, narrow-band Gaussian white noise can be represented by a complex Gaussian process when using the equivalent low-pass representation, where the modulus (absolute value) represents amplitude, and the argument (angle) represents carrier phase, therefore the integration of it over an interval, which arises in the decision variables of many demodulators, is a complex Gaussian variable.

Eb/N0 Vs BER for BPSK over Rayleigh Channel and AWGN Channel

he phenomenon of Rayleigh Flat fading and its simulation using Clarke’s model and Young’s model were discussed in the previous posts. The performance (Eb/N0 Vs BER) of BPSK modulation (with coherent detection) over Rayleigh Fading channel and its comparison over AWGN channel is discussed in this post.

We first investigate the non-coherent detection of BPSK over Rayleigh Fading channel and then we move on to the coherent detection. For both the cases, we consider a simple flat fading Rayleigh channel (modeled as a – single tap filter – with complex impulse response – h). The channel also adds AWGN noise to the signal samples after it suffers from Rayleigh Fading.

The received signal y can be represented as

y=hx+n

where n is the noise contributed by AWGN which is Gaussian distributed with zero mean and unit variance and h is the Rayleigh Fading response with zero mean and unit variance. (For a simple AWGN channel without Rayleigh Fading the received signal is represented as y=x+n).

Non-Coherent Detection:

In non-coherent detection, prior knowledge of the channel impulse response (“h” in this case) is not known at the receiver. Consider the BPSK signaling scheme with ‘x=+/- a’ being transmitted over such a channel as described above. This signaling scheme fails completely (in non coherent detection scheme), even in the absence of noise, since the phase of the received signal y is uniformly distributed between 0 and 2pi regardless of whether x[m]=+a or x[m]=-a is transmitted. So the non coherent detection of the BPSK signaling is not a suitable method of detection especially in a Fading environment.

Coherent Detection:

In coherent detection, the receiver has sufficient knowledge about the channel impulse response.Techniques like pilot transmissions are used to estimate the channel impulse response at the receiver, before the actual data transmission could begin. Lets consider that the channel impulse response estimate at receiver is known and is perfect & accurate.The transmitted symbols (‘x’) can be obtained from the received signal (‘y’) by the process of equalization as given below.

y^=yh=hx+nh=x+z

here z is still an AWGN noise except for the scaling factor 1/h. Now the detection of x can be performed in a manner similar to the detection in AWGN channels.

The input binary bits to the BPSK modulation system are detected as

r=real(y^)=real(x+z)d^=1,ifr>0d^=0,ifr<0

Theoretical BER:

The theoretical BER for BPSK modulation scheme over Rayleigh fading channel (with AWGN noise) is given by

Pb=12⎛⎝1−Eb/N01+Eb/N0−−−−−−−−−√⎞⎠

The theoretical BER for BPSK modulation scheme over an AWGN channel is given here for comparison

Pb=12erfc(Eb/N0−−−−−−√)

Simulation Model:

The following model is used for the simulation of BPSK over Rayleigh Fading channel and its comparison with AWGN channel

BPSK Modulation over Rayleigh and AWGN channel

Simulation Results:

The Simulated and theoretical performance curves (Eb/N0 Vs BER) for BPSK modulation over Rayleigh Fading channel and the AWGN is given below.

Eb/N0 Vs BER for BPSK over Rayleigh and AWGN Channel