Physical Communication ( ) – Contents lists available at ScienceDirect Physical Communication journal homepage: www.elsevier.com/locate/phycom Full length article Resource block Filtered-OFDM for future spectrally agile and power efficient systems✩ Jialing Li ∗, Erdem Bala, Rui Yang InterDigital Communications Inc., 2 Huntington Quadrangle, 4th Floor, South Wing, Melville, NY 11747, United States a r t i c l e i n f o Article history: Received 13 March 2013 Received in revised form 16 August 2013 Accepted 4 October 2013 Available online xxxx Keywords: Multicarrier modulation OFDM Filtered-OFDM Filter bank multicarrier modulation Spectrally agile systems Cognitive radio a b s t r a c t Spectrum sharing is a commonparadigm in future communication systems and a spectrally agile baseband waveform with minimal out-of-band emissions is a critical component. In this paper, we propose a new multicarrier modulation technique, called resource block Filtered-OFDM (RB-F-OFDM) and present the transceiver design. This waveform can be used over channels with non-contiguous spectrum fragments and exhibits very low adjacent channel interference, which is required for cognitive radio systems with multi channel carrier aggregation capabilities. As such, regulatory based very stringent adjacent channel leakage ratio (ACLR) and adjacent channel selectivity (ACS) requirements can be met. We show that the transceiver complexity may be reduced by utilizing an efficient polyphase implementation that is commonly used in the filter bank multicarrier (FBMC) modulation. In addition, some efficient peak-to-average power ratio (PAPR) reduction techniques can be naturally applied. The new design is backwards compatible with legacy OFDM based systems. Simulation results to evaluate the performance, including measured bit error rate (BER) in multipath channels, are provided. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Wireless data traffic has grown significantly since 3G communication networks were widely deployed. The growth rate is expected to be even higher when the 4G technologies become more mature, more data hungry wireless services become available, and affordable smart phones loaded with data demanding applications start to deeply penetrate the handset market [1–3]. It is clear that exclusive usage of the current commercial spectrum will not be sufficient to support such a growth. The spectrum scarcity will become a main roadblock for the expansion of the overall wireless industry. To meet the future market demand, regulators and government agencies are actively ✩ Part of the content in this paper has been presented in our conference paper [42].∗ Corresponding author. Tel.: +1 917 294 8070. E-mail addresses: jialing.li.phd2@gmail.com (J. Li), erdem.bala@interdigital.com (E. Bala), rui.yang@interdigital.com (R. Yang). seeking for new spectra and, at the same time, strongly en- couraging the industry and academia to develop technolo- gies that support efficient spectrum sharing [4–6]. Spectrum sharing can be characterized by the level of coordination in different layers of the system, and can be achieved in multiple dimensions such as frequency, time, space, code, and power [7]. Someof the legacymulti-access technologies used in current systems that rely on accurate synchronization may not be directly applicable in future air interface designs. To fully utilize the potential benefits of spectrum sharing, a new set of enabling technologies for channel access, resource management and spectrally agile air interfaces need to be developed. Cognitive radio (CR) has been developed as such a technology to achieve these goals [8]. The general framework of the CR technologies has been developed over the past fifteen years [9,10]. Some of the CR concepts have been utilized in wireless communication standards that use TV white space as the transmission spectrum [11]. Still, a few fundamental technology areas are yet to be developed so that truly cognitive systems 1874-4907/$ – see front matter© 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.phycom.2013.10.003 2 J. Li et al. / Physical Communication ( ) – can be standardized and commercialized. One such area is selection of the baseband waveform that would enable opportunistic transmission on any available spectrum fragments. This requires dynamic spectrum sharing at the level of almost arbitrary spectral granularity. To achieve this goal, in this paper, we focus on waveforms with multicarrier modulation (MCM). MCM techniques enable simultaneous transmissions of a set of data over multiple narrow band subcarriers. It can be shown that MCM combined with an advanced modula- tion and coding scheme (MCS) can achieve very high spec- tral efficiency in frequency selective channels [12]. As one of the simplest MCM schemes, Orthogonal Frequency Divi- sionMultiplexing (OFDM) has beenwidely used inmodern wireless communication systems with tightly controlled synchronization. Orthogonal Frequency Division Multiple Access (OFDMA) based on OFDM has been successfully used in LTE systems. However, OFDMA may not be suit- able for CR systems due to its high out-of-band emission (OOBE) originated from large side lobes of the rectangular pulse shape. In CR systems, different transmitter–receiver links between primary and secondary users or among sec- ondary users may not be synchronized. In this case, high OOBE of OFDMA results in significant inter-carrier interfer- ence (ICI) at victim users, especially when nonlinear power amplifiers (PA) are used at transmitters. Thus, victim users may suffer from significant adjacent channel interference (ACI), if their receivers do not have proper ACI rejection capability. Some of the common methods considered to reduce ACI are lowering the transmit power and creating large enough guard bands. Clearly, the former limits the coverage area and the latter leads to low spectral utiliza- tion. Other methods without sacrificing the coverage and spectral utilizationwould require advanced signal process- ing technologies. Filter bank multicarrier (FBMC) modulation is a family of MCM techniques proposed as an alternative to OFDM to reduce ACI. OFDM-Offset QAM (OFDM-OQAM) is a popu- lar implementation of FBMC schemes [13–20]. In OFDM- OQAM, adjacent subcarriers of the signal overlap to achieve a high spectral efficiency. Different from OFDM, the real and imaginary parts of the QAM symbols are processed separately with 2× symbol rate. Besides OFDM-OQAM, Filtered Multitone (FMT) [21,22], Cosine Modulated Filter Bank (CMFB) [23,24], and Exponentially Modulated Filter Bank (EMFB) [25], are also some of the commonly consid- ered FBMC schemes in literature. In FBMC, a prototype fil- ter needs to be carefully designed to minimize or zero out inter-symbol interference (ISI) and ICI while keeping the side lobes small. Even though FBMC schemes have much better spectral containment, their implementation in prac- tical systems poses several challenges such as high com- plexity and latency. Also, since FBMC schemes do not use a cyclic prefix (CP), they require more complex equaliza- tion algorithms thanOFDM, especially in doubly dispersive channels [26]. A more recently proposed MCM scheme with reduced OOBE is the generalized frequency division multiplex- ing (GFDM) [27,28] in which the signal transmitted on separate frequency chunks are filtered individually and summed before transmission. GFDM has a block based structure; a block consists of several symbols and a CP is added to a block instead of to each symbol as in OFDM. Tail biting is used at the transmitter to reduce the required CP length and convert the filtering operation into circular con- volution to achieve a