Amplitude-modulated, Cosine PE and WP Pulses: Theory and Applicability

RADIOENGINEERING, VOL. 30, NO. 2, JUNE 2021 443 Amplitude-modulated, Cosine PE and WP Pulses: Theory and Applicability Ioan E. LAGER 1, Martin STUMPF 2 1Faculty EEMCS, Delft University of Technology, Mekelweg 4, 2628 CD Delft, the Netherlands 2Department of Radioelectronics, Brno University of Technology, 616 00 Brno, Czech Republic i.e.lag[email protected], mar[email protected] Submitted February 12, 2021 / Accepted March 2, 2021 Abstract. The amplitude-modulated, cosine powerexponential (PE) and windowed-power (WP) pulses are discussed, by insisting on their time-domain normalization. Illustrative examples of signatures and their correspondent frequency-domain behavior are given. These examples compellingly demonstrate the possibility to replace non-causal pulses of prevalent use by causal, or even time-windowed, pulses with closely resembling signatures. Keywords Causality, time-domain analysis, pulse generation 1. Introduction Pulse shape (discrimination) is a topic of broad scientific interest, ranging from particle physics [1–3] (where it allows distinguishing between signals from neutrons and gamma rays) to ubiquitous technologies such as radar and (wireless) communication. When zooming in on wireless communications, the major challenge is responding to the exploding needs for data transfer, with ultra-wideband (UWB) technology being one of the most viable candidates for securing ultra-high data rates in conjunction with manageable power requirements [4]. Reliable and fast data transfer critically depends on preserving the pulse shape, with [5–7] focussing on the UWB figures of merit that are of relevance to this end. One of the important roadblocks in achieving high throughput and low bit errors is the intersymbol interference (ISI) [8, p. 43]. Here, dispersion and pulse ringing are the two channel-related causes of ISI, with ringing putting a cap on the pulse repetition rate and, thus, on the baud-rate. As a result, important efforts are invested in mitigating the pulse ringing, as evidenced by the (technological) approaches advocated in [9–12]. At this point, it is noted that all explorations in [5–7,9–12] rely on using (time-differentiated) Gaussian pulses, a patently non-causal signature1. Recently, [13] highlighted the need to ensure causality in electromagnetic (EM) investigations. That publication catalogued a number of strictly causal pulses, two of them having ringing shapes. This survey will examine the power-exponential (PE) and windowed-power (WP) amplitude-modulated, cosine pulses. While the PE variant was introduced in [14], the WP variant was not yet presented. A particular emphasis will be placed on the normalization of these pulses, that methodology being potentially applicable to the design of ringing shapes using different apodizations [15, p. 55]. Typical signatures and spectra of the advocated pulses will be provided. The pulses will then be shown to have the ability to accurately replicate time-domain (TD) signatures of pulses generated by existing circuitry, thus allowing the incorporation of causal, analytic copies of the relevant signals in (EM) models or within design strategies concerning complex systems. 2. Definition of Pulses The PE and WP unipolar prototype pulses were elaborately discussed in [14], [16], their TD expressions being PE(ν,tr,t)=t0νexp [−ν(t0−1)]H(t0)(1) and WP(ν,tr,t)=t0ν(2−t0)νH(t0)H(2−t0)(2) respectively. In these expressions ν⩾1is the pulse raising power, tris the pulse rise-time2,tis the time coordinate, t0=t/tris the normalized time coordinate and H(·) is the Heaviside step function. For practical reasons, νis taken to assume only integer values. Furthermore, taking ν⩾3 ensures the continuity of the pulses and their first two time derivatives at t0=0, and at t0=2for WP (see [14], [16]). 