For decades, the classical (unilateral) Laplace transformation (LT) has been offered as a mathematical tool for the calculation of signals at linear systems and for the solution of linear differential equations in the frequency domain. It was asserted - and widely believed - that LT principally has the following advantages, especially as compared to Fourier transformation (FT):
The third assertion also is "half true" - in an unfortunate sense. As will be shown below, the formalism allegedly provided by LT for incorporating a system's initial state is based on an artifact, i.e., on an inconsistency of LT's derivation theorem. The unfortunate aspect is that for many types of application the formalism works, although it fails in many other cases [77]. This has supported the belief that the theory of LT in principle is consistent and it is the user's fault when he runs into problems.
However, as it turns out, also the fourth assertion is wrong. LT cannot be regarded as mathematically consistent. This becomes particularly apparent when the derivation theorem and the shift theorem are more closely inspected. Below, I will demonstrate this principally for the derivation theorem.
The derivation theorem of LT says that to get the Laplace-transform of the (first) temporal derivative of a signal p(t) one must multiply the original signal's transform, P(s), by s and subtract the original signal's initial value p(0). So, the L-transform of the derivative is said to be sP(s)-p(0).
Many generations of students of electrical engineering and of other engineering sciences have been bothered with exercises on calculating system responses from the initial state, using LT. For some kinds of simple systems the method indeed works; however, for other systems it fails. The method is far from fool-proof. As a remedy of problems with "taking advantage" of the initial values, it has been recommended that, instead of the original signal's value at t=0, i.e., p(0), the value p(-0) should be used. However, one also frequently finds the advice that one should try p(+0). In a highly specialized, comprehensive book on LT I have found the recipe that one should use p(-0) when doing the calculations on an analogue computer, but p(+0) when using a digital computer. Professors who are wise and cautious give exercises to their students only with the presupposition that the system is in a non-excited initial state, such that the initial values can a priori be set to 0.
I have shown [70], [71], [77], that the initial value of the derivation theorem is an artifact, i.e., that the derivation theorem of unilateral LT, as it traditionally has been taught, is incorrect. The initial value of LT's derivation theorem emerges from the erroneous assumption that it were "allowed" - i.e., mathematically consistent - to confine observation and treatment of the signal to the time interval t > 0. Thus, when the derivative of a causal signal - the latter by definition being 0 for t < 0 - was considered, it was ignored what happens at t = 0. What happens is, that at t = 0 a delta impulse occurs whose energy is determined by the original signal's initial value p(0), i.e., the height of the step that in general occurs at t=0.
In any reasonable concept of Fourier-like transformation, that delta impulse must be regarded as pertinent to the derivative, such that the delta impulse must be taken into account when the derivation theorem is worked out. The original derivation theorem does not do this. Instead of referring to the true derivative, the theorem refers to the true derivative minus the delta impulse at t = 0. While the derivation theorem is correct with respect to the decapitated derivative, it is wrong with respect to the true one. The difference between the correct and the erroneous derivatives is the transform of the delta impulse, i.e. p(0). If the derivation theorem had been deduced from the true derivative signal, it would not include any initial value.
The reason why the proponents of unilateral LT confined consideration of the signal to the interval t > 0 was, that the step of the signal that in general occurs at t = 0, rigorously cannot be derivated. They obviously believed that they could circumvent this problem by just ignoring both the step and the delta impulse that by differentiation emerges from the step. The punishment for such hazardous thinking came along by
If certain conceptual implications of FT, and such LT as well, had been obeyed, one could hardly have come to thinking that one could confine consideration of the signal to the interval t > 0. This is because both FT and LT ("unilateral" or not) inevitably transform their argument signal function from the entire infinite time region, i.e., for -oo < t < +oo (see also topic Causal Fourier transformation). This is so because these transformations are continuous in terms of frequency. Therefore it is wrong to think that any transform P(s) could be treated just as if it were including information on a signal p(t) only for the time region t > 0 when the integral's lower limit was set to t = 0.
