A fundamental tool for calculating, analyzing, and designing time-frequency domain relationships between signals is a pulse shaping filter. Although it sounds like this might be a type of circuit, this is normally just a concept used as a mathematical function in designing communication channels.
There are many types of pulse shaping filters that can be implemented at a hardware, software, or firmware level. The three principle types of pulse shaping filters widely used in telecom are:. Sinc filter—this filter has a rectangular transfer function, and the time-domain impulse response is a sinc function.
Gaussian filter—as its name implies, this filter has a Gaussian transfer function. These filters and other filtration functions are centered at zero frequency and are symmetric. Other common pulse shaping filters are the square root raised cosine filter, Nyquist, and square-root Nyquist filters. Other filters can be custom-designed by engineering the desired impulse response or transfer function, but the filter design depends on the signal integrity goals in a communication channel. Unfortunately, all physical communication channels including transmission lines on a PCB have finite bandwidth.
However, for long links, total insertion loss and ISI can significantly distort a transmitted signal to the point where equalization in the receiver cannot recover it. The finite bandwidth of a real transmission line, coaxial cable, twisted pair cable, or other transmission medium means there is another form of interference seen at the receiver when a stream of digital pulses is sent down a transmission line: intersymbol interference, or ISI. The interaction between an input digital signal, the finite bandwidth of the channel, and the formation of ISI at the receiver is shown below.
The receive pulse is chosen calculated by Algorithm 2. The main simulation parameters have the same setting as in Table 3. As observed in Fig. Noise power level is set as the same in Fig. Moreover, comparing Figs. In this section, we provide several transceiver pulse pairs optimized according to Algorithm 3, both for time-invariant and time-varying channels.
Detailed simulation parameter setting is presented in Table 5 , in which two extreme noise power levels are selected. An interesting observation is that for the case of the low-noise-power level, the proposed pulses converge to the pulses used in conventional CP-OFDM. For the case of high noise power, Fig. Intuitive interpretation of this result is that since the SNR loss due to transceiver mismatching becomes dominating in such noise-limited region, matched filtering is desirable.
Propagation channels are commonly time-variant in practical communication systems. Figure 14 illustrates the derived pulse shapes for both low- and high-noise-power levels.
The optimized pulse pair for a doubly dispersive channel in the high SNR region is close to rectangular-shaped. For the channel with a high-noise-power level, Fig. In reality, pulse shapes need to be properly designed according to the system requirements and available resources and channel information.
Several exemplary design methods have been addressed in detail in this section. According to 3GPP current agreement, new waveform may be applied for new emerging services e. A new air interface design needs to provide means to adapt the physical layer parameters according to requirements for the different services and different frequency bands envisaged for 5G operation [ 26 ]. In the following, we elaborate on how the flexibility of pulse-shaped OFDM can be used to provide different PHY configurations through different parameterizations.
Our focus here is on two parameters of pulse-shaped OFDM, namely, pulse shape design and numerology design. Considering short packet transmission in URLLC service or TDD systems, short pulses are desirable to enable low-latency transmission of packets spread over very few symbols and fast switching between uplink and downlink. Long symbols, on the contrary, would yield long transitions times due to their symbol tails.
Given these circumstances, pulse-shaped OFDM with small overlapping factor K should be chosen, basically extending the symbol duration by up to half the symbol interval at maximum, i. Considering the numerology design for these cases, e. It should be noted here that a larger CP length allows for improving the spectral containment, similar as increasing the symbol length characterized by the overlapping factor K , as indicated in Table 8.
Hence, balancing between CP length and symbol length may be a useful consideration in some scenarios to allow finding the optimum solution. For the MTC service and frequency-division duplex FDD systems, long pulses should be chosen to offer more room for the robustness against time-frequency distortions, since requirements on time localization of the transmit symbols are not so stringent here.
In such cases, the overlapping factor of pulse-shaped OFDM can be chosen large, i. This parameter setting is beneficial for providing good TFL property, which enables the system to become robust against distortions caused by time-asynchronous transmission, which can be introduced by random movement of devices with sporadic data transmission of short bursts only—a typical MTC scenario. Thus, pulse-shaped OFDM becomes an enabler for asynchronous multiple access e.