1Interestingly, [6] already noted the non-causality of the employed signature, but surmised that the inevitable truncation errors do not significantly impact on the system performance. 2The pulse rise-time tr>0is defined as the time between the unipolar pulse’s onset and the instant when it reaches its maximum. In the case of the time-differentiated versions of the PE and WP pulses, tralso represents the zero-crossing time of the relevant pulses. DOI: 10.13164/re.2021.0443 SIGNALS 444 I. E. LAGER, M. STUMPF, AMPLITUDE-MODULATED, COSINE PE AND WP PULSES: THEORY AND APPLICABILITY The frequency-domain (FD) behavior of the PE pulse is [14] c PE(ν,tr,ω)=tr Γ(ν+1)exp(ν) (jωtr+ν)ν+1(3) in which ωis the angular frequency (ω=2πf, with fbeing the frequency), and Γ(·)denotes the Euler gamma function – since νis integer, Γ(ν)=(ν−1)!. The FD behavior of the PE pulse is [16] d WP(ν,tr,ω)=exp(−jωtr)tr2νν!√2πJν+½(ωtr) (ωtr)ν+½(4) where Jn+½is the Bessel function of the first kind and fractional order [17, Sec. 10.1]. Note that |c PE(ω)| =|c PE(−ω)| and |d WP(ω)| =|d WP(−ω)| since the pulses are real functions. The PE was used in [14] for constructing the ringing cosine PEcos pulse via the expression PEcos(ν,tr, κw,t)=PE(ν,tr,t)cos(2π˜ωwt0)(5) in which κwis a ‘wiggle factor’ that, for a harmonic carrier of frequency f0, counts the carrier cycles per tras κw=trf0, and ˜ωw=2πκw. By using the same strategy, a WP ringing cosine pulse is defined via WPcos(ν,tr, κw,t)=WP(ν,tr,t)cos(˜ωwt0).(6) The frequency-domain (FD) behavior of PEcos and WPcos can be easily derived, the relevant expresions being c PEcos(ν,tr, κw, ω) =1 2hc PE(ν,tr,ω −ω0)+c PE(ν,tr,ω +ω0)i(7) and d WPcos(ν,tr, κw, ω) =1 2hd WP(ν,tr,ω −ω0)+d WP(ν,tr,ω +ω0)i(8) in which ω0=2πf0=2πκw/tr. For studying (electronic) circuits in which no net electric charge is transported (as the ones in antenna engineering), time-differentiated versions of the thus far introduced pulses are needed. The relevant pulses are denoted as ∂tPE, ∂tWP, ∂tPEcos and ∂tWPcos. Their TD and FD expressions follow from the corresponding, original expressions (the FD behavior requiring simply a multiplication by jω). 3. Normalization of Pulse TD Signatures The normalization of the ∂tPE and ∂tWP was elaborately discussed in [14], [16] and will not be repeated here. When turning to the ringing cosine pulses, it is observed that the normalization could be applied to the pulse itself. However, such a pulse normalization does not lend itself to an analytic handling and, thus, is not considered here. Instead, our approach is to normalize the pulses’ envelopes. By that standard, PEcos and WPcos are readily normalized, since the expressions (1) and (2) are implicitly normalized. The time-differentiated, ringing cosine pulses need a more careful examination. For simplicity, H(·) will be omitted from the derived expressions, its effect being implicitly accounted for. The explicit mentioning of trand tas arguments of PE and WP will be also omitted. By time-differentiating (5), it follows that ∂tPEcos(ν,tr, κw,t)=∂t[PE(ν)cos(˜ωwt0)] =ω0PE(ν)[K1(t0)cos(˜ωwt0)−sin(˜ωwt0)](9) in which K1(t0)=β1 1−t0 t0=1 tan [α1(t0)]for t0>0(10) with β1=1/˜ωw. By substituting (10) in (9), and using some standard trigonometric identities it is found that ∂tPEcos(ν,tr, κw,t)=ω0F1(t0)cos [˜ωwt0+α1(t0)](11) in which F1(t0)=PE(ν) sin [α1(t0)] =PE(ν−1)exp(1−t0)ht02+β2 1(1−t0)2i½ .