# Review of Channel Estimation for Candidate Waveforms of Next Generation Networks
## 4 System Model Description of FBMC OFDM-OQAM and GFDM-OQAM
### OFDM-OQAM Transceiver
#### 4.1.2 OFDM-OQAM Transceiver: Non-PR design  
Transmitter Description  
In Non-PR conditions of filter banks, the impulse response of the AFB FIR prototype filter is different from that of the SFB. In Non-PR filter design, the analysis filter bank is not directly  
Figure 19: Structure of filter bank OFDM-OQAM receiver.  
a time-reversed version of the corresponding synthesis filter bank due to the difference in prototype filter impulse response \(\hat{p}[\hat{n}]\). To achieve this imperfection, an error \(\bar{\delta}\) is randomly introduced to the analysis filter FIR prototype filter such that \(-\bar{\Delta}<\bar{\delta}<+\bar{\Delta}\), where \(\bar{\Delta}=0.1\). Hence, the filter bank prototype filter according to Equation (7) is modified by the introduction of an error (\(\delta[k]\)) so as to yield the non-perfect reconstructed FIR analysis prototype filter bank formulated in Equation (23).  
\[\hat{p}[\hat{n}]=\hat{P}[0]+2\sum_{k=1}^{\frac{L_{p}-1}{2}}(-1)^{k}(\hat{P}[k]+ \bar{\delta}[k])cos\left(\frac{2\pi k}{KM}\left(\hat{n}+\frac{KM-(L_{p}-1)}{2} \right)\right) \tag{23}\]  
The transmitted filtered OFDM-OQAM symbol \(y[\hat{n},k]\) over the \(k\)th sub-carrier for the \(\hat{n}\)th symbol, assuming Non-PR, is obtained by convolving the synthesis filter bank impulse response (\(g[\hat{n},k]\)) with the up sampled transmitted signal (\(x[\hat{n},k]\)) as:  
\[y[\hat{n},k]=(x[\hat{n},k]*g[\hat{n},k])=\sum_{l=0}^{L_{p}-1}x[\hat{n}-l,k]g(l,k) \tag{24}\]  
Channel Description  
The channel assuming Non-PR is modelled in a similar way to the channel model presented in Section 4.1.1.2.  
Receiver Description  
Since the analysis filter bank is not directly a time-reversed version of the corresponding synthesis filter bank, the analysis filter bank impulse response (\(f[\hat{n},k]\)) obtained from the introduction of error is:  
\[\hat{f}[\hat{n},k]=\hat{p}[\hat{n}]exp\left(j\frac{2\pi k}{M}\left(\hat{n}- \frac{(L_{p}-1)}{2}\right)\right). \tag{25}\]  
The \(\hat{n}\)th filter bank OFDM-OQAM symbol received over the \(k\)th sub-carrier is therefore given as [29]:  
\[r[\hat{n},k]=y[\hat{n},k]h[\hat{n},k]+w[\hat{n},k]. \tag{26}\]  
In Equation (26), \(h[\hat{n},k]\) is the frequency selective Rayleigh fading channel impulse response and \(w[\hat{n},k]\) signifies the mutually independent zero-mean complex AWGN process with variance \(\sigma_{w,k}^{2}\) given as:  
\[\sigma_{w,k}^{2}=E\left[x^{2}[\hat{n},k]\right]10^{\frac{-\gamma_{AB}}{10}}. \tag{27}\]  
The received signal \(s[n,k]\), after \(r[\hat{n},k]\) has been processed with the filter bank, is given as [2]:  
\[s[n,k]=\left[r[\hat{n},k]*f[\hat{n},k]\right]_{\downarrow\frac{M}{2}}, \tag{28}\]  
\[s[n,k]=\left[x[\hat{n},k]q[\hat{n},k]+\eta[\hat{n},k]\right]_{\downarrow\frac {M}{2}}, \tag{29}\]  
where  
\[q[\hat{n},k]=\left[h[\hat{n},k]g[\hat{n},k]*\hat{f}[\hat{n},k]\right], \tag{30}\]  
and \(\eta[\hat{n},k]\) in Equation (29) is a Gaussian noise process expressed by:  
\[\eta[\hat{n},k]=\left[w[\hat{n},k]*f[\hat{n},k]\right]. \tag{31}\]