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Pulse shaping design for OFDM systems
EURASIP Journal on Wireless Communications and Networking volume 2017, Article number: 74 (2017)
Abstract
Spectrally contained OFDMbased waveforms are considered key enablers for a flexible air interface design to support a broad range of services and frequencies as envisaged for 5G mobile systems. By allowing for the flexible configuration of physical layer parameters in response to diverse requirements, these waveforms enable the inband coexistence of different services. One candidate from this category of waveforms is pulseshaped OFDM, which follows the idea of subcarrier filtering while fully maintaining the compatibility with CPOFDM. In this paper, we provide an overview of pulse shaping methods in OFDM systems and propose a new pulseshaped design method with arbitrary length constraint and good timefrequency localization property. Based on the pulse design, we discuss different receiver realizations and present a criterion for pulse shape evaluation. In addition, the parameterizations of OFDM system to address diverse requirements of the services envisaged for the 5G systems are described. Link and system performance results for selected scenarios show that a proper design of the OFDM numerologies and pulse shapes could substantially improve the performance under time and frequency distortions. Furthermore, pulseshaped OFDM is able to support asynchronous transmissions and reduce the signal sensitivity to Doppler distortions, rendering it beneficial for various applications from the context of vehicular communications and the Internetofthings.
Introduction
The next generation of mobile systems, the fifth generation (5G), is envisaged to accommodate a large variety of new scenarios and use cases, which impose diverse requirements to the system. More specifically, the three main services, enhanced mobile broadband (eMBB), ultrareliable lowlatency communication (URLLC), and massive machine type communication (mMTC), impose different requirements on the 5G air interface [1, 2], yielding new technical challenges.
As one of the key components, waveform design is considered a fundamental brick stone for enabling a flexible air interface design. Recent 3GPP has conducted comprehensive discussions on waveform design for 5G new radio (NR). According to the latest agreements [3, 4], orthogonal frequencydivision multiplexing (OFDM) and discrete Fourier transform spread OFDM (DFTsOFDM)based waveforms, including filtering and windowing for spectral containment, are the most promising candidates for 5G eMBB service, which is underpinned by their success in 4G LTE and many other systems. New waveforms targeting requirements of the novel services envisioned for NR are for further study.
In order to provide flexibility on physical layer, recent research has focused on enhancements of the OFDM waveform with respect to its supported numerologies and considering additional filtering components; for an overview, refer to [5–7]. Generally speaking, the new waveform proposals fall into two main categories: subcarrierwise filtering, comprising filter bank multicarrier (FBMC) [6], windowed OFDM [8] and pulseshaped OFDM (POFDM) [9], etc, and subband wise filtering, composed of universal filtered (UF)OFDM [10] and filtered OFDM [11], etc. FBMC in particular received a lot of attention in research during the past years [12], thanks to its favorable properties of not requiring a cyclic prefix (CP) and attaining very steep filter slopes, which can facilitate an excellent isolation of the signal power in frequency domain. However, these favorable properties are “bought” by a relaxed orthogonality (in fact, strict orthogonality holds for the realvalued signal field only), which requires a redesign of several algorithms developed for conventional OFDM systems. Due to this reason, it was hard for FBMC to get commonly accepted as a mature candidate for 5G, which set off the research on CPOFDM compatible waveforms with filtering, targeting to maintain as many of the favorable properties of FBMC as possible [5]. Table 1 reviews the transmit waveform specified by existing mobile system standards, spanning 2G to 5G communication systems. As flexibility and forwardcompatibility are considered vital properties for future 5G air interface design, we propose here the flexible pulseshaped OFDM waveform with configurable numerology sets and pulse shapes. It is compatible with the current stateoftheart OFDMbased communication system and multiantenna technologies, while the option for pulse shape design enables radio coexistence and improved robustness to timefrequency distortions.
Pulseshaped OFDM is closely related to windowed OFDM and filtered multitone (FMT) [6]. It exploits the pulse shape as an additional degree of freedom for multicarrier modulation systems. The principle of OFDM with pulse shaping has been introduced in [13]. One of the main criteria for pulse shape design is the timefrequency localization (TFL), which has been identified in [14] as an important target to achieve low outofband emissions and low interference induced in doubly dispersive channels. Furthermore, [15–17] investigated the pulse shape optimization framework considering realistic channel knowledge. However, these resulting pulse shapes usually span over several successive symbols, rendering them not well suited for selected scenarios like shortblock lowlatency transmission or fast time division duplex (TDD) uplinkdownlink switching. To address this practical issue, [18, 19] proposed some analytic solutions for the design of short pulse shapes for specific settings.
This paper is dedicated to the design and application of pulseshaped OFDM in future mobile radio systems. We show that through a proper pulse shape design, it is possible to substantially improve the robustness to timefrequency distortions, to provide better spectral containment and thus to enable flexible physical layer (PHY) configurations for selected subbands within a given system bandwidth, which can be tailored to particular servicespecific requirements. Specifically, the contributions of this paper are as follows: a comprehensive overview of pulse shaping methods in OFDM systems is provided, followed by a new pulse shape design method with arbitrary length constraint, which maintains orthogonality while providing good timefrequency localization property. Based on the designed pulse, we also discuss different receiver realizations and provide a criterion for evaluating pulse shapes. In addition, we describe the suitable parameterizations for the pulse shape design to address requirements of the diverse services envisaged for the 5G system. Finally, the implementation complexity of pulseshaped OFDM systems is analyzed.
The paper is organized as follows: Section 2 will introduce the system model with pulse shaped OFDM and stateofthe art OFDM systems. Section 3 gives the principles for OFDM pulse shape design and some practical methods. Section 4 evaluates the pulse shape examples designed in Section 3. Section 5 discusses the parametization of pulse shaped OFDM for the new services and challenges envisaged in future mobile systems, and Section 6 addresses the practical implementation and system impacts. Some application examples are illustrated in Section 7. Finally, Section 8 draws the conclusions.
OFDM system and pulse shaping
In this section, a generic OFDM system model with pulse shaping is introduced. The stateoftheart OFDM systems (including CPOFDM, windowed OFDM, timefrequencylocalized OFDM [14]) can be considered as a typical pulseshaped OFDM system. Their design methodology and the system impact on the OFDM numerology are briefly discussed.
System model
The transmit signal s(t) of an OFDMbased multicarrier system can be generally represented as follows [13, 15]
where a _{ m,n } is the information bearing symbol on the mth subcarrier of the nth symbol. M _{A} is the number of active subcarriers. The transmit filter bank g _{ m,n }(t) is a timefrequency shifted version of the transmit pulse shape (also known as prototype filter)^{1} g(t), i.e.,
with symbol period T and subcarrier spacing F. Note that subbandbased filtering can be used on top of s(t) with a bandpass filter, in order to further suppress the outofband (OOB) leakage.
At the receiver side, the demodulated symbol \(\tilde {a}_{m,n}\) is obtained by correlating the received signal r(t) with the receive filter γ _{ m,n }(t):
where (·)^{∗} denotes the complex conjugate operation, γ _{ m,n }(t) is a timefrequency shifted version of the receive pulse γ(t)^{2}
In short, a generic OFDMbased system with pulse shaping can be presented by the following steps: the transmit signal is first synthesized using (1), passed through propagation channels, and then analyzed at the receiver through (3).
If the pulses employed at the transmitter and the receiver are the same, i.e., g(t)=γ(t), the approach is matched filtering [7]. Alternatively, different pulses can be used at the transmitter and the receiver, i.e., g(t)≠γ(t), yielding the mismatched filtering. Generally, matched filtering aims at maximizing the signaltonoise ratio (SNR) in additive white Gaussian noise (AWGN) channel, while mismatched filtering allows for better balancing the effect of intersymbol interference (ISI) and intercarrier interference (ICI) experienced in doubly dispersive channels with the effect of noise enhancement.
Different from conventional CPOFDM where the pulse shape is fixed to the rectangular pulse, pulseshaped OFDM follows the idea of fully maintaining the signal structure of CPOFDM but allowing for the use of flexible pulse shapes to balance the localization of the signal power in time and frequency domain. The prototype filter pair g(t) and γ(t), together with the numerology parameters T and F, are the central design parameters for pulseshaped OFDM system.
A useful representation of numerology design is a lattice that contains the coordinates in the timefrequency plane. Assume the symbol period is T=N T _{ s } and subcarrier spacing is set to F=1/M T _{ s }, where T _{ s } is the sampling period and \(M, N\in \mathbb {N}\) denote fast Fourier transform (FFT) size and the number of samples constituting one symbol period, respectively. Figure 1 depicts the rectangular lattice representation for OFDM. The metric 1/T F can be considered as the data symbol density in rectangular sampling lattice and it is proportional to the spectral efficiency.
In this paper, we choose the numerology T and F such that T F=N/M>1 holds [15]. Under this condition, orthogonality can be guaranteed for the signal space, yielding the full compatibility with the current techniques developed for OFDM.
Pulseshaped OFDM allows the pulse shape to extend over the symbol period, rendering successively transmitted symbols to overlap or partially overlap. The overlap is characterized by the overlapping factor K, which is defined as the ratio of filter length L _{ g } and the symbol period, i.e., K=L _{ g }/T. The factor K can be set to any rational number in pulseshaped OFDM.
Transceiver of pulseshaped OFDM
As a typical uniform filter bank system, the overall transceiver structure of the pulseshaped OFDM system is given in Fig. 2. The pulse shaping can be efficiently realized by a polyphase network (PPN) [20] for arbitrary overlapping factor K. For short pulse shapes where K≈1, the PPN structure can be simplified to the “CP addition”/“CP removing”/“zeropadding” and “windowing” operations, etc. For K>1, PPN implementation can be considered as a realization of an “overlapadd” procedure.
Stateoftheart OFDM systems and numerology design
Numerology design for multicarrier systems, including the determination of symbol period T and subcarrier spacing F, is an essential part in the system design. Its design needs a comprehensive consideration of many aspects, such as spectrum efficiency or propagation channel characteristics. In this section, we will briefly introduce the numerology design of the stateoftheart OFDMbased systems; a detailed overview on the waveform candidates under discussion for 5G is provided in Appendix 1. All those waveform candidates can be considered as special cases in the pulse shaped OFDM framework.
CPOFDM
The derivation of OFDM numerology w.r.t. (T,F) can be carried out by the following steps:

