A Comparison of CP-OFDM, PCC-OFDM and UFMC for 5G Uplink Communications

Received October 18, 2019, accepted October 21, 2019, date of publication October 28, 2019, date of current version November 8, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2949792

GAYATHRI KONGARA 1, CUIWEI HE 1, LEI YANG 2, AND JEAN ARMSTRONG 1, (Fellow, IEEE) 1Department of Electrical and Computer Systems Engineering, Monash University, Melbourne, VIC 3800, Australia 2Faculty of Information Technology, Monash University, Melbourne, VIC 3800, Australia Corresponding author: Jean Armstrong ([email protected])

ABSTRACT Polynomial-cancellation-coded orthogonal frequency division multiplexing (PCC-OFDM) is a form of OFDM that has waveforms which are very well localized in both the time and frequency domains and so it is ideally suited for use in the 5G network. This paper analyzes the performance of PCC-OFDM in the uplink of a multiuser system using orthogonal frequency division multiple access (OFDMA) and compares it with conventional cyclic prefix OFDM (CP-OFDM), and universal filtered multicarrier (UFMC). PCC- OFDM is shown to be very much less sensitive to time and frequency offsets than either CP-OFDM or UFMC. For a given constellation size, PCC-OFDM in additive white Gaussian noise (AWGN) requires 3 dB lower signal-to-noise ratio (SNR) for a given bit-error-rate, and the SNR advantage of PCC-OFDM increases rapidly when there are timing and/or frequency offsets. For PCC-OFDM no frequency guard-band is required between different OFDMA users. PCC-OFDM is completely compatible with CP-OFDM and adds negligible complexity and latency, because it uses a simple mapping of data onto pairs of subcarriers at the transmitter, and a simple weighting-and-adding of pairs of subcarriers at the receiver. The weighting- and-adding step, which has been omitted in some of the literature, is shown to contribute substantially to the SNR advantage of PCC-OFDM. A disadvantage of PCC-OFDM (without overlapping) is the potential reduction in spectral efficiency because subcarriers are modulated in pairs, but this reduction is more than regained because no guard band is required and because, for a given channel, larger constellations can be used.

INDEX TERMS 5G, multiuser, uplink, OFDM, PCC-OFDM, UFMC, timing offset, frequency offset.

I. INTRODUCTION of many recent wireless communication systems. The many The design of the 5G mobile network presents many new advantages of CP-OFDM include robustness in the pres- challenges not faced by earlier generations of mobile access ence of multipath transmission, relative insensitivity to tim- technology. This is because of the wide range of diverse ing offsets, compatibility with multiple-input multiple-output services 5G will have to support [1]. There is the predicted (MIMO) systems and the ability to support multiple access exponential increase in demand for very high bandwidth in the form of orthogonal frequency division multiple access connection resulting from video streaming and other high (OFDMA). The well-known disadvantages of OFDM include data rate applications. At the same time, the emergence of high out-of-band (OOB) power, sensitivity to frequency off- the internet of things (IOT) will produce a very large number set, and high peak-to-average power ratio (PAPR) [12], [13]. of low speed users. Reconciling these very different types of In the current generation of mobile systems, workable communication is very challenging and is currently the topic solutions have been found which overcome the disadvantages of extensive research [2]–[11]. of CP-OFDM. The OOB power can be reduced by leaving some band-edge subcarriers unused and by windowing within A. CHALLENGES IN USING OFDM IN 5G the cyclic prefix. Sophisticated synchronization algorithms Orthogonal frequency division multiplexing (OFDM) in the overcome the problem of frequency sensitivity. A multitude form of CP-OFDM (cyclic prefix OFDM) has been the basis of solutions to the PAPR problem have been described in the literature [14]. In the LTE uplink a modified form of The associate editor coordinating the review of this manuscript and OFDM called DFT-spread OFDM has been developed to approving it for publication was Ding Xu . enhance capacity and cell-edge user throughput performance.

This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ 157574 VOLUME 7, 2019 G. Kongara et al.: Comparison of CP-OFDM, PCC-OFDM and UFMC for 5G Uplink Communications

The advantages and challenges of using DFT-spread OFDM PCC-OFDM reduced as research in MIMO systems exploded in 5G are discussed in [15]. But simple adaptions to these in response to the landmark papers on multiple antenna sys- techniques are not adequate for 5G. This has inspired exten- tems [34], [35]. However, as we will show the basic form of sive recent research into different waveforms [2]–[11], [15]. PCC-OFDM is ideally suited to the uplink in mMTC and has OFDMA [1], [16] using CP-OFDM may work well in the potential to be a key technology for 5G. the downlink and meet the demand for higher bandwidth, A few papers have recently been published on the appli- the requirements of massive MIMO, and the use of higher cation of PCC-OFDM to 5G. Some theoretical analysis and frequency bands. However, it is not suitable for the uplink simulation results for PCC-OFDM were presented in [36] of the massive machine-type communication (mMTC) which and [37], but these did not include the receiver weighting will result from the IOT. Wunder et al. [17] conclude that the and adding operation that is required to optimize perfor- strict synchronization required in OFDMA using CP-OFDM mance. We will show that the receiver processing gives will not be appropriate for the sporadic traffic generated by substantial extra benefit. Experimental work using a soft- the IOT for two reasons. Firstly many of these devices will ware defined radio testbed has confirmed the potential of be battery powered and will spend most of the time in a PCC-OFDM [38]–[40]. dormant state, awakening only periodically to transmit data, In PCC-OFDM data is mapped onto adjacent subcarriers and secondly because of the hardware, time, and energy, that using opposite polarities. Since the first publications on PCC- is required for strict synchronization. Hence the search for OFDM [21]–[24] a number of other techniques have been waveforms that can be used by these devices in the uplink, described which also map data onto pairs of OFDM sub- and which are tolerant to time and frequency offsets. The carriers. In symmetric cancellation coding, data is mapped waveforms which have been considered so far, are well local- onto two subcarriers symmetrically spaced in the data vec- ized in the time and frequency domain but are not necessarily tor [41]. Other schemes use a complex conjugate weighting orthogonal. Techniques which have been proposed include to map the data to pairs of adjacent or symmetrically spaced filterbank multicarrier (FBMC) [7], [9]–[11] universal fil- subcarriers [42]. A number of other related schemes have tered multicarrier (UFMC) [18], and generalized frequency also been described (see [30], [31] and references therein). division multiplexing (GFDM) [7]. These typically involve The various techniques have slightly different properties. some form of frequency domain filtering, or time domain Most are less sensitivity to frequency offset than conventional windowing, or some combination of both filtering and win- CP-OFDM and some reduce the sensitivity to phase offsets dowing. or I/Q imbalance. In general these techniques, unlike PCC- OFDM, do not result in a windowing effect on the transmitted B. BACKGROUND ON PCC-OFDM OFDM symbol. As a result, unlike PCC-OFDM, they are Polynomial cancellation coded OFDM (PCC-OFDM) is one very sensitive to time offsets and delay spread, unless a cyclic form of OFDM which is very well localized in both the time prefix is used. The advantage of these other techniques is that and frequency domain, and which is compatible with CP- in general their PAPR statistics are similar to CP-OFDM and OFDM and also with MIMO [19], [20]. The first papers so they have a lower PAPR than PCC-OFDM, but this is at on PCC-OFDM were published around two decades ago the cost of significantly increased sensitivity to time offsets if [21]–[24] and showed that PCC-OFDM is robust to frequency a cyclic prefix is not used, or higher energy-per-bit and lower offset [22]. A number of papers quickly followed showing spectral efficiency if a cyclic prefix is used. that PCC-OFDM is also robust to time offset [25], phase noise [26], and time-varying channels including Rayleigh C. CONTRIBUTIONS, NOVELTY AND SIGNIFICANCE fading channels [21], [27], [28], and has a very rapid spectral In this paper, we present a study of the performance of roll-off [29]. There has also been some recent work on PCC- PCC-OFDM in the context of multiuser uplink communica- OFDM extending the earlier work on PCC-OFDM in single tions for 5G. The work is highly significant as it presents a user systems [30], [31]. The simplest form of PCC-OFDM new technique which clearly outperforms other techniques involves mapping data to adjacent pairs of subcarriers. This that have been proposed for this important emerging 5G results in substantial cancellation of the frequency domain application. The contributions include: 1 sidelobes, and a form of windowing in the time domain. 1) The first comprehensive study of multiuser Its main disadvantage is that it is not spectrally efficient. PCC-OFDM. This is the first paper which takes into To overcome this, a form of PCC-OFDM using overlapping account both the transmitter and receiver processing. symbol periods was developed [32]. This uses a concept The receiver weighting-and-adding operation is cru- similar to the weighted-overlap-and-add (WOLA) technique cial, and the only previous work on multiuser PCC- that has recently been proposed for 5G [33]. Interest in OFDM [36], [37] does not include this receiver operation. As a result the earlier work very substantially 1One misconception that has occurred in the past is that PCC-OFDM is underestimates the performance of PCC-OFDM. The simply a form of repetition coding. While repetition coding would give a 3 dB contribution of this new work is significant because it gain improvement in an AWGN channel it would not provide the improved performance in the presence of time and frequency offsets that PCC-OFDM shows that, when correctly configured, PCC-OFDM demonstrates. substantially outperforms UFMC and CP-OFDM,

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FIGURE 1. OFDM and PCC OFDM transmitter (a) OFDM transmitter and (b) PCC mapping at transmitter.

