Constant Envelope Modulation Techniques for Limited Power Millimeter Wave Links

electronics

Article Constant Envelope Modulation Techniques for Limited Power Millimeter Wave Links

Yael Balal *, Monika Pinchas and Yosef Pinhasi Faculty of Engineering, Ariel University, Ariel 40700, Israel; [email protected] (M.P.); [email protected] (Y.P.) * Correspondence: [email protected]; Tel.: +972-3-9066272

Received: 18 November 2019; Accepted: 3 December 2019; Published: 11 December 2019

Abstract: The demand for increased capacity and link availability for mobile communications requires the utilization of higher frequencies, such as millimeter waves at extremely high frequencies (EHFs) above 30 GHz. In this regime of frequencies, the waves are subjected to high atmospheric attenuation and dispersion effects that lead to a degradation in communication reliability. The fact that solid-state millimeter and sub-millimeter wave sources are producing low power calls for effective signaling utilizing waveforms with a low peak to average power ratio (PAPR), such as constant envelope (CE) modulation. The CE techniques present a PAPR of 0 dB resulting in peak power transmission with high energy efficiency. The study of the performances of constant envelope orthogonal modulation techniques in the presence of co-channel interference is presented. The performance is evaluated in terms of the average symbol error rate (SER) using analytical results and simulations. The theory is carried out for the CE-M-ary time orthogonal (CE-MTO) and CE-orthogonal frequency division multiplexing (CE-OFDM), demonstrating comparable performances while leading to a simpler implementation than that of the CE-OFDM.

Keywords: co-channel interference (CCI); constant envelope OFDM (CE-OFDM); constant envelope MTO (CE-MTO); millimeter wave communications

1. Introduction Wireless communications require more bandwidth because of the necessity of the capacity increasing, the availability, and the reliability. As a result, new bands need to be searched for in the electromagnetic spectrum, reaching millimeter and sub-millimeter wavelengths. Recent technological developments have made extremely high frequencies (EHF) a candidate for wireless applications, such as for the ﬁfth generation (5G) of cellular communications (for which bands in the 24.25–29.5 GHz and 37–43.5 GHz are already allocated [1]), satellite communications [2,3], high resolution radars [4,5], and remote sensing [6,7]. The realization of wireless communications in the millimeter wavelength systems is becoming commercial, compact, and less expensive [8,9]. However, the fact that the atmospheric medium is not entirely transparent to millimeter waves (MMWs) requires careful considerations regarding the frequency selective absorption and dispersion eﬀects emerging in this band [10]. Moreover, low power transmissions and reduced receiver sensitivities lead to further degradation in the link performances [11]. These phenomena also apply to radars and remote sensing systems operating in the millimeter and Terahertz frequencies [12]. While the bands in the ultra-high frequency (UHF) spectrum are fully used, millimeter waves (the EHF spectrum) are relatively free of users, and broad bands of frequencies can be allocated for wireless communication applications. With the demand for more spectrums for the mobile communication infrastructure, additional bands within the millimeter waves regime are allocated for the future 5G of the cellular networks [13]. The extension of the spectrums enables ultra-reliable low-latency communication (URLLC) broadband services, with faster access and an increased capacity.