Set the CP length T _{cp} according to the channel characteristics, i.e., at least longer than the maximum channel excess delay τ _{max}.
$$ T_{\text{cp}} = T\frac{1}{F} \geq \tau_{\text{max}}. $$(5) 
Determine the minimal subcarrier spacing F such that the signaltointerference ratio (SIR) for the maximum Doppler frequency (ν _{max}) is above the minimum SIR requirement (SIR_{min}) for supporting the highest modulation requirement in the system.

Determine the approximate values of T and F based on the above two steps.

Quantize T and F according to the sampling rate and subframe numerology.
The above steps are based on the premise that CPOFDM can support reliable transmission without ISI and ICI if the maximum excess delay of the channel is smaller than the CP length. It is a pragmatic approach since the robustness of CPOFDM is pronounced in the time domain rather than in the frequency domain.
WOFDM
Windowed OFDM (WOFDM) is originally introduced as an enhancement to CPOFDM for reducing the OOB emission. Recently, 3GPP RAN1 agreed that windowing is one of the favored approaches for achieving spectral confinement.
Essentially, WOFDM is a pulseshaped OFDM system, where a window with smoothened edges is used instead of a rectangular one (as used in CPOFDM) to effectively reduce the side lobes. Some overlap of the window tails of the succeeding symbols is allowed, thereby from link performance perspective, WOFDM is aiming at trading off its robustness in the time domain (due to the relaxation of CP) for an improved robustness in the frequency domain. We will show later that for the typical operational range of mobile systems, a properly designed WOFDM system can outperform its CPOFDM counterpart and better fulfill timefrequency (TF) localization requirements.
TFlocalized OFDM
Given the same spectral efficiency, it has been shown that the link level performance can be improved over conventional CPOFDM and its pragmatic (T,F) numerology design [14, 15]. One solution is the TFlocalized OFDM aiming at minimizing the distortion resulting from timefrequency dispersive channels [14]. The numerology design of this waveform is comprised of the following steps:

Determine the ratio of T and F: In order to reduce ISI and ICI, the numerology T and F of the TFlocalized OFDM should be chosen in correspondence to the characteristic parameters of the doubly dispersive channel. Specifically, for the given maximal time delay τ _{max} and maximal Doppler spread ν _{max}, the choice of T and F should satisfy [14]
$$ \frac{T}{F}=\frac{\tau_{\text{max}}}{\nu_{\text{max}}}. $$(6) 
For a fair comparison with CPOFDM, the product of TF should be set to the same value as in CPOFDM, reflecting the relative CP overhead or spectral efficiency loss. Combined with the ratio of T and F specified above, parameters T and F can easily be obtained. Otherwise, if TF is not specified, the numerology needs to be determined as follows: First, initialize TF with some predefined number; then, calculate SIR using TFlocalized pulses in case of maximum excess delay and Doppler frequency; finally, adapt the numerology T and F to guarantee that the resulting SIR can support the transmission using the highest modulation format.
OFDM pulse shape design and proposed methods
Future mobile communication systems are envisioned to support the coexistence of multiple services with diverse requirements. PHY setting including waveform configuration is thus anticipated to be adapted to different requirements for each service. For example, URLLC requires low latency, and comparably short pulse is favorable. Narrowband internetofthings (NBIOT) service targets at good coverage extension and allows long pulse design. Machinetype communication (MTC) with mobility may require pulse design to be robust to asynchronicity and Doppler spread. In this section, we discuss several pulse shape design approaches and outline their features and applications.
Pulse shape categorization
In OFDM systems with pulse shaping, the ISI and ICI are determined by the transmit pulse g(t) and the receive pulse γ(t). In this paper, we use the pulse shape categorization according to the correlation property [7]:

Orthogonal pulse design is the pulse shaping scheme where perfect reconstruction condition is fulfilled (details given in Section 3.2.2) and matched filtering is employed.

Biorthogonal pulse design is the pulse shaping scheme where perfect reconstruction condition is fulfilled and mismatched filtering is employed.

Nonorthogonal pulse design is the pulse shaping scheme where perfect reconstruction condition is not fulfilled.
Design criteria
Depending on the specific criteria for pulseshaped OFDM systems, pulse shapes are constructed to satisfy diverse requirements. Herein, we discuss several commonly applied conditions that the pulse shape design needs to fulfill.
Length constraint
Length constraint is the primary design criterion for OFDM pulse shapes. Since many use cases in the eMBB and URLLC context require short processing latency and stringent timing for framed transmission, short pulse lengths comparable to one OFDM symbol duration (i.e., K≃1) are favorable here. In other scenarios such as mMTC or NBIOT, though, latency constraints may be relaxed, such that pulse lengths of several symbols (K≥2) can be allowed if they provide clear benefits addressing the needs of the corresponding service.
Nearperfect reconstruction condition
Assuming an ideal channel where r(t)=s(t), a perfect reconstruction (PR) condition holds if \(\left \langle g_{m^{\prime },n^{\prime }},\gamma _{m,n}\right \rangle =\) \(\delta _{m^{\prime } m}\delta _{n^{\prime } n}\), where g _{ m,n } and γ _{ m,n } follow the definition in (2) and (4), respectively. Due to the fact that a certain level of selfinterference can be tolerated for the reliable transmission of modulated signals in practice, we redefine the orthogonal and biorthogonal conditions as nearperfect reconstruction condition by slightly relaxing the conventional PR condition in this paper to allow for minor crosscorrelation, i.e.,
where ε is determined according to the error vector magnitude (EVM) and signaltointerferenceplusnoise ratio (SINR) requirement, as detailed in Appendix 2. The condition (7) is also named as biorthogonality condition assuming g(t)≠γ(t), since it is a prerequisite in biorthogonal division multiplexing (BFDM) systems for reconstructing \(\tilde {a}_{m,n}\) from r(t) [6]. Under the condition that matched filtering is employed, i.e., g(t)=γ(t), (7) reduces to the orthogonality condition.
Opposed to the orthogonal transceiver pulse where the SNR for AWGN channel is maximized, BFDM has a potential to further reduce ISI and ICI for dispersive channels at the cost of a noise enhancement. It has been shown in [14, 15] that a necessary condition to achieve perfect reconstruction (either orthogonal or biorthogonal) is T F≥1. Larger values of TF lead to larger spectral efficiency loss but provide more degrees of freedom for the orthogonal pulse design.
Timefrequency localization
The ISI and ICI can be reduced if the pulse shapes at the transmitter and receiver are jointly TF localized. The classical way to measure timefrequency localization (TFL) of a filter involves the Heisenberg uncertainty parameter [7, 14]. Filters with good TFL properties have a Heisenberg parameter ξ closer to 1. Assuming the center of gravity of g(t) is at (0,0), the “width” of g in the time and frequency domain is often measured using the secondorder moments defined as
where G(f) is the Fourier transform of g(t). Then, the Heisenberg uncertainty parameter ξ is given by
where equality holds if and only if g(t) is a Gaussian function, rendering such filter to have optimal TFL [14].
Note that the joint TFL of transceiver pulses considering channel dispersion is related to the TF concentration properties of both transmit and receive pulses. The work in [15] has asserted that excellent TFL characteristics can be simultaneously achieved by pulse pairs.
SIR/SINR optimization
In wireless communication systems, the essential goal is to transmit signals reliably in practical channels. Hence, the above criteria can be slightly relaxed to increase the design degree of freedom, as long as the link performance with pulse shaping is optimized relative to certain dispersive channels of interest.
One common criterion is SIR or SINR optimization, namely, the transceiver pulses are chosen to optimize the SIR/SINR under certain dispersive channels. The resulting pulse shapes may not exactly satisfy perfect reconstruction condition, but offer better ISI/ICI robustness in dispersive channels compared to orthogonal or biorthogonal design. In the following, we detail this optimization problem for both continuous and discrete channel models.
Continuous model
assuming a doubly dispersive fading channel satisfying widesense stationary uncorrelated scattering (WSSUS) property, its scattering function (channel statistics) can be described as [15]
where h(t,τ) is the timevarying impulse response at time instance t and delay τ, ν is the Doppler frequency, and \(\mathbb H\) indicates the random linear timevarying channel. We call the WSSUS channel h(t,τ) underspread, if the support of \(C_{\mathbb H}(\tau, \nu)\) is constrained in a rectangular region of {−τ _{max}≤τ≤τ _{max},−ν _{max}≤ν≤ν _{max}} with τ _{max} ν _{max}≪1. The SINR involving pulse shaping is represented by
where \(\sigma _{\mathrm {n}}^{2}\) is the noise variance, \(A_{g,\gamma }(\tau, \nu) = \int _{t} \gamma (t) g^{*}(t \tau)e^{j2\pi \nu tdt}\) is the cross ambiguity function of transceiver pulse pair 〈g(t),γ(t)〉 [15], and \(Q_{\mathbb H}^{(0)} (\tau, \nu)\) is defined as
The energy of the transmit symbols are assumed to be normalized to one. If the noise variance part \(\sigma _{\mathrm {n}}^{2}\) is omitted in (10), it reduces to the SIR metric.
Discrete model
Based on the sampling period T _{ s }, assume the discrete dispersive channel to have P paths, where the pth path is characterized by the path delay τ _{ p }, the Doppler frequency shift ν _{ p }, and the complex channel gain η _{ p }(t). Let the P channel gains be stacked into one vector η(t)=[η _{1}(t),…,η _{ P }(t)]^{T}.
Under the assumption of WSSUS property and applying it to the discrete model, the channel correlation function \(\mathbf {R}_{\mathbb {H}}=\mathbb {E}\{\boldsymbol {\eta }(t)\boldsymbol {\eta }(t)^{H}\}\) yields a diagonal matrix, with the diagonal elements indicating the power of the individual path gains and (·)^{H} denoting the Hermitian operation. Assuming transmit and receive filters being discretized as well and their power being normalized to one, both these filters can be represented by the vectors \(\mathbf {g}\in \mathbb {R}^{L_{g}\times 1}\), \(\boldsymbol {\gamma }\in \mathbb {R}^{L_{\gamma } \times 1}\) containing the discrete filter coefficients, where L _{ g } and L _{ γ } denote the filter lengths for transmit and receive filters, respectively. Using the above discrete expressions, the discrete model of the SINR is given by
where the energy of the transmit symbols are assumed to be normalized to one. G _{ m,n } is a matrix constructed from the filter vector g _{ m,n }, which is created analogously to its continuous counterpart (2), representing the filter used for subcarrier position m and symbol position n. Each column of matrix G _{ m,n } represents the filter vector g _{ m,n } transmitted through one of the P channel taps, i.e., the vector in the pth column is shifted by the corresponding path delay τ _{ p } and modulated by the Doppler frequency ν _{ p }. For symbols preceding or succeeding the symbol of interest, i.e., for n≠0, an additional time shift of n times the symbol duration N has to be considered. If the noise variance part \(\sigma _{\mathrm {n}}^{2}\) is dropped, (11) reduces to the SIR metric.
Design methods
Taking the abovementioned design criteria into consideration, the ultimate goal of pulse design is to have short pulses with maximal spectral efficiency, optimal timefrequency localization, minimized interference, and best SINR performance for arbitrary channels. Nevertheless, not all the requirements can be fulfilled simultaneously in reality, either due to contradictory conditions or practical constraints. Alternatively, in this section, we propose two approaches to design the transceiver pulse shapes for practical pulseshaped OFDM systems, where both can respect an arbitrarily given length constraint.

