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

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.


Introduction
Pulse shape (discrimination) is a topic of broad scientific interest, ranging from particle physics [1][2][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][6][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][10][11][12].
At this point, it is noted that all explorations in [5][6][7][9][10][11][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.

Definition of Pulses
The PE and WP unipolar prototype pulses were elaborately discussed in [14], [16], their TD expressions being and respectively. In these expressions ν 1 is the pulse raising power, t r is the pulse rise-time2, t is the time coordinate, t = t/t r is the normalized time coordinate and H(·) is the Heaviside step function. For practical reasons, ν is taken to assume only integer values. Furthermore, taking ν 3 ensures the continuity of the pulses and their first two time derivatives at t = 0, and at t = 2 for 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 t r > 0 is 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, t r also represents the zero-crossing time of the relevant pulses.
The PE was used in [14] for constructing the ringing cosine PE cos pulse via the expression PE cos (ν, t r , κ w , t) = PE(ν, t r , t) cos(2πω w t ) (5) in which κ w is a 'wiggle factor' that, for a harmonic carrier of frequency f 0 , counts the carrier cycles per t r as κ w = t r f 0 , andω w = 2πκ w . By using the same strategy, a WP ringing cosine pulse is defined via The frequency-domain (FD) behavior of PE cos and WP cos can be easily derived, the relevant expresions being PE cos (ν, t r , κ w , ω) and WP cos (ν, t r , κ w , ω) 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 ∂ t PE, ∂ t WP, ∂ t PE cos and ∂ t WP cos . Their TD and FD expressions follow from the corresponding, original expressions (the FD behavior requiring simply a multiplication by jω).

Normalization of Pulse TD Signatures
The normalization of the ∂ t PE and ∂ t WP 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, PE cos and WP cos 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 t r and t as arguments of PE and WP will be also omitted.
By time-differentiating (5), it follows that in which for t > 0 (10) with β 1 = 1/ω w . By substituting (10) in (9), and using some standard trigonometric identities it is found that Since The shape of F 1 (t ) is dictated by PE(ν − 1) that peaks at 1 for t = 1. It can now be concluded that the envelope of the ∂ t PE cos pulse is normalized by multiplying the time derivative by 1/ω 0 = (2π f 0 ) −1 , a fact already stated in [14,Eq. (15)].
By time-differentiating (6), it follows that in which (14) with β 2 = ν/(πκ w ). By substituting (14) in (13), and using some standard trigonometric identities it is found that (15) in which 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 t = 1 and takes the values β 2 for t = 0 and t = 2, and 1 for t = 1. The shape of F 2 (t ) is dictated by WP(ν − 1) that peaks at 1 for t = 1. It can now be concluded that the envelope of the ∂ t WP cos pulse is also normalized by multiplying the time derivative by 1/ω 0 = (2π f 0 ) −1 .

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.

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 PE cos pulses with t r = 1, κ w = 2, and ν assuming the values 3 and 6, respectively. The TD plots illustrate the effect of ν on the pulse's shape. The spectral behavior has a nonzero DC value and peaks, as expected, at f 0 3.
The ∂ t PE cos 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.   3The definition of f norm entails that f 0 / f norm = 1/ν, implying that the spectral diagrams' peaks correspond to the abscissas -0.47 and -0.78 for ν = 3 and 6, respectively.
The spectral behavior resembles that of the PE cos above f 0 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 ∂ t PE cos 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 WP cos pulses with t r = 1, κ w = 4, and ν = 4, 8, respectively. Again, the effect of ν on the pulse's shape is noticeable. The spectral behavior bears some overall resemblance  with that in Fig. 1, but the lobe around f 0 is 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 ν in these experiments). The corresponding ∂ t WP cos 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]).

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 S orig (t), while S repl (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 100 (17) in which T is the interval between the earliest among the onsets of S orig and S repl and the end S orig .
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 approach5 employing a grid-search over ν, t r and f 0 (and, implicitly, κ w ) 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 t r and f 0 by measuring some features of S orig . 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 ∂ t PE pulse with a high ν. Based on these tests, a PE with ν = 4 and a ∂ t PE with ν = 24 were selected. The next phase was a full grid search on ranges of t r and f 0 values centered around the initial guesses. The constructed pulses' magnitudes (V m ) 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) t shift time shifts with respect to the onsets in S orig . 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 V m = 3.94 V, ν = 4, t r = 0.52 ns, f 0 = 1.67 GHz (κ w = 0.87); the replica is delayed by t shift = 119 ps. The replication error amounts to Err % = 5.2%. The constructed S repl is superimposed on S orig in Fig. 5. 2. For the signature in [11,Fig. 2d]: A ∂ t PE pulse with V m = 2.35 V, ν = 24, t r = 1.06 ns, f 0 = 1.84 GHz (κ w = 1.95); the replica is delayed by t shift = 7 ps. The replication error amounts to Err % = 2.2%. The constructed S repl is superimposed on S orig in Fig. 6.
Another example of mimicking a measured signal via a ∂ t PE 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. 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.

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.