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The short wavelength of the millimeter waves enables the utilization of small equipment and antennas. The MMW cell will be a few hundredths of a meter, having multiple antennas communicating in a massive multiple-input multiple-output (MIMO) arrangement [14,15]. Beamforming techniques will allow for the operation of multiple antennas to direct the transmission to the operators [16–18]. The power produced by solid-state devices is limited to several tens of milli-watts in MMWs. This limitation calls for effective signaling utilizing constant envelope (CE) waveforms with a low peak to average power ratio (PAPR). The utilization of constant envelope techniques enables a non-linear class of operation, resulting in a better efficiency of energy consumption. This is particularly important for battery-powered mobile appliances because of their limited power resources [19–21]. Constant envelope (CE) orthogonal signaling is proposed as an efficient modulation technique for wireless communication systems [22,23]. The peak to average power ratio (PAPR) in a CE waveform is 0 dB. The popular orthogonal frequency division multiplexing (OFDM) signaling, which is commonly used in communications, presents envelope fluctuations, resulting in a high PAPR [24]. Several CE techniques have been suggested. In the literature [25,26], the information-bearing message signal is transformed into a constant envelope waveform by the utilization of the phase modulation. In the CE-OFDM discussed in the literature [25], the OFDM waveform is used to phase modulate the carrier, while in our proposed CE-M-ary time orthogonal (CE-MTO) technique described in [26], orthogonal waveforms in the time domain were used. In our previous work [26], we showed that the implementation of the CE-MTO technique is expected to be simpler than that of the CE-OFDM, and that the symbol error rate (SER) performances in the presence of additive white gaussian noise (AWGN) and fading channels are comparable for both techniques. Interference is a significant challenge of wireless communications that has a detrimental effect on the overall link performances. Although it cannot be completely mitigated, the interference effect can be reduced to a certain extent. Several types of interferences exist [27], one of which is co-channel interference (CCI) [28]. It occurs when the interfering signal overlaps with the desired signal in the frequency domain. In order to mitigate the CCI, many methods have been proposed [29–32]. Channel coding is a traditional and effective way to reduce the CCI, and also provides some robustness [33]. In this paper, we examine the symbol error rate (SER) in the presence of CCI and AWGN, for both CE-OFDM and CE-MTO techniques. An analytical expression is derived, from which one can estimate the degradation caused by the CCI in an MMW channel employing constant envelope signaling. In Section2, we review the link budgets for the required signal and the interfering one, both operating in the millimeter wave regime. Section3 presents an analytic study of the detection for a constant envelope modulated signal in the presence of co-channel interference. In Section4, additive white Gaussian noise is added to the analysis for calculating the symbol error rate. The simulation results are given in Section5 for di fferent CCI scenarios. A comparison is made with the results obtained from the analytical calculations. Section6 summarizes and concludes the paper.

2. Communication Links in Millimeter Waves When the electromagnetic radiation in the millimeter wavelengths and terahertz frequencies regime propagates through the atmosphere, it experiences selective molecular absorptions [34–40]. Several empirical and analytical models have been suggested for estimating the millimeter and sub-millimeter wave transmission of the atmospheric medium. The transmission characteristics of the atmosphere at the EHF band were calculated using the millimeter propagation model (MPM), developed by Liebe [41–44]. The inhomogeneous atmospheric transmission aﬀects the amplitude and phase of the signals transmitted in the millimeter and sub-millimeter wavelengths [45]. The transmission characteristics of the atmosphere are shown in Figure1. In this example, graphs are drawn for several values of relative humidity (RH = 0%, 25%, 50%, 75%, and 100%) at sea level, assuming clear skies and no fog or rain. The attenuation factor, α( f ) in dB/km, is drawn in Figure1a, revealing absorption peaks at 22 and 183 GHz, where the resonance absorption of water (H2O) occurs; as well as absorption peaks at 60 and 119 GHz, due to the absorption resonances of oxygen (O2)[46–48]. Electronics 2019, 8, x FOR PEER REVIEW 3 of 13 Electronics 2019, 8, 1521 3 of 13

(H2O) occurs; as well as absorption peaks at 60 and 119 GHz, due to the absorption resonances of Aoxygen minimum (O2) [46–48]. attenuation A minimum is obtained attenuation at atmospheric is obtained transmission at atmospheric “windows” transmission in the Ka- “windows” (35 GHz) and in W-bandsthe Ka- (35 (94 GHz) GHz), and as W-bands well in the (94 vicinity GHz), as of well 130 andin the 220 vicinity GHz [37of ].130 The and gray 220 bandsGHz [37]. represent The gray the bandsspectrum represent allocated the forspectrum the 5G. allocated In Figure for1b, the we 5G. show In Figure the incremental 1b, we show group the incremental delay (in ps /groupkm) caused delay (inby theps/km) phase caused variation by the in thephase frequency. variation The in atmosphericthe frequency. transfer The atmospheric characteristics transfer can be characteristics calculated for canany be value calculated of the relative for any humidityvalue of the (RH). relative humidity (RH).

(a)

(b)