1.
Orthogonal design without channel statistics: For the case that the system has no reliable knowledge on channel statistics for pulse optimization, we seek to apply (almost) orthogonal transceiver pulse pair with good timefrequency localization.

2.
Biorthogonal design with channel statistics: For the case that the system has reliable knowledge on channel statistics (e.g., scattering function) for pulse optimization, transceiver pulses are designed to achieve optimal link level performance (w.r.t. SIR/SINR) given such channel knowledge.
Orthogonal pulse design without channel statistics
As introduced before, the orthogonal pulse design employs matched filtering at the transceiver in order to achieve the maximum SNR for AWGN channels. In the absence of channel statistics, we suggest to use such orthogonal pulse design with good TFL characteristic. The TFL property is of vital importance in the pulse design since it affects the vulnerability to ISI/ICI in doubly dispersive channels. Note that T F>1 is assumed here to have sufficient degrees of freedom for the pulse design. In the following, a universal approach for producing orthogonal pulses with constrained length as well as good TFL will be proposed.
Before detailing the proposed method, we first review the orthogonal pulse generation in the literature, which provides a basis for our proposal. The classical approach in [14, 15] consists of the following steps.

Select an initial welllocalized pulse, e.g., a Gaussian pulse with a decaying factor α.
$$ g_{\text{\texttt{gauss}}}^{(\alpha)}(t)=\left(2\alpha\right)^{1/4} e^{\pi \alpha t^{2}} $$(12) 
Construct an orthogonal system \(\left (g_{\perp }^{(\alpha)}, T, F\right)\) based on g gauss(α):
$$ g_{\perp}^{(\alpha)}=\text{orth}\{g_{\text{\texttt{gauss}}}^{(\alpha)},T,F\} $$(13)
Orthogonalization can be constructed according to [14], or efficient numerical solution for orthogonalization can be obtained by matrix factorization methods [21, 22].
It is proven in [14] that by appropriately dilating or shrinking \(g^{(\alpha)}_{\text {\texttt {gauss}}}\), i.e., adjusting α, one can easily generate the optimal TFL pulses to match different channel dispersion properties.
The resulting orthogonal pulse \(g^{(\alpha)}_{\perp }\) usually is unconstrained in its temporal length, resulting in a large overlapping factor. As elaborated above, this is not desired in many use cases from the eMBB and URLLC context, where a timeconstrained short pulse is preferred. In order to generate such a pulse from \(g^{(\alpha)}_{\perp }\) given the desired filter duration D _{req}=K T with K≥1, the simplest approach is to directly perform soft or hard truncation on \(g^{(\alpha)}_{\perp }\). However, this approach leads to nonorthogonality and degrades the TFL properties.
For generating orthogonal prototype filters with fixed length close to the symbol duration, (i.e., K∼1), Pinchon et al. have derived two explicit expressions to compute the filter coefficients for two different optimization criteria: minimizing OOB energy and TFL [19]. Using the discretization illustrated in Fig. 1, the derivation requires the condition N _{0}=M _{0}+1 where N _{0}=N/gcd(N,M) and M _{0}=M/gcd(N,M). Such constraint renders the extension to more general cases not straightforward.
We propose a method that aims at generating orthogonal pulses with arbitrary length constraint and maintaining good TFL property and orthogonality [23]. Given an initial welllocalized pulse, by repeatedly performing orthogonalization and truncation, the overall process will converge under a given convergence criterion. A design example is described below. Details of the algorithm are described in Algorithm 1 which involves several essential steps.

Initialize the pulse g ^{(0)}: We choose a Gaussian pulse \(g_{\text {\texttt {gauss}}}^{(\alpha)}(t) = (2\alpha)^{1/4}e^{\pi \alpha t^{2}}\) as the initial pulse g ^{(0)} due to its optimal TFL [14]. This step is similar to the first step of the abovementioned standard method but described in a discrete manner. The factor α determines the TFL of g gauss(α)(t). In order to reduce ISI and ICI, it is suggested to choose α≈ν _{max}/τ _{max} [14]. In general, α can be adjusted to match different channel conditions.