(12) Since ν⩾3, PE(ν−1)is well-defined. The function F1(t0)is finite, with lim t0↓0 F1(t0)=0,F1(1)=1, and lim t0→∞ F1(t0)=0. The shape of F1(t0)is dictated by PE(ν−1)that peaks at 1 for t0=1. It can now be concluded that the envelope of the ∂tPEcos pulse is normalized by multiplying the time derivative by 1/ω0=(2πf0)−1, a fact already stated in [14, Eq. (15)]. By time-differentiating (6), it follows that ∂tWPcos(ν,tr, κw,t)=∂t[WP(ν)cos(˜ωwt0)] =ω0WP(ν)[K2(t0)cos(˜ωwt0)−sin(˜ωwt0)](13) in which K2(t0)=β2 1−t0 t0(2−t0)=tan [α2(t0)]for 0<t0<2(14) with β2=ν/(πκw). By substituting (14) in (13), and using some standard trigonometric identities it is found that ∂tWPcos(ν,tr, κw,t)=ω0F2(t0)cos [˜ωwt0+α2(t0)](15) in which F2(t0)=WP(ν) sin [α2(t0)] =WP(ν−1)ht04−4t03+(4+β2)t02−2β2 2t0+β2 2i½ .(16) RADIOENGINEERING, VOL. 30, NO. 2, JUNE 2021 445 Since ν⩾3, WP(ν−1)is well-defined. The multiplier of WP(ν−1)is a positive, smooth function that is symmetric with respect to t0=1and takes the values β2for t0=0and t0=2, and 1 for t0=1. The shape of F2(t0)is dictated by WP(ν−1)that peaks at 1 for t0=1. It can now be concluded that the envelope of the ∂tWPcos pulse is also normalized by multiplying the time derivative by 1/ω0=(2πf0)−1. 4. Numerical Experiments The practical utility of the introduced ringing signals is henceforth argued by examining some typical signatures and exploring their ability to replicate signals generated by existing circuitry. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.5 0 0.5 1 t/tr PEcos(ν, tr, κw, t) ν= 3 ν= 3 κw= 2 κw= 2 ν= 6 ν= 6 κw= 2 κw= 2 (a) Signatures −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 −100 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 log(f/fnorm) 20 log| c PEcos(ω)/ c PEcos(ω)max|(dB) ν= 3 ν= 3 κw= 2 κw= 2 ν= 6 ν= 6 κw= 2 κw= 2 fnorm =ν(κw/tr) fnorm =ν(κw/tr) c PEcos ν=3 asymptote c PEcos ν=3 c PEcos ν=6 asymptote c PEcos ν=6 (b) Spectral behavior Fig. 1. PEcos pulse with ν=3(dashed line) and ν=6(solid line), tr=1and κw=2. The solid patches in (a) represent the pulse envelopes, with the lighter patch corresponding to ν=6. The asymptotes of the spectral contents are shown in (b). 4.1 Examples Some typical signatures of ringing cosine pulses are now given. For each pulse, the signature, the envelope and the spectral behavior will be given. PE-type signals are firstly discussed. Figure 1 shows two PEcos pulses with tr=1,κw=2, and νassuming the values 3 and 6, respectively. The TD plots illustrate the effect of νon the pulse’s shape. The spectral behavior has a nonzero DC value and peaks, as expected, at f03. The ∂tPEcos performance is demonstrated in Fig. 2 for the same pulse parameters as those in Fig. 1. The differences in the TD signatures are small and they manifest primarily in the beginning of the pulse. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.5 0 0.5 1 t/tr ∂tPEcos(ν, tr, κw, t) ν= 3 ν= 3 κw= 2 κw= 2 ν= 6 ν= 6 κw= 2 κw= 2 (a) Signatures −5.5 −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 −100 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 log(f/fnorm) 20 log| d ∂tPEcos(ω)/ d ∂tPEcos(ω)max|(dB) ν= 3 ν= 3 κw= 2 κw= 2 ν= 6 ν= 6 κw= 2 κw= 2 fnorm =ν(κw/tr) fnorm =ν(κw/tr) d ∂tPEcos ν=3 asymptote d ∂tPEcos ν=3 d ∂tPEcos ν=6 asymptote d ∂tPEcos ν=6 (b) Spectral behavior Fig. 2. ∂tPEcos pulse with ν=3(dashed line) and ν=6(solid line), tr=1and κw=2. The solid patches in (a) represent the pulse envelopes, with the lighter patch corresponding to ν=6. The asymptotes of the spectral contents are shown in (b). 