Carrier Recovery Research Articles

High-Speed 2x1 Multiplexer with Carrier-Reservoir Semiconductor Optical Amplifiers

Leveraging the rapid carrier recovery times and minimal polarization sensitivity of carrier-reservoir semiconductor optical amplifiers (CR-SOAs), this study embeds them in a Mach–Zehnder interferometer (MZI) setup to emulate a 2x1 multiplexer (MUX) operating at 120 Gb/s. The focus is on incorporating AND logic gate functionalities into the CR-SOAs-based MZI structure to facilitate high-quality multiplexing. The proposed methodology utilizes the intrinsic gain and phase modulation capabilities of CR-SOAs-based MZI to effectively manipulate data streams. This innovative approach capitalizes on the unique properties of CR-SOAs, such as fast response times and low polarization sensitivity, to achieve optimal signal transmission quality and efficient multiplexing. To assess MUX performance, a quality factor metric is introduced as a comprehensive measure of signal integrity. Through exhaustive simulations and meticulous analysis, the study demonstrates the feasibility of achieving the desired data rate while maintaining superior signal transmission quality. The results underscore the efficacy of CR-SOAs-based MZI as versatile modules for high-speed multiplexing applications, offering unparalleled performance and efficiency. This research represents a significant advancement in understanding optical communication systems and provides valuable insights for optimizing signal quality and mitigating interference in practical real-world scenarios.

Enhanced frame synchronization and carrier recovery in coherent FSO communication: a pseudo-random and cyclic QPSK approach.

To alleviate the impact of atmospheric turbulence on receiver carrier recovery in free-space optical communication, this study introduces a novel frame synchronization and carrier recovery method. This method employs a blend of pseudorandom sequences and specific cyclic quadrature phase shift keying (QPSK) training sequences, facilitating frame synchronization and frequency offset (FO) estimation. The research designs a time-series metric curve to counteract the effect of side peaks, enhancing the accuracy of frequency offset estimation via QPSK spectrum features. Furthermore, the method incorporates a two-stage frequency offset estimation process, utilizing the inherent sign characteristics of the x and y polarization states. Applied to the quadrature amplitude modulation system under atmospheric turbulence conditions, the proposed algorithm demonstrates high precision under both strong and weak turbulence conditions. Furthermore, we confirmed the universality of the algorithm across multiple modulation formats.

Process intensification of tar chemical looping reforming based on ReaxFF reaction kinetics investigation

Traditional chemical processes currently produce tar as a by-product with low efficiency. This paper proposes a novel tar chemical looping reforming (t-CLR) process based on reactive force field (ReaxFF) reaction kinetics investigation to enhance its utilization efficiency. Firstly, the t-CLR process is initially established to determine the optimum operating conditions. Through ReaxFF molecular dynamics (ReaxFF-MD) simulations conducted under above conditions, the activation energies of the fuel reactor, steam reactor, and air reactor are determined to be 63.25 kJ/kmol, 107.85 kJ/kmol, and 159.53 kJ/kmol respectively. By implementing micro-dynamic parameters filling, oxygen carrier recovery, and heat integration, the optimal t-CLR process achieves a 97.60 % recovery rate of the oxygen carrier and reduces external utility consumption by 71.39 %. The comprehensive framework with macro and micro dimensions is established based on the above studies to promote the cleaner production of traditional processes. Finally, the utilization efficiency of tar is efficiently improved through t-CLR technology, which allows for an adjustable C/H ratio in syngas. The t-CLR technology can be applied to the chemical and energy industries by adding value to by-products and optimizing energy utilization. Overall, this work provides important guidance for process intensification, energy conservation, environmental protection, and waste utilization. The study further promotes the development of industrial production in a cleaner and more efficient direction.