The numerology should be designed according to service requirements and channel characteristics, followed by further adjustment of the applied pulse. Using the specification in Fig. For a detailed realization of the PPN structure, please refer to [ 20 ]. Recalling the definition of the overlapping factor K , the implementation of the state-of-the-art single and multi-carrier waveforms can be unified with the PPN structure, as shown in Table 6.
We remark that, alternatively, a system featuring multi-rate multi-pulse shaping synthesis and analysis could also benefit from the implementation with frequency sampled filter banks [ 27 , 28 ]. Furthermore, we exemplify the complexity comparison as follows. Taking the whole PHY-layer baseband processing into account, where multi-rate sampling and conversion, MIMO processing, coding, and decoding are considered, the complexity overhead for modulator and demodulator part due to the PPN implementation is rather marginal.
In practice, one can adopt a subband-wise low-pass filtering to shape and fit the transmit signal to the spectral mask, as long as the shaping does not lead to a considerable EVM loss [ 11 ].
Alternatively, subcarrier-filtering can also improve the spectral containment. In the following, we evaluate the PSD with both ideal power amplifier model and Rapp model. If the degrees of freedom for constructing the localized pulse shape are high, e. For small overlapping factor, e. For the evaluation of the spectral containment, the non-linearity of RF unit should be considered.
To model the non-linearity of a power amplifier PA , we use the Rapp model with smoothness factor equal to 3 and 8. The results are evaluated with 10 MHz bandwidth where data occupies 50 resource blocks RBs.
In Fig. The product TF is set to 1. For a more aggressive spectrum usage requiring minimum guard subcarrier overhead, additional subband-wise filtering can also be applied to pulse-shaped OFDM signal. In the section, we provide some applications of pulse-shaped OFDM and evaluate the link performance in the respective scenarios. Considering uplink transmission, due to radio propagation latency, timing misalignment occurs for the uplink signals at the base station, unless a closed-loop TA adjustment is performed.
For the case of massive machine connections, each UE sporadically needs to send a small data packet only, with a long period of silence following. The TA adjustment procedure run for each link would impose a huge overhead to the system, especially if UE mobility is considered. Consequently, TA-free multiple access would be desirable for MTC uplink transmissions, as the time-consuming setup procedure for timing adjustment could be omitted, saving a considerable amount of signaling and shortening the duty cycles.
A new PHY design to support such asynchronous transmission thus becomes a prerequisite to enable TA-free uplink transmission. The scenario is illustrated in Fig. A such designed pulse-shaped OFDM system is particularly useful to be combined with non-orthogonal multiple access schemes like SDMA, if the base station can barely fully synchronize with each user in the uplink at reasonable complexity [ 9 ]. We apply the pulse shape depicted in Fig. Detailed simulation assumptions are given in Table 9.
The energy of the symbols are assumed to be normalized to one. The simulation results shown in Figs. High-mobility scenarios become of great importance for future wireless communications. For 5G NR systems, vehicular-to-anything V2X service will enable safe driving and cooperative autonomous driving.
The determination of such parameter highly depends on the propagation channels and service requirement. Detailed simulation parameters are given in Tables 10 and 11 for link performance evaluation. This paper has summarized the pulse design methods for OFDM systems and provided a new design method taking into consideration an arbitrary length constraint, orthogonality, and good time-frequency localization.
We have also addressed different approaches for receiver realizations and provided a criterion for the evaluation of the pulse design, namely the SINR contour. To meet diverse requirements envisaged for future communication systems, physical layer configuration based on pulse-shaped OFDM has been addressed with suitable parameterizations in pulse design. Practical issues like implementation and complexity are also analyzed for pulse-shaped OFDM systems.
The flexibility of pulse-shaped OFDM multicarrier waveform is attributed to both its different numerology setting and to its transceiver pulse shapes. The numerology configuration mainly aims at defining the time-frequency operational range, while the design of pulse shapes is for further refining the time-frequency localization according to the system or service requirements.