PCC-OFDM and UFMC transmitters and receivers that are studied in this paper. In Section III, the time and frequency domain properties of the three waveforms are described, and the ICI caused by time and frequency offsets is analyzed. Detailed simulations of the three waveforms in single user systems in the presence of time and frequency offsets are presented in Section IV. In Section V the simulations are extended to consider the multiuser case. The many advan-

FIGURE 2. OFDM and PCC-OFDM receiver (a) OFDM receiver and (b) PCC tages and the possible disadvantages of PCC-OFDM are OFDM weighting and adding at receiver. discussed in Section VI, and finally conclusions are drawn in Section VI. which are two waveforms which have been considered for the multiuser uplink in 5G. 2) New analytical and simulation results which demon- II. MULTIUSER OFDM SYSTEMS strate the separate contributions of the transmit- In this section we describe the transmitter and receiver struc- ter and receiver processing. These results show the tures for the three forms of OFDM that we analyze and their important contribution of receiver processing to the application in a multiuser system. robustness of PCC-OFDM in the presence of timing Fig.1 shows the key elements of an OFDM transmitter, and frequency offsets. This is essential to understand- and the mapping function required for PCC-OFDM. In the i- ing why this work reaches different conclusions from th symbol period the data to be transmitted is mapped onto a i = i i ··· i ··· i earlier research [36], [37]. complex vector X [X0, X1 Xk XN−1] which is input 3) The first study of a multiuser OFDMA system using to an N-point-IFFT. The corresponding vector at the IFFT is i = i i ··· i ··· i PCC-OFDM that considers frequency offsets, and x [x0, x1 xm xN−1] where from the definition of the the combination of timing and frequency offsets. IFFT This is an important factor in an mMTC environment, N−1 1 X j2πkm where strict synchronization is not practicable. (The xi = √ X i exp . (1) m k N earlier papers [36], [37] considered only timing offset, N k=0 not frequency offset, or the combination of timing and frequency offset). Usually a cyclic prefix is appended to xi before parallel-to- 4) Detailed simulation results for the performance of serial conversion, digital-to-analog conversion, filtering, and PCC-OFDM in a multiuser context. These include upconversion to a radio frequency carrier. The corresponding results for much larger time and frequency offsets than OFDM receiver and PCC weighting-and-adding block are have been considered in single-user OFDM. They also shown in Fig. 2, where the q-th input to the receiver FFT is q = q q ··· q ··· q include the effect of interference from other users. The vector y y0, y1 yn yN−1 and the output vector is q = q q ··· q ··· q body of work on single-user OFDM typically considers Y Y0 , Y1 Yl YN−1 where only the small offsets that may remain after the receiver − has been synchronized to the received waveform. They 1 N 1 −j2πln q = √ X q also usually assume that there are no unsynchronized Yl yn exp . (2) N N interfering signals. n=0 5) A clear description of the aspects of earlier PCC- In many applications some subcarriers, for example the OFDM research that are relevant to this new work. band-edge subcarriers, are unused and the corresponding D. ORGANIZATION OF PAPER inputs are set to zero. The only difference between a The remainder of this paper is organized as follows. Section II PCC-OFDM transmitter and a simple CP-OFDM trans- describes the multiuser systems and the CP-OFDM, mitter shown is the mapping of data onto adjacent pairs

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FIGURE 3. UFMC Transmitter and Receiver.

FIGURE 4. Configuration of multi-user uplink.

2 i of subcarriers. Each input Dk0 is mapped onto adjacent pairs the overall sensitivity of PCC-OFDM to frequency and i = i i = − i of subcarriers, so that X2k0 Dk0 and X2k0+1 Dk0 . In a time offset. In the simulations we will show that no PCC-OFDM receiver, after equalization, the two subcarriers cyclic preﬁx is required in PCC-OFDM in this appli- in each pair are combined as shown in Fig. 2(b) to give cation, but one can be used if necessary, for example i i if the same equipment is used for both CP-OFDM and (Y 0 − Y 0 ) i = 2l 2l +1 PCC-OFDM. Zl0 . (3) 2 In this paper, we consider the uplink of a multi-user OFDM Because the noise in the two subcarriers is independent, based system, where multi-user access is achieved by using weighing and adding the two subcarriers in the receiver frequency-division-multiplexing with different users being improves the signal-to-noise ratio (SNR) by 3 dB (see allocated different subbands, and where each subband is a Appendix A). We will show that it also further reduces group of adjacent OFDM subcarriers. Groups of 12 subcarriers are often considered in the literature, and this is what 2To simplify the discussions in the paper, we consider the case of mapping onto pairs of subcarriers where the lower index is even. This is not necessary. is used in our simulations. Each user is allocated one or Any adjacent pairs of subcarriers will provide the same performance gains. more subbands. To limit interuser interference, guard bands

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FIGURE 5. Subband spectra for two users for (a) CP-OFDM, (b) PCC-OFDM and (c) UFMC for N = 256, NCP = 32 and L = 33. of unused subcarriers may be inserted between the subcarriers A. SPECTRA OF CP-OFDM, PCC-OFDM AND UFMC allocated to each user. In this paper we consider two cases: Fig. 5 shows the spectra of two 12-subcarrier subbands sepa- systems with guard bands of 12 subcarriers, and systems with rated by a guard band of 12 subcarriers for systems with IFFT no guard bands. size, N = 256, for the three waveforms we consider. Fig. 6 The system description for UFMC is more complicated shows the band-edge of the spectra in more detail. Fig. 5(a) than for CP-OFDM or PCC-OFDM as a separate IFFT and shows the case of conventional CP-OFDM with CP length, a separate time-domain filter are required for each subband. NCP = 32. The ripple in the pass band of CP-OFDM and the Fig.3 shows a UFMC transmitter and receiver. Each trans- absence of deep nulls in the OOB are due to the CP [12], [43]. mitter will in general require multiple N-point IFFTs: one The corresponding spectrum for PCC-OFDM (with no CP) is IFFT for each subband. To reduce the spectral leakage into shown in Fig. 5(b). The spectral roll-off is much more rapid other subbands, each subband is separately filtered before for PCC-OFDM. This rapid roll-off is an important advantage the filtered signals representing each subband are combined in subband multiplexing, as it means that there is much less and transmitted. At the receiver the signal is zero-padded interference between users if the frequencies are not precisely before input to a 2N-point FFT. It can be shown that the aligned. Fig. 5(c) shows the spectra for UFMC for a Dolph- wanted signals appear on alternate outputs of the IFFT [18]. Chebyshev filter of length, L = 33. Comparing PCC-OFDM These outputs are then equalized before sub-band demapping. and UFMC it can be seen that the OOB power of PCC- A major disadvantage of UFMC is the requirement for multi- OFDM initially falls off more quickly than that of UFMC, ple FFTs and filters. These increase both the complexity and but is higher for frequencies further from the subband. It can latency of UFMC systems. be seen that for CP-OFDM, if timing or frequency offsets Fig. 4 shows the block diagram that describes the multi- disrupt the strict orthogonality between subcarriers, there is user configuration for the three waveforms that we analyze. potentially significant interference between subbands. While A number of different transmitters transmit on different sub- there is more overlap for PCC-OFDM than UFMC, the over- bands. In this paper we consider the case where all users all performance also depends on the receiver processing. The within a system use the same waveform (i.e. CP-OFDM, combining of adjacent subcarriers in PCC-OFDM results in PCC-OFDM or UFMC). At each transmitter a subband map- further interference reduction so that, as will be shown in per maps the data onto the subbands allocated to that user. The Section IV, in the overall system PCC-OFDM outperforms received signal is the sum of the received signals from each UFMC. The spectrum of UFMC depends on the length of the of the users plus additive white Gaussian noise (AWGN). filter used. A longer filter will result in more rapid spectral The receiver demodulates the signal and then, based on their roll-off, at the cost of some loss in overall spectral efficiency allocated subbands, the subband demapper separates the data and increase in signal processing complexity. transmitted by each user. The rapid roll-off of PCC-OFDM can be understood by comparing the power spectral density (PSD) of an individual III. TIME AND FREQUENCY DOMAIN PROPERTIES OF OFDM subcarrier with that of a PCC-OFDM subcarrier pair. CP-OFDM, PCC-OFDM AND UFMC It is well known that the Fourier transform on an OFDM The relative performances of CP-OFDM, PCC-OFDM and subcarrier has the form of a sinc function. In [29] it is shown UCMC in a multiuser system depend very much on the time both mathematically and graphically (see [29] Fig. 2) that and frequency domain properties of the three waveforms, and the sidelobes of adjacent subcarriers are very similar and so the level of ICI that any time or frequency offset causes. In this the [+1, −1] PCC mapping results in substantial cancellation section we describe these properties and the ICI that results of the sidelobes. This was then used to show that, subject to a from time and frequency domain offsets. number of assumptions and conditions, the PSD of an OFDM