Metallic layered VSe2 saturable absorber based single- and dual-wavelength ultrafast fiber laser

Vanadium selenide (VSe2) is a special two-dimensional (2D) transition metal dichalcogenide (TMD) material with metallic properties similar to graphene, which has been proved to have saturable absorption properties and used in mode-locked fiber lasers. As an excellent saturable absorber (SA) material, VSe2 has advantage of ultrafast carrier recovery time and broadband optical response range. However, the potential application of VSe2 in the field of ultrafast photonics has not been fully developed so far. In this work, the as-prepared VSe2 nanosheets were deposited on tapered fiber by optical deposition to obtain VSe2 SA. By inserting VSe2 SA into an Er-doped fiber laser, the mode-locked pulses with a central wavelength of 1574.4 nm can be obtained. The pulse width of the laser is measured to be ∼680 fs which is narrowest in the reported VSe2 SA-based lasers. In addition, the unique dual-wavelength mode-locked pulse is also obtained by further adjusting intracavity dispersion and polarization. This kind of dual-wavelength pulse is valuable in high order harmonic generation enhancement, terahertz wave generation, dense wavelength division multiplexing, dual-frequency comb and other applications. Our results show that metallic VSe2 is of great potential value in ultrafast fiber laser.

Difference of carrier dynamics in a semiconductor saturable absorber mirror with and without B+ ion-implantation.

Three samples whose growth temperatures were 450°C, 500°C, and 560°C for S E S A M 1, S E S A M 2, and S E S A M 3, respectively, were tested by femto-second time-resolved transient absorption spectroscopy. The results indicate that the carrier dynamics of excited state absorption were dominant, and the lifetimes of carriers trapped by defect levels were about tens of pico-seconds. To further study the influence of carrier dynamics and recovery time of samples by ion-implantation, B + ions of 80 and 130 KeV were implanted into the samples with dose of 1014/c m 2. The modified samples showed a dominance of ultra-fast carrier dynamics of ground-state bleaching and direct recombination, which lasted for hundreds of femto-seconds, over excited state absorption. Additionally, carrier fast trapping was observed to be competitive with the excited state absorption process. After ion-implantation, the carrier dynamics of carrier trapping were enhanced, which contributed to forming an ultra-short laser, while the carrier dynamics of absorption of the excited state were suppressed. The conclusion that defect levels were partially eliminated by B + ion-implantation can be drawn.

Hierarchical blind phase search for correcting imperfect phase rotation in QAM signals synthesized via optical coherent superposition.

Coherent superposition has been proposed to synthesize high-order quadrature amplitude modulation (QAM) by coherently superposing low-order QAMs in the optical domain. These approaches could effectively relax the digital-to-analog converter resolution and reduce the complexity of the driving electronics. However, in the superposition process, imperfect phase rotations (IPRs) in low-order QAMs will be transferred to the resultant high-order QAM. Importantly, the induced IPR cannot be compensated for by conventional linear equalizers and carrier recovery methods. To combat the induced IPR, herein, we propose a hierarchical blind phase search (HBPS) algorithm to compensate for the IPRs in synthesized high-order QAMs. The proposed HBPS can match the generation mechanism of the IPRs in coherent superposition, by tracing back and estimating the IPR in the QPSK-like constellation of each hierarchy and finally correcting the induced IPRs. Simulation and experimental results verify that this algorithm could effectively compensate for the IPR in the resultant 16-QAMs synthesized using coherent superposition approaches. The proposed HBPS shows significant optical signal-to-noise ratio (OSNR) gains compared to the conventional blind phase search (BPS) method for high-order QAMs coherently superposed using optical signal processing (OSP) and tandem modulators (TMs). Specifically, at the BER of 2.4e-2, the HBPS achieves a 1.5-dB OSNR sensitivity enhancement over the BPS in either OSP or TMs-based schemes, even with an imperfection rotation of up to 20∘.

Long-distance continuous-variable quantum key distribution over 100-km fiber with local local oscillator.

Quantum key distribution (QKD) enables two remote parties to share encryption keys with security based on the laws of physics. Continuous-variable (CV) QKD with coherent states and coherent detection integrates well with existing telecommunication networks. Thus far, long-distance CV-QKD has only been demonstrated using a highly complex scheme where the local oscillator is transmitted, opening security loopholes for eavesdroppers and limiting potential applications. Here, we report a long-distance CV-QKD experiment with a locally generated local oscillator over a 100-kilometer fiber channel with a total loss of 15.4 decibels. This record-breaking distance is achieved by controlling the phase noise-induced excess noise through a machine learning framework for carrier recovery and optimizing the modulation variance. We implement the full CV-QKD protocol and demonstrate the generation of keys secure against collective attacks in the finite-size regime. Our results mark a substantial milestone for realizing CV quantum access networks with a high loss budget and pave the way for large-scale deployment of secure QKD.