Here, we detail this relation as follows. The pulse shape can be flexibly adjusted. The pulse shape and length are not specified. The pulse shaping is carried out with a circular convolution, which corresponds to periodically time-varying filters. The transmit pulse g t can be considered as the Dirichlet sinc function. In [ 25 ], EVM indicates a measurement of the difference between the ideal and measured symbols after equalization.
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PP Vaidyanathan, Multirate digital filters, filter banks, polyphase networks, and applications: a tutorial. An efficient algorithm for the discrete Gabor transform using full length windows Marseille, A method for constructing localized pulse shapes under length constraints for multicarrier modulation Nanjing, Download references.
The authors would like to acknowledge the contributions of their colleagues in the project, although the views expressed in this contribution are those of the authors and do not necessarily represent the project.
The source files of the manuscript will be available on www. The detailed simulation codes cannot be shared publicly due to company policy. All the authors contribute to the ideas, the developments of the methods, and the results in this manuscript.
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You can also search for this author in PubMed Google Scholar. Correspondence to Xitao Gong. Reprints and Permissions. Zhao, Z. Pulse shaping design for OFDM systems.
J Wireless Com Network , 74 Two optical modes can be coupled efficiently by acousto-optic interaction in the case of phase matching. If there is locally only one spatial frequency in the acoustic grating, then only one optical frequency can be diffracted at a position z. The incident optical short pulse is initially polarized onto the fast axis polarization of the birefringent crystal. Every optical frequency travels a certain distance before it encounters a phase matched spatial frequency in the acoustic grating.
At this position z , part of the energy is diffracted onto the slow axis polarization. The pulse leaving the device onto the extraordinary polarization will be made up all the spectral components that have been diffracted at various positions. Since the velocities of the two polarizations are different, each optical frequencies will see a different time delay given by:. The amplitude of the output pulse, or diffraction efficiency, is controlled by the acoustic power at position z.
More precisely, it has been shown Tournois , for low value of acoustic power density, to be proportionnal to the convolution of the optical input and of the scaled acoustic signal:.
Thus by generating the proper function, one can achieve any arbitrary convolution with a temporal resolution given by the inverse of the available filter bandwidth. A more detailed analysis is given in the following part based on a first order theory of operation, and second order influence will then be estimated. The acousto-optic crystal considered in this part is Paratellurite TeO 2. The propagation directions of the optical and acoustical waves are in the P-plane which contains the [] and [] axis of the crystal.
The acoustic wave vector K makes an angle a with the [] axis. Because of the strong elastic anisotropy of the crystal, the K vector direction and the direction of the Poynting vector are not collinear. An extraordinary optical wave polarized in the P-plane with a wave vector k d is diffracted with an angle d relative to the [] axis.
To maximize the interaction length for a given crystal length, and hence to decrease the necessary acoustic power, the incident ordinary beam is aligned with the Poynting vector of the acoustic beam, i.
Figure 10 shows the k-vector geometry related to the acoustical and optical slowness curves. V and V are the phase velocities of the acoustic shear waves along the [] axis and along the [] axis respectively.
Acoustic and optic slowness curves and k-vector diagram. The single frequency solution of the coupled mode theory for plane waves Yariv and Yeh allows to relate the diffracted light intensity to the incident light intensity and to the acoustic power density P present in the interaction area by the formula:. L being the interaction length along the optical wave vector k 0 , the wavelength of the light in vacuum, the density of TeO 2 crystal, p an elasto-optic coefficient, and M 2 the merit factor given by:.
From eq. As the interaction is longitudinal or quasi-collinear the efficiency of diffraction is excellent. If is the incident optical bandwidth, the number of programming points N and the estimation of the acoustic power density to maximally diffract the whole bandwidth will be:.
When the goal is to control the spectral phase and amplitude in the largest possible bandwidth, to obtain the shortest possible pulse, the diffraction efficiency has to be maximized and hence P 0 minimized Wide Bandc cut. When the goal is to shape the input pulse width with the higher resolution, the optimization is a trade-off between the spectral resolution and the diffraction efficiency High Resolution cut.