low-complexity transceiver. To fully utilize the advantages of OFDMwith CP in terms of robust- ness to multipath fading and implementation simplicity, several other OFDM-based techniques with lower OOBE have also been developed [29]. Among them, a simple and commonly used technique is to filter the OFDM signal, and is referred to as Filtered-OFDM. However, when available spectrum fragments are not contiguous, filtering becomes challenging since a separate filter needs to be dynamically designed and used for each fragment. Combining the benefits of FBMC and Filtered-OFDM, in this paper, we propose a new MCM transceiver technique. In this technique, the available spectrum fragments are divided into chunks of contiguous subcarriers, referred to as resource blocks (RB). And the signal transmitted on each RB is generated and filtered individually. The proposed MCM is called RB Filtered-OFDM (RB-F-OFDM). It has the advantage of being modular and scalable since the same transmit (or receive)module is used for all RBs. In addition, some of the peak-to-average power ratio (PAPR) reduction techniques [30–32] can be naturally adapted to the RB-F- OFDM design. The above features are very important for efficient implementation of the signal processing blocks in hardware and modem power consumption. We show that the complexity of the proposed transce- iver can be reduced by utilizing the polyphase implemen- tation that is also used in FMT. Moreover, orthogonality is still maintained between subcarriers, and RB-F-OFDM modulates each data symbol in the same time and same subcarrier as in OFDM. Therefore, one can also show that the RB-F-OFDM modulated signal can be directly demod- ulated by the legacy OFDM receiver, and that its receiver could also demodulate legacy OFDM signals. This feature makes the system based on RB-F-OFDM backwards com- patible with the legacy OFDM systems. It should be noted that although RB-F-OFDM is sim- ilar to GFDM in the sense that both schemes divide the available bandwidth into frequency chunks and filter each chunk individually, there are fundamental differences be- tween the two. One important difference is that orthog- onality between subchannels is not maintained in GFDM and a more complex receiver is required to cancel the created self-interference, and as such, it is not backwards compatible to legacy OFDM systems. In addition, due to the block based structure and tail biting, GFDM is suitable for packet based wireless systems with low mobility. Finally, tail biting causes a significant increase in the OOBE since the filtered signal is distorted. Padding zeros to end of a source block may be used to reduce the increased OOBE at the expense of a spectral efficiency loss. Note that our RB-F-OFDM idea was first presented in [42]. Much more details in the RB-F-OFDM design, es- pecially in the aspects of complexity reduction and PAPR reduction, are described in this paper. This paper is orga- nized as follows: In Section 2, we introduce the system model. Section 3presents a general introduction of the pro- posed RB-F-OFDM scheme with PAPR reduction. An effi- cient polyphase implementation of the RB-F-OFDM is also presented in this section. In Section 4,wepresent a detailed evaluation of the performance of RB-F-OFDM. Specifically, J. Li et al. / Physical Communication ( ) – 3 Fig. 1. Non-contiguous spectrum divided into RBs. the proposed scheme’s spectral containment, bit error rate (BER) and PAPR performance is evaluated and complexity and latency analysis is presented. Finally, Section 5 draws the conclusions. Notations. 0a×b signifies an all zero matrix of dimension a × b. max(a[n]) and E(a[n]) stand for the maximum and expectation of a sequence a[n], respectively. ⌊a⌋ and ⌈a⌉ stand for the maximum integer not greater than real value a and the minimum integer not smaller than real value a, respectively. lcm(a, b) stands for the least common multiple of two integers a and b. a mod b stands for the remainder of the Euclidean division of a by b. δ[n] is the Dirac delta function. 2. Systemmodel In an MCM system, usually a group of contiguous subcarriers is assumed to be the smallest dynamically available transmission band (the smallest granularity of frequency resource). All the assigned transmission bands or opportunistically detected frequency bands are consist- ing of one or multiple of such a group of contiguous sub- carriers. We call such a group of contiguous subcarriers as resource block (RB). Therefore, the entire spectrum that a system or device detects or utilizes could be considered uniformly divided into RBs. As depicted in Fig. 1, a non- contiguous spectrum may be divided into RBs of the same size. Assume each RB is consisting of D subcarriers. The RBs are label the 0th, 1st, 2nd, . . . , kth, . . . , (K − 1)th RBs, where K is the maximum number of available RBs. The maximum number of subcarriersM = KD. The normalized center frequency of the kth RB is fk = mk/L, where mk is the ‘‘center subcarrier index’’ of the kth RB and is not necessary an integer. L is a power of 2, and is related to the sampling rate of the transmit signal fs = L/T , where T is the symbol duration. Define the nth data symbol vector (anM × 1 vector) as S[n] = [S0[n] S1[n] · · · SK−1[n]]T (1) where the nth data symbol vector for the kth RB is a D× 1 vector Sk[n] = [Sk0[n] Sk1[n] · · · Sk,D−1[n]]T . (2) When the kth RB is available for transmission and has data loaded, Sk[n] ≠ 0D×1. Otherwise, Sk[n] = 0D×1. In this paper, we assume that through perfect feedback in control channel, both the transmitter and receiver have the knowledge of the available RB list assigned or detected. In practice, each RB can have its own properties, such as power loading, MCS, and hybrid automatic repeat request (HARQ). 3. Resource block Filtered-OFDM (RB-F-OFDM) In this section, we present our RB-F-OFDM design for spectrally agile and power efficient systems. 3.1. General description The design criteria of RB-F-OFDM are as follows: Firstly, it provides a spectral containment improvement of OFDM and Filtered-OFDM for contiguous and non-contiguous spectrums. Secondly, its transmit signal is compatible with the existing OFDM receiver in current standards and systems. And the RB-F-OFDM receiver could demodulate OFDM signals in current systems. Lastly, the RB-F-OFDM transceiver enables phase rotation based PAPR reduction techniques to improve the power efficiency of the system. The block diagrams of the proposed RB-F-OFDM transmit- ter and receiver are illustrated in Figs. 2 and 3, respectively. The main idea of the RB-F-OFDM is to generate the spectrally contained per-RB filtered CP-OFDM signal inde- pendently and obtained the transmit signal as their phase rotated sum. The phase rotation enables PAPR reduction. Fig. 2. RB-F-OFDM transmitter block diagram. 