## 3 Guard Interval Analyses
### General Multi-Order IIR Channel  
The guard interval can be obtained as  
\[\overline{s}_{v}(k) =\sum_{i_{0}=0}^{k}a(i_{0})y_{v}(k-i_{0})+\sum_{i_{1}=k+1}^{l_{g}} a(i_{1})y_{v-1}(k-i_{1}+l_{g}) \tag{23}\] \[=\frac{1}{\sqrt{N}}\Bigg{[}\sum_{i_{0}=0}^{k}\sum_{n_{0}=0}^{N-1} a(i_{0})\frac{d_{n_{0},v}}{A_{i_{0}}}e^{j2\pi(k-i_{0}-l_{g})n_{0}/N}\] \[+\sum_{i_{1}=k+1}^{G}\sum_{n_{1}=0}^{N-1}a(i_{1})\frac{d_{n_{1},v }}{A_{n_{1}}}e^{j2\pi(k-i_{1})n_{1}/N}\Bigg{]}.\]  
Classical OFDM benefits from the FFT, which reduces its complexity from \(\mathcal{O}(N^{2})\) to \(\mathcal{O}(Nlog_{2}(N))\). The guard interval of IIR-OFDM is different from classical OFDM, which can the circular convolution of the received signal and channel coefficients. Therefore, the guard interval requires additional complexity \(\mathcal{O}((l_{g}+1)^{2}+(l_{g}+1))=\mathcal{O}(l_{g}^{2}+3l_{g}+2)\) for design. Note that the complexity of the guard interval is related to the length of loops.  
The guard interval consists of a linear superposition of the transmitted symbol and is closely related to the coefficient of the IIR channel. Suppose the expectation of guard interval is \(p_{v}(k)=\mathbb{E}[\overline{s}_{v}(k)\overline{s}_{v}(k)^{*}]\), which can be expressed as  
\[p_{v}(k) =\frac{1}{N}\Bigg{[}\sum_{i_{0}=0}^{k}\sum_{i_{1}=0}^{k}a(i_{0})a( i_{1})^{*}\sum_{n_{0}=0}^{N-1}\sum_{n_{1}=0}^{N-1}\frac{\mathbb{E}[d_{n_{0},v-1}d _{n_{1},v-1}^{*}]}{A_{n_{0}}A_{n_{1}}^{*}}e^{j2\pi[(k-l_{g})(n-0-n_{1})-n_{0}i_ {0}+n_{1}i_{1}]/N}\] \[\quad+\sum_{i_{0}=0}^{k}\sum_{i_{1}=k+1}^{l_{g}}a(i_{0})a(i_{1})^{* }\sum_{n_{0}=0}^{N-1}\sum_{n_{1}=0}^{N-1}\frac{\mathbb{E}[d_{n_{0},v-1}d_{n_{1 },v-1}^{*}]}{A_{n_{0}}A_{n_{1}}^{*}}e^{j2\pi[k(n-0-n_{1})-n_{0}i_{0}+n_{1}i_{1 }]/N}\] \[\quad+\sum_{i_{0}=k+1}^{l_{g}}\sum_{i_{1}=k+1}^{l_{g}}a(i_{0})a( i_{1})^{*}\sum_{n_{0}=0}^{N-1}\sum_{n_{1}=0}^{N-1}\frac{\mathbb{E}[d_{n_{0},v-1}d_{n_{1 },v}^{*}]}{A_{n_{0}}A_{n_{1}}^{*}}e^{j2\pi[k(n-0-n_{1})-n_{0}i_{0}+n_{1}i_{1 }]/N}\Bigg{]}. \tag{24}\]  
After simplification, we have  
\[p_{v}(k) =\frac{\sigma_{d}^{2}}{N}\sum_{n=0}^{N-1}\Bigg{[}\frac{\sum_{i_{ 0}=0}^{k}\sum_{i_{1}=0}^{k}a(i_{0})a(i_{1})^{*}e^{j2\pi(i_{0}-i_{1})n/N}}{A_{ n}A_{n}^{*}} \tag{25}\] \[\qquad\qquad\qquad+\frac{\sum_{i_{0}=k+1}^{l_{g}}\sum_{i_{1}=k+1} ^{l_{g}}a(i_{0})a(i_{1})^{*}e^{j2\pi(i_{0}-i_{1})n/N}}{A_{n}A_{n}^{*}}\Bigg{]}.\]  
The guard interval of IIR-OFDM has a higher complexity than the CP of classical OFDM, and its expectation value increases with the coefficient of the IIR channel. Inaddition, a major problem encountered in IIR-OFDM design is to ensure the system stability of the IIR channel model under multiple loops. The unstable IIR channel cannot be demodulated and equalized at the receiver. Generally, loop signals are considered harmful, so increasing the loss of loops and suppressing the amount of multiple paths in loops have been proposed. When in the case of multipath loops, it is required that the loop loss is able to ensure that the IIR channel satisfies the \(z\)-transformed poles are all within the unit circle.  
It is worth noting that IIR-OFDM signals occupy the same bandwidth and transmitter bits as classical OFDM. Therefore, IIR-OFDM has the same spectral efficiency as classical OFDM. For power consumption, IIR-OFDM has the same transmission power as classical OFDM when the guard interval and CP are ignored. However, from (24), the guard interval expectation of IIR-OFDM is higher than OFDM symbols. Therefore, the total transmission power is the same, and the number of transmitted symbols of IIR-OFDM will be less than classical OFDM. IIR-OFDM has a higher delay required to process data at the transmitter than CP due to the complexity of the guard interval. At the receiver, the IIR-OFDM equalization process is like OFDM without additional delay. However, it is worth noting that IIR-OFDM is specifically proposed for under IIR channels.