Figure 1.1. TheThe eﬀeffectect of of the the atmosphere atmosphere on (ona) attenuation,(a) attenuation,α( f ) inα dB()f /km, in anddB/km, (b) group and ( delayb) group increment, delay ∆τd( f ) in psΔ/km.τ () increment, d f in ps/km. When carrying out a link budget, the atmospheric attenuation must be taken into account. In millimeterWhen carrying wave links,out a thelink antennas budget, arethe usuallyatmospheric directive, attenuation resulting must mainly be taken in line-of-sight into account. (LOS) In millimeter wave links, the antennas are usually directive, resulting mainly in line-of-sight (LOS) paths. When a transmitter (T; with an eﬀective isotropic radiated power (EIRPT)) and an interferer (I; paths. When a transmitter (T; with an effective isotropic radiated power ( EIRP )) and an interferer (I; with EIRPI) transmit a signal to a receiver in their LOS (see Figure2), the correspondingT power from withthe transmitting EIRPI ) transmit station a signal is as follows: to a receiver in their LOS (see Figure 2), the corresponding power from the transmitting station is as follows: 2 λ 2α( f )dTR PSignal = GR 2 e− EIRPT (1) 4πdλTR · − α() · PG=⋅⋅ e2 fdTR EIRP (1) Signal Rπ T and from the interferer station, it is as follows:4 dTR and from the interferer station, it is as follows: 2 λ 2α( f )dIR PInter f erence = GR e− EIRPI (2) 4πdIR 2 · · λ − α() =⋅⋅2 fdIR PGInterference RI eEIRP (2) G = c πf where R is the receiver antennas gain, λ /4 isdIR the wavelength of the transmitted signal (c is the speed of light, f is the frequency), dTR and dTI are the distances between the transmitter or interferer

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λ = where GR is the receiver antennas gain, cf/ is the wavelength of the transmitted signal (c is Electronics 2019, 8, 1521 4 of 13 the speed of light, f is the frequency), dTR and dTI are the distances between the transmitter or interferer to the receiver, respectively, and 2α ()f is the power attenuation factor caused by the toatmospheric the receiver, medium. respectively, The signal and 2α to( f )interferenceis the power po attenuationwer ratio at factor the receiver caused bysite the is atmosphericgiven by the medium.following: The signal to interference power ratio at the receiver site is given by the following:

!22 P P ddEIRP−⋅α () − EIRP Transmitter==⋅⋅Transmitter IRIR2α (dddTRTR IRdIR) T T SIR = SIR= e e− · − (3)(3) P d · · EIRP InterPd fInterference erence TRTR EIRP I I

Figure 2. Illustration of the signal transmission in the presence of interference. EIRP—effective isotropic Figure 2. Illustration of the signal transmission in the presence of interference. EIRP—effective radiated power. isotropic radiated power. 3. Error Rate Degradation in the Presence of Co-Channel Interference 3. Error Rate Degradation in the Presence of Co-Channel Interference Co-channel interference occurs between two transmitters that are using the same frequency. The co-channelCo-channel interference interference can severely occurs aff betweenect the performance two transm ofitters the that symbol are errorusing rate the (SER).same frequency. In this section, The theco-channel SER performance interference properties can severely of the affect CE-MTO the perf andormance the CE-OFDM of the symbol in the presenceerror rate of (SER). co-channel In this interferencesection, the areSER studied. performance properties of the CE-MTO and the CE-OFDM in the presence of co- channelThe signalinterference arriving are at studied. the receiver from the transmitter is a constant envelope phase-modulated waveform.The signal Its complex arriving amplitude at the receiver at baseband from the is giventransmitter by the is following a constant [25 envelope]: phase-modulated waveform. Its complex amplitude at baseband is given by the following [25]: ( ) = jφ(t) s t ()A=ce jtφ() (4) s tAec (4) where j = √ 1 and Ac is the (constant) signal envelope. φ(t) is the information-bearing message =−− φ() phase.where Assumingj 1 and an in-band Ac is the interference, (constant) itssignal respective envelope. complex t amplitudeis the information-bearing is as follows: message phase. Assuming an in-band interference, its respective complex amplitude is as follows: A ι(t) = c ejθ(t) (5) √ Ac jtθ () ι ()te= SIR (5) SIR with the power given by the variance, as follows: with the power given by the variance, as follows: h i A2 E ι2(t) = cA2 (6) Etι2 () SIR= c (6) SIR where we define SIR as the signal to interference power ratio at the receiver site, as in Equation (3). Thewhere interfering we define signal SIR phase, as theθ signal(t), is ato stochastic interference process powe thatr ratio is uniformly at the receiver distributed site, as( inπ, Equationπ). Both the(3). θ ()t − ()−ππ, signalThe interfering and interference signal phase, are received , is simultaneously, a stochastic process resulting that is in uniformly a composite distributed waveform, written. Both as follows:the signal and interference are received simultaneously, resulting in a composite waveform, written as follows:

φθ() 11() 1 ()=+=⋅ ()ιφθφθ () jt + jt =() +() + () + () (7) rstttAecc e A cos t cos  t jAt c sin  sin  t SIR SIR SIR (7) 144444424444443144444424444443 It() Qt ()

where its phase is as follows:

Electronics 2019, 8, 1521 5 of 13 where its phase is as follows:

 1  " #  sin[φ(t)] + sin[θ(t)]  Q(t)  √SIR  φˆ (t) = arctan = arctan . (8) I(t) cos[φ(t)] + 1 cos[θ(t)]   √SIR 