4 J. Li et al. / Physical Communication ( ) – Fig. 3. RB-F-OFDM receiver block diagram. Fig. 4. Per-RB CP-OFDM transmit module and receive module. To lower the computational complexity, the per-RB CP- OFDM signal is generated in low rate and upsampled. The per-RB transmit filter with good frequency selectivity not only eliminates the spectral images due to upsampling but also provides good spectral containment to the per-RB sig- nal. The detailed ideas of RB-F-OFDM are as follows: Firstly, in the RB-F-OFDM transmitter (‘‘RB-F-OFDM Tx’’) in Fig. 2, the data of each RB, Sk[n], k = 0, 1, . . . , K−1, is generated independently to form a per-RB modulated signal xk[m]. The transmit signal is the phase rotated sum of all the per-RB signals. It is given as x [m] = K−1 k=0 ejφkxk [m] . (3) In this way, the transceiver structure could enable phase rotation based PAPR reduction techniques, such as partial transmit sequences (PTS) [30,31] and selective mapping (SLM) [32]. The signal of each RB (resp., the signal of groups of RBs), instead of the signal of each subcarrier (resp. the signal of groups of subcarriers), are dealt with in SLM (resp. PTS). At the transmitter, by properly optimizing the phase rotations, φk, k = 0, 1, . . . , K−1, the PAPR of the transmit signal could be minimized. At the receiver, the complex conjugates of the phase rotations e−jφk , k = 0, 1, . . . , K−1 are applied. If PAPR reduction is disabled, ejφk = 1, k = 0, 1, . . . , K − 1. The details in PAPR reduction will be described in Section 3.3, and for simplicity, ignored in the following discussions. Secondly, module based direct implementations could be used at transmitter and receiver. At the transmitter, a per-RB transmit module (the red shaded block in Fig. 2) could be used to modulate Sk[n] (the nth data symbol vector in the kth RB) into xk[m] (the modulated signal of the kth RB) for all RBs. In the per-RB transmitmodule, Sk[n] is first modulated as a CP-OFDM signal of possibly lower sampling rate fs1 = N/T , where N is a power of 2, and N ≤ L. This is done as in the per-RB CP-OFDM transmit module in Fig. 4 (i.e., the ‘‘OFDM Tx’’ block in Fig. 2) by using an N-point inverse fast Fourier transform (IFFT), CP insertion, and a parallel-to-serial (P/S) conversion. And then, as in Fig. 2, the low-rate CP-OFDM signal is upsampled by Q = L/N , and filtered by a lowpass per-RB transmit filter pT [n]. After filtering, the RB modulation modulates the filtered signal into the frequency band of the kth RB according to the RB’s ‘‘central subcarrier index’’ mk, to form the per-RB signal xk [n] = m bk [m] pT [n−mQ ] ej 2πmkn L . (4) On the other hand, at the RB-F-OFDM receiver (‘‘RB-F- OFDM Rx’’), a per-RB Filtered-OFDM receive module (the red shaded block in Fig. 3) has ‘‘reverse’’ operations of the per-RB Filtered-OFDM transmit module. It could be used in a module based direct implementation to demodulate Sˇk[n] (the channel and filter corrupted nth data symbol vector in the kth RB) from y[n] (the received signal) for J. Li et al. / Physical Communication ( ) – 5 Fig. 5. PSD of upsampled signal, per-RB transmit filter, and per-RB signal. Fig. 6. RB-F-OFDM transmitter and receiver block diagram. all RBs. In the per-RB receive module, y[n] is demodulated from frequency band of the kth RB to baseband according to the RB’s ‘‘central subcarrier index’’ mk, to form an RB demodulated signal. Then, the RB demodulated signal goes through the per-RB receive filter pR[n] to reject out-of-RB signals. After filtering, the filtered signal is downsampled by Q . The resulting low-rate signal is demodulated as a CP-OFDM signal, to form the demodulated streams Sˇk[n]. This is done in the per-RB CP-OFDM receive module in Fig. 4 (i.e., the ‘‘OFDM Rx’’ block in Fig. 3) by using a serial- to-parallel (S/P) conversion, CP removal, and N-point fast Fourier transform (FFT). Thirdly, the per-RB transmit and receive filters are properly designed to achieve good spectral agility in both contiguous and non-contiguous spectrums. As in the power spectral density (PSD) plot in Fig. 5, a properly de- signed per-RB transmit filter should eliminate all spec- tral images due to the upsampling operation and mini- mize OOBE. The signal overlap between RBs does not cre- ate ICI due to bi-orthogonality between subcarriers in CP- OFDM systems in different RBs. Therefore, the minimum requirement of pT [n] to maintain orthogonality between RBs is that its passband bandwidth is D subcarriers, i.e., BWpass = D/T , each side of transition band has the band- width less than the bandwidth of N − D subcarriers, i.e., BWtrans = (N − D)/T , and the stopband has attenuation of at least 55 dB. In addition to this minimum requirement, we require that each per-RBmodulated signal only has sig- nal overlapping its adjacent RBs but not the RBs beyond its adjacent RBs, i.e., xk[m] only has overlap with xk−1[m] and xk+1[m] in the frequency domain. The signal leakage of a per-RB signal to its non-adjacent RBs is negligible. In this way, the RB-F-OFDM could achieve low OOBE for both contiguous and non-contiguous spectrums. The per- RB receive filter is also crucial in the sense that it should have good frequency selectivity to reject out-of-RB signals. Therefore, the minimum requirement of pR[n] is that it is a lowpass filter with passband bandwidth of D subcarriers, i.e., BWpass = D/T . For simplicity, in this paper, we assume that the per-RB transmit and receive filters are identical, i.e., pT [n] = pR[n] = p[n]. The filter design is not the in- terest of this paper. We simply assume to use equal ripple filters with passband attenuation of 0.75 dB, stopband at- tenuation of 58 dB, and stopband slop of 20. Lastly, the RB-F-OFDM transmitter essentially modu- lates each data symbol to the same subcarrier as it is mod- ulated in CP-OFDMusing an L-point IFFT, subject to a phase modulation. This phase modulation is due to the fact that CP insertion in RB-F-OFDM is before the RB modulation, while CP insertion in CP-OFDM is after the subcarrier mod- ulation. The phase modulation in RB-F-OFDM for the nth data symbols in the kth RB is e−j2πmkLcpn/L, where Lcp is the number of CP samples in sampling rate fs. A CP-OFDM receiver could be used to demodulate the signal. The chan- nel estimation (CHEST) in CP-OFDM could be used to es- timate the equivalent channel, including the transmit and receive filtering and multipath channel. And the 1-tap fre- quency domain equalizer (FDE) in CP-OFDM could be used to equalize the equivalent channel and perform phase de- modulation (i.e., applying a multiplier of ej2πmkLcpn/L for the nth data symbols in the kth RB). Alternatively, the phase demodulation may be applied to the source symbols prior 6 J. Li et al. / Physical Communication ( ) – to the RB-F-OFDM transmitter. Note that similarly, the RB- F-OFDM receiver with phase modulation, along with the CHEST and 1-tap FDE, as in Fig. 6, could demodulate OFDM signals in current systems as well. The realistic CHEST and more advanced equalizer designs are not of interest in this paper. 3.2. Efficient polyphase implementation In the above direct implementation of RB-F-OFDM, the computational complexity could be very high since it scales by the number of available RBs. Note that the RB-F-OFDM transmitter could be represented as the per- RB CP-OFDM signals going through a synthesis filter bank (SFB) of the FMT modulation. On the other hand, the RB- F-OFDM receiver could be represented as that received signal passes through an analysis filter bank (AFB) of the FMT demodulation followed by the per-RB CP-OFDM receive modules. Note that the orthogonality between the subchannels in the SFB-AFB is maintained due to the bi- orthogonality in CP-OFDM systems. In this section, we derive the efficient polyphase imple- mentation of the RB-F-OFDM transmitter and receiver to further reduce the complexity when the number of avail- able RBs is large. Note that in this section, it is assumed that PAPR reduction is disabled, i.e., ejφk = 1, k = 0, 1, . . . , K− 1. Without loss of generality, let mk = kD. And we further assume thatD, the number of subcarriers per RB, is a power of 2, so that C = L/D is a power of 2, and the RB modu- lation and demodulation multipliers become e±j2πkDn/L = e±j2πkn/C . In the next two sub-sections, we derived the effi- cient polyphase implementations of the SFB and AFB based on [33–35]. 