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FIGURE 6. Band-edge detail of spectra for CP-OFDM, PCC-OFDM and UFMC, for N = 256, NCP = 32 and L = 33. subcarrier falls off with frequency, f , as 1/(f 2N 2), while for N 1, the PSD of a subcarrier pair in PCC-OFDM falls off as 1/(f 4N 4). For OFDM, it was assumed that the complex data mapped onto the OFDM subcarriers comprised independent zero-mean random variables [29], and that there was no CP. Rectangular pulse shaping and perfect interpolation were also implicitly assumed. For PCC-OFDM the complex data mapped to the PCC-OFDM subcarrier pairs was assumed to comprise independent zero-mean random variables. The overall spectrum of an OFDM signal has been shown to depend on a number of aspects of the transmitter design, including N, NCP, the number of unused band-edge subcarriers, the pulse shaping, and the interpolation ﬁlter- ing [43]. (Note: the comments about OOB power in [29] are FIGURE 7. Time domain envelopes for (a) CP-OFDM, (b) PCC-OFDM and inaccurate). (c) UFMC for N = 256, NCP = 32 and L = 33.

B. TIME-DOMAIN ENVELOPES OF CP-OFDM, which is a complex exponential of frequency, k, multiplied PCC-OFDM AND UFMC by a complex window (1 − exp (j2πm/N )). It is important Fig.7 shows the time domain envelopes for the three wave- to note that the window function is independent of k, so is forms we consider. CP-OFDM has the familiar square win- the envelope of every PCC pair and hence of the entire dow and has a symbol length of N + NCP = 288. The PCC-OFDM symbol. The magnitude of the window function formula for the PCC-OFDM envelope shown in Fig. 7(b) can is given by be calculated as follows. From the definition of the FFT, x p m |1 − exp (j2πm/N )| = 2 (1 − cos (2πm/N )). (7) is given by N−1 For PCC, as no CP is used, the symbol length is equal to 1 X j2πkm = xi = √ X i exp (4) the FFT size, which in this case is N 256. For UFMC m k N the envelope depends on the filter length. Fig. 7(c) shows the N k=0 result for N = 256 and L = 33, which results in an overall i i where the component of xm due to the input Xk is symbol length of N + L − 1 = 288. The filtering in UFMC results in an envelope which tapers at the start and end of i 1 i j2πkm x = √ X exp . (5) the symbol, and this reduces the sensitivity of UFMC to time m,k N k N offsets. i So the component of xm due to the pair of PCC coded inputs, i i i = − i Xk and Xk+1, where Xk+1 Xk is C. ANALYSIS OF INTERCARRIER INTERFERENCE CAUSED BY TIME OFFSET xi + xi m,k m,k+1 In this section the effect of timing offset on OFDM and + 1 i j2πkm i j2π(k 1)m PCC-OFDM is analyzed in detail, and the importance of = √ Xk exp − Xk exp N N N receiver processing in PCC-OFDM is explained. This extends 1 j2πkm j2πm the analysis in [29] to include the derivation of expressions = √ X i exp 1 − exp (6) N k N N for the PCC-OFDM signals before and after receiver weight-

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FIGURE 8. ICI and ISCHI as a function of time offset for N = 256 (a) CP-OFDM, (b) PCC-OFDM no receiver weighting and adding and (c) PCC-OFDM with receiver weighting and adding. ing and adding, and for intercarrier as well as intersymbol of the energy transmitted on a given subcarrier is received on interference. the same subcarrier. When the time offset increases, the ICI increases, and there is substantial ICI even in quite distant 1) ICI IN OFDM subcarriers. For CP-OFDM, as long as the there is no frequency offset, and any timing offset is less than the length of the CP, the received 2) ICI IN PCC-OFDM WITH TRANSMITTER WEIGHTING ONLY subcarriers are orthogonal, and can be recovered without any When the data at the IFFT input is PCC encoded, then ICI or intersymbol interference (ISI). However when this Xk+1 = −Xk , and the l-th output of the receiver FFT is condition is not satisfied the performance degrades rapidly. given by the sum of the components due to Xk and Xk+1. To understand the effect of timing offset consider a sequence In Appendix C, it is shown that i i i i i of two IFFT output vectors x = x , x , x ··· x − and 0 1 2 N 1 i i i i − − h + + + + i , + , N 1 p i+1 = i 1 i 1 i 1 ··· i 1 Yl,k Yl,k+1 1 X j2πn (k − l) x x0 , x1 , x2 xN−1 and one FFT input vector = exp i q = q q q ··· q X N N y y0, y1, y2 yN−1 . If the receiver is perfectly syn- k n=0 = chronized, the channel is flat, there is no noise, and q i, j2π (n + p) q = i q = i × 1 − exp . (10) then y x and Y X . Now consider the effect of a time N offset of p samples where p ≥ 1, and for NCP = 0, so h i Like the result for conventional OFDM given in (9), the mag- i = i i i ··· i y y0, y1, y2 yN−1 nitude of a component of ICI depends only on the time offset, h i p, and (k − l), and does not depend on k or l separately. = xi ··· xi , xi+1 ··· xi+1 . (8) p N−1 0 p−1 i,i + i,i i = Fig.8(b) shows Yl,k Yl,k+1 /Xk for N 256. Again, It can be seen that yi is a function of both Xi and Xi+1. Both Xi like Fig. 8(a), for p = 0 and k = l there is no ICI: all of and Xi+1 cause ICI, but Xi+1 also causes ISI. Consider first the energy transmitted on a given subcarrier is received on the components of yi and Yi due only to Xi, which we denote the same subcarrier, however because the data is transmitted i,i i,i i,i h i i i on both the k-th and the (k + 1)-th subcarriers, there are two by y , and Y respectively, so y = x ··· x − , 0 ··· 0 . p N 1 p = In Appendix B, it is shown that non-zero values for 0. When the time offset increases, the ICI increases, but for all offsets, p, the ICI is less than

i,i N−1−p in Fig. 8(a). Yl,k 1 X j2πn (k − l) = exp (9) X i N N k n=0 3) ICI IN PCC-OFDM WITH BOTH TRANSMITTER AND RECEIVER PROCESSING where Y i,i is the component of Y i,i due to X i . Equation (9) l,k l k We now analyze the effect of time offset on the overall PCC- i,i i shows that the magnitude of a component of ICI, Yl,k /Xk , OFDM system including both the transmitter mapping, and depends on the time offset, p, and (k − l), the difference the receiver weighting-and-adding operation. In this case the between the subcarrier indices. The magnitude of the ICI does i i performance depends on how Dk0 varies as a function of Zl0 , not depend on the separate indices, k and l directly. Fig. 8(a) and as the time offset is varied. Denoting the components

i,i i = i i i,i i,i shows Yl,k /Xk for N 256 as a function of time offset, of Z due to D by Z , and the component of Zl0 due i,i i 0 = 0 − 0 p/N , and (k − l). For p = 0 and k = l, there is no ICI: all to Dk0 as Zl0,k0 , and using Zl Y2l Y2l +1 /2 and

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FIGURE 9. ICI and ISCHI due to ISI as a function of time offset for N = 256 (a) CP-OFDM, (b) PCC-OFDM no receiver weighting and adding and (c) PCC-OFDM with receiver weighting and adding.

Dk0 = X2k0 = −X2k0+1, it is shown in Appendix D that Similarly for PCC-OFDM encoding at the transmitter only, the ICI is given by (14), as shown at the bottom of this i,i N−1−p Zl0,k0 1 X j2πn page. Performing a similar analysis for PCC-OFDM with = 1 − exp i D 0 2N N both transmitter mapping and receiver weighting-and-adding, k n=0 i,i i+1 i,i+1 and denoting the component of Z 0 due to D 0 as Z 0 0 , 0 0! l k l ,k j4πn k − l j2π (n + p) the interference is given by (15), as shown at the bottom of × exp 1 − exp . N N this page. These results are shown in Fig. 9. It can be seen that (11) as the time offset increases, the ICI due to ISI also increases.

