DIFFERENT MODELS OF FADING ENVIRONMENTS

Multipath fading is due to the constructive and destructive combination of randomly delayed, reflected, scattered, and diffracted signal components. This type of fading is relatively fast and is therefore responsible for the short-term signal variations. Depending on the nature of the radio propagation environment, there are different models describing the statistical behavior of the multipath fading envelope……………..1 .Rayleigh fading 2.Rician fading 3.Nakagami –q (hoyt) model 4.Nakagami-n(rice)model 5.Nakagami –m model 6.Two ray Rayleigh fading model.According to each model we get different distribution function and as well as different PDF(Probability Density Function),moment generation function(MGF)and amount of fading(AF) etc. Depending on these we can calculate the particular situation of a particular environment. The description of all types are given bellow.

RAYLEIGH FADING:

The Rayleigh distribution is frequently used to model multipath fading with no direct line-of-sight (LOS) path. The Rayleigh fading model has an AF equal to 1 and typically agrees very well with experimental data for mobile systems, where no LOS path exists between the transmitter and receiver antennas . It also applies to the propagation of reflected and refracted paths through the troposphere and ionosphere and to ship-to-ship radio links.Nakagami –q (hoyt) model:

AF of the Nakagami-q distribution is therefore given by

and hence ranges between 1 (q = 1) and 2 (q = 0). The Nakagami-q distribution spans the range from one-sided Gaussian fading (q = 0) to Rayleigh fading (q = 1). It is typically observed on satellite links subject to strong ionospheric scintillation . one-sided Gaussian fading corresponds to the worst-case fading or, equivalently, the largest AF for all multipath distributions considered in our analyses.

Nakagami-n(rice)model:

The Nakagami-n distribution is also known as the Rice distribution. It is often used to model propagation paths consisting of one strong direct LOS component and

many random weaker components. The AF of the Nakagami-n distribution is given by

and hence ranges between 0 (n=1) and 1 (n =0). The Nakagami-n distributionspans the range from Rayleigh fading (n = 0) to no fading ( constant amplitude)(n=1). This type of fading is typically observed in the first resolvable LOS paths of microcellular urban and suburban land-mobile, picocellular indoor and factory environments. It also applies to the dominant LOS path of satellite and ship-to-ship radio links.Nakagami –m model:

AF of

Hence, the Nakagami-m distribution spans via the m parameter the widest range of AF (from 0 to 2) among all the multipath distributions considered in this book. For instance, it includes the one-sided Gaussian distribution (m = 12) and the Rayleigh distribution (m = 1) as special cases. In the limit as m=+α, the Nakagami-m fading channel converges to a nonfading AWGN channel. Furthermore, whenm < 1, we obtain a one-to-one mapping between the m parameter and the q parameter, allowing the Nakagami-m distribution to closely approximate the Nakagami-q (Hoyt) distribution, and this mapping is given by

Similarly, when m > 1 we obtain another one-to-one mapping between the m parameter and the n parameter (or, equivalently, the Rician K factor), allowing the Nakagami-m distribution to closely approximate the Nakagami-n (Rice) distribution, and this mapping is given by

Finally, the Nakagami-m distribution often gives the best fit to landmobileand indoor-mobile multipath propagation, as well as scintillating ionospheric radio links .

The logical representation of error probability performance of receiver of such signals is very much important. The receiver performance over AWGN channel is analytically more desirable for the fading channel too. There are two main tool to developing this mathematical representation.1. Gaussian Q-function.2. Marcum Q-function.Gaussian and Marcum Q-functions that have been around for many decades and to this day still dominate the literature dealing with error performance evaluation have an intrinsic value in their own right with respect to their relation to well-known probability distributions. What we aim to show, however, is that aside from this intrinsic value, these canonical forms suffer a major disadvantage in situations where the argument(s) of the functions depend on random parameters that require further statistical averaging. Such is the case when evaluating average error probability on the fading channel as well as on many other channels with random disturbances. Herein lies the most significant value of the alternative representations of these functions: namely, their ability to enable simple and in many cases closed-form evaluation of such statistical averages.GAUSSIAN Q -FUNCTION:

One-Dimensional Case:

The one-dimensional Gaussian Q-function (often referred to as the Gaussian probability integral), Q(x), is defined as the complement (with respect to unity) of the cumulative distribution function (CDF) corresponding to the normalized (zero mean, unit variance) Gaussian random variable (RV) X. The canonical representation of this function is in the form of a semi-infinite integral of the corresponding probability density function (PDF), namely

………………………(1)

In principle, the representation of (1) suffers from two disadvantages. From a computational standpoint, this relation requires truncation of the upper infinite limit when using numerical integral evaluation or algorithmic techniques. More important, however, the presence of the argument of the function as the lower limit of the integral poses analytical difficulties when this argument depends on other random parameters that ultimately require statistical averaging over their probability distributions. For the pure AWGN channel, only the first of the two disadvantages comes into play which ordinarily poses little difficulty and therefore accounts for the popularity of this form of the Gaussian Q-function in the performance evaluation literature. However, for channels perturbed by other disturbances, in particular the fading channel, the second disadvantage plays an

important role since, as we shall see later, the argument of the Q-function depends, among other parameters, on the random fading amplitudes of the various received

signal components. Thus, to evaluate the average error probability in the presence of fading, one must average the Q-function over the fading amplitude distributions. It is primarily this second disadvantage, namely, the inability to average analytically over one or more random variables when they appear in the lower limit of an integral, that serves as the primary motivation for seeking alternative representations of this and similar functions. Clearly, then, what would be more desirable in such evaluations would be to have a form for Q(x) wherein the argument of the function is in neither the upper nor the lower limit of the integral and furthermore, appears in the integrand as the argument of an elementary function (e.g., an exponential). Still more desirable would be a form wherein the argument-independent limits are finite. In what follows, any function that has the two properties above will be said to be in the desired form.