3.2.1. Synthesis filter bank (SFB) The SFB is the component with inputs of b0[n], b1[n], . . . , bK−1[n], and outputs of x[n]. Substituting (4) into (3), the RB-F-OFDM transmit signal could be expressed as x [n] = K−1 k=0 m bk [m] p [n−mQ ] ej 2πkn C (5) which is exactly the expression of an FMT transmit signal, whereC ≥ Q ,C ≥ K , and lcm(Q , C) = C . After exchanging the order of summation, we obtain x [n] = m K−1 k=0 bk [m] ej 2πkn C p [n−mQ ] . (6) Express n as n = n C C + w, w = 0, 1, . . . , C − 1. (7) Substituting (7) into (6), we obtain x [n] = m uw [m] p [n−mQ ] (8) where the signal uw [m] = K−1 k=0 bk [m] ej 2πkw C , w = 0, 1, . . . , C − 1 (9) could be obtained through a C-point IFFT. Re-express n as n = ρQ + v, ρ = n Q , v = 0, 1, . . . ,Q − 1. (10) We define the vth polyphase component (with respect to Q ) of filter p[n] as pQ ,v [n] = p [nQ + v] , v = 0, 1, . . . ,Q − 1, n = 0, 1, . . . (11) and obtain Q such polyphase filters of p [n]. Therefore, substituting (10) into (8), we obtain x [n] = m uw [m] p [(ρ −m)Q + v] = uw [n] ∗ pQ ,v [n] (12) which is the convolution of the IFFT output sequence uw[n] and the vth polyphase filter pQ ,v[n], and the values of w and v depend on n, and are from (7) and (10), respectively. Since lcm(Q , C) = C , we obtain v = w mod Q , v = 0, 1, . . . ,Q − 1. (13) Therefore, each IFFT output sequence uw[n] corresponds to a unique polyphase filter pQ ,v[n]. The value of the pair (w, v) changes periodicallywith a period of C . Thus, amul- tiplexer (MUX) could be used to choose the samples from the polyphase filtered sequences, based on the sample in- dex. The resulting polyphase implementation of the SFB and the overall RB-F-OFDM transmitter block diagram are depicted in Fig. 7. 3.2.2. Analysis filter bank (AFB) Similar to the SFB, the polyphase implementation of the AFB could also be derived. Define the C × 1 output vector from the AFB in themth instance as a [m] = a0 [m] a1 [m] · · · aK−1 [m] aK [m] · · · aC−1 [m]T (14) where ak[m] is the input to the per-RB CP-OFDM receive module in the kth RB. Note that only the first K sequences are the useful outputs, and the last C − K sequences are discarded. The input to the CP-OFDM receivemodule in the kth RB could be expressed as ak [m] = y [n] e−j 2πkn C ∗ p [n] n=mQ = l y [l] e−j 2πkl C p [mQ − l] . (15) Express l as l = γ C + z, γ = l C , z = 0, 1, . . . , C − 1. (16) Substituting (16) into (15), we obtain ak [m] = γ C−1 z=0 y [γ C + z] e−j 2πkzC p [mQ − γ C − z] (17) J. Li et al. / Physical Communication ( ) – 7 Fig. 7. RB-F-OFDM transmitter block diagram using polyphase implementation of SFB. which is exactly the expression of an FMT received signal at the kth subchannel. After exchanging the order of sum- mation, we obtain ak [m] = C−1 z=0 γ y [γ C + z] p [mQ − γ C − z] e−j 2πkz C = C−1 z=0 uˆz [m] e−j 2πkz C (18) where the signal is uˆz [m] = γ y [γ C + z] p [mQ − γ C − z] = γ yz [γ ] p [mQ − γ C − z] (19) with the C-downsampled received signal yz [γ ] = y [γ C + z] , z = 0, 1, . . . , C − 1. (20) Therefore, the output vector from the AFB could be ob- tained through a C-point FFT, where each FFT input se- quence uˆz[m] comes from the convolution of yz[γ ] with a possible time varying filter. The C-downsampled received signal yz[γ ] could be obtained by using an S/P conversion from y[n]. Let mQ − z = ζC + c, ζ = mQ − z C , c = (mQ − z) mod C, m ∈ Z . (21) Also, define the cth polyphase component (with respect to C) of filter p[n] as pC,c [n] = p [nC + c] , c = 0, 1, . . . , C − 1, n = 0, 1, . . . (22) and obtain C such polyphase filters of p[n]. Therefore, sub- stituting (21) and (22) into (19), after some math manipu- lations, we obtain uˆz [m] = γ yz [γ ] p [(ζ − γ ) C + c] = yz [ζ ] ∗ pC,c [ζ ] (23) where the values of ζ and c depend onm, and are from (21). Therefore, the polyphase filter pC,c[ζ ] for the FFT input se- quence uˆz[m] is a time-varying filter that changes period- ically with a period of q = C/Q . For notation simplicity, define the periodic time-varying filter for the FFT input se- quence uˆz[m] as g(β)z [n] = pC,(βQ−z) mod C [n] , β = m mod q. (24) The resulting polyphase implementation of theAFB and the overall RB-F-OFDM transmitter block diagramare depicted in Fig. 8. Each time-varying filter could be implemented using the tap-delayed line, filtering in each branch, and a MUX to choose the output signal, as in Fig. 9. 3.3. Peak-to-average power ratio (PAPR) reduction PAPR is a measure of the envelope variation of a wave- form and is the peak amplitude of the waveform divided by the root-mean-square value of the waveform. In gen- eral, MCM transmit signals have large PAPR, which re- quires PAs to have a very large linear range. Otherwise, the nonlinearity leads to signal distortion, which causes larger OOBE and BER. A lot of PAPR reduction methods in litera- ture for MCM systems could be applied to the RB-F-OFDM. In particular, the distortionless phase rotation based PAPR reduction techniques, e.g., PTS [30,31] and SLM [32] (a spe- cial case of PTS when there is 1 RB/group), are natural choices for RB-F-OFDM, as depicted in Fig. 2. Note that if the efficient polyphase implementation of RB-F-OFDM trans- mitter is used, the sum signal of each RB group could be generated using one efficient implementation of the RB-F- OFDM transmitter. In PTS, K per-RB modulated signals are divided into ⌈K/ng⌉ groups, eachhaving atmostng RBs. The samephase rotation is used in each group, while the 0th group has phase rotation value of 1. The phase rotations of ⌈K/ng⌉ groups form one phase vector. The side information of ⌈K/ng⌉ − 1 phase rotations may be sent to the receiver. In this case, the resulting spectral efficiency loss depends on the number of groups, the nature of the phase vector (e.g., quantized or not), and the time interval of generating phase vectors. Alternatively, the phase rotation could be 8 J. Li et al. / Physical Communication ( ) – Fig. 8. RB-F-OFDM transmitter block diagram using polyphase implementation of AFB. Table 1 Modified LTE-like MCM systems. System parameters CP parameters Design parameters OFDM Symbol duration: T = 1/∆f Subcarrier spacing:∆f = 15 kHz Number of samples per symbol duration: L = 1024 Number of subcarrier per RB: D = 12 Maximum number of active RBs: K = 50 Maximum number of active subcarriers: M = KD One subframe: 2 slots/subframe One slot: 7 data symbols (7T ) and half additional symbol duration (0.5T ) used for CP, with CP duration Tcp1 = 80T/1024 (i.e., Lcp1 = 80L/1024) for the first symbol and Tcp2 = 72T/1024 (i.e., Lcp2 = 72L/1024) for the other symbols. Average CP length: Lcp = L/14 N/A Filtered-OFDM Transmit and receive filter: Root Raised Cosine (RRC) filter Filter length: Lf = 35 RB-F-OFDM Per-RB FFT size and equal ripple filter length: N = 128 & Lp = 53 Low-rate CP length: ⌊Lcp/Q ⌋ Fig. 9. Time-varying filter block diagram at AFB. considered part of the equivalent multipath channel and detected in CHEST. The phase rotation is recovered at the receiver, possibly in the 1-tap FDE. Therefore, the computational complexity is only added to the transmitter. In this case, enough pilot symbols in each RB group are required to enable reliable CHEST. If the pilot density in each RB group is not high enough, the CHEST may rely on interpolation between consecutive RB groups. In such a case, a pilot based phase estimator can be utilized for each RB group. Thus, spectral efficiency loss may occur due to additional pilots. In practical systems, each phase vector is used for a few consecutive ‘‘OFDM’’ symbols. This results in an inherent delay in optimizing the phase vector. The phase vector optimization has been studied exten- sively in literature [30,31,36–39]. We consider the sim- ple random PTS (RPTS) and quantized PTS (QPTS). In RPTS, besides the all one vector, nr phase vectors are randomly generated and each phase value is uniformly dis- tributed in [0, 2π). The output signal for each phase vec- tor is obtained, and the one achieves the lowest maximum PAPR value in a certain time interval is chosen. On the other hand, QPTS is similar to RPTS, except that each phase value is randomly chosen from a size-npts candidate set {0, 2π/npts, 4π/npts, . . . , 2π(npts − 1)/npts}. To reduce computational complexity in optimizing the phase vector, a PAPR threshold could be set to allow early termination of search when the PAPR threshold is achieved. 