It can be seen from (11) that the interference for PCC after D. ANALYSIS OF INTERCARRIER INTERFERENCE CAUSED both receiver and transmitter processing, depends only on p BY FREQUENCY OFFSET and k0 − l0 and not on k0 or l0 separately. Fig.8(c) shows The ICI due to frequency offset in PCC-OFDM was analyzed i,i i = Zl0,k0 /D2k0 for N 256 as a function of time offset in [22]. In this section we present new results which demon- p/N and k0 − l0. Comparing Fig.8(c) with Fig. 8(b) it can strate the separate contributions of transmitter and receiver be seen that receiver weighting-and-adding step reduces the processing in PCC-OFDM. A detailed analysis is given in interference considerably. Appendix F. For the case where the difference in frequency between the frequency of the receiver local oscillator and the 4) ICI DUE TO INTERSYMBOL INTERFERENCE carrier frequency of the received signal is 1f , and the phase Expressions for the components of interference caused by offset at the start of the i th symbol is θ0 [22], and assuming intersymbol interference from the i + 1 th symbol can be no timing offset, a ﬂat channel and no noise, then it can be derived in a similar way. In this case shown that i,i N−1 h + + i Yl,k 1 X j2πn (k − l + 1fT ) yi,i+1 = 0 ··· 0, xi 1 ··· xi 1 (12) = exp . (16) 0 p−1 X i N N k n=0 and it can be shown in Appendix E that for OFDM From (16) it can be seen that the magnitude of ICI due

i,i+1 N−1 to frequency offset depends only on (k − l), the differ- Yl,k 1 X j2πn (k − l) = exp . (13) ence between the subcarrier indices and 1fT , where 1fT X i+1 N N k n=N−p is the frequency offset normalized to the subcarrier spacing.

i,i+1 + i,i+1 N−1 Yl,k Yl,k+1 1 X j2πn (k − l) j2π (−N + p + n) = exp 1 − exp (14) X i+1 N N N k n=N−p

i,i+1 N−1 0 0! Zl0,k0 1 X j2πn j2πn 2k − 2l j2π (−N + p + n) = 1 − exp exp 1 − exp (15) Di+1 2N N N N k n=N−p

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FIGURE 10. ICI and ISCHI as a function of frequency offset for N = 256 (a) CP-OFDM, (b) PCC-OFDM no receiver weighting and adding and (c) PCC-OFDM with receiver weighting and adding.

For PCC-OFDM considering only the transmitter weighting, it is shown in Appendix F that

i,i + i,i N−1 Yl,k Yl,k+1 1 X j2πn (k − l + 1fT ) = exp X i N N k n=0 j2πn × 1 − exp . (17) N For PCC-OFDM when the receiver weighting-and-adding step is included

i,i N−1 2 Zl0,k0 1 X j2πn = 1 − exp i X 0 2N N 2k n=0 0 0 ! j2πn 2k − 2l + 1fT × exp . (18) N FIGURE 11. BER in an AWGN as a function of Eb/N0 for CP-OFDM, PCC-OFDM and UFMC with 4-QAM, 16-QAM and 64-QAM constellations. Fig. 10 shows the ICI as a function of frequency offset for CP-OFDM and PCC-OFDM with and without weighting assuming θ0 = 0. The normalized frequency offset is varied for 0 ≤ 1fT ≤ 2, which corresponds to two subcarrier spacings. For CP-OFDM it can be seen in Fig 10(a) that, as 1fT varies from 0 to 1, the ICI grad- ually shifts from one subcarrier to the next, and that there is signiﬁcant interference across a number of subcarriers. Fig. 10(b) shows the ICI as a function of frequency offset for PCC-OFDM without the receiver weighting-and-adding step, while Fig 10(c) shows the ICI in PCC-OFDM with receiver weighting-and-adding. It can be seen that, as for time offset, both the transmitter mapping and the receiver weighting- and-adding contribute to the reduction in interference in PCC-OFDM.

FIGURE 12. BER in an AWGN as a function of Eb/N0 for PCC-OFDM with IV. PERFORMANCE OF CP-OFDM, PCC-OFDM AND and without weighting and adding in the receiver for 4-QAM, 16-QAM and 64-QAM constellations. UFMC - SINGLE-USER CASE We now show how the different properties of the three waveforms affect their sensitivity to timing and frequency offsets importance of the receiver weighting-and-adding operation and to noise in a single-user scenario, and demonstrate the in the PCC-OFDM receiver.

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FIGURE 13. BER in an AWGN as a function of E /N for with time offset FIGURE 16. Eb/N0 required for waveforms to achieve a target BER of b 0 − τ = 0.05 for CP-OFDM, PCC-OFDM and UFMC with 4-QAM, 16-QAM and 10 2 as a function of normalized time offset τ. 64-QAM constellation.

FIGURE 17. BER comparison of three waveforms for varying constellation FIGURE 14. BER in an AWGN as a function of Eb/N0 for with time offset sizes affected by a normalized frequency offset, 1fT = 0.05. τ = 0.05 for PCC-OFDM with and without weighting and adding in the receiver for 4-QAM, 16-QAM and 64-QAM constellations.

FIGURE 18. BER comparison of three waveforms for varying constellation = FIGURE 15. BER in an AWGN as a function of Eb/N0 for with time offset sizes affected by a normalized frequency offset, 1fT 0.2. τ = 0.2 for PCC-OFDM with and without weighting and adding in the receiver for 4-QAM, 16-QAM and 64-QAM constellations. offset between transmitter and receiver. Fig. 11 shows BER A. PERFORMANCE IN AN AWGN CHANNEL results for the three waveforms as a function of Eb/N0 where We ﬁrst consider the case of single user in an additive white Eb is the energy per bit and N0 the single-sided noise spec- Gaussian noise (AWGN) channel with no timing or frequency tral density. Results are given for 4-QAM, 16-QAM and

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FIGURE 19. Eb/N0 required for waveforms to achieve a target BER of 10−2 as a function of normalized frequency offset, 1fT .

FIGURE 21. Two user BER results for normalized timing offset of User 2, τ = 0.05. User 2 received power 10 dB greater than User 1 (a) guard band of 12 subcarriers, (b) no guard band.

pairs of subcarriers reduces the spectral efficiency, so that for a given constellation size PCC-OFDM carries only slightly more than half the data of CP-OFDM. This means that to compare systems of approximately equal spectral efficiency the plots for 16-QAM PCC-OFDM should be compared with the plots for 4-QAM CP-OFDM and UFMC in Fig. 11, Fig. 13 and Figs. 15-29. For CP-OFDM, Eb depends on the length of the cyclic prefix. The results are for the case of N = 256 and NCP = 32. Similarly for FBMC, Eb depends on the filter length and the results in Fig. 11 are for N = 256 and L = 33. The results for FBMC and CP-OFDM are very similar because L = NCP + 1, so the overhead is the same for each system. As expected, for each type of waveform the Eb/N0 required for a given BER increases as the constellation FIGURE 20. Two user BER results for normalized timing offset of User 2, size increases. τ = 0.05 (a) guard band of 12 subcarriers, (b) no guard band. Fig. 12 demonstrates the importance of the weighting-and- adding operation in a PCC-OFDM receiver. It shows the BER 64-QAM constellations. For each constellation size, PCC- performance for PCC-OFDM without weighting and adding, OFDM requires the lowest Eb/N0 of the three waveforms. that is the result if data estimation is based on the received This is because at the PCC-OFDM receiver the noise in signal on only one of the two PCC subcarriers, and also with different subcarriers is independent, so combining subcarrier weighting and adding of the subcarrier pair. For each size pairs at the receiver reduces the required Eb/N0 by 3 dB. The of constellation it can be seen that the weighting-and-adding disadvantage of PCC-OFDM is that the mapping of data onto operation reduces the required Eb/N0 by 3 dB.

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FIGURE 22. Two user BER results for normalized timing offset of User 2, FIGURE 23. Two user BER results for normalized timing offset of User 2, τ = 0.2 (a) guard band of 12 subcarriers, (b) no guard band. τ = 0.2. User 2 received power 10 dB greater than User 1 (a) guard band of 12 subcarriers, (b) no guard band.

