Causal Fourier transformation - CFT

Causal Fourier transformation, CFT, is a new mathematical method for calculating signals at linear, time-invariant systems in the frequency domain, and for solving linear differential equations by operator calculus [68], [104] p. 71-75. It can be used as a replacement of conventional Fourier transformation (FT) and, in particular, of Laplace transformation (LT).

The advantages of CFT, as compared to FT and LT, are the following.

• For physically possible causal signals p(t), CFT does not have any restrictions concerning convergence.
• The transformation of any time signal p(t) into the corresponding frequency function P(s) is simpler than for FT and LT as there is no need to determine any limit value for t®¥.
• CFT does not suffer from certain inconsistencies such as those of LT (see topic Laplace transformation).
• The complex frequency functions P(s) obtained by CFT are genuine Fourier spectra, i.e., dependent on imaginary frequency s = jw, as opposed to complex frequency such as for LT.
• As the frequency functions are genuine Fourier spectra, the frequency function of a linear system's impulse response, i.e., the system's transmission function, is identical to the system's "frequency response" (frequency characteristic).
• The frequency functions obtained by CFT are formally identical to those obtained by unilateral LT (only their implications are somewhat different). Therefore the comprehensive tables of correspondences of LT can be taken advantage of in using CFT.
• CFT provides a clean and safe theoretical basis to the method of operator calculus, as opposed to both FT and LT. (FT fails for many important types of signal by lack of convergence; LT fails because of its inherent inconsistencies.)

The CFT is a by-product of my endeavours to achieve deeper conceptual insights into Fourier transformation [68]. CFT is causal in two respects:

• CFT by definition operates exclusively on causal signals, i.e., signals that are zero previous to a certain finite instant t₀.
• CFT was designed for employment on causal systems, i.e., systems that can respond only to stimuli that previously have occurred.

Obviously, for real-life (analogous) signals and systems, these two confinements do not impose any restriction. It is these confinements that provide for the definition of CFT to be legitimate.

This is the definition of CFT:

Here p(t), P(s) denote the (real) signal and its (complex) transform, respectively; t₀ denotes the instant of time at which the signal starts to be different from 0, and s = jw. The particular notation of the integral in Eq.(1) is meant to indicate that from the integral only the lower limit must be taken, i.e., with the argument t₀, and with a negative sign. So the crucial trick involved here is, that the contribution given by the integral's upper limit (which in conventional FT is +¥) a priori is discarded. This makes immediately apparent that, indeed, there is no problem of convergence and no need to be worried about determination of the limit value for t®¥.

If the contribution of the integral's upper limit were not discarded, the above transformation just were conventional Fourier-transformation of a causal signal. So, as compared to conventional FT, the CFT in fact is reduced, i.e., to the contribution of the integral's lower limit alone. This is why in [68] I have termed it RFT (Reduced Fourier transformation). However, the label CFT obviously fits better.

So, when one lets alone for an instant the question what are the implications of discarding the upper limit, it is apparent that the inverse transformation to CFT is identical to that of conventional FT, i.e.

On first sight, the definition of CFT Eq.(1) may look like dirty mathematics. However, in fact it is just reasonable mathematics; it takes advantage of conceptually understanding essential implications of Fourier-transformation.

The crucial notion is that, for obtaining a legitimate transform, formal convergence for t®¥ of the transformation integral is not required. One can merely define that the integral's upper limit converges; even more specifically, that for t®¥ it converges to zero - such that, indeed, it does not contribute to the transform.

To understand this, note that for the determination of causal signals at a linear causal system one can employ a kind of Fourier transformation that integrates only to a finite instant t₁, i.e.,

This in fact is conventional FT (i.e., FT with infinite limits at the integral) of a causal signal f(t) that starts at t₀, ends at t₁, and is identical to zero outside the interval t₀ < t < t₁. Though F(s) is different from the conventional FT transform of the same but infinitely ongoing causal signal (provided that that transform exists), there is no difference with respect to the time interval t < t₁. This is why the transformation Eq.(3) is legitimate and yields for causal linear systems mathematically correct results in the time interval t < t₁.

The transformation Eq.(3) obviously does not have any problem of convergence, since the limits of the integral both are finite. Ordinarily the value of the integral's upper limit is not zero; it depends, besides the signal f(t), on t₁. Therefore, in general the arbitrary parameter t₁ is included in the transform F(s) and any terms dependent on it render the frequency function more complicated than actually is necessary. The contribution of the upper limit to F(s) indeed turns out to be redundant. By inverse transformation one can show that the portion of F(s) that is dependent on t₁ contributes to the recovered signal only in the interval t > t₁.