Note that when SIR , the resulted phase is φˆ (t) = φ(t), equal to that of the required signal, → ∞ as expected. In order to calculate the interference component in the phase, we assume φ(t) = 0. In this case, the phase ﬂuctuation due to the interferer is as follows:      sin[θ(t)]  ξ(t) = arctan . (9)  √SIR + cos[θ(t)] 

Further analysis considers the detection of orthogonal modulation techniques (CE-OFDM [25] and CE-MTO [26] techniques). In the receiver (see Figure3), after an analog to digital conversion, the samples, r[n], are sent to the phase demodulator. The output of the phase demodulator in the continuous-time presentation is φˆ (t) = φ(t) + ξ(t), where ξ(t) is the phase interference component given by Equation (9). The amplitude is kept constant by a limiter. A set of matched ﬁlters calculate the correlations, as follows:

N N Xchip Xchipn o Q[k] = φˆ [n]qk[n] = φ[n] + ξ[n] qk[n] = S[k] + I[k] (10) n=1 n=1 ( ) PN where φ[n] = 2πhC Ik qk[n] , h is the modulation index, and C is a constant used to normalize the k=1 · variance of the resulted phase, namely: I N 1, 3, ... , (M 1) , where M is the pulse-amplitude { k}k=1 ∈ ± ± ± − modulation (PAM) constellation, Ik is the N real valued data symbols, and qk[n] is the discrete orthogonal waveforms. In the CE-OFDM technique, qk[n] is the orthogonal subcarriers’ quadrature components, while in CE-MTO qk[n] is the orthogonal series generated by the Hadamard matrix. S[k] is the signal component, as follows:

N N   Xchip Xchip XNsa  2  S[k] = φ[n]q [n] = 2πhCx[n]q [n] = 2πhC q [n]I = 2πhCqIˆ = dI (11) k k  k  k k k n=1 n=1 n=1 p where d = 2πhCqˆ, and for the CE-MTO case, C = 3/N(M2 1) and qˆ = N , where N − chip chip is the number of chips contained in the orthogonal waveforms, qk[n]. For the CE-OFDM case, N p 2 chip C = 6/N(M 1) and qˆ = , where N = NDFT is the number of points in the inverse discrete − 2 chip Fourier transformation (IDFT). The interference component in Equation (10) is as follows:

XNsa I[k] = ξ[n]qk[n]. (12) n=1

As ξ[n] is the independent random values, the sum of Equation (12) makes I[k] approximately Gaussian, having I[k] N 0, σ2 . Based on the literature [25], we may approximate SER as follows: ∼ I ! M 1 d SER 2 − Q (13) ≈ M σI where in our case, we have the following: Electronics 2019, 8, x FOR PEER REVIEW 5 of 13

φθ() + 1 () sintt sin  φˆ ==Qt() SIR . ()ttantanarc arc  (8) It() φθ+ 1 cos()tt cos () SIR

Note that when SIR →∞, the resulted phase is φφˆ()tt= (), equal to that of the required signal, as expected. In order to calculate the interference component in the phase, we assume φ ()t = 0 . In this case, the phase fluctuation due to the interferer is as follows:

sin θ ()t Electronics 2019, 8, x FOR PEER REVIEW ξ ()ttan= arc  6 of 13 + θ () . (9) SIRcos  t NN Electronics 2019, 8, 1521 chip chip Nsa 6 of 13 Further analysis[]== considersφπ [] [] the detection [] of []orth =ogonal π modulation2 [] ==tec πhniquesˆ (CE-OFDM [25] S k n qkkkkkk n22 hCx n q n hC  q n I 2 hCqI dI (11) and CE-MTO [26] techniques).nn==11 In the receiver (see Figure 3), after n = 1 an analog to digital conversion, the rn[] samples, , are sent to the phase demodulator.s The output2 of the phase demodulator in the where dhCq= 2π ˆ , and for the CE-MTOq case, CNM=−31()h andi qNˆ = , where N is the ˆ 2 1 2 chip chip continuous-time presentation is σφφξI ()=ttt=+qEˆ ()[ξ [n ()]] , whereqˆ ξE()ιt [ n is] .the phase interference component (14) ≈ 2 2Ac qn[ ] numbergiven by of Equation chips contained (9). The amplitude in the orthogonalis kept consta waveforms,nt by a limiter. k A set. For of matchedthe CE-OFDM filters calculate case, The substitution of Equation (6) in Equation (14), and σ and d in Equation (13), results, for both the correlations,2 as follows:N chip I CNM=−61() qˆ = NN= CE-MTO and CE-OFDM, and in the following:, where chip DFT is the number of points in the inverse discrete 2 NNchip chip []==+=+Ιφφξˆ [] []{} [] [] [] [] [] Fourier transformation (IDFT).QSk The interference nqnkk component ns nqnin Equation k (10) kis as follows: (10) nn==11M 1  6NchipSIR  SER 2 − QNsa2πh . (15) Ι=M[]ξ [] []( 2 )  ≈ knqn k N M. 1  (12) n=1 −

