# Performance of Differential-Phase OFDM Systems over Selective Multipath Fading Channels
## 2 Channel Specifications  
In a typical terrestrial wireless channel, the transmitted signal propagates over multipath channels which can be modeled as a time-varying random process. The impulse response of the multipath channel, \(g(t;\tau)\), can be expressed as  
\[g(t;\tau) = \sum_{p=1}^{p}g_{p}(t;\tau)\delta(\tau-\tau_{p}(t)) \tag{2}\] \[= \sum_{p=1}^{p}\alpha(t;\tau)e^{i\phi(t;\tau)}\delta(\tau-\tau_{p} (t))\]  
where \(g_{p}(t;\tau)\) and \(\tau_{p}(t)\) are the \(p\)-th multipath component impulse response and time delay, respectively, \(\tau\) is a variable time delay, and \(P\) represents the total number of paths in the multipath environment. The phase term \(\phi(t;\tau)\) represents the total phase shift encountered in the channel due to free space propagation. Practically, the number of the multipath (scatters) is considered to be large enough to represent the channel impulse response, \(g(t;\tau)\), as a zero-mean complex-valued Normal random process, according to the Central Limit Theorem. Over a small-scale time, the channel impulse can be modeled as a wide-sense-stationary uncorrelated scattering (WSS-UCS) process. Therefore, the correlation function is given by  
\[\mathbb{R}_{gg}(\Delta t;\tau) = \mathbb{E}[g(t;\tau)g^{*}(t+\Delta t;\tau)] \tag{3}\] \[= \sum_{p=1}^{p}\mathbb{E}[g_{p}(t;\tau)g_{p}^{*}(t+\Delta t;\tau)]\]  
Over a short distance interval, the delay \(\tau_{p}(t)\) is relatively independent of the time \((=\tau_{p})\) and it is a uniformly distributed random variable in the range of \([0,\tau_{max}]\). \(\tau_{max}\) is the maximum excess delay of the channel given by \(\tau_{max}=P\Delta\tau\) and \(\Delta\tau\) is called the bin time delay width given by \(\Delta\tau=\tau_{p+1}-\tau_{p}\)[13]. The power delay profile, \(\rho(\tau)\), which represents the signal intensity received over a multipath channel, is obtained as the spatial average of the complex baseband channel impulse response as \(\Delta t=0\), such as  
\[\mathbb{R}_{gg}(0;\tau) = \sum_{p=1}^{p}\mathbb{E}[|g_{p}(t;\tau)|^{2}] \tag{4}\] \[= \sum_{p=1}^{p}\rho(\tau)\delta(\tau-\tau_{p})\]  
where \(\rho(\tau_{p})=\mathbb{E}[\alpha^{2}(\tau_{p})]\) represents the power delay profile of the channel. In OFDM systems, the symbol duration, \(T_{s}\), is very small when compared with the channel coherence time (\(T_{s}\ll T_{C}\)). This case is called slow fading in which the envelope, \(|g_{p}(t;\tau)|\), remains almost constant over a period of several data symbols; thus, \(\alpha(t)=\alpha\) and will be treated as a random variable with Nakagami-\(m\) probability density function (pdf). Nakagami-\(m\) best fit in land mobile to model dense scatters which results in delay times that are relatively close to each other [14; 15]. The pdf of the random variable \(\alpha\) can be given by  
\[f_{\alpha}(\alpha)=\frac{2}{\Gamma(m)}\left(\frac{m}{\Omega}\right)^{m}\alpha^ {2m-1}\exp\left(-\frac{m\alpha^{2}}{\Omega}\right),\hskip 28.452756 pt\alpha\geq 0, \tag{5}\]  
where \(\Gamma(. )\) is the gamma function. Nakagami-\(m\) distribution represents a general model that can fit the practical measurements of the envelope of the channel impulse response which has a mean square of \(\Omega=\mathbb{E}[\alpha^{2}]\). The parameter \(m\) is called the severity parameter and it measures how much the envelope \(\alpha\) suffers from fading. In this case, less fading severity is associated with larger values of the fading parameter \(m\). A special case is when \(m=0.5\), which represents the one-sided Gaussian fading that suffers from more severe fading than the case of \(m=1\), which represents the Rayleigh fading. As \(m\) approaches infinity, the channel statistics correspond to nonfading conditions. Other fading distributions that may be approximated from the Nakagami-\(m\) pdf are the Rician and log-normal fading distributions under certain conditions.