Orthogonalize g ^{(n−1)}[l] using the standard method, namely, by computing
$$\begin{array}{*{20}l} g^{(n)} = \text{orth}\{ g^{(n1)}, N, M \}. \end{array} $$(14) 
Truncation is applied using a truncation window g _{W}. The width of the window L _{W} corresponds to the desired pulse length. Common windows include rectangular (RECT), raisedcosine (RC(β)), and root raisedcosine (RRC(β)) windows, where β is the rolloff factor. For β→0, RC(β) and RRC(β) converge to RECT.

Orthogonalization and truncation are iteratively applied by
$$\begin{array}{*{20}l} g^{(n)} = \big(\text{orth}\{g^{(n1)}, N, M\} \big)\cdot g_{\text{W}} \end{array} $$(15)until \(\frac {\g^{(n)}  g^{(n1)}\}{\g^{(n1)}\} \leq \varepsilon \). The coefficient ε can be interpreted as a tradeoff between orthogonality and TFL. Small ε leads to a higher number of iterations and improved orthogonality; large ε leads to pulses with better TFL. Here, ε is set to 10^{−4}.
Both a fixed window g _{w} or an iterationvarying window can be used in the algorithm.
To illustrate the algorithm procedure, we first discuss the relationship between orthogonality and the number of iterations for a specific example in which the orthogonality is measured by SIR. The essential parameter settings are listed in Table 2. As depicted in Fig. 3, it is obvious that by increasing the number of iterations, i.e., setting a small ε, the orthogonality of g can be improved. Moreover, if taking the convergence time into consideration, confining the number of iterations to less than ten is reasonable as well, as the SIR is already more than 80 dB after the first few iterations.
Figure 4 presents the time and frequency impulse responses for the initial Gaussian pulse, optimized pulse after the first iteration, and the final result, which indicates how the number of iterations influences the time and frequency localization properties for the obtained pulse shapes.
Biorthogonal design with channel statistics
Biorthogonal pulse design allows using different pulses at the transceiver sides to maximize the link performance. It employs mismatched filtering to balance the robustness against ISI/ICI in doubly dispersive channels with the noise enhancement. In general, biorthogonal design capitalizes on more degrees of freedom compared to the orthogonal design, which may lead to better performance in practice, especially in selfinterferencelimited scenarios.
Given that the channel statistics are available, there are two common approaches for biorthogonal design: first, fixing the transmit filter and design the optimal receiver filter and second, joint transmit and receive pulse design. SINR is applied as a typical measure for the design optimization.
Optimized receiver filter design
With regard to the predetermined transmit pulse g, we now derive the optimized receive pulse γ to maximize link performance with taking SINR as the optimization measure. SINRg,γ D in (11) can be reformulated as
where A and B are Hermitian matrices given respectively by
Note that (16) is defined as a generalized Rayleigh quotient, which is associated with a generalized eigenvalue problem A γ=ζ B γ [16, 17]. The maximum SINR target ζ _{max} corresponds to the maximum generalized eigenvalue of A and B, when the receive filter γ is chosen as the corresponding generalized eigenvector γ _{max} i.e., A γ _{max}=ζ _{max} B γ _{max}. Detailed implementation is presented in Algorithm 2.
Joint transmitter and receiver design
Considering the joint optimization of the transmit and receive filters w.r.t. the provided channel statistics for WSSUS channels, [16] showed that the primal problem is a nonconvex problem. An efficient alternating algorithm has been proposed to achieve a local optimum. Its detail implementation is listed in Algorithm 3. In general, this algorithm calculates the transmit and receive pulses alternatingly until the overall process converges.
SINR evaluation of pulse design based on receiver realizations
In Section 3, we have introduced two exemplary methods for designing a pair of transmit and receive pulse shapes. In practical communication systems, one may encounter a fixed transmit pulse that cannot be changed further, so that only the receive pulse can be subject to optimization. Taking this aspect into account, we propose in this section different solutions for the receiver design that depend on the usage of statistical channel knowledge and evaluate the pulse design using the SINR contour as measure.
Evaluation metric: SINR contour
For any doubly dispersive channel, the achievable SINR of given transceiver pulse pair can be computed by (10) or (11). Therefore, we can draw a SINR contour w.r.t. time delay spread τ and Doppler spread ν for WSSUS channels. Such contour is important to visualize the link performance. Basically, the point SINR(τ,ν) on the contour indicates the selfinterference plus noise level when the signal modulated with a pulse pair is undergoing a TF dispersion with a delay region [−τ,τ] and Doppler region [−ν,ν]. To compute SINR via (11) for SINR contour plot, the channel scattering function need to be a priori known. In practice, however, accurate channel statistic is not available but only channel characteristics such as maximum delay τ _{max} and maximum Doppler frequency ν _{max}. Without further specification, in this section, we assume that the “default” support region of the underspread WSSUS channel is an origincentered rectangle shape [13], whose side lengths are equal to 2τ _{max} and 2ν _{max}, respectively. The diagonal entries of channel correlation function \(\mathbf {R}_{\mathbb {H}}\) are set to be equal.
SINR evaluation based on receiver realizations
Given a transmit pulse optimized according to the orthogonal or biorthogonal methods, two receive pulse designs are considered here: socalled naive receiver which is designed without channel information or maxSINR receiver which takes channel information into the design procedure.
Naive receiver without channel knowledge
Transmit pulse based on orthogonal design
Provided the transmit pulse optimized by the orthogonal method, naive receiver refers to the receive pulse which adopts a symmetric shape of the transmit pulse generated by Algorithm 1, i.e., γ(t)=g(t). Herein, we provide several design examples in this section.
A necessary condition for generating orthogonal pulses is to fulfill T F>1. On the other hand, larger TF leads to smaller spectral efficiency. As a compromise, TF is set to be slightly larger than 1. We choose T F=1.07 and T F=1.25 (same as normal/extended CP overhead in LTE) and α=1. Table 3 lists the key parameters in Algorithm 1.
Figure 5 illustrates the pulse shapes for overlapping factor set to K=1.07. Solid line and dashed line indicate the optimized pulse in this paper and [19], respectively. Both results are close to the pulse shapes used in windowed OFDM. For the case of T F=1.25 (Fig. 5 b), the optimized pulse in this paper converges to the analytically derived pulse shape with the optimal TFL in [19]. Given the transmit pulse with K=1, the proposed pulse shapes are depicted in Fig. 6. For T F=1.25, \(g^{K=1}_{2}(t)\) coincides with the pulse proposed in [19], which aim at minimizing the OOB leakage.
This transceiver pulse design method is suitable to the scenario requiring good frequency localization, i.e., one symbol per transmission time interval (TTI) transmission, which is the extreme case of time division duplex (TDD) transmission requiring the lowest roundtrip time (RTT). In such scenario, owing to the guard periods inserted between the uplink and downlink, there is no ISI, and hence, pulses that are welllocalized in the frequency domain are favored. From the design perspective, we select g gauss(α) with α≪1 as the initial pulse, as in this case, we only need to consider suppressing ICI when designing the short pulses [24].
Allowing long pulse, the exemplary orthogonal design results with K=4 are given in Fig. 7. To compare with CPOFDM, the SIR contour with pulse pair \(g^{K=4}_{1}/\gamma ^{K=4}_{1}\) (dashed) and g _{rect}/g _{rect} (solid), as well as \(g^{K=4}_{2}/\gamma ^{K=4}_{2}\) (dashed) and g _{rect}/g _{rect} (solid) are depicted in Fig. 8. The number on contour line indicates the lowest achievable SINR level that a pulse pair could support within the closed region. In particular, compared with CPOFDM, the proposed design possesses the strong robustness against time synchronization errors while maintaining similar support in frequency domain, which could potentially enable timing advance (TA)free transmission in uplink or support downlink multipoint transmission with large coverage. In particular, the proposed design for T F=1.25 supports a similar T−F contour region for highorder modulation (e.g., 64 QAM) while achieving overall larger T−F contour support for lower modulation (e.g., QPSK and 16 QAM). Thus, the T F=1.25 multicarrier waveform is more robust in challenging dispersive scenarios, such as highspeed vehicular transmission, to achieve high reliability.
Transmit pulse based on biorthogonal design
Naive receiver for biorthogonal design indicates adopting mismatched pulse shape of the transmit one without exploiting channel knowledge. To exemplify the receiver realization in this case, transmit pulse is fixed as two options: conventional rectangular pulse g _{RECT} and the raisedcosine (RC) shaped pulse g _{RC}, which is commonly used in WOFDM systems. We remark that other transmit pulse obtained from biorthogonal design is applicable.
For performance evaluation, g _{RC} is generated by the convolution with a window w with length N _{0} and a rectangular window with length N. According to [6], any pulse shape satisfying \(\sum _{i=0}^{N_{0}1}w_{i}=1\) can be selected as a window. Without further specification, we choose h as Hanning windowing and set N _{0}=N _{CP}/2 with N _{CP}=N−M. All the essential parameters are listed in Table 4. Note that the noise power is normalized according to the average transmit signal power, which is assumed to be equal to one.
Figure 9 shows the RC transmit pulse combined with the rectangular receive pulse γ=γ _{RECT} and raisedcosine receive pulse γ=γ _{RC}, respectively. The ratio N/M is set to 1.1. The SINR contour with the pair g _{RECT}/ γ _{RECT} (solid) and g _{RC}/ γ _{RECT} (dashed) are depicted in Fig. 10. The xaxis denotes the delay τ normalized to symbol period T in the time domain, while the yaxis represents the Doppler ν normalized to subcarrier spacing F in the frequency domain. It can be observed that in noiselimited scenario, given rectangular receive pulse, g _{RC} achieves stronger robustness to asynchronization in the time domain and meanwhile supports similar dispersion g _{RECT} in the frequency domain. For the interferencelimited scenario, i.e., noise variance equal to −31 dB in Fig. 10 a, g _{RECT}/γ _{RECT} and g _{RC}/γ _{RECT} for 28 dB SINR level have similar regions in contour plot, e.g., to support 256 QAM on physical downlink shared channel in LTE.
MaxSINR receiver with channel statistical knowledge
With the assumption of a rectangularshaped channel scattering function, we evaluate the performance with transmit pulse from both orthogonal and biorthogonal design and its corresponding maxSINR receive pulse.
Transmit pulse based on orthogonal design
Choosing the transmit pulse for \(g_{1}^{K=1.07}\) shown in Fig. 5 a, we evaluate its SINR operational range w.r.t. double dispersion and make a comparison to g _{cpofdm}. 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. 11, compared with g _{cpofdm}, \(g_{1}^{K=1.07}\) and its respective maxSINR receive pulse are more robust to time dispersion in highnoisepower regions, i.e., noise variance equal to −25, −22, and −19 dB. For the case when \(\sigma _{\mathrm {n}}^{2}\) is −31 dB, the performance of \(g_{1}^{K=1.07}\) on the level of 28 dB is worse than g _{cpofdm}, thus making it an undesirable choice for enabling 256 QAM in such case.
Transmit pulse based on biorthogonal design
We analyze in this section the SINR contours of g _{RECT} and g _{RC} with its corresponding receive pulse calculated by Algorithm 2 according to channel statistics. Noise power level is set as the same in Fig. 11, and parameter settings are given in Table 3.
As depicted in Fig. 12, given maxSINR receiver, g _{RC} outperforms g _{RECT} w.r.t. robustness to timing misalignment, while maintaining comparable robustness to frequency dispersion. Moreover, comparing Figs. 12 and 10, the optimized receiver is more robust against the frequency misalignment than the naive one, especially when the time shift close to zero.