Mid-temperature chemical looping methane reforming for hydrogen production via iron-based oxygen carrier particles

Chemical looping steam methane reforming (CL-SMR) via iron-based oxygen carriers is a promising method for efficient hydrogen production. To overcome challenges such as high reaction temperatures (>850 °C) and scarcity of low-cost, durable oxygen carriers (OCs), we have developed iron-based particles mixed with various ratios of nickel-based particles to achieve remarkable performance in CL-SMR at 600 °C. The mixed particles showed 85.23% methane conversion and 3.47 and 1.01 mL/min/gOC hydrogen production rates in the reduction and steam oxidation steps, respectively, in the two-step CL-SMR reaction. In the three-step CL-SMR reaction, air oxidation led to full recovery of oxygen carriers, enhancing methane conversion to 93.30% and elevating hydrogen production rate to 1.41 mL/min/gOC during steam oxidation. Precise control over methane conversion and hydrogen production in the three-step CL-SMR system is achievable by manipulating the mixing ratios of iron-based to nickel-based OC particles. Comprehensive experimental tests were conducted, covering practical aspects like support materials, gas velocity, and steam-to-carbon ratios. The outstanding cyclic stability of OC particles was confirmed over 200 consecutive redox cycles at 600 °C. The mid-temperature iron-based oxygen carrier particles, integrated with chemical looping demonstration project, might provide a powerful approach toward more efficient and scalable hydrogen production.

Design of Mode-Locked Semiconductor Laser Comb-Based Analog Coherent Links

Multi-wavelength analog coherent links using mode-locked laser (MLL) frequency combs as transmitter and local oscillator (LO) sources are proposed. Carrier recovery (CR) in all wavelength channels is achieved using only two optical phase-locked loops (PLLs), while polarization demultiplexing and static phase offset removal are performed using cascaded optical phase shifters. A three-section Fabry-Perot semiconductor laser structure is proposed for the comb sources. Phase-error performance in a 2.6 Tb/s system using 56-Gbaud dual-polarization quadrature phase-shift keying on 13 channels is studied. For optical and microwave beat linewidths of 2 MHz and 1 kHz, respectively, achieving phase-error penalties below 1.5 dB requires PLL delays below 400 ps. A symmetric CR scheme is shown to achieve better phase-error performance than an asymmetric CR scheme. In 13-channel analog coherent links, the MLL comb-based design is projected to consume 38 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> less power than a resonator-enhanced electro-optic comb-based design and 20 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> less power than a design using arrays of single-wavelength lasers as transmitter and LO sources, excluding modulator driver power, which is identical for the three designs.

Why Constant-Composition Codes Reduce Nonlinear Interference Noise

A time-domain perturbation model of the nonlinear Schrödinger equation is used to study wave-length-division multiplexed communication over a single polarization. The model explains (a) why constant-composition codes offer an improvement in signal-to-noise ratio compared with independent and uniform selection of constellation points and (b) why similar gains are obtained using carrier recovery algorithms even without using constant-composition codes.

All-optical matching structure for multi-order modulation formats

With the development and popularity of the Internet, network security has become a popular topic. The traditional solution for screening the security of network information is to convert the signal transmitted in the optical fiber into an electrical signal through an optical conversion. Then we perform symbol matching within the electronic firewall to determine whether the signal is a risk. However, due to the limitation of carrier recovery time in the semiconductor, the photoelectric converter cannot handle signals with too high rate, and it also causes a large power consumption. To improve the efficiency and scalability of signal matching, we propose an all-optical matching structure. This structure can identify the position of the target sequence in the input data sequence in the all-optical environment, and the structure is suitable for any amplitude and phase modulation format. The matching result of each bit of the target sequence and input data sequence is obtained by phase delay and interference. Then the matching result of each bit of the target sequence and input data sequence is integrated by logic AND gate, regenerator, and delay line. Finally, the matching result between the whole target sequence and the input data sequence is obtained. This structure can avoid photoelectric transformation and processes the signal directly at the optical domain, which greatly reduces the consumption of time and energy. We use VPItransmissionMaker software to simulate and verify the structure. The length of the input signal during the simulation is 8 bits, the length of the target signal is 4 bits, and the transmission rate of the signal is 100 GBaud. We simulated the binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), 8-phase shift keying (8PSK), and 16 quadrature amplitude modulation (16QAM) modulation formats and analyzed the noise performance of the system. The results show that the structure can achieve all-optical matching of high-speed optical signals in BPSK, QPSK, 8PSK, and 16QAM modulation formats and has good anti-noise performance.