A number of years ago, Craig [1] cleverly showed that evaluation of the average probability of error for the two-dimensional AWGN channel could be considerably simplified by choosing the origin of coordinates for each decision region as that defined by the signal vector as opposed to using a fixed coordinate system origin for all decision regions derived from the received vector. This shift in vector space coordinate systems allowed the integrand of the two-dimensional integral describing the conditional (on the transmitted signal) probability of error to be independent of the transmitted signal. A by-product of Craig’s work was a definite integral form for the Gaussian Q-function, which was in the desired form. In particular, Q(x) of (1) could also now be defined (but only for x >= 0) by

Two-Dimensional Case:

The normalized two-dimensional Gaussian probability integral is defined by

N.B. : Ergosic and quasi-static are different. In the quasi-static case the shannon capacity does not exist. THey have to use the outage capacity instead while in the ergodic channel, the shannon capacity is well define since the shannon capacity is the average concept. Hence, the channel must be varied in some senses. To make thing simple...

Ergodic capacity -> you get a number as a capacity such as 10 bps/HzQuasi-static capacity -> you get a cdf/ccdf plot of the capacity and you get only a probability to communicate at a given rate.

Definition of Ergodic: A stochastic process is ergodic if no sample helps meaningfully to predict values that are very far away in time from that sample.

- Ergodic channel

* The channel gain process is ergodic, i.e., the time average is equal to the ensemble average. In other words,

the randomness of the channel gain can be averaged out (removed) over time. So long-term constant bit rates can be supported.

- Non-ergodic channel

* The channel gain is a random variable and does not change with time. The channel gain process is stationary but not ergodic, i.e., the time average is not equal to the ensemble average. In other words, the randomness of the channel gain can not be averaged out (removed) over time. So long-term constant bit rates can not be supported

SOMETHING EXTRA DEPENDS ON CHANNEL CAPACITY:

One of the major performance characteristics of a MIMO channel is its capacity.

For time variant fading channel there are multi capacity definitions including:1. Mean [ergodic] capacity[relay on ergodic channel]2. Outage capacity[relay on non-ergodic channel]3. Delay-limited capacity

These can be use for three broad cases-----------

1. CSI is known to transmitter and receiver.2. CSI is known to receiver only.3. CSI is not known to transmitter and receiver.

MIMO channel capacity:When we study about the MIMO channel capacity then we have to give attention that the channel matrix may be of different types. And each of them has a channel capacity. These are ………………………………

1. H may be a deterministic matrix :

2. H may be a ergodic random matrix:

3. H may be a non ergodic matrix chosen randomly in the beginning and held constant for all the time.[here the classical sennon capacity formula is not exist.

A non ergodic process is a quasi-static process and the outage capacity is determined.]:

Where Fc(R) is a commutative distribution function and R is the maximum achievable information rate.

CORRELATION STRUCTURE OF THE MIMO CHANNEL:

Definition 1:

1. A random MIMO channel represent bye matrix H is called rayleigh –fading if the elements of H is jointly distributed zero mean circular symmetric Gaussian.

2. If its mean is not zero then it is known as rician fading channel.

Definition 2:

if

Definition 3:

Г is the MIMO channel co relation matrix

If Г =cI and I is a identity matrix and c is a normalization constant then the channel is uncorrelated otherwise it is said to be correlated.

Г represent a full description of the MIMO channel matrix H. Still it is somehow difficult to analysis the MIMO channel with arbitrary Г . So another model is there known as “KRONEKAR MODEL” . It significantly simplifies the analysis and simulation of correlated channels by allowing independent model at the transmitter and receiver side as………………………

Symmetric Gaussian i.i.d entries with unit varience. This model describes that when Rt=I and Rr= I then that is a semi correlated channel and when Rt≠I and Rr≠ I then the channel is double correlated. This model is successfully applicable when the antenna separation is more with each other.

When in case of small antenna separation the “KRONEKAR MODEL” is failed then another model was propose and that was more accurate then another.

In this model the join correlation of both the transmitter and receiver end has been proposed.

To evaluate the effect of correlation in an explicit form their there are some popular parametric correlation models are there.

1.uniform correlation model:

Correlation between any pair of antenna at transmitting or receiving side is equal and real.

2.exponential correlation model:

Here the elements of R are represented through a single complex correlation parameter r.

This model is successfully implemented in various cases. This model shows that correlation decrease with increase of antenna

spacing.

3.Quadetric exponential correlation model:

It is a physically motivated single parameter correlation matrix model.

It is used in IEEE 802.11 n wireless communication. This model shows that correlation decrease faster with increase of

antenna spacing.

4.Tri-digonal Model:

When correlation is significant only among adjacent antenna then this model is used.

CAPACITY OF RAYLEIGH FADING CHANNEL:

Uncorrelated Rayleigh fading channel.

HαURHWUTH

Identically distributed HW with unit variance[zero mean/jointly distributed]

Semi-correlated MIMO fading channel.

Double correlated Rayleigh fading channel.

IMPORTANT NOTE:

1. If the frequency selective mimo fading channel impulse response has no inter tap correlation then its ergodic capacity is same as that of frequency flat Rayleigh fading channel. If there is any correlation then its ergodic capacity is less that of frequency flat Rayleigh fading channel.

2. A slow time varying and flat time varying channel have the same ergodic capacity but different capacity distribution. The large the time variation that higher and narrower the PDF curves and the large outage capacity at acceptably low outage probabilities.

ASYMPTOTIC CAPACITY DISTRIBUTION OF MIMO CHANNEL: