## 3 Background
### Multi-Carrier Modulation  
FOFDM is a multi-carrier modulation technique in which a high data rate substream is demultiplexed into lower data rate substreams to increase the duration of each substream so that inter-symbolinterference (ISI) can be reduced. The orthogonal subcarriers are generated using sine/cosine bases and the orthogonality is achieved in a time window of width equal to the duration of the symbol. Therefore, FOFDM is not band limited. Each subcarrier produces side lobes that in turn create inter-carrier interference (ICI), which can be increased due to multipath channel effect that also cause an increase in ISI. Cyclic prefix (CP)/Guard Interval (GI) is added to each FOFDM symbol to avoid this problem at the cost of transmission efficiency degradation.  
WOFDM is another multi-carrier modulation technique with lower computational complexity than FOFDM [22]. This technique is also a strong candidate for high data rate communication systems [23], and therefore will be explained in more detail. The orthonormal wavelets in WOFDM can be generated using symmetric or asymmetric multistage tree structure of Quadrature Mirror Filter (QMF) bank. The symmetric multistage synthesis and analysis side QMF bank is shown in Figure 1(a,b) respectively. The equivalent structure of WOFDM modulator and de-modulator using noble identities is shown in Figure 2. By using symmetric structure, the orthonormal wavelets are given by the following equation:  
\[f_{k}(n)=\prod_{p=1}^{P}\sqrt{2}\,t_{k,p}\left(\frac{n}{2^{p-1}}\right) \tag{1}\]  
where \(\prod\) represents the convolution operation, \(P\) is the number of levels of this structure, \(k\in\{0,1,2,3,...,2^{P}-1\}\), and \(t_{k,p(n)}\in\{f_{l(n)},f_{h(n)}\}\) is the filter impulse response corresponding to \(k_{th}\) sub-channel at \(p_{th}\) level. \(f_{l(n)}\)and \(f_{h(n)}\)are impulse responses of the low-pass, and high-pass filters respectively, for perfect reconstruction of QMF bank. The high pass filter can be derived from the low pass filter by the relation: \(f_{h}(n)=(-1)^{n}f_{l}(U-1-n)\), where \(U\) is the length of the filter [24].  
Figure 1: Symmetric multistage WOFDM modulator and demodulator. (**a**) Symmetric multistage synthesis side QMF bank; (**b**) Symmetric multistage analysis side QMF bank.  
The output \(y(n)\) of WOFDM modulator can be expressed as:  
\[y(n)=\sum_{k=0}^{2^{P}-1}\sum_{m}x_{k}\left(m\right)f_{k}(n-2^{P}m) \tag{2}\]  
where \(x_{k}(n)\) is the \(k_{th}\) sub-channel input of WOFDM modulator. For WOFDM demodulation, the orthonormal wavelet bases are generated using symmetric analysis side QMF bank as follows:  
Figure 1: _Cont._  
Figure 2: Equivalent structure of WOFDM modulator and demodulator using noble identities. (**a**) A \(2^{P}\) sub-channel WOFDM modulator; (**b**) A \(2^{P}\) sub-channel WOFDM demodulator.  
\[h_{k}(n)=\prod_{p=1}^{p}\frac{1}{\sqrt{2}}\,u_{k,p}\left(\frac{n}{2^{p-1}}\right) \tag{3}\]  
where \(u_{k,p(n)}\in\left\{g_{l(n)},g_{h(n)}\right\}\) is the filter impulse response corresponding to \(k_{th}\) sub-channel at \(p_{th}\) level, \(g_{l(n)}\) and \(g_{h(n)}\) are time reversals of \(f_{l(n)}\), and \(f_{h(n)}\), respectively [25].  
From Figure 3, it can also be observed that the constellation of FOFDM and BPSK-FOFDM is two-dimensional while that of WOFDM and BPSK-WOFDM is one-dimensional. Due to this reason, RF section of FOFDM and BPSK-FOFDM as well as \(M\)-ary QAM, \(M\)-ary DQPSK and \(M\)-ary OQPSK which shares a similar transmitter and receiver architecture, is potentially more complex as compared to that of WOFDM and BPSK-WOFDM.