3The definition of fnorm entails that f0/fnorm =1/ν, implying that the spectral diagrams’ peaks correspond to the abscissas –0.47 and –0.78 for ν=3and 6, respectively. 446 I. E. LAGER, M. STUMPF, AMPLITUDE-MODULATED, COSINE PE AND WP PULSES: THEORY AND APPLICABILITY The spectral behavior resembles that of the PEcos above f0 but, evidently, has no DC component. From these signatures it can be inferred that the PE-modulated pulses are opportune for modeling ringing signatures with a sharp increase and a long tail, with ∂tPEcos having the capability to reflect the situations when no net electric charge is transported in a circuit. Upon now turning to WP-type signals, Figure 3 shows two WPcos pulses with tr=1,κw=4, and ν=4,8, respectively. Again, the effect of νon the pulse’s shape is noticeable. The spectral behavior bears some overall resemblance 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.5 0 0.5 1 t/tr WPcos(ν, tr, κw, t) ν= 4 ν= 4 κw= 4 κw= 4 ν= 8 ν= 8 κw= 4 κw= 4 (a) Signatures 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.5 0 0.5 1 t/t r WP cos (ν, t r ,κ w ,t) ν=4 ν=4 κ w =4 κ w =4 ν=8 ν=8 κ w =4 κ w =4 a −5.5 −5−4.5 −4−3.5 −3−2.5 −2−1.5 −1−0.5 0 0.5 1 −100 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 log(f/fnorm) 20 log|  WPcos(ω)/  WPcos(ω)max|(dB) ν=4 ν=4 κw=4 κw=4 ν=8 ν=8 κw=4 κw=4 fnorm =ν(κw/tr) fnorm =ν(κw/tr)  WPcos ν=4 asymptote  WPcos ν=4  WPcos ν=8 asymptote  WPcos ν=8 b (b) Spectral behavior Fig. 3. WPcos pulse with ν=4(dashed line) and ν=8(solid line), tr=1and κw=4. The solid patches in (a) represent the pulse envelopes, with the lighter patch corresponding to ν=8. The asymptotes of the spectral contents are shown in (b). with that in Fig. 1, but the lobe around f0is narrower and the DC level is lower, which concurs with the very low spectral leakage of the WP prototype proven in [16] (to a lesser degree, these effects are also referable to the higher νin these experiments). The corresponding ∂tWPcos performance following from using the same pulse parameters is exemplified in Fig. 4. The same observations as in the case of the PE-type signals can be made. The presented signatures recommend the WP-type pulses for situations requiring a time-windowed harmonic function. A typical such application can be found in the field of timed antennas [18] (the advantages of using the WP apodization were elaborately explained in [13], [16]). 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −1 −0.5 0 0.5 1 t/tr ∂tWPcos(ν, tr, κw, t) ν= 4 ν= 4 κw= 4 κw= 4 ν= 8 ν= 8 κw= 4 κw= 4 (a) Signatures −5.5−5−4.5 −4−3.5 −3−2.5−2 −1.5 −1 −0.5 0 0.5 1 −100 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 log(f/fnorm) 20 log| d ∂tWPcos(ω)/ d ∂tWPcos(ω)max|(dB) ν= 4 ν= 4 κw= 4 κw= 4 ν= 8 ν= 8 κw= 4 κw= 4 fnorm =ν(κw/tr) fnorm =ν(κw/tr) d ∂tWPcos ν=4 asymptote d ∂tWPcos ν=4 d ∂tWPcos ν=8 asymptote d ∂tWPcos ν=8 (b) Spectral behavior Fig. 4. ∂tWPcos pulse with ν=4(dashed line) and ν=8(solid line), tr=1and κw=4. The solid patches in (a) represent the pulse envelopes, with the lighter patch corresponding to ν=8. The asymptotes of the spectral contents are shown in (b). RADIOENGINEERING, VOL. 30, NO. 2, JUNE 2021 447 4.2 Replication A second investigation concerns the ability of the advocated pulses to mimic signatures that are effectively generated by existing circuitry. The signatures in [11, Fig. 2c,d] are selected to this end – note that those signatures were constructed by starting from non-causal Gaussian-type