Joint transmitter and receiver design with channel statistical knowledge
In this section, we provide several transceiver pulse pairs optimized according to Algorithm 3, both for timeinvariant and timevarying channels. Detailed simulation parameter setting is presented in Table 5, in which two extreme noise power levels are selected.
Figure 13 a,b depicts the computed pulse shapes respectively for low and highnoisepower levels in timeinvariant channels, where the normalized maximum frequency shift is ν _{max}/F=0 and the normalized maximum time delay is τ _{max}/T=10%. An interesting observation is that for the case of the lownoisepower level, the proposed pulses converge to the pulses used in conventional CPOFDM. This result makes sense since CPOFDM is known to be optimal in the high SNR scenario with low Doppler spreads. For the case of high noise power, Fig. 13 b shows the transceiver pulses are close to a matched pulse pair. Intuitive interpretation of this result is that since the SNR loss due to transceiver mismatching becomes dominating in such noiselimited region, matched filtering is desirable.
Propagation channels are commonly timevariant in practical communication systems. To evaluate the performance in this case, we select τ _{max}/T=5% and ν _{max}/F≈1.6% by assuming that an object moves at a relatively high velocity in a medium delay spread environment, as characterized, for example, in the extended vehicular A (EVA) channel model [25]. Figure 14 illustrates the derived pulse shapes for both low and highnoisepower levels. The optimized pulse pair for a doubly dispersive channel in the high SNR region is close to rectangularshaped. However, due to the frequency shifts, both g and γ have some irregular shaping at the filter head and tail, which are visible as “steps” in the figure. For the channel with a highnoisepower level, Fig. 14 b shows that g and γ are nearly matched, as can be explained analogous to Fig. 13 b.
Ideally, pulse shape optimization aims at fulfilling the orthogonal condition, achieving good TFL and SIR/SINR performance. 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.
Air interface PHY design based on POFDM
According to 3GPP current agreement, new waveform may be applied for new emerging services (e.g., URLLC, MTC) other than eMBB in 5G NR systems.
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 pulseshaped OFDM can be used to provide different PHY configurations through different parameterizations. Our focus here is on two parameters of pulseshaped OFDM, namely, pulse shape design and numerology design.
Considering short packet transmission in URLLC service or TDD systems, short pulses are desirable to enable lowlatency 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, pulseshaped OFDM with small overlapping factor K should be chosen, basically extending the symbol duration by up to half the symbol interval at maximum, i.e. K∈ [ 1;1.5]. If K∼1 is chosen, the solution reduces to WOFDM. Considering the numerology design for these cases, e.g., subcarrier spacing, symbol interval, and symbol overhead, the methodology for OFDM/WOFDM systems described in Section 2.3 can be adopted, followed by an optimization of the designed pulse shape. 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 frequencydivision duplex (FDD) systems, long pulses should be chosen to offer more room for the robustness against timefrequency distortions, since requirements on time localization of the transmit symbols are not so stringent here. In such cases, the overlapping factor of pulseshaped OFDM can be chosen large, i.e., up to K=4. This parameter setting is beneficial for providing good TFL property, which enables the system to become robust against distortions caused by timeasynchronous transmission, which can be introduced by random movement of devices with sporadic data transmission of short bursts only—a typical MTC scenario. Thus, pulseshaped OFDM becomes an enabler for asynchronous multiple access (e.g., frequency/space division multiplexing access), facilitating grantfree and timingadvancefree communication—for details, refer to [9]. The numerology should be designed according to service requirements and channel characteristics, followed by further adjustment of the applied pulse.
Implementation and system impact
Implementation and complexity
Using the specification in Fig. 1 for symbol period T=N T _{s} and subcarrier spacing F=1/M T _{s}, the transmit and receive signal can be efficiently synthesized and analyzed using a PPN implementation (e.g., Fig. 2). For a detailed realization of the PPN structure, please refer to [20]. Recalling the definition of the overlapping factor K, the implementation of the stateoftheart single and multicarrier waveforms can be unified with the PPN structure, as shown in Table 6. We remark that, alternatively, a system featuring multirate multipulse 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. Assuming a symmetric transceiver pulse design, namely, g(t)=γ(t), M=2048, and T F=1.07, the number of operations including complex multiplications and additions for implementing different waveforms are summarized in Table 7. As seen from the table, the overall complexity overhead introduced by the PPNbased implementation for pulseshaped OFDM is minor compared to CPOFDM. Taking the whole PHYlayer baseband processing into account, where multirate 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.
Spectrum confinement and coexistence
Conventional CPOFDM suffers from strong OOB leakage of its power spectral density (PSD) due to the slow frequency decay property of the rectangular pulse. In practice, one can adopt a subbandwise lowpass 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, subcarrierfiltering can also improve the spectral containment. In the following, we evaluate the PSD with both ideal power amplifier model and Rapp model.
For properly designed pulse shapes, the PSD of pulseshaped OFDM surpasses CPOFDM. If the degrees of freedom for constructing the localized pulse shape are high, e.g., for an overlapping factor K=4 (see in Fig. 15 a, b), the resulting PSD of pulseshaped OFDM is satisfactory even without any additional spectral mask filtering, i.e., incurring no EVM loss. For small overlapping factor, e.g., K=1.07≈1 (see in Fig. 15 c, d), the spectral containment in frequency domain becomes slightly worse; however, a satisfactory PSD can still be achieved, resulting in still a small number of guard subcarriers for spectral coexistence.
Assuming −50 dBc/Hz as the required spectral leakage, the required number of guard sucarriers based on the above PSD results is summarized in Table 8. The results are based on the LTE setting of 15 kHz subcarrier spacing for 20 MHz bandwidth.
For the evaluation of the spectral containment, the nonlinearity of RF unit should be considered. To model the nonlinearity of a power amplifier (PA), we use the Rapp model with smoothness factor equal to 3 and 8.3 dB output backoff. The results are evaluated with 10 MHz bandwidth where data occupies 50 resource blocks (RBs). In Fig. 16 a, b, the PSD performance (before and after the PA) of OFDM, pulseshaped OFDM, and OFDM with subbandfiltering are shown, respectively. The product TF is set to 1.07 for the first two waveforms, and K=1.07 is used for pulseshaped OFDM, while OFDM with subbandfiltering employs a halfsymbol length FIR filter. We observe that pulseshaped OFDM achieves comparable spectral containment as OFDM systems with subbandfiltering, both significantly outperforming conventional CPOFDM systems. If taking the PA nonlinear effects into account, pulseshaped OFDM still offers similar performance as OFDM with subband filtering in OOB emission, which is slightly better than that of OFDM systems.
For a more aggressive spectrum usage requiring minimum guard subcarrier overhead, additional subbandwise filtering can also be applied to pulseshaped OFDM signal. However, the tradeoff between EVM, OOB leakage, and particularly the linearity for RF unit (cost and power efficient) at both base station (BS) or user equipment (UE) sides should be carefully reviewed.
Application examples
In the section, we provide some applications of pulseshaped OFDM and evaluate the link performance in the respective scenarios.
Uplink timing advance (TA)free access
Considering uplink transmission, due to radio propagation latency, timing misalignment occurs for the uplink signals at the base station, unless a closedloop TA adjustment is performed. For example, if the cell radius is 1732 m, TA misalignment could be in a range of 0∼13μs. 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, TAfree multiple access would be desirable for MTC uplink transmissions, as the timeconsuming 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 TAfree uplink transmission.
The scenario is illustrated in Fig. 17, showing two UEs transmitting to one BS with different timing offsets in a spatial division multiple access (SDMA) manner. From the SINR contour in Fig. 8, we observe that pulseshaped OFDM with long pulse (K=4) can support large timing offset, rendering it suitable for uplink TAfree (or relaxed TA) transmissions. A such designed pulseshaped OFDM system is particularly useful to be combined with nonorthogonal 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. 7 a with overlapping factor K=4. 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. 18 and 19 confirm the advantages of pulseshaped OFDM over CPOFDM, exhibiting substantial link performance gains of 3∼5 dB.
HST/V2X with high mobility
Highmobility scenarios become of great importance for future wireless communications. For example, highspeed train (HST) has already been considered in LTE as one important new use case for MBB service. For 5G NR systems, vehiculartoanything (V2X) service will enable safe driving and cooperative autonomous driving. The HST and V2X scenarios are illustrated in Fig. 20.
For the PHY configuration based on pulseshaped OFDM, we need to derive a reasonable product of TF in a pulseshaped design. The determination of such parameter highly depends on the propagation channels and service requirement. In this scenario, as highmobile objects are involved, the channels are often characterized as “doubly dispersive.” Based on the modeling report [29–31], the maximum path delay and Doppler shift are summarized in Fig. 21. From the channel modeling, we consider that the (T,F) lattice should be adjusted best between 60 and 75 kHz for an isotropic design with T F=1.25 for guaranteed performance in many scenarios, especially in extreme high velocity cases (for reference, LTE uses 15 kHz with T F=1.07, IEEE 802.11p uses 156 kHz with T F=1.25). We apply the pulse shape depicted in Fig. 7 b with overlapping factor K=4. Detailed simulation parameters are given in Tables 10 and 11 for link performance evaluation. From the BLER performance depicted in Figs. 22 and 23 (solid—ideal channel estimation, dash—least square (LS)based channel estimation), we see about 1∼3 dB performance gain by pulseshaped OFDM due to the welllocalized pulse shape design.
Conclusions
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 timefrequency 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 pulseshaped OFDM has been addressed with suitable parameterizations in pulse design. Practical issues like implementation and complexity are also analyzed for pulseshaped OFDM systems.
The flexibility of pulseshaped 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 timefrequency operational range, while the design of pulse shapes is for further refining the timefrequency localization according to the system (or service) requirements. We have shown that exploiting pulse shaping as an additional degree of freedom in OFDM system design can be used beneficially to improve the system’s robustness against time and frequency distortions and the spectrum coexistence capabilities, facilitating efficient fragmented spectrum access and machinetype communications.
Endnotes
^{1} Pulse shape and prototype filter are used interchangeably throughout the paper.
^{2} We assume the energy of both transceiver pulses g(t) and γ(t) are normalized to one.
^{3} For simplifying the analysis, causality of the system is firstly ignored, and thus CPOFDM is modeled as halfprefixed and halfsuffixed OFDM.
Appendix 1
Overview of different candidate waveform proposals
Pulseshaped OFDM generalizes several stateoftheart OFDMbased waveform candidates for future mobile systems. Here, we detail this relation as follows.