4. Performance evaluation In this section, the performance of RB-F-OFDM is evaluated and comparedwith OFDM and Filtered-OFDM in terms of complexity, latency, spectral containment, PAPR, andBER. The performance comparison ofMCMs is based on the modified LTE-like systems specified in Table 1, if not otherwise specified. Note that in Table 1, the CP lengths of OFDM, Filtered-OFDM, and RB-F-OFDM are the same. Therefore, they have the same spectral efficiency. 4.1. Computational complexity In this sub-section, computational complexity is evalu- ated in terms of number of real multiplications per ‘‘OFDM J. Li et al. / Physical Communication ( ) – 9 Table 2 Complexities of other MCMs. MCM Number of real multiplications per ‘‘OFDM symbol’’ (M complex data symbols) OFDM µOFDM = 2(L log2 L− 3L+ 4)+ 4M Filtered-OFDM µFiltered-OFDM,tx only = 2(L log2 L− 3L+ 4)+ 2LLf + 4M µFiltered-OFDM,tx&rx = 2(L log2 L− 3L+ 4)+ 2LLf + 2(L+ Lcp)Lf + 4M symbol’’ (i.e., at most M complex data symbols) for each MCM system. But multiplications with ±1 and ±j are not included since they are merely flips of sign and/or flips of real and imaginary parts. The normalized complexity of one MCM is the complexity of that MCM divided by that of OFDM. A pair of Lx-point FFT and IFFT (via Split Radix FFT [40]) with complexity µFFT&IFFT(Lx) = 2(Lx log2 Lx − 3Lx + 4) (25) for Mx (where Mx ≤ Lx) QAM symbols (subcarriers) is used as the component in the efficient implementations of all MCMs, where Lx is a power of 2. A 1-tap FDE with complexity µ1-tap FDE(Mx) = 4Mx (26) forMx QAM symbols (subcarriers) is assumed for allMCMs. Therefore, the complexities of OFDM and Filtered-OFDM (either using transmit filter only or using both transmit and receive filters), including the 1-tap FDE, are summarized in Table 2. In Filtered-OFDM in [29], it is assumed that the transmit filtering and adding CP could be combined such that the filtering is only performed once for the CP samples. 4.1.1. RB-F-OFDM complexity In the direct implementation, similar to Filtered-OFDM in [29], it is assumed that the transmit filtering and adding CP could be combined such that the filtering is only performed once for the CP samples; it is also assumed that the receive filtering, CP removal, and downsampling could be combined such that filtering is performed only for the needed samples to yield the downsampled signals. Note that the per-RB filter is a real filter. The complexities of different components for one RB are summarized in Table 3. In the polyphase implementation of RB-F-OFDM, the complexity of the SFB and AFB is due to the C-point IFFT, C- point FFT, and 4q sets of polyphase filters of p[n]. Assume that the number of active RBs is nRB, where nRB ≤ K . The complexities of different components are summarized in Table 4. The total complexities of different implementations of RB-F-OFDM are summarized in Table 5. 4.1.2. Evaluation and comparison with other MCMs Examples of modified LTE-like MCM systems using the parameters in Table 1 are considered here. But we change the number of subcarriers per RB from D = 12 in LTE to D = 2d, where d is 4, 5, or 6. Consequently, the maximum number of RBs in a 10 MHz bandwidth system should be reduced from K = 50 RBs to K = 640/D RBs. The normalized complexities of the OFDM and Filtered- OFDM are summarized in Table 6. And the normalized complexities of RB-F-OFDM with different RB sizes D and different number of RBs nRB are summarized in Table 7 for different implementations. From the comparison between these tables, we have the following observations: Firstly, in RB-F-OFDM, the direct implementation is more efficient when nRB is small, while the polyphase implementation is more efficient when nRB is large. Secondly, by increasing D (decreasing the frequency granularity), the complexity of RB-F-OFDM for a fixed total bandwidth could be highly reduced. Thirdly, when there are 16 subcarriers/RB, the complexity of RB-F-OFDMwith polyphase implementation is 17–20 times the complexity of OFDM, depending on nRB. It is much higher than the complexity of OFDM, and is up to twice that of Filtered-OFDM with both transmit and receive filters. Yet it has much better spectral containment than Filtered-OFDM in non-contiguous spectrum. Lastly, when there are 32 or 64 subcarriers/RB, the complexity of RB-F-OFDM could be reduced to less than that of Filtered- OFDM with both transmit and receive filters. 4.2. Latency The latency discussed in this sub-section is the inherent latency introduced in the MCM’s structure from the input of the QAM symbols to the output of the estimated QAM symbols. The latency due to the arithmetic operations (e.g., multiplication) is ignored since they depend on hardware or digital signal processing (DSP) implementation. Let the sample duration be Ts = T/L. Note that Filtered-OFDM andRB-F-OFDMessentially have the same latency asOFDM with CP. In OFDM, latency comes from the L : 1 P/S and Table 3 Complexities of different components in the direct implementation of RB-F-OFDM for one RB. N-point FFT/IFFT Tx/Rx filtering 1-tap FDE RB modulation/demodulation µFFT&IFFT(N) = 2(N log2 N − 3N + 4) µfiltering = 2(2N + ⌈Lcp/Q ⌉)Lp µ1-tap FDE(D) = 4D 8(L+ Lcp) Table 4 Complexities of different components in polyphase implementation of RB-F-OFDM. Per-RB processing Non per-RB processing N-point FFT/IFFT 1-tap FDE # RB SFB/AFB # Sample µFFT&IFFT(N) = 2(N log2 N−3N+4) µ1-tap FDE(D) = 4D nRB µSFB+AFB = µFFT&IFFT(C)+4qLp = 2(C log2 C−3C+4+2qLp) N + ⌊Lcp/Q ⌋ 10 J. Li et al. / Physical Communication ( ) – Table 5 Complexities of different implementations of RB-F-OFDM. MCM Number of real multiplications per ‘‘OFDM symbol’’ (M complex data symbols) Direct µRB-F-OFDM,direct (D,N, Lp, nRB) = nRB[2(N log2 N − 3N + 4)+ 2(2N + ⌈Lcp/Q ⌉)Lp + 8(L+ Lcp)+ 4D] Polyphase µRB-F-OFDM,polyphase(D,N, Lp, nRB) = 2nRB(N log2 N − 3N + 4+ 2D)+ 2(N + ⌈Lcp/Q ⌉)(C log2 C − 3C + 4+ 2qLp) Table 6 Normalized complexities of other MCMs. MCM OFDM Filtered-OFDM Parameters Tx only Tx & Rx Normalized complexity 1 5.24 9.78 1 : L S/P conversion pair, which is LTs = T , and from CP (of duration Tcp). Therefore, the total latency is τOFDM = τFiltered-OFDM = τRB-F-OFDM = T + Tcp. (27) The latencies of OFDM, Filtered-OFDM, and RB-F-OFDM for different CP durations used in 3GPP LTE are shown in Table 8. 4.3. Spectral containment In this section, we evaluate the spectral containment of RB-F-OFDM and compare it with OFDM and Filtered- OFDM. The spectral containment could be evaluated in terms of the PSD, out-of-band (OOB) power versus the OOB frequency (fout in MHz), as well as the required minimum guard band to meet certain OOB power requirement, in systems with 5 MHz bandwidth (i.e., using 300 subcar- riers). We consider the required OOB power to be either −50 dBr or −55 dBr. The OOB power is measured with 20 kHz spacing, and each OOB power of a particular OOB frequency is the average measured power of a 100 kHz bandwidth centered at that OOB frequency. 