# Performance of Differential-Phase OFDM Systems over Selective Multipath Fading Channels
## 3 Problem Statement
### The Received OFDM Signal over Multipath Fading Channels  
The transmitted signal, \(s(t)\), will experience a delay, a multipath fading propagation while passing through an AWGN channel. Therefore, the received signal, \(r(t)\), can be expressed as  
\[r(t)=\int_{-\infty}^{\infty}s(t-\tau)g(\tau;t)d\tau+n(t) \tag{13}\]  
where \(n(t)\) represents the complex-valued AWGN process. Substituting (10) and (2) in (13), the received signal can be written as  
\[r(t) = \frac{1}{\sqrt{N}}\sum_{i=1}^{\infty}\sum_{n=0}^{N-1}\sum_{p=1}^{ p}g_{p}(t;\tau)\;u_{n,i}\,e^{2\pi f_{n}(t-\tau_{p}-iT_{f})}f(t-\tau_{p}-iT_{f})+n(t) \tag{14}\]  
The analog-to-digital converter in the OFDM block diagram takes a sample from the received baseband signal every \(t=mT_{s}+iT_{f}\) seconds at a sampling rate of \(f_{s}=1/T_{s}\). Let the delay in the received signal be \(\tau_{p}=N_{p}T_{s}\) and the guard period is \(T_{g}=N_{g}T_{s}\). Recall that  
Figure 1: Block diagram of the transmitter and receiver for the differential-phase OFDM system. \(T_{u}=NT_{s}\), \(T_{f}=N_{f}T_{s}=(N+N_{g})T_{s}\), and \(f_{n}=n/NT_{s}\). Therefore, the discretized received signal is \(r(mT_{s}+iT_{f})=r_{m}\), and it is given by  
\[r_{m} = \frac{1}{\sqrt{N}}\sum_{i=1}^{\infty}\sum_{n=0}^{N-1}\sum_{p=1}^{ P}g_{p}(m)\,u_{n,i}\,f(m)\,e^{j2\pi n(m-N_{p})/N}+n(m) \tag{15}\]  
where \(g(m)=g(mT_{s}+iT_{f})\), \(n(m)=n(mT_{s}+iT_{f})\), and \(f(m)=f(mT_{s}-N_{p}T_{s})\) are the discrete channel model, the discrete version of the AWGN, and the discrete rectangular window, respectively, and  
\[f(m)=\left\{ \begin{array}{ll}1&-N_{g}+N_{p}\leq m\leq N+N_{p}\\ 0&\mbox{elsewhere}\end{array} \right. \tag{16}\]  
In slow fading channels, \(g_{p}(m)=\alpha_{p}\,e^{i\phi_{p}}\) (where \(\alpha_{p}=\alpha(\tau_{p})\)). The GP in the received frame is removed by eliminating the leading \(N_{g}\) samples from the \(N_{f}\) clock. The FFT of \(r_{m}\) has a size of \(N\). The output of the \(k\)-th branch of the FFT processor is  
\[r_{k}=\frac{1}{\sqrt{N}}\sum_{m=0}^{N-1}r_{m}\,e^{-j2\pi km/N} \tag{17}\]  
Substituting (15) into (17) results in  
\[r_{k} = \frac{1}{N}\sum_{i=1}^{\infty}\sum_{p=1}^{P}\sum_{n=0}^{N-1}\sum_ {m=0}^{N-1}\alpha_{p}\,e^{i\phi_{p}}\,u_{n,i}\,f(m)\,e^{-j2\pi nN_{p}/N}\,e^{j2 \pi(n-k)m/N}+\bar{n}_{k} \tag{18}\]  
where \(\bar{n}_{k,i}\) is the FFT-processed version of the AWGN. The output of the FFT processor from the \(k\)-th branch from the \(i\)-th OFDM frame is  
\[r_{k}=\sum_{i=1}^{\infty}r_{ki} \tag{19}\]  
where  
\[r_{ki} = \frac{1}{N}\sum_{p=1}^{P}\sum_{n=0}^{N-1}\sum_{m=0}^{N-1}\alpha_{ p}\,e^{i\phi_{p}}\,u_{n,i}\,f(m)\,e^{-j2\pi nN_{p}/N}\,e^{j2\pi(n-k)m/N}+ \bar{n}_{k,i} \tag{20}\]  
Since FFT is a linear process, the FFT-processed AWGN in (20) has the same distribution with zero mean as  
\[\mathbb{E}[\bar{n}_{k,i}]=\frac{1}{\sqrt{N}}\sum_{m=0}^{N-1}\mathbb{E}[n(m)]e ^{-j2\pi km/N}=0 \tag{21}\]  
and variance  
\[\mathbb{E}[\bar{n}_{k,i}\bar{n}_{k,i}^{*}] = \frac{1}{N}\sum_{m=0}^{N-1}\sum_{l=0}^{N-1}\mathbb{E}[n(m)n^{*}(l )]e^{-j2\pi k(m-l)/N} \tag{22}\] \[= \frac{1}{N}\sum_{m=0}^{N-1}\sum_{l=0}^{N-1}\sigma_{k}^{2}\delta( m-l)e^{-j2\pi k(m-l)/N}\] \[= \sigma_{k}^{2}\]  
where \(\sigma_{k}^{2}\) is the AWGN one-sided power spectral density \(\sigma_{k}^{2}=N_{o}\). The OFDM signal is received from the \(P\) multipath components with propagation delay \(\tau_{p}\) where \(p=1,2,3,\cdots,P\). Some of the propagation delays are less than the guard period, say \(P_{1}\) components, and the rest have propagation delays greater than the guard period. The received signals from the \(P_{1}\) paths have delays of \(\tau_{p}<T_{g}\) and will not suffer from ISI and ICI, as shown in Figure 2. The other \(P\)-\(P_{1}\) received signals with delay of \(\tau_{p}>T_{g}\) will incur ISI and ICI as shown in Figure 3[19; 20]. The ISI occurs between the adjacent OFDM frames in two successive signaling intervals, and the ICI problem is introduced due to the absence of orthogonality between the adjacent subcarriers.