Analysis of coherent-OFDM modulation for FSO communication over distribution with pointing errors

In this paper, we have analyzed a proposed coherent-orthogonal frequency division multiplexing (Co-OFDM) free space optical (FSO) communication system under ▪ distribution channel model. The proposed Co-OFDM FSO system contains a heterodyne mixer at the receiver section, which is used to convert the optical frequency to radio frequency (RF). After that, a carrier recovery circuit is used and then a synchronization circuit is used to correct the timing and phase of the received signal. Here, we derived an ABER expression and outage probability expression for the Co-OFDM FSO communication system under ▪ distribution channel model. We analyzed the results for different atmospheric turbulence (AT) conditions, pointing errors and the number of subcarriers. We have analyzed the system outage probability with varying received optical power and pointing error for different SNR thresholds and also analyzed for different AT conditions. In the results, we observed that the ABER performance improves as AT conditions improve, as the number of subcarriers decreases and as the pointing error decreases. Also, we observed that when AT conditions get worse, then to get same outage probability, we have to increase the SNR threshold. We have compared the Co-OFDM FSO system with popular modulation techniques PAM and PPM under ▪ distribution channel model.

Highly stable femtosecond pulse generation enabled by the indium tin oxide nanocrystals

The ultrafast third-order optical nonlinearity of indium tin oxide nanocrystals (ITO NCs) and their application in femtosecond laser generation have been investigated experimentally. The ITO NCs exhibit a large modulation depth of ∼25%, nonlinear refractive behavior with self-focusing, ultrafast carrier recovery time of ∼300 fs, and high damage threshold of ∼1.16 TW cm−2, and the stable femtosecond Er-doped fiber laser can be delivered with a signal-to-noise ratio over 80 dB modulated by the ITO NCs successfully. The experimental results indicate that the ITO NCs can be excellent ultrafast nonlinear optical materials for developing highly stable photonic and optoelectronic devices.

A Low-Complexity Joint Compensation Scheme of Carrier Recovery for Coherent Free-Space Optical Communication

In this paper, a low-complexity joint compensation scheme of carrier recovery (JCSCR) for coherent free-space optical (CFSO) communication is proposed. We applied the carrier recovery joint compensation approach to a CFSO communication system in the quadrature phase shift keying (QPSK) modulation format. A signal-preprocessing stage, which effectively avoided the repetitive operations found in traditional carrier recovery schemes, was proposed. Unlike in existing carrier recovery algorithms, the modulated phase of the received signal could be accurately removed using only the sum and subtraction of real absolute values in the signal-preprocessing stage, greatly reducing the complexity of the operation. Since this algorithm avoids the traditional fourth operation, the system’s complexity is reduced while additional noise generated by fourth cross-terms would be prevented and system noise immunity would be greatly enhanced. In addition, this algorithm uses joint compensation of phase errors in the final compensation stage, further reducing the complexity of the computation of the whole algorithmic scheme. A 10 Gbps QPSK CFSO communication transmission experiment was conducted in an atmospheric turbulence channel to verify the proposed technique and improvement in receiver sensitivity.