As ξ [n] is the independent random values, the sum of Equation (12) makes Ι[k] approximately

2 Gaussian, having Ι[kN]  ()0,σΙ . Based on the literature [25], we may approximate SER as follows:

M −1 d SER≈ 2 Q σ (13) M Ι where in our case, we have the following:

1 σξ=≈qE垐 22[] n q E  ι[] n I 2 . (14) 2Ac

The substitution of Equation (6) in Equation (14), andσ Ι and d in Equation (13), results, for both CE- MTO and CE-OFDM, in the following:  M −1 6N chip SIR SER≈ 22 Qπ h . (15) FigureFigure 3.3. ConstantConstant envelopeenvelope M-aryM-aryM timetime orthogonalorthogonalNM() (CE-MTO)2(CE-MTO)−1 blockblock diagram.diagram.  AA comparison comparison is is made madeN between between the the analytical analytical results results obtained obtained using using Equation Equation (15) (15) and and those those φπ[]nhCIqn=⋅2 [] C obtainedobtainedwhere from from simulationssimulations kk ofof the the CE-MTO CE-MTO, h is and the and the modulation the CE-OFDM CE-OFDM demodulationindex, demodulation and in theis ina presence constantthe presence of externalused of to k=1 interference. Figure4 shows the graphs of the SER as a function of the SIR. The correlations between external interference. Figure 4 shows the graphs of the SER asN a function of the SIR. The correlations the graphs are revealed. The graphs were drawn assuming{}IMN∈±{}=1, ±16 3,...,chips= ± ( for − 1 each ) of the N = 14 betweennormalize the thegraphs variance are revealed. of the resulted The graphs phase, were namely: drawn assumingk k=1chip Nchip 16 chips, forwhere each M of is the the orthogonal waveforms. Several PAM constellation orders, M = 4, 8, and 16, wre considered. In order Npulse-amplitude = 14 orthogonal waveforms.modulation Severa(PAM)l PAMconstellation, constellation I is orders, the N realM = valued4, 8, and data 16, wresymbols, considered. and qn In[] to keep the bandwidth restrained, the modulation indexk is set to 2πh = 0.4, 0.5, and 0.6. k order to keep the bandwidth restrained, the modulation index is set to 2πh = 0.4,0.5,and[] 0.6. is the discrete orthogonal waveforms. In the CE-OFDM technique, qnk is the orthogonal [] subcarriers’ quadrature components, while in CE-MTO qnk is the orthogonal series generated by the Hadamard matrix. S []k is the signal component, as follows:

(a)

Figure 4. Cont.

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(b)

