Probably the most common and well understood performance measure characteristic of a digital communication system is SNR. In simple math- ematical terms, if γ denotes the instantaneous SNR (a random variable) at the receiver output that includes the effect of fading, then
¯ γ =
∞
0 γpγ(γ)dγ, (1.4)
is the average SNR, where pγ(γ) denotes the probability density function (PDF) of γ.
4.1.2 Outage Probability
Another standard performance criterion characteristic of diversity systems operating over fading channels is the so-called outage probability - denoted by Pout. The outage probability defined as the probability that the instanta-
Performance Analysis of Wireless Systems
neous error probability exceeds a specified value or equivalently the prob- ability that the output SNR, γ, falls below a certain specified threshold, γth. Mathematically speaking, we have
Pout=
γth
0 pγ(γ)dγ, (1.5)
which is the cumulative distribution function (CDF) of γ, evaluated at γ = γth.
4.1.3 Average SER
The third performance criterion and undoubtedly the most difficult of the three to compute is the average SER1. On the other hand, it is the one
that is most revealing about the nature of the system behavior and the one most often illustrated in documents containing system performance evaluations. Thus, it is of primary interest to have a method for its evalua- tion that reduces the degree of difficulty as much as possible. The primary reason for the difficulty in evaluating average SER lies in the fact that the conditional (on the fading) SER is, in general, a nonlinear function of the instantaneous SNR, as the nature of the nonlinearity is a function of the modulation/detection scheme employed by the system. Thus, for exam- ple, in the multichannel case, the average of the conditional SER over the fading statistics is not a simple average of the per channel performance measure as was true for average SNR. Nevertheless, the moment generat- ing function (MGF)-based approach can be used in simplifying the analysis and in a large variety of cases allows unification under a common frame- work [56].
Averaging the conditional probability of error, which is the probability of error over AWGN channels, the average SER can be shown as
Pe= ∞
0 P (E|γ)pγ(γ)dγ, (1.6)
where P (E|γ) is the conditional SER. When characterizing the performance of coherent digital communications, the generic form of the expression for the error probability involves the Gaussian Q-function with an argument proportional to the square root of the instantaneous SNR of the received
1The discussion that follows applies, in principle, equally well to average BER. Further-
more, the terms bit error probability (BEP) and symbol error probability (SEP) are often used in the literature as alternatives to BER and SER.
signal. In the case of communication over a slowly fading channel, the in- stantaneous SNR (γ) is a time-invariant random variable with a PDF, pγ(γ). To compute the average error probability, one can evaluate an integral wh- ose integrand consists of the product of the above-mentioned Gaussian Q-function and fading PDF, that is [56, Eq. (5.1)]
Pe= ∞
0 c Q (
√gγ) pγ(γ)dγ, (1.7)
where c and g are constants that depend on the specific modulation/detec- tion combination. Furthermore, it can be shown that (1.7) can be rewritten as [56, Eq. (5.3)] Pe= ∞ 0 Mγ −g 2 sin2θ pγ(γ)dγ, (1.8)
where Mγ(s) is the MGF of γ ,i.e., Mγ(s) ∞
0 esγpγ(γ)dγ, which is the
Laplace transform of pγ(γ) with the exponent reversed in sign. Since tables of Laplace transforms are readily available, the desired form of the Gaus- sian Q-function therefore allows evaluation of Pe in the simplest possible way, in most cases resulting in a single integral on θ (when the Laplace transform is available in closed form).
4.2 Multipath Fading
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 prop- agation environment, there are different models describing the statistical behavior of the multipath fading envelope. In this thesis, we consider the Rayleigh, Rician, and Nakagami-m channel models, which are mostly used for analyzing wireless systems:
4.2.1 Rayleigh
The Rayleigh distribution is frequently used to model multipath fading with no direct line-of-sight (LOS) path. In this case, the received instan- taneous SNR per symbol of the channel, i.e., γ, is distributed according to pγ(γ) =1 γexp −γ γ , (1.9)
Performance Analysis of Wireless Systems
where γ =Esσh
N0 is the average SNR per symbol, Esis the energy per symbol,
σh is the link path-loss or variance of the channel coefficient h, and the additive noise at all receiving terminals is modeled as zero-mean complex Gaussian random variables with varianceN0.
Rayleigh fading is viewed as a reasonable model for tropospheric and ionospheric signal propagation as well as the effect of heavily built-up ur- ban environments on radio signals [57].
4.2.2 Rician (Nakagami-n)
The Rician distribution is also known as the Nakagami-n distribution [58]. It is often used to model propagation paths consisting of one strong direct LOS component and many random weaker components. Here, the received SNR follows the distribution [56, Eq. (5.10)]
pγ(γ) =(K + 1) γ e −K+(K+1)γ γ I0 2 K(K + 1)γ γ , (1.10)
where K is the Rician factor of the source-destination link, which is de- fined as the ratio of the power in the LOS component to the power in the other (non-LOS) multipath components [56]. In (1.10), I0(·) is the modified
Bessel function of first kind and order zero, andγ =E[γ]. The Rician distri- bution spans the range from Rayleigh fading (K = 0) to no fading (constant amplitude) (K =∞). This type of fading is typically observed in the first re- solvable LOS paths of microcellular urban and suburban land-mobile [59], picocellular indoor [60], and factory [61] environments. It also applies to the dominant LOS path of satellite [62] and ship-to-ship [63] radio links.
4.2.3 Nakagami-m
The Nakagami-m PDF, which is in essence a central chi-square distribu- tion, is given by [56, Eq. (5.14)]
pγ(γ) =m mγm−1 γmΓ (m)e
−mγ
γ , (1.11)
where m is the Nakagami-m fading parameter, which is a real number ranging from 12 to +∞, Γ (·) is the gamma function, and γh=E[γh].
The Nakagami-m distribution spans via the m parameter the wide range of the multipath distributions. 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→ +∞, the Nakagami-m fading channel converges to a non-fading AWGN channel. Furthermore, when m > 1, there is a one-to-one mapping between the m parameter and the Rician K factor (see e.g., [56, Eq. (2.26)]), allowing the Nakagami-m distribution to closely approximate the Rician distribution.
The Nakagami-m distribution often gives the best fit to land-mobile [64] and indoor-mobile [65] multipath propagation, as well as scintillating iono- spheric radio links [66].