Harvesting Plasmonic Near-Infrared Photons by Hot Hole Transfer in Nonstoichiometric-Semiconductor Plasmonic Heterojunctions

Infrared light is an untapped energy source that accounts for half of solar energy. Thus, harvesting near-infrared photons for different applications needs further development for efficient utilization of the whole solar spectrum. Herein, we report harvesting infrared photons by hot hole transfer from a nonstoichiometric plasmonic semiconductor (Cu2–xSe) to an adjacent semiconductor (CdSe) in the heterostructure (HNCs). We found a decrease in transient signal intensity and hole–phonon scattering time in Cu2–xSe/CdSe HNCs compared to the pure Cu2–xSe nanocrystals, attributed to plasmon-induced hot hole transfer. A similar kind of TA kinetics has been observed for 800 and 930 nm laser pulse excitation for Cu2–xSe/CdSe HNCs. The spectroscopic results reveal anomalous carrier populations and slow carrier recovery dynamics at the 1P state of the CdSe phase in Cu2–xSe/CdSe HNCs, further establishing the plasmonic hot hole transfer mechanism. The estimated plasmon-induced hot hole transfer time from the nonstoichiometric semiconductor Cu2–xSe NC system is determined and found to be ∼177 fs. Fabricated plasmonic thin-film devices exhibit an excellent conductivity (0.15 A at 2 V) and photoconductivity (0.2 A at 2 V) under near-infrared light illustration (λlight = 1200 nm) in a plasmonic HNC system. These results reveal a novel approach for harvesting light photons from the visible to near-infrared region of the solar spectrum and provide a new avenue for developing hot carrier-based device applications of plasmonic semiconductor-based nanomaterials.

NbS2-nanosheet-based saturable absorber for 1.5 µm and 2 µm ultrafast fiber lasers

Niobium disulfide is a kind of transition metal dichalcogenides which has the advantages of large modulation depth, short excited carrier recovery time and high carrier mobility, theoretically being very suitable for the applications in ultrafast photonics. Up to now, there are few reports on the application of NbS2 in ultrafast optical devices, especially in the devices with long working wavelength, such as 2 µm. In this work, few-layer NbS2 nanosheets were prepared by liquid exfoliation method and the NbS2-SA was prepared by covering the nanosheets on tapered fiber. The device NbS2-SA exhibits good nonlinear saturable absorption property with a modulation depth of 10.2 % at 1550 nm. Based on the NbS2-SA, both 1.5 µm and 2 µm ultrafast fiber lasers are realized. The 1.5 µm mode-locked fiber laser has a pulse width of 753 fs and the 2 µm mode-locked fiber laser has an estimated pulse width of 5.77 ps and a maximum output power of 5.56 mW. All the results prove that NbS2 is an excellent broadband saturable absorption material.

Carrier recovery for satellite-to-ground coherent laser communication systems using double feedback loop and Viterbi–Viterbi feedforward cascade structure

High dynamic and wide range frequency offset in satellite-to-ground coherent optical communication system caused by Doppler frequency shift, transceiver laser frequency offset and laser frequency stabilization technology based on perturbation is one of the important differences compared with terrestrial fiber-optic coherent transmission. The design of a reliable frequency offset acquisition and fine carrier tracking algorithm is critical to ground optical stations. Two-stage feedback loop and Viterbi–Viterbi (V–V) feedforward cascaded carrier recovery algorithm is presented in this work. Pre-decision-based angle differential estimator (PADE) is adopted as the residual frequency offset (RFO) estimator in the first stage to estimate the high dynamic and wide range of frequency offset. Then V–V based RFO estimator in the second stage is employed to accurately compensate the residual frequency. V–V algorithm is used to compensate phase offset, which share the same logic resource with V–V based RFO estimator. A real-time 10-Gbps data rate PM-QPSK experiment based on FPGA is performed to evaluate the algorithm performance. The proposed cascaded algorithm exhibits excellent performance in terms of dynamic tracking speed and normalized mean-square-error (MSE) of the FOE.

Nonlinear Absorption Response of Zirconium Carbide Films.