Paradoxically, the proponents of unilateral LT were inconsequent on their own decision to ignore the time interval t <= 0. If they had apprehended what the derivation theorem's initial value p(0) formally means, they should consequently have ignored it. The formal meaning of p(0) - such as that of any real constant in the frequency domain of LT - is that of a delta impulse at t = 0. So, why didn't they discard the initial value for being outside LT's range of definition? They ignored the delta impulse in the time domain but they failed on ignoring its "mirror image" in the frequency domain.
For my criticism on the derivation theorem of unilateral LT I earned quite contradictory responses. A prominent expert on systems theory entirely rejected my arguments, saying that the formalism of unilateral LT was mathematically correct. (So I was wrong). By contrast, from another expert, in his review of a paper on the topic that I had submitted to a prominent US journal, I got a rejection, saying that I had not offered anything new and that I was "... beating a dead horse." (So I was right, though too late.) This happened 15 years ago. When today I look into the contemporary literature on signals and systems, and into the contents of lectures on the topic, I find the horse alive and well. If there was any obituary on the horse, it does not appear to have been noticed. The belief in the benefit of the initial values still is prevalent. This strongly reminds me of the tale about the emperor's new dress.
What regards the view of formalists to whom "mathematical correctness" as such is more important than the message conveyed by the mathematical code, I am inclined to reply that mathematical correctness is (or should be) more than just formal correctness, i.e., that formal correctness does not necessarily imply mathematical correctness. This was very well expressed by one of history's greatest mathematicians, C.F. Gauss:
Es ist der Charakter der Mathematik der neueren Zeit .., daß durch unsere Zeichensprache und Namengebung wir einen Hebel besitzen, wodurch die verwickeltsten Argumentationen auf einen gewissen Mechanismus reduziert werden... Wie oft wird jener Hebel eben nur mechanisch angewandt, obgleich die Befugnis dazu in den meisten Fällen gewissen stillschweigende Voraussetzungen impliziert. Ich fordere, man soll bei allem Gebrauch des Kalküls, bei allen Begriffsverwendungen sich immer der ursprünglichen Bedingungen bewußt bleiben, und alle Produkte des Mechanismus niemals über die klare Befugnis hinaus als Eigentum betrachten...
From the misconception on which unilateral LT is based there inevitably originates another inconsistency, namely that of the shift theorem. While the causal signal's step in t=0 was so carefully hidden outside LT's range of definition, it became hard to ignore it any longer when the signal was somewhat delayed (i.e., right-shifted). When, on the other hand, a causal signal is left-shifted (negative delay), it becomes impossible to describe it correctly by LT, as the latter doesn't know anything about the time region t <= 0. I wonder if anyone has ever taken advantage of the complicated formalism that Doetsch (1967a) has offered to account for a negative delay.
The inconsistency of LT also becomes drastically apparent when one combines shift with derivation.When by a positive delay the step of the causal signal becomes visible within the scope of LT, nobody being in his right mind can deny that the derivative of the delayed signal includes a delta impulse. When the derivation theorem for the delayed signal is worked out, the transform of the delta impulse, i.e. p(T), cancels out the initial value of the orignial theorem, i.e. -p(T), where T is the delay. Consequently, the derivation theorem for the delayed signal will no longer include an "initial value". So we have the paradoxical situation that LT actually has two different derivation theorems, i.e., one for the undelayed causal signal and another for the same signal with a positive delay. The alleged advantage of LT, i.e., that it enables accounting for a system's initial state, disappears as soon as the signal is delayed by an infinitesimally small amount. That's not quite a reliable advantage, is it?
In summary: Unilateral LT is not tenable and should quickly be abandoned.
It obviously has existed not without considerable benefit. The theory of
correspondences and the tables that have been created will continue to
be an invaluable heritage. I believe that Causal Fourier
transformation is a most promising replacement for LT.