B. PERFORMANCE WITH TIMING OFFSET Fig. 14 again shows the importance of the weighting-and- We now consider the effect of time offsets between the trans- adding operation in the PCC-OFDM receiver. It shows that it mitter and receiver for the single user case. Fig. 13 shows the reduces the sensitivity to timing offset as well as providing a BER in AWGN for a receiver offset of τ = 0.05 where τ is 3 dB improvement in SNR, it gives an improvement of up to measured as a fraction of the OFDM symbol period excluding a 6 dB for 64-QAM with a timing offset of τ = 0.05. the CP, and positive τ represents a delay in the receiver Fig. 15 shows the effect of increasing τ to 0.2 which is timing relative to the transmitter. As N = 256 this represents greater than the length of the CP. For UFMC and CP-OFDM a delay of 13 samples. Comparing Fig. 13 with Fig. 11 it the BER plateaus at levels about 10−2 whereas the BER can be seen that this offset does not change the BER for for PCC-OFDM is much lower, with 4-QAM and 16-QAM CP-OFDM. This is expected as the offset is less than the CP showing relatively little degradation. We now explore the length of 32 samples. In contrast the time offset increases the effect of varying the timing offset. Fig. 16 shows how the BER for UFMC. This is most clearly seen from the 64-QAM performance of each waveform varies as a function of time results where UFMC now has a higher BER than CP-OFDM. offset, τ. It compares the required Eb/N0 for CP-OFDM, This is because the receiver window is not aligned with the PCC-OFDM and UFMC for a target BER of 10−2 for 4-QAM maximum of the time domain envelope shown in Fig. 7. For and 16-QAM. For CP-OFDM the BER is constant as long PCC-OFDM there is a very slight increase in BER because as the delay is within the cyclic prefix length, but increases the first part of each symbol is missed, so not all of the rapidly for τ < 0, or τ > 0.125. The plots for CP-OFDM received symbol energy is used to recover the data, but this are not symmetric about τ = 0 because the simulations are effect is less than for UFMC as the time domain envelope based on the use of a cyclic prefix (not postfix), so for timing does not change so quickly. This result is consistent with the errors within the length of the CP there is no ICI, or ISI. effect of timing offset shown in Fig. 8 (c). In contrast PCC-OFDM is much less sensitive to time offset

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FIGURE 24. Two user BER results for normalized frequency offset of FIGURE 25. Two user BER results for normalized frequency offset of = User 2, 1fT = 0.05 (a) guard band of 12 subcarriers, (b) no guard band. User 2, 1fT 0.05, User 2 received power 10 dB greater than User 1 (a) guard band of 12 subcarriers, (b) no guard band. and degrades only slowly. Even offsets of τ = ±0.2 require an increase of less than 2 dB to maintain a BER of 10−2. The 4-QAM UFMC, and for small offsets has approximately performance of UFMC is relatively constant over the smaller equal performance. This means that the loss in spectral effi- range −0.08 < τ < 0.08 but degrades rapidly for larger time ciency of PCC-OFDM due to the mapping of data onto two offsets. subcarriers, can potentially be regained by using bigger constellations. C. PERFORMANCE WITH FREQUENCY OFFSET - SINGLE USER CASE V. PERFORMANCE OF CP-OFDM, PCC-OFDM Frequency offsets between transmitter and receiver will also AND UFMC - TWO-USER CASE degrade the BER performance. Fig. 17 shows the effect of a As we are ultimately interested in the application of normalized frequency offset, 1fT = 0.05. Comparing Fig. 17 PCC-OFDM for uplink multiple access, we now investigate with the AWGN results shown in Fig. 11, it can be seen that the sensitivity of the three waveforms to time and frequency frequency offset has negligible effect on PCC-OFDM for all offsets in a two-user scenario. A number of configurations are constellation sizes, while the performance of CP-OFDM and possible, but here we calculate the BER for User 1 for the case UFMC are significantly degraded, with the BER plateauing where the receiver is synchronized to User 1, so that there is for 64-QAM constellations. The effect of further increasing no timing or frequency offset between User 1 and the receiver, 1fT is demonstrated in Fig. 18 showing an even greater per- but User 2 may have a timing or frequency offset. Whereas formance advantage of PCC-OFDM. Fig. 19 shows the effect in the previous section the ICI is ‘same user ICI’: the ICI was of varying frequency offsets. Both UFMC and CP-OFDM caused by subcarriers allocated to one user, in this section degrade rapidly as frequency offset increases, while even the impairment may also be due to ‘other user ICI’, which normalized frequency offsets of 0.2 have little effect on is the ICI caused by subcarriers allocated to a different user. PCC-OFDM. Similarly for ISI. As can be seen in Fig. 8, the level of ICI A point to note is that both for large time offsets and is strongly dependent on the spacing between the subcarriers large frequency offsets 16-QAM PCC-OFDM outperforms concerned, so we consider both cases, that is where there is

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FIGURE 26. Two user BER results for normalized frequency offset of FIGURE 27. Two user BER results for normalized frequency offset of User 2, 1fT = 0.2 (a) guard band of 12 subcarriers, (b) no guard band. User 2, 1fT = 0.2, User 2 received power 10 dB greater than User 1 (a) guard band of 12 subcarriers, (b) no guard band. a guard band between the subcarriers allocated to each user and cases where there is no guard band.

A. PERFORMANCE OF TWO-USER SYSTEM IN AWGN CHANNEL Simulations were performed for two users for 12 subcarrier subbands with a 12 subcarrier guard band and an AWGN channel. The BER results were identical to the single user AWGN case shown in Fig. 11, and this was also the case when there was no guard band. This was the expected result for CP-OFDM and PCC-OFDM because in the absence of time or frequency offset the subcarriers are strictly orthogonal. However there was also no observable degradation for UFMC. In general the powers of the signals received from dif- FIGURE 28. Two user BER results for normalized frequency offset of ferent users will not be equal, to explore the effect of a strong User 2, 1fT = 0.05, and normalize time offset of User 2, τ = 0.05. interfering signal, the received power of User 2 was increased No guard band. by 10 dB. There was still no observable degradation: the BER plots were the same as Fig. 11. is a very slight degradation in the performance of 64-QAM UFMC. Removing the guard band (Fig. 20 (b)) causes no B. PERFORMANCE OF TWO-USER SYSTEM observable increase in degradation of UFMC. Fig. 21 shows WITH TIMING OFFSET the effect of increasing the received power of User 2 by 10 dB. Fig. 20 shows the BER performance when User 2 has a The key point is that even with no guard band and a higher time offset of τ = 0.05. For a 12 subcarrier guard band power interfering signal there is no observable change in comparing Fig. 20(a) with Fig. 11 it can be seen that there the performance of PCC-OFDM. There is also no change

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the performance of PCC-OFDM. Fig. 26 shows the effect of increasing the frequency offset. Now even with a guard band (Fig. 26 (a)) the frequency offset degrades the performance of CP-OFDM and UFMC. Fig. 27 shows that only for large frequency offset and for a high power interfering signal does PCC-OFDM show any degradation, and this is only observable for the largest constellation.

D. PERFORMANCE OF TWO-USER SYSTEM WITH TIME AND FREQUENCY OFFSET Finally we investigate the performance when User 2 has both time and frequency offset. Fig. 28 shows that all three waveforms perform well for small time and frequency offsets even with no guard band. Comparing Fig. 24, which shows the results for 1fT = 0.05, τ = 0.05, with Fig. 11 it can be seen that only 64-QAM CP-OFDM and 64-QAM UFMC show any change. Similarly, Fig. 29 shows that even for 1fT = 0.2, τ = 0.05 and with no guard band PCC-OFDM shows no degradation.