4.3.1. RB-F-OFDM with different parameters In this sub-section, we investigate how the RB-F-OFDM transmit signal’s spectral containment depends on the per- RB FFT size N and equal ripple filter length Lp. The investi- gation allows us to find tradeoffs between the guard band size (to meet OOB power requirement) and other perfor- mance measures that depend on the per-RB FFT size N and filter length Lp. In Fig. 10, the OOB power (in dBr) is plotted versus the OOB frequency (fout in MHz) for RB-F-OFDM transmit sig- nals with different per-RB FFT sizes and filter lengths. And Table 8 OFDM, Filtered-OFDM, RB-F-OFDM latencies. Tcp 0 5.2 µs 4.7 µs 16.7 µs τOFDM = τFiltered-OFDM = τRB-F-OFDM 0.0667ms 0.0719ms 0.0714ms 0.0834 ms the minimum guard bands for the RB-F-OFDM transmit signals tomeet the OOB power thresholds are summarized in Table 9. We have the following observations: Firstly, the stop- band attenuation of the transmit signal depends mainly on the per-RB FFT size. Due to the equal ripple filter de- sign with fixed passband and stopband deviations require- ments, the stopband attenuation of all equal ripple filters of different lengths are very close. Therefore, as in Fig. 10, with the same per-RB FFT size but different filter lengths, the stopband attenuations of RB-F-OFDM transmit signals are very close. When the per-RB FFT size is larger, the up- sampling factor Q is smaller, and thus, in each RB, the spectral images due to upsampling are further away from the desired passband and have larger signal attenuation after filtering. Therefore, the stopband attenuation of the transmit signal is larger. When the per-RB FFT size is large enough, e.g., 512, andwhen a properly designed per-RB fil- ter, e.g., the 37-tap or 53-tap equal ripple filter, is used, all spectral images could be almost eliminated and the OOB power of the transmit signal could be as small as −90 dB. Secondly, the transmit signal’s transition band behavior depends mainly on the equal ripple filter length. Shorter filter has slower roll off and larger transition band, while longer filter has faster roll off and smaller transition band, as long as all spectral images fall in the stopband of the per- RB transmit filter. Therefore, in the design of RB-F-OFDMsystems, the per- RB FFT size and filter length could be jointly chosen to meet the OOB power requirement, and to provide tradeoffs between OOBE and other performance measures. For example, as in Table 9, if the design requirement of spectral containment is to have guard bands no greater than Table 7 Normalized complexities of RB-F-OFDM. J. Li et al. / Physical Communication ( ) – 11 Fig. 10. OOB power of RB-F-OFDM transmit signals versus OOB frequency. Table 9 Minimum guard bands for 5 MHz RB-F-OFDM systems. Per-RB FFT size 32 64 128 128 256 512 512 1024 Filter length 85 69 53 69 53 37 53 37 Guard band for−50 dB 370 kHz 328 kHz 416 kHz 309 kHz 394 kHz 565 kHz 400 kHz 567 kHz Guard band for−55 dB >1600 kHz 957 kHz 466 kHz 356 kHz 439 kHz 634 kHz 443 kHz 637 kHz 500 kHz for OOB power threshold −55 dB, RB-F-OFDM systems with (N, Lp) being (128, 53), (128, 69), (256, 53), and (512, 53) all satisfy the requirement. Complexity wise, the onewith (128, 53) achieves the lowest complexity, and may be chosen in the design. 4.3.2. Comparison with other MCMs In Figs. 11 and 12, respectively, the OOB power is plot- ted versus the OOB frequency (fout in MHz) for different MCM systems without or with a nonlinear PA that is mod- eled using a 5th order polynomial. The minimum guard bands for differentMCM systemswithout nonlinear PA are as in Table 10. We have the following observations: Firstly, OFDM is most robust to nonlinearity of PA, because its OOBE after the nonlinear PA does not highly increase, com- pared to its OOBE without the nonlinear PA. But due to its large OOBE, it always has the worst spectral containment. Secondly, even though Filtered-OFDM with a transmit fil- ter could largely reduce the OOBE, compared to OFDM, this is only true if PA nonlinearity is not taken into account. Filtered-OFDM is quite sensitive to PA nonlinearity. When there is a nonlinear PA, Filtered-OFDM’s OOBE is much larger, and increasing the transmit filter length does not help to reduce OOBE. Lastly, compared to Filtered-OFDM, RB-F-OFDM is more robust to PA nonlinearity. It only has around 10 dB OOBE increase when a nonlinear PA is ap- plied. This is in part due to the fact that PA nonlinearity largely increases the power in transition band and stop- band of a filtered signal but not the power in passband. As in Fig. 5, the filtered spectral images of the passband are the dominant interference in the stopband of a per-RB sig- nal in RB-F-OFDM. Even though PA nonlinearity increases the power in transition bands and stopbands between the spectral images, the stopband of the entire RB-F-OFDM transmit signal would not have large power increase. We conclude that RB-F-OFDM still provides consider- able small OOBE when a nonlinear PA is applied. It is much better thanOFDMand Filtered-OFDM. Certain PA lineariza- tion, e.g., digital pre-distortion (DPD), could be used to fur- ther reduce the OOBE increase due to nonlinear PA. In Fig. 13, the PSDs of OFDM, Filtered-OFDM, and RB- F-OFDM are compared in a system with 5 MHz band- width when the available spectrum is fragmented and non-contiguous, and it illustrates the advantage of RB-F- OFDM over Filtered-OFDM in this scenario. In this figure, two available spectrum fragments are separated by an un- available frequency band that is occupied by others. When Filtered-OFDM is used with a filter over the entire fre- quency band, we can see that the OOBE to the unavail- able frequency band between the available fragments is the same as that of OFDM.However, RB-F-OFDMcanutilize the available fragments with less OOBE to the unavailable frequency band in the middle. RB-F-OFDM needs a much 12 J. Li et al. / Physical Communication ( ) – Fig. 11. OOB power of MCM transmit signals versus OOB frequency without nonlinear PA. Fig. 12. OOB power of MCM transmit signals versus OOB frequency with nonlinear PA. Table 10 Minimum guard bands for 5 MHz MCM systems without nonlinear PA. MCM OFDM Filtered-OFDM RB-F-OFDM Parameters N/A 18-tap RRC 35-tap RRC 52-tap RRC 128-point per-RB FFT and 53-tap equal ripple filters Guard band for−50 dB >1600 kHz 465 kHz 279 kHz 357 kHz 416 kHz Guard band for−55 dB 1142 kHz 608 kHz 435 kHz 466 kHz J. Li et al. / Physical Communication ( ) – 13 Fig. 13. PSDs of MCM schemes with fragmented and non-contiguous spectrum. smaller guard band to reach the OOBE threshold of−50 dB or−55 dB. 4.4. PAPR performance In this section, we compare the PAPR performance of RB-F-OFDM with OFDM and Filtered-OFDM, and evaluate the PAPR performance of RB-F-OFDMafter PAPR reduction. The PAPR performance is evaluated in terms of the comple- mentary cumulative distribution function (CCDF) of PAPR for MCM signals. All results are generated for 10 MHz sys- tems with 600 subcarriers and 10,000 subframes. In general, the PAPR observation interval can be very long. But in practice, it is usually chosen as the effective symbol duration. In this paper, the PAPR measure for the nth ‘‘OFDM’’ symbol is defined as PAPRn = max |x [l]| 2 E |x [l]|2 , l ∈ n L+ Lcp , n L+ Lcp + 1, . . . , (n+ 1) L+ Lcp− 1 . (28) Note that for the filtering based MCM, e.g., Filtered-OFDM and RB-F-OFDM, due to the overlapping of symbols in time, the observed PAPR values in the beginning and end of a continuous transmission are discarded for statistics purpose. 4.4.1. Without PAPR reduction In Fig. 14, we observe that the CCDF curves of PAPR of theOFDM, Filtered-OFDM, andRB-F-OFDMsignals are very close when no PAPR reduction techniques are applied. 