## 1 Introduction  
Orthogonal Frequency Division Multiplexing (OFDM) is a modulation technique commonly used in wireless communication systems. It works by dividing the information signal into multiple sub-channels, transmitting data in an orthogonal manner over each sub-channel, thereby enhancing spectral efficiency and combating multipath interference. The key feature of classical OFDM is the introduction of a cyclic prefix (CP), i.e., a replicated segment of the symbol tail is added before each OFDM symbol. This design effectively eliminates inter-symbol interference (ISI) and simplifies the processing at the receiver, allowing it to better handle time delays caused by multipath transmission [1]. Therefore, OFDM is widely used in modern wireless communication standards such as 4G and 5G, and its excellent resistance to interference and high spectral utilization make it a crucial technology for achieving high-speed data transmission [2]. Generally, a CP of an appropriate length is inserted before each OFDM symbol to prevent multipath interference. However, the channel response in some scenarios is very long, although it has a finite impulse response (FIR), and a CP of a sufficient length can severely degrade the spectral efficiency. If the length of the CP is reduced to improve the spectral efficiency of the OFDM system, the system will inevitably be affected by ISI and inter-carrier interference (ICI). When an OFDM system is subjected to interference and continues to use a single-tap equalization approach, it can lead to a severe bit error rate (BER). Therefore, researchers have conducted studies on OFDM systems with short or free CPs. In [3], a new OFDM waveform with low CP overhead (SR-OFDM) for narrow-bandInternet of Things (NB-IoT) systems is proposed. Compared to classic OFDM, SR-OFDM employs the concept of symbol repetition across multiple consecutive OFDM symbols, where multiple SR-OFDM symbols occupy only one CP. This significantly reduces CP overhead and improves the spectral and energy efficiency of the NB-IoT system. The precoded vector OFDM scheme is proposed in [4]. By designing a suitable precoding matrix, frequency-valued zeros in the frequency-selective channel band are removed for interference suppression without knowing the channel state information (CSI). Ref. [5] proposes a CP-free OFDM scheme with successive multipath interference cancellation (SMIC) that does not require the CP and uses stored feedback equalization (SFE) to remove the ISI prior to the Fast Fourier Transform (FFT) at the receiver. The SFE operation leaves CP gaps between adjacent OFDM blocks and regenerates the CP using the estimated signal. Ref. [6] proposes a scheme to maximize the channel impulse response (CIR) energy within a given window and shorten the effective CIR, thus reducing the CP length. In addition, the strategy of modeling the channel into infinite impulse response (IIR) with blind estimation is presented in [7]. For channels with long CIR, an IIR filter is applied at the receiver side to eliminate ISI. However, not all channel responses are FIR, and channels may be IIR in specific electromagnetic environments, such as resonant walls with great reflections or full-duplex communication [8]. Classical OFDM schemes with shortened or free CPs suffer from ISI under IIR channels. Under IIR channels, the signal is affected by previous signals from the loop and the resulting ISI. The results of the study demonstrated the feasibility of communication under IIR channels by mitigating loop interference through a combination of passive suppression and active cancellation of the loop interference [9; 10]. The main purpose of passive suppression is to reduce the loop signal power at the receiver, such as antenna isolation, coupling networks, and cross-polarization. Active suppression, on the other hand, utilizes the node's knowledge of its own transmitted signal to eliminate loop interference [11].