VI. ADVANTAGES AND DISADVANTAGES OF PCC-OFDM FOR 5G In addition to the advantages of extreme insensitivity to time and frequency described above, because PCC-OFDM is a form of OFDM, it also retains many of the advantages of CP-OFDM. For example, single tap equalizers can be used in PCC-OFDM to correct for frequency selective fading, and PCC-OFDM, like CP-OFDM is well suited to use in MIMO systems. Although not considered in the simulations FIGURE 29. Two user BER results for normalized frequency offset of in this paper, PCC-OFDM has also been shown to be very User 2, 1fT = 0.2, and normalize time offset of User 2, τ = 0.05. (a) guard insensitive to Doppler spread [21]. The additional mapping band of 12 subcarriers, (b) no guard band. onto subcarriers at the transmitter and the weighting-and- adding step at the receiver mean that PCC-OFDM adds for CP-OFDM as this time offset is within the cyclic pre- negligible complexity and latency relative to CP-OFDM fix. However the performance of 64-QAM UFMC degrades and has significantly lower complexity and latency than significantly. A slightly surprising result is that for this case UFMC. the no guard band 64-QAM UFMC has better performance One important potential disadvantage of the form of PCC- than 64-QAM UFMC with a guard band. This is because OFDM described in this paper is the potential loss in spectral with no synchronization errors the ICI in UFMC falls on the efficiency resulting from the mapping of data onto pairs of odd subcarriers but timing offsets disrupt this. The effect of subcarriers rather than onto single subcarriers. It is likely increasing τ to 0.2 is shown in Fig. 22 and Fig. 23. In all that much of this loss can be recovered due to a com- cases PCC-OFDM substantially outperforms UFMC and bination of the other properties of PCC-OFDM, including CP-OFDM. the fact that no frequency guard band is required between users, and that the improved performance may allow larger C. PERFORMANCE OF TWO-USER SYSTEM WITH constellations to be used. For low data rate applications a FREQUENCY OFFSET small loss in spectral efficiency may be acceptable. For high Fig. 24 shows the BER performance when User 2 has a data rate applications, particularly in the downlink, PCC- frequency offset of 1fT = 0.05. For a 12 subcarrier guard OFDM with overlapped symbol periods may offer a good band comparing Fig. 24 (a) with Fig. 11 it can be seen solution [32]. that there is a very slight degradation in the performance A second potential disadvantage of PCC-OFDM is its high of 64-QAM CP-OFDM. With no guard band (Fig. 24 (b)) PAPR, which is higher than CP-OFDM [44]. An interest- the performance of 64-QAM CP-OFDM degrades further and ing area of future research would be to investigate PAPR 64-QAM UFMC also shows some change. Fig. 25 shows the mitigation schemes for PCC-OFDM. An alternative solution effect of increasing the received power of User 2 by 10 dB. might be to use no PAPR reduction. High PAPR is a problem The key point is that even with no guard band and a higher in many OFDM applications particularly high power broad- power interfering signal there is no observable change in cast television and radio applications because, unless linear

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i amplifiers are used in the transmitters, the peaks result in signal-to-noise ratio (SNR) of Yk is given by a distorted signal which contains out-of-band spectral com- n 2o ponents which may interfere with other signals. The dis- E X i i k advantage of linear amplifiers is that they are not power SNR Yk = . (20) n i 2o efficient. However recent research [45] has shown that E Wk when the power consumption of the digital signal processing is considered there is no overall power advantage in This is also the SNR for PCC-OFDM before combining. The applying PAPR reduction techniques for low power appli- receiver output for PCC-OFDM after combining, and taking cations. The overall system advantages or disadvantage of into account the PCC weighting at the transmitter is given by PCC-OFDM compared with other 5G solutions when all i i Y − Y + of the properties of insensitivity to time and frequency Z i = k k 1 k 2 offset and the consequent reduction in signal processing X i + W i − X i − W i required are considered will be an interesting area of future = k k k+1 k+1 research. 2 1 h i i i i i = Xk + Xk + Wk + Wk+1 . (21) VII. CONCLUSION 2 This paper has evaluated PCC-OFDM as a contender for 5G So the SNR of Z i is given by systems and compared it with well-known waveforms such as k CP-OFDM and UFMC. It has been shown that PCC-OFDM n i 2o E 2Xk performs substantially better than CP-OFDM and UFMC in i = SNR Zk n o i + i 2 the presence of time and/or frequency offsets. Large offsets E Wk Wk+1 that would make transmission with CP-OFDM or UFMC n i 2o impossible, cause very little degradation in PCC-OFDM. For 4E Xk = . (22) example time offsets of ±0.2 of a symbol period require n 2o E W i + W i only a 2 dB increase in SNR to achieve a given target BER. k k+1 Similarly frequency offsets of ±0.2 of a subcarrier spacing As the noise is white, the noise component on each received also require an increase of only 2 dB. It has been shown that subcarrier is independent so the weighting-and-adding step in the PCC-OFDM receiver contributes significantly to the performance. Because of the 2 2 2 i + i = i + i mapping of data onto two subcarriers, weighting and adding E Wk Wk+1 E Wk E Wk+1 improves the performance in AWGN by 3 dB but it improves 2 2 = i + i the performance by much more than this in the presence of E Wk E Wk time and frequency offsets. The performance of PCC-OFDM 2 in a multiple access system using OFDMA has been eval- = i 2E Wk . (23) uated. It has been shown that because PCC-OFDM is very well-localized in both the time and frequency domain it is Substituting this in (22) gives extremely robust in a multiuser system and as a result no n 2o guard band is required between users. Only for very large 4E X i i k constellations, very large frequency offsets, and when the SNR Zk = n i 2o power of the interfering user was 10 dB higher than that 2E Wk of the user for which the BER was being calculated was n i 2o any degradation observable. In conclusion PCC-OFDM is a 2E Xk = very strong contender for application in 5G as it substantially n 2o E W i outperforms other better known waveforms in the presence of k timing and frequency offsets. i = 2 × SNR Yk . (24)

APPENDIX A This shows that the weighting and adding in the receiver ANALYSIS OF CP-OFDM AND PCC-OFDM improves the SNR by a factor of 2, or equivalently 3 dB. IN AN AWGN CHANNEL The graphs in Section IV use the conventional form of BER For the case of an AWGN channel and no timing or frequency curves in terms of energy-per-bit rather than SNR. For a given offset, then for both PCC-OFDM and CP-OFDM average transmit power, the energy per bit for PCC-OFDM is twice that for OFDM without a CP, because twice as many Y i = X i + W i (19) k k k subcarriers are used to transmit a given amount of data. For i where Wk is the white noise component of the k-th output of CP-OFDM the energy used in the cyclic preﬁx has to be taken i the receiver FFT in the i-th symbol period. Wk are zero mean into account. Both of these factors are included in the results identically distributed Gaussian random variables. So the presented in Section IV.

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APPENDIX B which after some rearranging gives ANALYSIS OF ICI IN OFDM DUE TO TIMING OFFSET N−1−p 1 j2πpk j2πn (k − l) To analyze the effect of timing offset in OFDM, we consider i,i = i X i i i i i Yl,k Xk exp exp a sequence of two IFFT outputs x = x , x , x ··· x − N N N h i 0 1 2 N 1 n=0 i+1 = i+1 i+1 i+1 ··· i+1 and x x0 , x1 , x2 xN−1 and one FFT input (31) yq = yq, yq, yq ··· yq . For q = i, and if the receiver 0 1 2 N−1 so is perfectly synchronized, the channel is flat and there is no noise then, yq = xi, and Yq = Xi. For an unsynchronized i,i N−1−p Yl,k 1 j2πpk X j2πn (k − l) p p ≥ = exp exp system with a time offset of samples where 1, and for X i N N N k n=0 NCP = 0 i h i i i i i (32) y = y0, y1, y2 ··· yN−1 h i Noting that |exp (j2πpk/N )| = 1, this can be simplified to = i ··· i i+1 ··· i+1 xp xN−1, x0 xp−1 (25) i,i N−1−p Yl,k 1 X j2πn (k − l) i i+1 = exp . (33) which is a function of both X and X . The components of X i N N yi and Yi due only to Xi are given by k n=0 i,i h i i i y = xp ··· xN−1, 0 ··· 0 (26) APPENDIX C ANALYSIS OF ICI IN PCC-OFDM DUE TO TIMING OFFSET and WITH TRANSMITTER WEIGHTING ONLY N−1 1 X −j2πln For PCC-OFDM Xk+1 = −Xk and the l-th output of the Y i,i = √ yi,i exp l n N receiver FFT is the sum of the components due to Xk and N n=0 Xk+1, and is given by (34), as shown at the bottom of this N−1−p 1 X −j2πln page. This can be simplified to give (35), as shown at the bot- = √ xi exp . (27) n+p N tom of this page and (36), as shown at the bottom of this page. N n=0 APPENDIX D From (4) ANALYSIS OF ICI IN PCC-OFDM DUE TO TIMING OFFSET N−1 i 1 X i j2πk (n + p) WITH BOTH TRANSMITTER AND RECEIVER PROCESSING x + = √ X exp . (28) n p N k N When both the transmitter mapping, and the receiver k=0 weighting-and-adding operation are included the perfor- Substituting this in (27) gives mance depends on how Dk0 varies as a function of Z l0 and as the time offset is varied. Denoting the components of Zi due N−1−p N−1 i i,i i,i i i,i i,i 1 X 1 X i j2πk (n + p) D by Z and the component of Zl0 due to Dk0 as Zl0,k0 , and Y = √ √ Xk exp l N N N using Zl0 = Y2l0 − Y2l0+1 /2 and Dk0 = X2k0 = −X2k0+1 n=0 k=0 gives (37) as shown at the bottom of the next page, which −j2πln × exp (29) after some rearrangement gives N N−1−p 1 j4πpk0 j2πn i,i i i,i i X and the component of Y due to X is Z 0 0 = D 0 exp 1 − exp l k l ,k 2N k N N n=0 N−1−p 0 0! i,i 1 X j2πk (n + p) −j2πln j4πn k − l j2π (n + p) Y = X i exp exp × exp 1 − exp l,k N k N N n=0 N N (30) (38)

