# Performance of Differential-Phase OFDM Systems over Selective Multipath Fading Channels
## 1 Introduction  
Orthogonal Frequency Division Multiplexing (OFDM) is a multicarrier digital technology that has been continuously developed over the past five decades due to the high demand for robust digital broadcasting over wireless mobile channels. OFDM offers many advantages over conventional single carrier systems such as robustness against multipath delay spread and frequency-selective fading channels at a low receiver implementation cost [1]. In conventional single carrier (SC) modulation techniques, all symbols are modulated and transmitted over a single frequency carrier. A symbol pulse consisting of \(M\)-bits has duration of \(T_{s}=T_{b}\log_{2}M\), where \(T_{b}\) is the bit duration. The transmission bit rate is defined as \(R_{b}=1/T_{b}\) (bits/second). When the transmitted signal propagates over a terrestrial multipath broadcasting channel, the received signal may experience a delay over the different multipath components. Let the \(p\)-th path have a delay of \(\tau_{p}\) and the longest path has a delay of \(\tau_{max}\). The received signal from the longest path may experience an Inter-Symbol-Interference (ISI) and interfere with the previous \(\tau_{max}/T_{b}\) bits from the shortest path [2]. For example, if a single quadrature-modulated carrier is used with a 20 MHz IEEE 802.11a standard characterized by \(R_{b}=40\) Mbps, the received bit from the longest path, with \(\tau_{max}=1\) us, will interfere with the previous \(\tau_{max}R_{b}=40\) bits [3]. The increase in the bit transmission rate will increase the effect of ISI. In order to remove the interference and extract the original information from the received signal, complex equalization techniques are necessary. Multi carrier (MC) systems with \(N\) carriers split the original data stream into \(N\) parallel substreams [4]. Each substream is modulated and transmitted througha different single carrier and has a lower bit rate of \(R_{sc}=R_{b}/N\). However, the entire MC system has a bit transmission rate of \(R_{b}=1/T_{b}=NR_{sc}\). Since these symbols are modulated and transmitted in a multiplexed parallel fashion, the system is called a parallel transmission system. Accordingly, the received bit from the longest path will interfere with the previous \(\tau_{max}R_{sc}\)=\(\tau_{max}R_{b}/N\) bits. MC systems have proven to be more immune to ISI when compared to SC systems at the same transmission rate [5; 6]. This is because MC systems have multi-subchannels, \(N\), with relatively smaller bandwidth per subchannel. Using the example of the IEEE 802.11a system above shows that the ISI can be reduced significantly when \(N=64\) subcarriers are employed [7; 8]. This will result in \(\tau_{max}R_{b}/N=0.625\); that is, the bit received from the longest path will interfere only with the previous bit [9]. In multi carrier systems, the entire available channel bandwidth \(W\) is divided into \(N\) parallel simultaneous subchannels, each with a bandwidth of \(\Delta f=W/N\). To better utilize the entire channel bandwidth, high spectral efficiency is achieved by allowing overlap between the subchannels. Through the overlapping strategy, signals can be separated at the receiver side if the subchannels' center frequencies are chosen to be orthogonal. Two frequencies \(f_{n}\) and \(f_{k}\) are considered to be orthogonal if they satisfy the following condition  
\[\int_{0}^{T_{u}}f_{n}(t)f_{k}^{*}(t)dt = \frac{1}{T_{u}}\int_{0}^{T_{u}}e^{2\pi(f_{n}-f_{k})t}dt \tag{1}\] \[= \frac{1}{T_{u}}\int_{0}^{T_{u}}e^{2\pi(n-k)t/T_{u}}dt= \begin{cases} 1&n=k\\ 0&n\
eq k\end{cases} \]  
Therefore, orthogonality can be achieved by choosing the \(n\)-th subchannel passband center frequency as \(f_{n}^{\prime}=f_{c}+f_{n}\) for \(n=0,1,2,...,N-1\), where \(f_{n}=n/T_{u}\), \(f_{c}\) is the smallest center frequency among the \(N\) orthogonal frequencies, and \(T_{u}=NT_{s}=1/\Delta f\) is the OFDM frame duration. The frame is a multi-tone block in which all modulated symbols are OFDM encoded.