4.4.2. With PAPR reduction In this sub-section, we evaluate the performance of RB- F-OFDM with RPTS and QPTS using different numbers of randomly generated phase vectors nr (e.g., 8, 16, 32, 64) and different RB group sizes ng . For QPTS, npts = 8 quan- tized phase values with the candidate set {0, π/4, π/2, . . . , 7π/4}, or npts = 4 quantized phase values with the candidate set {0, π/2, π, 3π/2} are used. ‘‘RPTS (nr , ng )’’ means RPTS with nr random phase vectors and ng RB/ group; ‘‘QPTS (nr , ng , npts)’’ means QPTS with nr random phase vectors, ng RB/group, and npts quantized phase val- ues. The CCDF curves of PAPR for RB-F-OFDM, without PAPR reduction (black solid line), with RPTS (colored solid lines) and QPTS (dotted lines for npts = 4) are plotted in Figs. 15 and 16, for nr = 8 and nr = 64, respectively. The PAPR values for RPTS and QPTS at CCDF of 10−4 are summarized in Tables 11–13, respectively, for different combinations of parameters. We have the following observations for both RPTS and QPTS: Firstly, for the same number of randomly generated phase vectors, smaller RB group size (i.e., more RB groups) leads to lower PAPR. The PAPR performances of 1, 2, or 5 RB/group are very close, and they are relatively close to that of 10 or 13 RB/group. This suggests that large RB group size and less RB groups could be used to achieve relatively good PAPR performance while keeping minimum amount of side information. Secondly, for the same RB group size, larger number of random phase vectors inefficiently leads to lower PAPR. Thus, to lower the computational complexity in PAPR reduction, the number of random phase vectors could be kept to a relatively small number, e.g., 8 or 16. Thirdly, tradeoffs could be made to achieve same PAPR performance while minimizing the complexity and amount of side information. Lastly, with theminimum complexity increase and amount of side information in our investigation, i.e., by using 8 random phase vectors and 25 RB/group, the PAPR values at CCDF of 10−4 are 10.85 dB for RPTS (8, 25), 10.83 dB for QPTS (8, 25, 8), and 10.94 dB for QPTS (8, 25, 4), and have at least 1 dB gain, compared to the case without PAPR reduction (see Fig. 14). 14 J. Li et al. / Physical Communication ( ) – Fig. 14. CCDF curves of PAPR of different MCM signals. Fig. 15. CCDF of PAPR of RB-F-OFDM signals using RPTS/QPTS with nr = 8. Comparing RPTS and QPTS, when the RB group size is small (i.e., the number of RB groups are large), or when the number of random phase vectors is small, their PAPR per- formances are similar. Onlywhen the number of RB groups is small and when the number of random phase vectors is large enough, e.g., (nr , 25, npts) where nr = 16, 32, or 64 and npts = 8 or 4, and (nr , 17, 4) where nr = 32 or 64, QPTS has higher PAPR than RPTS. In particular, in this case, larger quantized phase candidate set provides better PAPR reduction in QPTS. 4.5. Bit error rate (BER) performance In this section, we evaluate the BER performance of RB- F-OFDM and compare it with OFDM and Filtered-OFDM. The BER is evaluated versus the Eb/N0 (the energy per bit to J. Li et al. / Physical Communication ( ) – 15 Fig. 16. CCDF of PAPR of RB-F-OFDM signals using RPTS/QPTS with nr = 64. Table 11 PAPR of different RB-F-OFDM signals with RPTS (nr , ng ) at CCDF of 10−4 . nr ng 1 2 5 10 13 17 25 8 10.05 10.09 10.07 10.26 10.33 10.53 10.85 16 9.80 9.77 9.82 9.92 10.11 10.22 10.58 32 9.57 9.60 9.71 9.74 9.87 10.04 10.49 64 9.45 9.47 9.53 9.65 9.74 9.95 10.41 Table 12 PAPR of different RB-F-OFDM signals with QPTS (nr , ng , 8) at CCDF of 10−4 . nr ng 1 2 5 10 13 17 25 8 10.10 10.07 10.08 10.19 10.31 10.42 10.83 16 9.79 9.84 9.82 9.95 10.14 10.23 10.76 32 9.58 9.61 9.69 9.79 9.86 10.09 10.71 64 9.48 9.47 9.51 9.65 9.73 10.02 10.68 Table 13 PAPR of different RB-F-OFDM signals with QPTS (nr , ng , 4) at CCDF of 10−4 . nr ng 1 2 5 10 13 17 25 8 10.03 10.05 10.04 10.14 10.32 10.45 10.94 16 9.80 9.79 9.87 9.98 10.11 10.38 10.87 32 9.59 9.63 9.66 9.85 9.87 10.26 10.96 64 9.45 9.45 9.51 9.63 9.78 10.19 10.94 noise power spectral density ratio) in dB. The moderately selective ‘‘EVA5’’ channel model (Extended Vehicular A channel model with maximum Doppler frequency 5 Hz) from [41] is used. It is assumed that perfect channel knowledge is known at the receiver. The 1-tap minimum mean square error (MMSE) FDE is used at the receiver of all MCMs. If not otherwise specified, only the results using 16QAM modulation is presented here due to page limit. Each BER point is obtained by averaging over 10,000 subframes. We investigate two cases. In the case without ACI, there is only one transmitter and one receiver, and no interferers in adjacent frequency bands. On the other hand, in the case with ACI, except for the desired transmitter and receiver, there is an interfering transmitter transmitting in an (non-overlapping) adjacent frequency band of the same bandwidth as the desired system. The two transmitters are perfectly synchronized in time. The edge subcarriers of the two systems are as close as spacing∆F being 1.5 subcarrier spacing away from each other. The ACI power ∆P in dB is defined as the difference between the average power of the interfering system’s passband and that of the desired system’s passband, and ∆P is a parameter in simulations. The two systems are as depicted in Fig. 17. At the desired receiver, the ACI is treated as noise, and the ACI power is not measured or accounted for in the MMSE equalization. Except for Section 4.5.1, 128-point per-RB FFT and 53- tap equal ripple per-RB filter is used in RB-F-OFDM. 4.5.1. RB-F-OFDM with different parameters In this section, we present the BER performance of RB- F-OFDMwith various per-RB FFT sizes and filter lengths in the EVA5 channel. ‘‘RB-F-OFDM (N, Lp,tx, Lp,rx)’’ means RB- F-OFDMwithN-point per-RB FFT, Lp,tx-tap per-RB transmit filter, and Lp,rx-tap per-RB receive filter.When Lp,rx is ‘‘N/A’’, it means that an L-point CP-OFDM receiver is used instead of the RB-F-OFDM receiver and there is no receive filtering. In Fig. 18, the raw BER performance of RB-F-OFDM systemswith 10MHzbandwidth (i.e.,K = 50RBs andM = 600 subcarriers) and without ACI is shown. Almost all RB- F-OFDM curves have some BER degradation, compared to 16 J. Li et al. / Physical Communication ( ) – Fig. 17. Simulation scenario with ACI. OFDM, since their filters contribute to the selectivity of the equivalent channel. The BER performance mainly depends on the filter length. RB-F-OFDMwith shorter overall filters provide better BER performances, while the ones with longer overall filters (compared to CP length) have larger degradation inmedium tohigh signal-to-noise-ratio (SNR). Note that when the per-RB FFT size is large enough, e.g., 128 or above, for the same filter length, larger per-RB FFT does not show advantage in BER performance. In Fig. 19, the raw BER performance of RB-F-OFDM systems with 5 MHz bandwidth (i.e., K = 25 RBs and M = 300 subcarriers) and with ACI ∆P = 0 dB is shown. Note that RB-F-OFDMwith 32-point per-RB FFT and 85-tap transmit filter and a CP-OFDM receiver does not show any performance gain over OFDMwhen ACI is present. Receive filtering is needed to reject ACI, since the CP removal breaks the continuity of the signal, resulting inmuch larger OOBE of the interfering signal. This phenomenon also holds for Filtered-OFDM. For RB-F-OFDM systems with both transmit and receive filters, their performance depend on the per-RB FFT sizes as well as the filter lengths. When the ACI power is 0 dB, only RB-F-OFDMwith 53-tap or 37- tap filters show performance gains over OFDM since the performance is more sensitive to the equivalent channel selectivity than ACI. Note that they also have quite close performance in the [10−2 10−1] raw BER range, which is the most meaningful raw BER range to achieve the highest throughput when channel coding is used in practical systems. Taking into account the spectral containment and complexity perspectives, the one with 128-point per-RB FFT and 53-tap transmit and receive filters is chosen in the design and evaluated in the next sub-sections. 