FigureFigure 4. 4. ComparisonComparison of of the the symbol symbol error error rate rate (SER) (SER) as as a afunction function of of the the signal signal to to interference interference (SIR) (SIR) power ratio for the interfered link, N = = 16, and N= = 14:(a)M = 4, 8, and 16,π 2π=h = 0.6, and (b) power ratio for the interfered link, Nchipchip 16 , and N 14 : (a) M = 4, 8, and 16, 20.6h , and (b) M M = 4, 2πh = 0.4, 0.5, 0.6. CE-MTO—constant envelope M-ary time orthogonal; OFDM—orthogonal = 4, 2π h = 0.4,0.5,0.6 . CE-MTO—constant envelope M-ary time orthogonal; OFDM—orthogonal frequency division multiplexing. frequency division multiplexing. 4. Performance of AWGN Channel in the Presence of CCI 4. Performance of AWGN Channel in the Presence of CCI In the presence of additive white Gaussian noise, n(t), in addition to the interference (see Figure3), the receivedIn the presence signal can of beadditive written white as follows: Gaussian noise, nt(), in addition to the interference (see Figure 3), the received signal can be written as follows: r(t) = s(t) + ι(t) + n(t). (16) rs()tttnt=++ ()ι () (). (16) Due to the fact that ι(t) and n(t) are independent random processes, that is, σ2 = σ2 + σ2 , where the I+N I 222N ι () nt() σσσ=+ DueAWGN to the standard fact that deviation t and is based on are the independent following [26 ]:random processes, that is, Ι+NN Ι , where the AWGN standard deviation is basedtv on the following [26]: qˆ2 σ = 2 . (17) N E ˆ σ = 2 b Nqlog M N N0 2 . Eb (17) 2logNM2 Here, Eb is the energy per bit of the transmitted signalN0 and N0 is the power spectral density (PSD) of the AWGN. In order to calculate the SER analytical expression of this new scenario, the expression of Here, E is the energy per bit of the transmitted signal and N is the power spectral density (PSD) (13) is used,b leading to: 0 of the AWGN. In order to calculate the SER analytical expression of this new scenario, the expression    uv  of (13) is used, leading to:  tu Eb    6NchipSIR log M  M 1  d  M 1  N0 2  SER 2 − Q q  = 2 − Q2πh . (18) M   M  2 Eb  ≈  2 2   (M 1) N logEbM + N SIR   σ + σ  6logNSIRN0 2 Mchip −−I N − chip N 2 MdM11 0 SER≈=222 Q  Qπ h . (18) MM22  A comparison is made betweenσσ+ the  analytical results obtained2 Eb using Equation (18) and the Ι N −+  ()MNMNSIR1log2 chip N simulation results of the CE-MTO and the CE-OFDM techniques in the presence0 of external interference and AWGN. Figure5 shows the simulated results for the SER as a function of the Eb/N0. It is A comparison is made between the analytical results obtained using Equation (18) and the noticeable that there is a high correlation between the simulated results for both modulation techniques simulation results of the CE-MTO and the CE-OFDM techniques in the presence of external (CE-OFDM and CE-MTO). interference and AWGN. Figure 5 shows the simulated results for the SER as a function of the ENb 0 . It is noticeable that there is a high correlation between the simulated results for both modulation techniques (CE-OFDM and CE-MTO).

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FigureFigure 5. 5.Comparison Comparison of of the the SER SER as as a a function function of ofE bEN/bN0 0for for interfered interfered link link in in the the present present of of additive additive white Gaussian noise (AWGN). N ==16, N = =14, M = 4, 2πhπ= 0.6= and SIR = 15 dB. white Gaussian noise (AWGN). chipNchip 16 , N 14 , M = 4, 20.6h and SIR = 15 dB.

The graphs in Figure5 have been created for N = 14 orthogonal waveforms, each containing The graphs in Figure 5 have been created for N =14 orthogonal waveforms, each containing Nchip = 16 chips. The PAM constellation order is M = 4 and the modulation index is kept at 2πh = 0.6. N =16 chips. The PAM constellation order is M = 4 and the modulation index is kept at 20.6π h = Thechip signal to interference power ratio is assumed to be SIR = 15 dB. The “Analytical AWGN only” curve. Therefers signal to to the interference analytical power expression ratio obtainedis assumed in ourto be previous SIR = 15 work dB. The [26 ]“Analytical in Equation AWGN (21) for only” the casecurve of therefers AWGN to the channel analytical only. expression The “Analytical obtained Interference in our previous only” work and the [26] “Analytical in Equation Interference (21) for the andcase AWGN” of the AWGN curves channel refer to Equationsonly. The “Analytical (15) and (18), Interference respectively. only” Figure and5 theshows “Analytical that for theInterference case of SIRand= AWGN”15 dB, M curves= 4, and refer2π hto= Equations0.6, the SER (15) forand the (18), “Analytical respectively. Interference Figure 5 shows only” isthat approximately for the case of π = equalSIR = to 15 0.025. dB, M Please = 4, and note the20.6h “Analytical, the SER AWGN for only”the “Analytical and the “Analytical Interference Interference only” is approximately only” curves functionequal to as 0.025. an asymptote Please note for the “Simulated”“Analytical AWGN and “Analytical only” and Interference the “Analytical and AWGN” Interference graphs only” in whichcurves the function interference as an and asymptote AWGN exists. for the “Simulated” and “Analytical Interference and AWGN” graphs in which the interference and AWGN exists. 5. Results for Different CCI Scenarios 5. Results for Different CCI Scenarios The simulation results for the different CCI scenarios are presented in the following. First, it is assumedThe that simulation the interferer results transmits for the different at the very CCI same scenar frequencyios are presented as the required in thetransmission. following. First, Then, it is aassumed frequency that shift, the∆ interfererf , is introduced transmits to the at the interferer very same carrier. frequency as the required transmission. Then, a frequencyThe analytical shift, EquationΔf , is introduced (18) for the to the SER interferer in the presence carrier. of interference and AWGN depends on the signalThe toanalytical interference Equation (SIR) (18) power for the ratio SER at in the the receiver, presence on of the interference modulation and index AWGN (2πh depends), and the on constellationthe signal to order interference (M). The (SIR) case whenpower the ratio carrier at the frequencies receiver, on of the transmittermodulation and index the ( interferer2π h ), and are the identicalconstellation is studied. order Note(M). thatThe forcase Figure when6 ,the the carrier “Analytical frequencies AWGN of only” the transmitter curve refers and to the the situation interferer whereare identical the PAM is constellationstudied. Note order that isfor M Figure= 4 and 6, thethe modulation“Analytical indexAWGN is 2onπhly”= curve0.6. In refers Figure to6 a,the wesituation show the where dependence the PAM on constellation the SIR parameter. order is AsM expected,= 4 and the a highermodulation SIR results index inis a20.6 lowerπ h = SER.. In FigureFigure6b 6a, shows we show the SER the dependence on the SIR modulation parameter. index, As expected,2πh. As thea higher value SIR of 2 resultsπh is increased, in a lower theSER. SER Figure decreases. 6b shows Finally, the FigureSER dependence6c demonstrates on the the modulation dependence index, on the2π constellationh . As the value M. of Smaller 2π h is constellationsincreased, the show SER adecreases. better performance. Finally, Figure 6c demonstrates the dependence on the constellation M. SmallerInterferer constellations shift in frequency show a better results performance. in phase θ( t) = 2π∆ f t, where ∆ f is the difference in carrier frequencies. Figure7 presents the spectrum of the CE-OFDM, CE-MTO, and the CW interference. Note that for the CE-MTO, Tchip is the time duration of a series “chip”, while for the CE-OFDM, Tchip = Tsymbol/N, where N is the number of subcarriers. The inspection of Figure7 reveals that the CE-MTO spectrum is null for the ∆ f T = integer. The interference is located at ∆ f T = 2. · chip · chip Comparing the two modulation techniques reveals a slight difference between the spectrum of the CE-OFDM and that of the CE-MTO.