1.
CPOFDM is a special case of pulseshaped OFDM, where g(t) and γ(t) are rectangular pulse shapes with the overlapping factor K=1. Specifically, prototype filters g _{cpofdm}(t) and γ _{cpofdm}(t) are given by^{3}
$$\begin{array}{*{20}l} g_{\text{cpofdm}}(t)&=\begin{cases}\begin{array}{lcl} \frac{1}{\sqrt{T}}\qquad\quad & & \text{for}\ t\in\left[\frac{T}{2},\frac{T}{2}\right] \\ 0 & & \text{otherwise} \end{array}\end{cases} \\ \gamma_{\text{cpofdm}}(t)&=\begin{cases}\begin{array}{lcl} \frac{1}{\sqrt{TT_{\text{cp}}}} & & \text{for}\ t\in\left[\frac{TT_{\text{cp}}}{2},\frac{TT_{\text{cp}}}{2}\right] \\ 0 & & \text{otherwise} \end{array}\end{cases} \end{array} $$(18)and the spectral efficiency is proportional to 1/T F=(T−T _{cp})/T. Note that applying the transmit and receive pulses g _{cpofdm}(t) and γ _{cpofdm}(t) are equivalent to the “CP addition” and “CP removal” operations in CPOFDM technology. In an (AWGN) channel, due to the discrepancy of transmit and receive pulses, namely, g _{cpofdm}≠γ _{cpofdm}, there is a mismatching SNR loss following CauchySchwarz inequality. Using the common setting in LTE systems with 7 or 25% CP overhead, the mismatching SNR loss is about 0.3 dB for T F=1.07, while about 1 dB for T F=1.25.