Since Paratellurite crystals are dispersive, the acoustic to optic frequency ratio depends on the wavelength through the spectral dispersion of optical anisotropy. For limited bandwidth , the dispersion of the crystal can be compensated b y programming an acoustic wave inducing an inverse phase variation in the diffracted beam. This self-compensation is, however, limited by the maximum group delay variation given by:.
The maximum bandwidth of self compensation depends upon the central wavelength and the crystal type cf table 2. If the bandwidth of operation is larger than this maximum bandwidth , it is necessary to use an outside compressor. The major component of the dispersion in TeO 2 is the second order. Self-compensation bandwidth , second order dispersion and higher order limited bandwidth 1. The first order theory is a good approximation despite strong hypothesis of acoustic and optic plane waves, acoustic and optic single frequencies.
The validity of these two hypothesis is studied in the following parts. The multi-frequencies general approach Laude is complex and not actually required for the simulation of the AOPDF Oksenhendler In the AOPDF crystal geometry, as only one diffracted mode can exist, the coupled-wave equation can be simplified and expressed in a matrix notation such as:. This equation can be solved independently of the number of acoustic frequencies considered. The solutions are:. The spectral phase is conserved even in the saturated or over saturated regime because it comes directly from the phase matching condition fig.
Simulation of acousto-optic diffraction for a spectral amplitude, b spectral phase. The first order can then be used to precompensate the saturation within few percents but exact pulse shaping requires to monitor and loop on the spectral amplitude. The spectral phase is automatically conserved through the Bragg phase matching condition. This coupled-wave analysis considers plane waves. Due to the size of the beam relatively to the wavelength, the acoustic wave cannot be considered as a single plane wave.
The acoustic beam finite dimension D a results in the limitation in spatial aperture of each wave that allows to represent the acoustic field in the components of angular spectrum as:. Due to the strong anisotropy of the crystal Zaitsev , the phase matching condition or Bragg synchronism condition can be rewritten as:.
The intensity of the diffrated field can be expressed as the superposition of the plane wave contribution with propagation angle :. As expected from the first order theory, any divergence of the beams decreases the resolution of the device.
While optical beam direction modifies mostly linearly the peak diffraction position in frequency, the acoustic direction has a quadratic dependance in the AOPDF configuration which modifies the symetry of the diffracted field intensity profile versus the frequency or wavelength as shown on figure 8.
Without initial divergence, the resolution of the device is not affected by such a divergence. This effect can be combined with the multi-frequencies through the momentum mismatch k of optical and acoustic wave vectors:.
The main physical effect not already considered is the walk-off of the diffracted beam and of the acoustic beam. These walk-offs are due to the anisotropy of the crystal. The figure 13 illustrates the two walk-offs and their consequences on the output diffracted beam. These effects combine each other also with the diffracted beam direction dispersion and finally result in a diffracted beam whose angular chirp is compensated by an adequate output face orientation but the spatial chirp illustrated on fig.
The effect is only a variation of the position of the different frequencies spatially. The maximum value corresponds to the walk-off over the complete crystal length and is given in table 1 for the different crystals.
By opposition to the 4f-line, this effect is not a coupling between optical frequencies and beam direction but rather a coupling between optical frequencies and beam position. The consequence of this coupling on a focal spot is very small. Illustration of optical walk-off a , acoustic walk-off b. The spatio-temporal caracteristics of the pulse shaping are directly include in the filter response function H x,.
For each x, calculus of the acoustic delay ac and determination of its longitudinal position in the crystal : Z x, ac. The filter optical function is calculated from H ac x, including saturation of the diffraction and its correction: H x,. This model strongest hypothesis is the localization of the diffraction at a specific position Z x, ac. As long as the bandwidth is large enough, this hypothesis is valid.
In the extreme case, a monochromatic acoustic wave fills completely the crystal. There are no specific position but in the same time there are no chromatic displacement at the output because of the monochromaticity!
Figure 10 shows simulation of a chirp, a delay and two pulses for the temporal intensity. AOPDF simulation including walk-off, temporal intensity in logarithmic scale and spatio-temporal vizualisation inset for a a fs delay, b a chirp fs 2 , c two pulses delayed by fs.