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(a)

(b)

(c)

Figure 6. Comparison of the SER as a function of EN for the interfered link, N =16 , and Figure 6. Comparison of the SER as a function of Eb/N0 forb the0 interfered link, Nchip = 16, andchip N = 14. (a)MN ==144;. 2(πah) =M0.6 = ;4; and 20.6 SIRπ h == 10,; 15, and and SIR 20 dB.= 10, (b )M15, =and4; SIR 20= dB.20 dB;(b) andM 2=π h4;= SIR0.4, = 0.5, 20 anddB; 0.6and. (c)20.4,0.5,and0.6 2ππhh== 0.6; SIR = 20 dB;. (c and) 20.6π Mh = 4, 8,; SIR and = 16. 20 dB; and M = 4, 8, and 16.

Interferer shift in frequency results in phase θπ()tft=Δ2 , where Δf is the difference in carrier frequencies. Figure 7 presents the spectrum of the CE-OFDM, CE-MTO, and the CW interference.

Note that for the CE-MTO, Tchip is the time duration of a series “chip”, while for the CE-OFDM, = TTchip symbol N, where N is the number of subcarriers. The inspection of Figure 7 reveals that the CE-

Electronics 2019, 8, x FOR PEER REVIEW 10 of 13 Electronics 2019, 8, x FOR PEER REVIEW 10 of 13 Δ⋅ Δ⋅ = MTO spectrum is null for the f Tchip = integer. The interference is located at fTchip 2 . Comparing Δ⋅ Δ⋅ = theMTO two spectrum modulation is null techniques for the ref vealsTchip = a integer. slight differenceThe interference between is located the spectrum at fT ofchip the2 CE-OFDM. Comparing andthe that two of modulation the CE-MTO. techniques reveals a slight difference between the spectrum of the CE-OFDM Electronicsand that2019 of, the8, 1521 CE-MTO. 10 of 13