2.
ZPOFDM is also a special case of pulseshaped OFDM, where the transmit pulse g _{zpofdm}(t) is a rectangular pulse of length T−T _{ zp } and the receive filter γ _{zpofdm}(t) is also rectangular shaped with length T. The overlapping factor is K=1.

3.
WindowedOFDM can be also considered as a special case within the pulseshaped OFDM framework, with overlapping factor 1<K<2 (usually K is slightly larger than 1). The pulse shape can be flexibly adjusted.

4.
Filtered multitone (FMT) is a pulseshaped OFDM system where the pulse shapes do not overlap in frequency domain [7]. The pulse shape and length are not specified. Different from FMT, pulseshaped OFDM allows for the overlapped filters in time domain or/and in frequency domain.

5.
DFTsOFDM is a special case of pulseshaped OFDM where a single carrier modulation is used (M=1). The pulse shaping is carried out with a circular convolution, which corresponds to periodically timevarying filters. The transmit pulse g(t) can be considered as the Dirichlet sinc function. The kth DFT spreading block is upsampled with N _{IDFT}/N _{DFTs,k} where N _{IDFT} and N _{DFTs,k} are the number of subcarriers of IDFT block and the size of DFT spreading block, respectively.

6.
Zerotail DFTspread OFDM (ZTDFTsOFDM) [32] is an extended single carrier modulation (M=1) based on “DFTsOFDM,” where the transmit pulse g(t) can be considered also as a N _{zp}expanded Dirichlet sinc function. Similar to DFTsOFDM, the upsampling ratio for kth DFT spreading block is N _{FFT}/N _{DFTs,k}+N _{zp}.
Appendix 2
EVM requirements for mobile communications
In [25], EVM indicates a measurement of the difference between the ideal and measured symbols after equalization. Following its definition, relationship between the required EVM and SIR (in linear scale) is given by
The limit of the EVM of each EUTRA carrier for different modulation schemes on Physical downlink shared channel (PDSCH) [25] along with the associated minimum SINR are summarized in the second and the third columns of Table 12, respectively.
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Acknowledgements
This work has been performed in the framework of the Horizon 2020 project FANTASTIC5G (ICT671660) receiving funds from the European Union. 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.
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Zhao, Z., Schellmann, M., Gong, X. et al. Pulse shaping design for OFDM systems. J Wireless Com Network 2017, 74 (2017). https://doi.org/10.1186/s1363801708498
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DOI: https://doi.org/10.1186/s1363801708498
Keywords
 OFDM
 Pulse shaping
 Multicarrier
 Filterbank