pulses. Hereafter, original signals will be generically denoted as Sorig(t), while Srepl(t)will stand for the constructed replica. The replication accuracy is assessed based on an energy percentage replication error that is calculated with the expression Err%=∫TSorig(t)−Srepl(t)2dt ∫TS2 orig(t)dt 100 (17) in which Tis the interval between the earliest among the onsets of Sorig and Srepl and the end Sorig. The original signals in [11, Fig. 2c,d] are shown in Figs. 5 and 6, respectively4. Since both signatures feature a long tail, a PE-type replication is selected based on the observations in Sec. 4.1. The mimicking was effectuated via a heuristic approach5employing a grid-search over ν,trand f0(and, implicitly, κw) that aimed at minimizing the error in (17). This procedure was implemented in Matlab®. Since the goal of this exercise was to obtain a good match and it was intended as a singular experiment, providing information on computational effectiveness is deemed irrelevant. To begin with, initial guesses were made for trand f0 by measuring some features of Sorig. It was then observed that the signature in [11, Fig. 2c] lends itself to a replication via a PE pulse with a low ν, whereas that in [11, Fig. 2d] lends itself to a replication via a ∂tPE pulse with a high ν. Based on these tests, a PE with ν=4and a ∂tPE with ν=24 were selected. The next phase was a full grid search on ranges of trand f0values centered around the initial guesses. The constructed pulses’ magnitudes (Vm) were determined by matching the maxima in the originals and the replicas. While these searches did yield some optimal values, the entailed Err%was considered too large. In order to improve the accuracy, the replicas’ onsets were allowed to move by some (small) tshift time shifts with respect to the onsets in Sorig. This choice entailed a third grid-search parameter that eventually, led to extremely satisfactory results. Specifically, the following replicas were constructed: 1. For the signature in [11, Fig. 2c]: A PE pulse with Vm=3.94 V,ν=4,tr=0.52 ns,f0=1.67 GHz (κw=0.87); the replica is delayed by tshift =119 ps. The replication error amounts to Err%=5.2%. The constructed Srepl is superimposed on Sorig in Fig. 5. 2. For the signature in [11, Fig. 2d]: A ∂tPE pulse with Vm=2.35 V,ν=24,tr=1.06 ns,f0=1.84 GHz (κw=1.95); the replica is delayed by tshift =7 ps. The replication error amounts to Err%=2.2%. The constructed Srepl is superimposed on Sorig in Fig. 6. Another example of mimicking a measured signal via a∂tPE signature can be found in [14, Sec. VI.A]. These experiments cogently prove the possibility to obtain extremely accurate causal, analytic copies of physically generated signals – these functions can then be used in models or within design strategies concerning complex systems. 77 77.5 78 78.5 79 79.5 80 80.5 −3 −2 −1 0 1 2 t(ns) V(t) (V) Original PEcos replica Fig. 5. Replication of the signature in [11, Fig. 2c] via a PEcos pulse with parameters: Vm=3.94 V,ν=4,tr=0.52 ns, f0=1.67 GHz (κw=0.87). The replica’s onset succedes the original signal’s onset by tshift =119 ps. The replication error amounts to Err%=5.2%. 77 78 79 80 81 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 t(ns) V(t) (V) Original ∂tPEcos replica Fig. 6. Replication of the signature in [11, Fig. 2d] via a ∂tPEcos pulse with parameters: Vm=2.35 V,ν=24, tr=1.06 ns,f0=1.84 GHz (κw=1.95). The replica’s onset precedes the original signal’s onset by tshift =7 ps. The replication error amounts to Err%=2.2%. 