Figure 7. Comparison of the spectrum as a function of the normalized frequency. Figure 7. Comparison of the spectrum as a function of the normalized frequency. Figure 7. Comparison of the spectrum as a function of the normalized frequency. Figure 8a–c shows the graphs of the SER as a function of Eb N 0 . The results in Figure 8a–c have Figure 8a–c showsN =14 the graphs of the SER as a functionN =16 of E N . The results in Figure 8a–c have been obtainedFigure8a–c for shows the orthogonal graphs of the waveforms SER as a functionof chip of E chips.b/b N0.0 The resultsorder constellation in Figure8a–c is haveM = = = 4been andbeen obtainedthe obtained modulation for forN =N index1414orthogonal is orthogonal 20.6π h = waveforms waveforms. It is assumed of Nofchip Nthatchip= 16 the16chips. signal chips. The to The interference order order constellation constellation power isratioM is = Mis4 = π = Δ⋅ Δ⋅ = SIRand4 and= the 15 the modulationdB modulation for different index index isfrequency2π ish =20.60.6h shift. It is .values, assumedIt is assumed thatf Tchip the that. signalIn the Figure signal to interference 8a, to whereinterference powerfTchip power ratio0.3 is ratio, SIR the= is 15SIR dB = for 15 di dBﬀerent for frequencydifferent shiftfrequency values, shift∆ f Tvalues,chip. In FigureΔ⋅f T 8a,. whereIn Figure∆ f 8a,Tchip where= 0.3 ,Δ⋅ thefT performance =0.3 , the performance of CE-MTO is revealed to be ·better than thatchip of CE-OFDM.· However, in chipFigure 8b, of CE-MTOΔ⋅ is revealed = to be better than that of CE-OFDM. However, in Figure8b, where ∆ f Tchip = 0.6, whereperformance fTchip of0.6 CE-MTO, the CE-OFDM is revealed demonstrates to be better a thanbetter that performance of CE-OFDM. than However,that of CE-MTO. in· Figure The 8b, thewhere CE-OFDM Δ⋅fT demonstrates =0.6 , the CE-OFDMΔ⋅ a better = performance demonstrates than a better that of performance CE-MTO. The than curves that in of Figure CE-MTO.8c refer The curves in Figurechip 8c refer to fTchip 1. In this scenario, the performance of CE-MTO and that of CE- to ∆ f Tchip = 1. In this scenario, the performance of CE-MTO and that of CE-OFDM are identical. curves· in Figure 8c refer to Δ⋅fT =1. In this scenario, the performance of CE-MTO and that of CE- OFDMThe results are obtainedidentical. inThe Figure results8a–c chip areobtained as expected in Figure when 8a–c examining are as theexpected spectrum when as given examining in Figure the7. spectrumOFDM areas given identical. in Figure The 7.results obtained in Figure 8a–c are as expected when examining the spectrum as given in Figure 7.

(a) (a) Figure 8. Cont.

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(b)

(c)

Figure 8. Comparison of the SER as a function ofEEbN/N0 for the interfered link: (aΔ⋅) fT∆ f Tchip =0.3= 0.3, Figure 8. Comparison of the SER as a function of b 0 for the interfered link: (a) chip· , (b) (Δ⋅b) ∆ f =Tchip = 0.6, andΔ⋅ (c) ∆ f =Tchip = 1. fTchip· 0.6 , and (c) fTchip · 1. 6. Summary and Conclusions 6. Summary and Conclusions Small PAPR modulation schemes are being used for better power efficiency. In CE-OFDM, the carrierSmall PAPR is phase-modulated modulation schemes with the are OFDM being signaling used for tobetter generate power a constantefficiency. envelope In CE-OFDM, waveform. the carrierCE-MTO is phase-modulated is a new suggested with technique the OFDM for constant signalin envelopeg to generate modulation, a constant which envelope is based waveform. on orthogonal CE- MTOwaveforms is a new in suggested the time domain, technique and for with constant the implementation envelope modula expectedtion, which to be simpleris based thanon orthogonal that of the waveformsCE-OFDM. in An the analytical time domain, expression and with for thethe SERimplemen performancetation expected degradation to be caused simpler by than the co-channelthat of the CE-OFDM.interference An in analytical different scenarios expression is presented.for the SER Itperformance is shown that degradation the SER decreases caused by in the presenceco-channel of interferenceinterference in when different the modulation scenarios indexis presented. is increased. It is shown The SER that grows the SER the higherdecreases the constellationin the presence order. of interference when the modulation index is increased. The SER grows the higher the constellation Author Contributions: Conceptualization, Y.P. and M.P.; Formal analysis, Y.B.; Investigation, Y.B.; order.Writing—original draft, Y.B.; Writing—review & editing, Y.P. and M.P. AuthorFunding: Contributions:This research Conceptualization, received no external Y.P. funding. and M.P.; Formal analysis, Y.B.; Investigation, Y.B.; Writing – original draft, Y.B.; Writing – review & editing, Y.P. and M.P.Funding: This research received no external Conflicts of Interest: The authors declare no conflict of interest. funding.

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