4.5.2. Without adjacent channel interference (ACI) In Fig. 20, the rawBERperformance of theMCMsystems with 5MHz bandwidth (i.e., K = 25 RBs andM = 300 sub- carriers) is shown in EVA5 channel without ACI. All MCMs perform identically at low to medium SNR, including the meaningful [10−2 10−1] raw BER range. At high SNR, OFDM performs the best as long as no ACI is present. Filtered- OFDM suffers slight BER degradation at high SNR, because its 35-tap transmit and receive filters contribute to the se- lectivity of the equivalent channel. As for RB-F-OFDM, it has largest BER degradation at high SNR, because the over- all equivalent channel is much longer than CP. 4.5.3. With ACI In Figs. 21 and 22, respectively, the raw BER perfor- mance of the MCM systems with 5 MHz bandwidth (i.e., K = 25 RBs and M = 300 subcarriers) is shown in EVA5 channel with ACI powers 10 and 30 dB, respectively, for 16QAM modulation. In Fig. 23, the raw BER performance of the MCM systems is shown in EVA5 channel with ACI power 30 dB for 64QAMmodulation. We have the following observations: Firstly, OFDM is most vulnerable to ACI. When ACI other than channel se- lectivity becomes a more dominant factor for performance degradation, Filtered-OFDM and RB-F-OFDM all outper- form OFDM, because they could mitigate certain levels of ACI. Secondly, the relatively short filters enable Filtered- OFDM to outperform RB-F-OFDM in the moderately se- lective EVA5 channel, when channel selectivity is a more dominant factor for performance degradation. However, Fig. 18. BER performance of RB-F-OFDM systems in EVA5 channel without ACI. J. Li et al. / Physical Communication ( ) – 17 Fig. 19. BER performance of RB-F-OFDM systems in EVA5 channel with ACI 0 dB. Fig. 20. BER performance of MCM systems in EVA5 channel without ACI for 16QAMmodulation. Filtered-OFDM exhibits to be relatively more sensitive to ACI power increase, compared to RB-F-OFDM, because the RB-F-OFDM receiver has better ACI rejection capability. In Fig. 11, compared to RB-F-OFDM, Filtered-OFDM (with 35- tap transmit filter) transmit signal has smaller transition band. But this does not imply that Filtered-OFDM have better BER performance under ACI. At the receiver, the per-RB receive filter of RB-F-OFDMhasmuch smaller pass- band and transition band, and could reject ACImuch better than the loose 35-tap receive filter used in Filtered-OFDM. Therefore, the RB-F-OFDM receiver structure, which filters each RB individually with amuch narrower per-RB filter, is more capable of ACI rejection.WhenACI power is relatively large, e.g., 30 dB, RB-F-OFDM outperforms Filtered-OFDM. The difference is more profound for higher order symbol modulation, e.g., 64QAM. In fact, preliminary results show that by adding very small guard bands, RB-F-OFDM could mitigate ACI much better and provide much better BER performance. This part of investigation, along with the investigation of the performance in non-contiguous spectrum, will be left for future work. 5. Conclusion and future work In this paper, we have presented a new MCM scheme called RB-F-OFDM, and presented an efficient polyphase implementation of the transceiver. The performance of the proposed MCM scheme has been compared with OFDM and Filtered-OFDM in the moderately selective EVA5 channel under various ACI conditions. Without ACI, OFDM offers the best performance while the performance of Filtered-OFDM and RB-F-OFDM degrades at high SNR compared to OFDM due to the fact that the CP length is not enough to remove the ISI completely when filters are used. With ACI, the RB-F-OFDM receiver structure, which 18 J. Li et al. / Physical Communication ( ) – Fig. 21. BER performance of MCM systems in EVA5 channel with ACI 10 dB for 16QAMmodulation. Fig. 22. BER performance of MCM systems in EVA5 channel with ACI 30 dB for 16QAMmodulation. Fig. 23. BER performance of MCM systems in EVA5 channel with ACI 30 dB for 64QAM. J. Li et al. / Physical Communication ( ) – 19 filters each RB individually with a much narrower per- RB filter, is more capable of ACI rejection. Therefore, the performance of the proposed RB-F-OFDM is least sensitive to ACI power increase, and as the ACI power increases, RB- F-OFDM eventually outperforms Filtered-OFDM. When compared to Filtered-OFDM, the proposed RB- F-OFDM enables utilization of non-contiguous spectrum due to very low achievable OOBE between the utilized channels, and more effective ACI rejection at the receiver. Therefore, RB-F-OFDM is more spectrally agile. More- over, the RB-F-OFDM transmit signal is backwards com- patible with legacy OFDM receivers and the RB-F-OFDM receiver could also demodulate legacyOFDMsignals. In ad- dition, we have shown that the PAPR of the proposedMCM scheme can significantly be reduced by utilizing SLM and PTS based techniques. 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Jialing Li received her Bachelors of Engineering in Electronic Engineering from University of Sci- ence and Technology of China in 2005, Masters of Science in Electrical Engineering from Poly- technicUniversity in 2008, andPh.D. in Electrical Engineering from Polytechnic Institute of New York University (formerly Polytechnic Univer- sity) in 2011. She is currently a Senior Engineer in InterDigital Communications Inc. Her current research interests are mainly with regard to Coordinated Multipoint (CoMP) Transmission/ Reception, interference mitigation in heterogeneous networks, advanced waveformdesign for spectral agile and power efficient systems, and phys- ical layer design of future spectrally efficient systems. Erdem Bala got his B.Sc. andM.Sc. from Bogazici Univeristy, Istanbul, Turkey and his Ph.D. from the University of Delaware; all in Electrical En- gineering. He has been with InterDigital Com- munications, NY as a research engineer since 2007. He has worked on several projects includ- ing the standardization of 3GPP LTE and LTE-A, advanced relaying schemes, dynamic spectrum access and more recently next generation air in- terface systems. His previous work experience includes positions as software design engineer at the Turkey and UK R&D labs of Nortel Networks, and internship at Mit- subishi Research Labs, MA. Rui Yang received his M.S. and Ph.D. degrees in Electrical Engineering from the University of Maryland in 1987 and 1992. He has 14 years experience in research and development of wireless communication systems. Since joined InterDigital Communications in 2000, he has led several product development and research projects. His interests include digital signal pro- cessing and air interface design for wireless de- vices. He holds more than 10 patents in those areas. He is currently a senior engineering man- ager at InterDigital Innovation Lab, leading a project on baseband and RF waveforms for future wireless communication systems. Resource block Filtered-OFDM for future spectrally agile and power efficient systems Introduction System model Resource block Filtered-OFDM (RB-F-OFDM) General description Efficient polyphase implementation Synthesis filter bank (SFB) Analysis filter bank (AFB) Peak-to-average power ratio (PAPR) reduction Performance evaluation Computational complexity RB-F-OFDM complexity Evaluation and comparison with other MCMs Latency Spectral containment RB-F-OFDM with different parameters Comparison with other MCMs PAPR performance Without PAPR reduction With PAPR reduction Bit error rate (BER) performance RB-F-OFDM with different parameters Without adjacent channel interference (ACI) With ACI Conclusion and future work References