Different orientations and crystal lengths give either higher diffraction efficiency or higher resolution from 3ps, 0. The principal limitations are :. The crystal cuts optimization is a trade-off between the temporal window maximization and the efficiency of diffraction.
The efficiency is determined by the acoustic power for each wavelength. This power depends upon the acoustic pulse shape and the maximal RF power acceptable by the transducer for the acoustic generation. This power is in the range of 10W. With an optimal chirp acoustic pulse high efficiency of diffraction can be achieved over large bandwidth.
This effect can decrease the efficiency of diffraction by an important factor. For example compensation of third order spectral phase influences the efficiency of diffraction. The jitter on the synchronization has a direct incidence on the absolute phase of the diffracted pulse. The delay in acoustic generation can be directly linked to the optical delay by the acoustic and optical temporal windows lengths.
This effect will be review on the CEP control off the pulse. As the crystal length equivalent acoustic duration is about 30s, a single acoustic wave can be synchronized perfectly with a single optical pulse only for laser repetition rate below 30kHz. For higher repetition rate, laser pulses will be diffracted by a same acoustic pulse at different position in the crystal leading to distortions.
As seen on the Fig. The main effect is to spread the different wavelength at different positions. Depending upon the crystal caracteristics, the maximal displacement is in the range of 0.
This effect can be completely nullify by a double pass configuration as shown in the experimental implementations. In this section, we compare the results obtain with the two technologies simulated with the models described in the previous part on identical pulse shaping examples. The ultrashort pulse considered is 20fs gaussian shape with a 2. The table 3 sums up the parameters and caracteristics of the different pulse shapers used in this part at nm.
In this part, we simulate the compression obtained for 1 a pulse but stretched by fs 2 and fs 3 spectral phase with a gaussian shape of 50nm full-width at half-maximum, 2. The compressed pulse for each pulse shaper will be caracterized on for its ps contrast, and the comparison between ideal compressed pulse energy distribution around the focus and the simulated one. The energy distribution is represented by plotting the energy distribution in the focal spot area and its difference with the ideally compressed pulse.
The initial pulse is stretched in time over about fs by the chirp and with a trailing edge due to the third order spectral phase on one ps at 10 The compression of this pulse by the. Due to pixelization, phase wraps and smoothing, the SLM and SLM 4f-pulse shapers create pulse replicas at a level 10 -3 and 10 -4 respictevely. These pulses mainly due to gaps, pixelization and phase wraps are on top-off a background at 10 -6 due to pixel smoothing.
This pedestal is due to a cut of the acoustic wave which is slightly longer than the crystal itself. On the focal spot the effect of the walk-off is more than 10 times smaller than for 4-f pulse shaper. High contrast compression of pulses clearly requires no pixelization, gaps, phase wraps or smoothing and thus is better achieved by AOPDFs.
Temporal square pulses are of interest both for Free Electron Laser electron bunch seeding and for optical parametric amplification pumping. The very first idea to obtain such a pulse is to apply a flat temporal phase and Fourier transform the square root of this temporal intensity profile. This leads to a spectral amplitude with a sinus cardinal shape.
This shape has a large pedestal bandwidth and a relatively narrow central peak. Considering that the initial pulse to shape has a gaussian shape with 50nm fwhm, obtaining a 2. If no temporal phase shape is required, the optimal shape is a mix of amplitude and phase shaping. The fast rise time will be obtain by linear phase on the sides, the flat top by a chirp. Such pulse shaping is shown on the fig. The effects of pixelization and smoothing clearly modify the temporal shape for the 4-f pulse shapers.
The effect of the beam size is not taken into account in this simulation. The central peak for the SLM pulse shaper can be compensated as proposed by Wefers and Nelson but only by an experimental feedback loop.
This kind of feedback loop depends upon the measurement technique and accuracy. To evaluate the quality of the pulse shaping for compressed pulse at the focal spot, three parameters are considered: the temporal intensity integrated over the focal volume considered; the spatial overlap integrals, within the focal volume, between frequencies; the relative strength of multiphoton transition probabilities, integrated across the focal volume.