4The signatures in Figs. 5 and 6 intentionally exclude some small oscillations preceding the onset that were present in the signals given in [11] (those oscillations do not actually pertain to the intended pulses and are deemed as irrelevant for replication purposes). 5More sophisticated strategies could also have been employed. Nonetheless, this experiment aimed at demonstrating the similarity between the original signatures and some of the advocated pulses, and not at obtaining an optimal replication of it. The selected heuristic approach serves this purpose. 448 I. E. LAGER, M. STUMPF, AMPLITUDE-MODULATED, COSINE PE AND WP PULSES: THEORY AND APPLICABILITY 5. Conclusions Two types of causal, normalized, ringing pulses, namely the amplitude-modulated, cosine power-exponential (PE) and windowed-power (WP) pulses were discussed. Their definition makes use of a small set of intuitive parameters. The pulses’ ease of use is augmented by their Fourier transform being analytical. Some typical signatures and spectral diagrams were presented, these illustrating the pulses’ ability to mimic either ringing signatures with a sharp increase and a long tail (PE) or time-windowed harmonic functions (WP). A combination of PE and time-differentiated PE pulses was shown to accurately replicate time-domain behavior of pulses generated by existing circuitry, thus creating the conditions for incorporating realistic signatures in complex models or design frameworks via causal, analytic signatures. References [1] VARGA, L. Pulse shape discrimination. Nuclear Instruments and Methods, 1961, vol. 14, p. 24–32. DOI: 10.1016/0029-554X(61)90047-7 [2] ROUSH, M. L., WILSON, M. A., HORNYAK, W. F. Pulse shape discrimination. Nuclear Instruments and Methods, 1964, vol. 31, p. 112–124. DOI: 10.1016/0029-554X(64)90333-7 [3] LANGEVELD, W. G. J., KING, M. J., KWONG, J., et al. 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IEEE Antennas and Wireless Propagation Letters, 2017, vol. 16, p. 1659–1662. DOI: 10.1109/LAWP.2017.2662205 [17] ABRAMOWITZ, M., STEGUN, I. A. Handbook of Mathematical Functions. Mineola (USA, NY): Dover Publications, 1968. ISBN: 9780486612720 [18] FRANCESCHETTI, G., TATOIAN, J., GIBBS, G. Timed arrays in a nutshell. IEEE Transactions on Antennas and Propagation, 2005, vol. 53, no. 12, p. 4073–4082. DOI: 10.1109/TAP.2005.859765 About the Authors . . . Ioan E. LAGER was born in 1962. He received his engineer degree (M.Sc.) from the University of Braşov, Romania, and Ph.D. degrees from Delft University of Technology, Netherlands (1996) and from the University of Braşov (1998). His research focuses on applied electromagnetics, especially on time-domain phenomena, and nonuniform array antennas. Martin ŠTUMPF received his Ph.D.degree in Electrical Engineering from the Brno University of Technology (BUT), Brno, Czech Republic, in 2011. After his Ph.D. research, he spent a year and a half as a Post-Doctoral Fellow with the ESAT-TELEMIC Division, Katholieke Universiteit Leuven, Leuven, Belgium. He is currently an Associate Professor with the Department of Radioelectronics, BUT. During a three-month period in 2018, he was a Visiting Professor at the UAq EMC Laboratory, University of L’Aquila, Italy. He has authored the books “Electromagnetic Reciprocity in Antenna Theory” (Wiley–IEEE Press, 2017), “Pulsed EM Field Computation in Planar Circuits: The Contour Integral Method” (CRC Press, 2018), and “Time-Domain Electromagnetic Reciprocity in Antenna Modeling” (Wiley–IEEE Press, 2019). His main research interests include modeling of electromagnetic wave phenomena with an emphasis on EMC and antenna engineering.