N−1−p N−1−p 1 j2πpk X j2πn (k − l) 1 j2πp (k +1) X j2πn (k +1−l) Y i,i + Y i,i = X i exp exp − X i exp exp l,k l,k+1 N k N N N k N N n=0 n=0 (34) N−1−p 1 j2πpk X j2πn (k − l) j2π (n + p) Y i,i + Y i,i = X i exp exp 1 − exp (35) l,k l,k+1 N k N N N n=0

i,i + i,i N−1−p Yl,k Yl,k+1 1 X j2πn (k − l) j2π (n + p) = exp 1 − exp (36) X i N N N k n=0

157590 VOLUME 7, 2019 G. Kongara et al.: Comparison of CP-OFDM, PCC-OFDM and UFMC for 5G Uplink Communications

and so Performing a similar analysis for PCC-OFDM with both i,i N−1−p transmitter mapping and receiver weighting-and-adding, and + Zl0,k0 1 X j2πn i,i i+1 i,i 1 = 1 − exp denoting the component of Zl0 due to Dk0 as Zl0,k0 , it can i D 0 2N N be shown that k n=0 − − 0 0 − 0! i,i+1 1 i+1 j2π (N p) 2k j4πn k l j2π (n + p) Z 0 0 = D exp × exp 1 − exp . l ,k 2N k N N N N−1 0 0! X j2πn j2πn 2k −2l (39) × 1−exp exp N N n=N−p APPENDIX E j2π (−N + p + n) ANALYSIS OF ICI DUE TO INTERSYMBOL INTERFERENCE × 1 − exp (45) N Expressions for the components of interference caused by intersymbol interference from the i + 1 th symbol can be and derived in a similar way. In this case i,i+1 N−1 h i Zl0,k0 1 X j2πn yi,i+1 = 0 ··· 0, xi+1 ··· xi+1 (40) = 1 − exp 0 p−1 Di+1 2N N k n=N−p and it can be shown that for OFDM 0 0! j2πn 2k − 2l + 1 − (N − p) k × Y i,i 1 = X i+1 exp exp l,k N k N N

N−1 X j2πn (k − l) j2π (−N + p + n) × exp (41) × 1 − exp . (46) N N n=N−p and APPENDIX F i,i+1 N−1 Yl,k 1 X j2πn (k − l) ANALYSIS OF ICI CAUSED BY FREQUENCY OFFSET = exp . (42) i+1 N N The ICI due to frequency offset in PCC-OFDM was analyzed Xk n=N−p in [22]. Here we present a more detailed analysis which Similarly, for PCC-OFDM encoding at the transmitter only, analyzes the separate contributions of transmitter and receiver it can be shown that processing in PCC-OFDM. i,i+1 i,i+1 1 + −j2π (N − p) k Consider the case where the difference in frequency Y + Y = X i 1 exp l,k l,k+1 N k N between the frequency of the receiver local oscillator and the  N−1 carrier frequency of the received signal is 1f and the phase X j2πn (k − l) ×  exp offset at the start of the i-th symbol is θ0 [22]. Assuming no N n=N−p timing offset, a ﬂat channel and no noise, then  h i i,i i i i i j2π (−N + p + n) y = y0, y1 ··· , yn ··· yN−1 (47) × 1 − exp  N where (43) i i j2πn1fT yn = exp (jθ0) xn exp . (48) so N

i,i+1 + i,i+1 N−1 So Yl,k Yl,k+1 1 X j2πn (k − l) = exp i+1 N N N−1 Xk n=N−p i,i 1 X i Y = √ exp (jθ0) x l N n j2π (−N + p + n) n=0 × 1 − exp . N j2πn1fT −j2πln × exp exp . (49) (44) N N

i,i 1 i,i i,i 1 i,i i,i Z 0 0 = Y 0 0 +Y 0 0 − Y 0 0 +Y 0 0 l ,k 2 2l ,2k 2l ,2k +1 2 2l +1,2k 2l +1,2k +1   0 N−1−p 0 − 0! 1 i j2πp2k X j2πn 2k 2l j2π (n + p) = D 0 exp  exp 1 − exp  2N k N N N n=0   0 N−1−p 0 − 0 + ! 1 i j2πp2k X j2πn 2k 2l 1 j2π (n + p) − D exp  exp 1 − exp  (37) 2N k N N N n=0

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And using (4) gives Now considering PCC-OFDM, including the receiver weighting-and-adding step, gives N−1 N−1 i,i 1 X 1 X j2πkn = √ √ i i,i 1 i,i i,i Yl exp (jθ0) Xk exp = + N Zl0,k0 Y2l0,2k0 Y2l0,2k0+1 N n=0 N k=0 2 1 j2πn1fT −j2πln − i,i + i,i × exp exp . (50) Y2l0+1,2k0 Y2l0+1,2k0+1 . (57) N N 2 Substituting (55) in (57), and with some rearrangement gives Rearranging gives N−1 2 i,i 1 i X j2πn N−1 N−1 Z 0 0 = exp (jθ ) X 0 1 − exp 1 j2πn (k − l + 1fT ) l ,k 0 2k i,i = X X i 2N N Yl exp (jθ0) Xk exp . n=0 N N ! n=0 k=0 j2πn 2k0 − 2l0 + 1fT (51) × exp (58) N So, for conventional OFDM where the subcarriers are inde- and pendently modulated i,i N−1 2 Zl0,k0 1 X j2πn − = 1 − exp N 1 i 1 X j2πn (k − l + 1fT ) X 0 2N N i,i = i 2k n=0 Yl,k exp (jθ0) Xk exp N N 0 0 ! n=0 j2πn 2k − 2l + 1fT × (52) exp . (59) N and REFERENCES − i,i N 1 [1] M. Shafi, A. F. Molisch, P. J. Smith, T. Haustein, P. Zhu, P. Silva, Yl,k 1 X j2πn (k − l + 1fT ) = F. Tufvesson, A. Benjebbour, and G. Wunder, ‘‘5G: A tutorial overview i exp . (53) X N N of standards, trials, challenges, deployment, and practice,’’ IEEE J. Sel. k n=0 Areas Commun., vol. 35, no. 6, pp. 1201–1221, Jun. 2017. From (53) it can be seen that the magnitude of ICI due [2] P. Banelli, S. Buzzi, G. Colavolpe, A. Modenini, F. Rusek, and A. Ugolini, − ‘‘Modulation formats and waveforms for 5G networks: Who will be the to frequency offset depends only on (k l), the difference heir of OFDM?: An overview of alternative modulation schemes for between the subcarrier indices and 1fT , where 1fT is the improved spectral efficiency,’’ IEEE Signal Process. Mag., vol. 31, no. 6, frequency offset normalized to the subcarrier spacing. pp. 80–93, Nov. 2014. [3] Z. E. Ankarali, B. Peköz, and H. Arslan, ‘‘Flexible radio access beyond For PCC-OFDM, considering only the transmitter 5G: A future projection on waveform, numerology, and frame design weighting, principles,’’ IEEE Access, vol. 5, pp. 18295–18309, 2017. [4] Q. Bodinier, F. Bader, and J. Palicot, ‘‘On spectral coexistence of CP- i,i i,i OFDM and FB-MC waveforms in 5G networks,’’ IEEE Access, vol. 5, Y + Y + l,k l,k 1 pp. 13883–13900, 2017. N− 1 1 j2πn (k − l + 1fT ) [5] A. Ijaz, L. Zhang, M. Grau, A. Mohamed, S. Vural, A. U. Quddus, i X M. A. Imran, C. H. Foh, and R. Tafazolli, ‘‘Enabling massive IoT in 5G and = exp (jθ0) Xk exp N N beyond systems: PHY radio frame design considerations,’’ IEEE Access, n= 0 vol. 4, pp. 3322–3339, 2016. N−1 1 X j2πn (k + 1 − l + 1fT ) [6] Y. Medjahdi, S. Traverso, R. Gerzaguet, H. Shaiek, R. Zayani, D. Demmer, i R. Zakaria, J.-B. Doré, M. B. Mabrouk, and D. Le Ruyet, ‘‘On the road to − exp (jθ0) Xk exp N N 5G: Comparative study of physical layer in MTC context,’’ IEEE Access, n=0 vol. 5, pp. 26556–26581, 2017. (54) [7] N. Michailow, M. Matthé, I. S. Gaspar, A. N. Caldevilla, L. L. Mendes, A. Festag, and G. Fettweis, ‘‘Generalized frequency division multiplexing which can be simplified to for 5th generation cellular networks,’’ IEEE Trans. Commun., vol. 62, no. 9, pp. 3045–3061, Sep. 2014. Y i,i + Y i,i [8] G. Fettweis, M. Krondorf, and S. Bittner, ‘‘GFDM—Generalized fre- l,k l,k+1 quency division multiplexing,’’ in Proc. IEEE 69th Veh. Technol. Conf., N− 1 1 j2πn (k − l + 1fT ) Apr. 2009, pp. 1–4. i X [9] S. Premnath, D. Wasden, S. K. Kasera, N. Patwari, and B. Farhang- = exp (jθ0) X exp N k N Boroujeny, ‘‘Beyond OFDM: Best-effort dynamic spectrum access = n 0 using filterbank multicarrier,’’ IEEE/ACM Trans. Netw., vol. 21, no. 3, j2πn pp. 869–882, Jun. 2013. × 1 − exp . (55) N [10] F. Schaich and T. Wild, ‘‘Waveform contenders for 5G—OFDM vs. FBMC vs. UFMC,’’ in Proc. 6th Int. Symp. Commun., Control Signal Process. Therefore, (ISCCSP), 2014, pp. 457–460. [11] B. Farhang-Boroujeny and H. Moradi, ‘‘OFDM inspired waveforms for 5G,’’ IEEE Commun. Surveys Tuts., vol. 18, no. 4, pp. 2474–2492, i,i + i,i N−1 Yl,k Yl,k+1 1 X j2πn (k − l + 1fT ) 4th Quart., 2016. = exp X i N N [12] J. Armstrong, ‘‘OFDM for optical communications,’’ J. Lightw. Technol., k n=0 vol. 27, no. 3, pp. 189–204, Feb. 1, 2009. [13] T. Jiang and Y. Wu, ‘‘An overview: Peak-to-average power ratio reduction j2πn × 1 − exp . (56) techniques for OFDM signals,’’ IEEE Trans. Broadcast., vol. 54, no. 2, N pp. 257–268, Jun. 2008.