The last parameters have been introduced by Sussman [ ] to be relevant criteria in multiphoton experiments. The normalized overlap integrals measure the spatial variation of different colors through the focus:. The values of the overlap integrals O 12 may vary considerably.
A value of 1 indicates that two colors have a complete overlap throughout the focal volume. Variations below 1 are significant for any multiphoton experiments. The probability for a vertical n-photon transition having no intermediate resonances is given by the nth-order power spectrum:.
Without space-time coupling, this transition probability is proportionnal to the peak intensity I at a point: P n I n. In order to quantify this effect, this ratio is integrated over the focal volume, normalized and compare with the non space-time coupled one as.
In the absence of space-time coupling, these values would be zero. However, since the spectral content at each point is modified, the ratios are not constant and their deviation from 1 can be considerable. The global value is integrated over the volume of interest, for example the focal volume. The influence of the spatio-temporal coupling is caracterized both by the r 21 ratio and the energy and power difference maps.
These effects can be very important when the pulse shaper is used to optimize a non linear effect at the focus on a non strictely homogeneous media.
Indeed, modification of the spatial profile instead of the temporal one can be the predominant effect in the optimization. Depending upon the relevant parameters of a pulse shaping experiment, the pulse shaper has to be adapted. An adequate simulation of the pulse shaper should estimate the minimum requirements. Also experimental implementation and alignement tolerances that are beyond the scope of this chapter should be taken into account.
Feedback loop can be used to optimize the temporal shape. But some inherent defaults of the pulse shaping technology can also compromise the experimental results and cannot be compensated by any feedback loop. Moreover the accuracy and dynamic of the measurement that should be used for the loop is already an experimental challenging part. This part gives examples of experimental implementations of pulse shaping with their advantages and limitations. As pulse shapers can be applied to a wide range of applications, experimental implementations are reviewed in the scope of the laser source from oscillators to multi-TeraWatt laser systems.
Example of feedback loop with measurement devices will also be given. Applications of pulse shaping with oscillators are direct pulse shaping of the output train of pulses. Among them are for examples, multiphoton microscope imaging, white light optimization for spectroscopy. The pulse shaper is used directly before the experiment. For 4-f pulse shaper there are no modification of the pulse shaping response function due to the high repetition rate of the laser.
The mask can be considered as fix. An imaging relay optics should be used to avoid magnification of the space-time coupling effects [ Tanabe ]. Many different kind of experiments have benefit from optimization by feedback loop such as multiphoton microscopy or coherent control fig. This feedback can be either used to optimize the pulse shape [ Coello ], or to directly optimize an experimental result by blind algorithms [ Assion , Brixner ]. Higher modulation refreshing rate can be obtain by using two lines LC SLM and switching from one line to the other.
Therefore, the acoustic pulse is moving in the crystal from pulse to pulse. Synchronization of the measurement system with the acoustic wave is then needed to eliminate the measurement with a partial acoustic wave in the crystal. In standard 25mm crystals, the complete acoustic time window is about 30s. Thus depending upon the duration of the acoustic wave t a used for the shaping, this acoustic wave is totally include in the crystal during 30s-t a.
The measurement has to be gated on during this time and off in between to consecutive acoustic pulse t a fig. This drawback of AOPDF can be overcomed by its higher refreshing rate up to 30kHz that can be used for differential measurements between two pulses shapes [Ogilivie ] eventually with heterodyne detection. New implementations of these two techniques for multiphoton or CARS microscopy, coherent control are currently published demonstrating higher efficiency or sensitivity.
In amplified system, depending upon the damage threshold energy and non linearities in the pulse shaper, the device is inserted in the laser chain or at the output. The irradiance limit W. To avoid any significative distorsion, this limit is defined as an upper value of cumulative effect of self-phase modulation B-integral :. Depending upon the tolerances on spatial and temporal distorsions due to this slef-phase modulation, the B-integral limit is set to 0.
This value includes the dispersion of the pulse by the crystal itself. This implies that a 4f-pulse shaper can be used at the output of an amplified system at the milliJoule level. The figure 15 details the different position where the pulse shaper can be inserted.
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