## 4 Simulation Results  
In this section, we validate IIR-OFDM from the bit error rate (BER). Suppose the signal-to-noise ratio (SNR) in the simulation is \(\sigma_{d}^{2}/\sigma_{w}^{2}\). BER indicates the probability of a bit being incorrectly demodulated, while SNR represents the power ratio of a useful symbol to the noise it carries. On the transmitter, the number of subcarriers is 128, transmitting one frame containing two symbols at a time, i.e., \(v=2\). In addition, the simulation is based on IIR-OFDM symbols modulated with quadrature phase shift keying (QPSK) in different IIR channels. The loop response of the IIR channel follows a complex zero-mean Gaussian Random Variable. IIR-OFDM is compared to classical OFDM, where both schemes carry the guard interval of length \(l_{g}\). Note that the guard interval for classical OFDM is replicated from the tail sample of the signal. The details of the simulation parameters are as follows. * Modulation: QPSK. * Number of subcarriers: \(N=128\). * Number of transmitted symbols: \(V=2\). * The loop response of the IIR channel: complex zero-mean Gaussian Random Variable. The loss of loops is assumed to be \(\beta\) dB, which is to maintain the stability of the IIR channel system under multiple loops. In addition, residual loops remain due to the difficulty of complete loop suppression in practice. Figure 1 shows the amplitude of the IIR-OFDM signal at values of different \(\beta\); the red and black lines are the amplitudes of the guard interval and OFDM signal, respectively. The number of loops is \(l_{g}=8\), so the length of the guard interval is 8. When \(\beta=10\) dB, the poles of the system appear outside the unit circle, the system is unstable, and the amplitude of the guard interval is larger than the OFDM signal. Too large a guard interval will result in the Radio Frequency side of the transmitter being unable to amplify linearly, and an unstable system will result in the receiver being unable to demodulate. However, when \(\beta=20\) dB, the guard interval amplitude is similar to the OFDM signal since the system is stable. The BER comparison under the IIR channel of the multipath loop is shown in Figure 2. The number of loops is \(l_{g}=8\) and follows a complex zero-mean Gaussian Random Variable with \(\sigma_{w}^{2}=1\). Suppose that \(h_{0}=1\) to avoid the effects of the forward path, i.e., \(a(0)=1\). The IIR channel system with \(l_{g}=8\) is stable when \(\beta>20\) dB. When the guard interval is ignored, the OFDM signal transmission power is the same for both schemes. Note that the BER performance of both schemes increases with increasing loss of loops. However, the BER of IIR-OFDM outperforms that of classical OFDM regardless of the loop loss. It is demonstrated that IIR-OFDM signals can resist ISI from multipath loops. Figure 3 and Figure 4 show the IIR-OFDM under the first-order IIR channel and the special delay IIR channel, respectively. Suppose that \(h_{0}=1\) for both IIR channels to avoid the effects of the forward path, i.e., \(a(0)=1\). In addition, the delay channel is a two-order IIR channel with \(a(1)=0\). The loops for both IIR channels follow a complex zero-mean Gaussian Random Variable with \(\sigma_{w}^{2}=1\). The loop loss is increased by active and passive suppression, so both IIR channel systems are stable when \(\beta>15\) dB. IIR-OFDM outperforms classical OFDM in terms of BER performance in both channels. When \(\beta=15\) dB, IIR-OFDM has a BER gain of about 0.8 dB compared to classical OFDM at a BER of \(10^{-3}\) over the first-order IIR channel and the special delay IIR channel. However, when \(\beta=20\) dB, the BER gain of IIR-OFDM is about 0.2 dB at a BER of \(10^{-3}\). Note that the performance gap between IIR-OFDM and classical OFDM decreases as the loop loss increases. When loops are completely eliminated, both performances become identical. It is well known that the loop signal causes ISI to the received signal at the receiver. Therefore, many active and passive suppression techniques for loops have been proposed.