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Lasanen, O. Apilo, A. Hekkala, C. Cassan, J.-P. Verdeil, Proc. IEEE Global Telecommun. Conf., vol. 5, Nov. 1998, pp. 2771–2776. J. David, and L. Pichon, ‘‘Power consumption trade-off between power [22] J. Armstrong, ‘‘Analysis of new and existing methods of reducing intercar- amplifier OBO, DPD, and clipping and filtering,’’ in Proc. 26th Int. Tele- rier interference due to carrier frequency offset in OFDM,’’ IEEE Trans. traffic Congr. (ITC), Sep. 2014, pp. 1–5. Commun., vol. 47, no. 3, pp. 365–369, Mar. 1999. [23] Y. Zhao and S.-G. Haggman, ‘‘Sensitivity to Doppler shift and carrier frequency errors in OFDM systems-the consequences and solutions,’’ in Proc. IEEE 46th Veh. Technol. Conf., vol. 3, May 1996, pp. 1564–1568. [24] Y. Zhao and S. G. Haggman, ‘‘Intercarrier interference self-cancellation scheme for OFDM mobile communication systems,’’ IEEE Trans. Com- mun., vol. 49, no. 7, pp. 1185–1191, Jul. 2001. [25] J. Shentu and J. Armstrong, ‘‘Blind estimation of time and frequency offset for PCC-OFDM,’’ in Proc. IEEE 53rd Veh. Technol. Conf., vol. 1, May 2001, pp. 702–706. [26] J. Shentu, K. Panta, and J. Armstrong, ‘‘Effects of phase noise on perfor- GAYATHRI KONGARA received the Ph.D. degree mance of OFDM systems using an ICI cancellation scheme,’’ IEEE Trans. in electrical and electronic engineering from the Broadcast., vol. 49, no. 2, pp. 221–224, Jun. 2003. University of Canterbury, in 2010. She was a [27] W. T. Ng and V. K. Dubey, ‘‘Analysis of PCC-OFDM systems for general Research Engineer with the Wireless Research time-varying channel,’’ IEEE Commun. Lett., vol. 9, no. 5, pp. 394–396, May 2005. Center (WRC), NZ, until 2013. In 2014, she [28] A. S. Alaraimi and T. Hashimoto, ‘‘Performance analysis of polyno- joined Monash University. She specializes in the mial cancellation coding for OFDM systems over time-varying Rayleigh design and implementation of signal processing fading channels,’’ IEICE Trans. Commun., vol. 88, no. 2, pp. 471–477, algorithms for 5G wireless communication sys- 2005. tems. Her current research interests include MIMO [29] K. Panta and J. Armstrong, ‘‘Spectral analysis of OFDM signals and its receiver implementation, physical layer for 5G, improvement by polynomial cancellation coding,’’ IEEE Trans. Consum. and software defined radios. Electron., vol. 49, no. 4, pp. 939–943, Nov. 2003. [30] T. Hashimoto, A. S. Alaraimi, and C. Han, ‘‘ICI mitigation schemes for uncoded OFDM over channels with Doppler spreads and frequency offsets—Part I: Review,’’ J. Commun., vol. 7, no. 9, pp. 676–685, 2012. [31] T. Hashimoto, A. S. Alaraimi, and C. Han, ‘‘ICI mitigation schemes for uncoded OFDM over channels with Doppler spreads and frequency offsets—Part II: Asymptotic analysis,’’ J. Commun., vol. 7, no. 9, pp. 686–700, 2012. [32] J. Armstrong, T. Gill, and C. Tellambura, ‘‘Performance of PCC-OFDM with overlapping symbol periods in a multipath channel,’’ in Proc. IEEE Global Telecommun. Conf., vol. 1, Nov. 2000, pp. 87–91. CUIWEI HE received the B.E. degree in telecom- [33] R1-162199: Waveform Candidates, Qualcomm Incorporated, San Diego, munication engineering from the Central South CA, USA, 2016. University of Forestry and Technology, Changsha, [34] G. J. Foschini and M. J. Gans, ‘‘On limits of wireless communications China, in 2010, the M.E. degree in digital signal in a fading environment when using multiple antennas,’’ Wireless Pers. processing from the University of the Ryukyus, Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. Okinawa, Japan, in 2013, and the Ph.D. degree [35] İ. E. Telatar, ‘‘Capacity of multi-antenna Gaussian channels,’’ Eur. Trans. in optical wireless communication from Monash Telecommun., vol. 10, no. 6, pp. 585–595, 1999. [36] S. Wang, J. S. Thompson, and P. M. Grant, ‘‘Closed-form expres- University, Melbourne, Australia, in 2017. He has sions for ICI/ISI in filtered OFDM systems for asynchronous 5G been a Research Fellow with the Department of uplink,’’ IEEE Trans. Commun., vol. 65, no. 11, pp. 4886–4898, Electrical and Computer Systems Engineering, Nov. 2017. Monash University, since September 2017. His current research interests [37] S. Wang, J. Armstrong, and J. S. Thompson, ‘‘Waveform performance for include digital signal processing, optical wireless communication, multiple- asynchronous wireless 5G uplink communications,’’ in Proc. IEEE Pers., input multiple-output (MIMO) technology, and optical orthogonal frequency Indoor, Mobile Radio Commun. (PIMRC), Sep. 2016, pp. 1–6. division multiplexing (OFDM).

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LEI YANG received the B.E. and Ph.D. degrees in JEAN ARMSTRONG received the B.Sc. degree electrical and computer system engineering from (Hons.) in electrical engineering from The Uni- Monash University, in 2008 and 2012, respec- versity of Edinburgh, Edinburgh, U.K., and the tively. In 2011, he joined Nokia Networks as a M.Sc. degree in digital techniques from Heriot- Senior Algorithm and System Engineer, where he Watt University, Edinburgh, U.K. She is currently worked on 4G/5G key technologies such as CoMP a Professor with Monash University, Melbourne, and MIMO. Since 2017, he has been involved in VIC, Australia, where she leads a research group new 5G waveform and optical wireless communi- working on topics including optical wireless, radio cation research with Monash University. frequency, and optical ﬁber communications. She has published numerous articles including more than 70 on aspects of OFDM for wireless and optical communications and has six commercialized patents. Before emigrating to Australia, she was a Design Engineer with Hewlett-Packard Ltd., U.K. In Australia, she has held a range of academic positions at Monash University, The University of Melbourne, and La Trobe University. During her career, she has received numerous awards, including a Carolyn Haslett Memorial Scholarship, a Zonta Amelia Earhart Fellowship, the Peter Doherty Prize for the Best Commercialization Opportunity in Australia (joint winner), induction into the Victorian Honour Roll of Women, the 2014 IEEE Communications Society Best Tutorial Paper Award, and the 2016 IET Mountbatten Medal. She is a Fellow of IEAust. She has served on the Aus- tralian Research Council (ARC) College of Experts and the ARC Research Evaluation Committee.

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