So, when one splits the transform F(s) defined by Eq.(3) into the two parts associated with the integral's limits, i.e.,

one can conclude from the above that F₁(s) as such represents a causal signal f₁(t) that starts at t₁; and F₀(s) represents a causal signal f₀(t) that starts at t₀. The role played by f₁(t) merely is to cancel the ongoing signal f₀(t) from the instant t₁ on. Therefore, in the interval t < t₁ it is the signal f₀(t) alone that provides the transform. This is why F₁(s) can a priori be discarded.

The redundancy of the transformation integral's upper limit may once more be demonstrated in the following way. From the above notions it is apparent that, in the transformation integral of conventional FT, the signal has to be correct only in the interval t < t₁, where t₁ denotes an arbitrary finite instant that either indicates the end of the observation interval or is beyond that. In the interval t > t₁ one can assign any desired structure to the signal, i.e., different from the original one, without doing any harm to the results for the interval t < t₁. In particular, one can assign a structure to the signal for t > t₁ that enforces convergence of the intergral to zero for t®¥. Provided that one can be sure that such a structure does exist for any given physical signal p(t), one does not even have to specify it in any particular case. If it is possible in this way to eliminate the contribution of the integral's upper limit, one can indeed just ignore that contribution, as is implied in the definition of CFT, Eq.(1). In fact there is at least one apparent way to enforce convergence of the upper limit to zero: One can make the complex signal p(t)exp(-st), like it is given in the interval t₀ < t < t₁, be proceeded by the same but sign-inverted signal, i.e. extending through the interval t₁ < t < 2t₁-t₀.

Finally, there is yet another, most significant proof; it is provided by unilateral LT, i.e., the transformation

where s = s+jw, and s > 0. The L-transform exists (which means that the integral converges at the upper limit), if for any real, positive, and finite constants (C, a) the condition

is fulfilled. Thus, for the L-transform of any causal signal p(t) to exist, it is required that the integral

converges for t®¥. As one can discern from Eq.(7), this is the case if, and only if, s > a. In that case, however, convergence inevitably is convergence to zero. So, indeed, if the transformation integral of LT converges at all for t®¥, it converges to zero. In the literature on LT I have not found any mentioning of this remarkable fact.

This fact puts Laplace transformation into the position of being just an awkward method to attain the same goal that in CFT is attained by definition, i.e., making the contribution of the transformation integral's upper limit zero. Moreover, the detour taken in LT to attain that goal imposes unnecessary restrictions as to the types of signal that can be transformed. CFT does not have any such restrictions.

It is now obvious that, and why, the CFT transforms are formally identical to those of LT, i.e., for to=0. What is different is only, that

• in CFT the frequency variable is purely imaginary, i.e., s = jw;
• CFT transforms are not conceptualized as being unilateral, i.e., valid only in the interval t > 0;
• CFT transforms do exist even of signals for which the LT integral does not converge.

In fact, it is important to apprehend that in CFT the signals are a priori conceptualized as being defined in the entire interval -¥ < t < +¥, just as in conventional FT. The fact that the signals are presupposed to be zero in the interval t < t₀ does not make them unilateral - at least if one is prepared to concede that a signal that is defined to be zero indeed is a signal.

The practical advantage of the formal identity of LT and CFT transforms is obvious: On can take advantage of the wealth of premanufactured correspondences of LT that is available in tables. So, when the LT transform of a given signal is found in the table, it can be immediately interpreted as the CFT transform for to=0. The transform for any non-zero value of to is obtained by employing the shift theorem of FT, i.e., multiplication by exp(-sto).

As CFT actually is a Fourier transformation, i.e. the causal one, all theorems known for FT do apply. In particular, the derivation theorem is just as for FT, i.e., not including any initial value (see topic Laplace transformation). Those who have believed in the benefit of the "initial values" included in LT's derivation theorem may ask how the initial state of a system can be accounted for when CFT is employed. The answer is that it can be done even more readily than with LT, because there are no a priori formal constraints. The basic idea is, that when a linear causal system at the beginning of observation includes energy (i.e., is in a non-zero "initial state"), this can originate only from the fact that the system previously somehow has been excited. When, in addition, another excitation signal is applied, the system's response merely is the sum of the responses to the two excitations. The details and the relationships of this approach to the conventional method of LT were outlined in [77].

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