Zirconium carbide (ZrC), a novel representative of the MXene family, has attracted considerable interest because of its outstanding physicochemical properties and potential applications in optoelectronic devices. For improving its performance as an optical modulator for ultrashort lasers, there is a call to continue studying the nonlinear optical behavior of MXene ZrC. Herein, for the first time, MXene ZrC films were fabricated on fused silica by magnetron sputtering deposition technology and used as a saturable absorber (SA) optical modulator in a passive Q-switched Nd:YAG laser. The saturation absorption behaviors of the prepared ZrC films were characterized by the Z-scan method. Their morphology, band structure, damage threshold, carrier recovery time, and saturation absorption properties were analyzed. The experimental results show that the MXene ZrC SA films exhibit excellent nonlinear optical characteristics, with a saturation intensity of 48.4 MW/cm2, a large modulation depth of 6.9%, and an ultrashort recovery time of 2.72 ps. In addition, the damage threshold of MXene ZrC SA films was estimated to be greater than 0.2516 J/cm2. By integrating the ZrC SA film optical modulator into the oscillator of the Nd:YAG laser, we achieved stable operation of the Q-switched laser with a central wavelength at 1.06 μm, with the shortest pulse width of 78 ns. The results of this study demonstrate the potential use of MXene ZrC SA films as optical modulators in ultrashort lasers.

A 4.6-pJ/b 200-Gb/s Analog DP-QPSK Coherent Optical Receiver in 28-nm CMOS

Coherent detection with polarization multiplexing is widely used because of its high spectral efficiency. However, it utilizes analog-to-digital converters (ADCs) and digital signal processing (DSP), which consume a large amount of power and thereby hinder its application in power-sensitive short-reach links. In this article, we present a 200-Gb/s analog <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> -band dual-polarization quadrature phase shift keying (DP-QPSK) coherent optical receiver (RX), which uses four-way time-interleaved sampling to relax the internal bandwidth limit. The RX utilizes a chromatic dispersion (CD) equalizer, a polarization demultiplexer (Pol. DEMUX), and carrier recovery (CR) to eliminate the intersymbol interferences induced by fiber impairments and the carrier frequency offset. A subsampling scheme is adopted for low-power adaptation. The decrease in the frequency offset detection range caused by subsampling is prevented using a consecutive slice sampling technique. Measurement results show that our RX supports equalization for impairments equivalent to a 10-km fiber and CR for a residual carrier frequency offset of 12 MHz. A bit error rate (BER) of less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1e-10$ </tex-math></inline-formula> is achieved with an energy efficiency of 4.6 pJ/b.

EVALUATION OF THE VALUE OF EQUIVALENT ENERGY LOSSES DUE TO THE QUALITY OF FREQUENCY SYNTHESIS FUNCTIONING IN DIGITAL COMMUNICATION SYSTEMS WITH QUASI-COHERENT RECEPTION OF SIGNALS WITH QUADRATURE AMPLITUDE KEYPAD

Useful transmitted information when using digital transmission systems with quadrature amplitude shift keying lies in the change in the amplitude and phase of the signal. This circumstance causes significantly higher requirements for the magnitude of destabilizing factors, namely the phase noise of signal packages at the input of the demodulating device of signals with quadrature amplitude keying, which affect the useful signal, resulting in a decrease in the reliability of reception. In this article, we restrict ourselves to considering the requirements for short-term frequency stability of synthesizers of digital transmission lines, which use signals of multi-position quadrature-amplitude keying in conjunction with quasi-coherent reception. The carrier wave is extracted on the receiving side directly with the help of a carrier recovery device from the received QAM signal by demodulating it, followed by narrow-band filtering. The phase-locked loop in this case functions as a narrow-band tunable filter for extracting the carrier wave. An increase in the quality of demodulation and signal filtering processes leads to a decrease in the phase fluctuation of the restored carrier, a decrease in the level of additive noise, and a decrease in the magnitude of energy losses when using quasi-coherent reception compared to the ideal one, which requires the complete absence of a phase error of the selected reference signal. Such an approximation of the noise immunity of real coherent QAM demodulators to the theoretical one limits the phase noise of the signal at the demodulator input, which is associated with the non-ideal operation of frequency synthesizers on the transmitting and receiving sides of digital communication lines. The research methods used in the article are based on the theories of potential noise immunity, demodulator synchronization, and phase locked loop systems. For the carrier recovery device, the noise band value was chosen in the range from to (where T – is the symbol duration), since with the indicated ratios, the distribution law of the phase error can be considered normal. It was assumed in the work that the devices for automatic gain control and clock synchronization in the radio receiving system function ideally, and the frequency response of the channel corresponds to the Nyquist condition.