Discontinuities and high-frequency band filters

Let $L(\omega)$ be the spectral characteristics of a high frequency band filter. We suggest that $L(\omega)\simeq 0$ at $\vert\omega\vert<\omega_{c}$ and that at $\vert\omega\vert\gt\omega_{c}$,Fourier-transformation of an input signal may be described as

$$F(\omega)\simeq {A_{r} \over {(i\omega)}^{r+1}}.$$

Then Fourier-transformation of the output signal g(t) will be equal to $G(\omega) \simeq {A_{r}L(\omega)\over {(i\omega)}^{r+1}}$. It means that

$$g(t)\simeq A_{r}l_{r+1}(t)$$

where l_r+1(t) is (r+1)^th integration of the impulse response. We have obtained two facts: (a) the only feature of the input signal that influences the shape of output signal is the order of its discontinuity, and (b) the amplitude of the discontinuity coincides with the amplitude of output signal. We conclude that the output signal is a response to the discontinuity. This conclusion may be interpreted from the point of view of the theory of ordinary differential equations. Let us assume that we have a filter which is described by the equation  

$$a_{n}y^{(n)}+ \ldots + a_{0}y=0.$$ (17)

If the input is $\sin \omega _{0}t \cdot H(t)$, then the output is y(t)=y₀(t)+y₁(t) where

$$y_{1}(t)=\vert L(\omega _{0})\vert \sin (\omega _{0}t-\varphi (\omega_{0}))$$

represents the particular solution with $L(\omega )={1 \over a_{n}(i \omega )^{n}+ \ldots a_{0}}$.y₀(t) describes the free oscillation that is the solution to a Cauchy problem for the equation (17) with conditions

y^(k)(0+)=A_k

(here $A_{0}=0, A_{1}= \omega _{0}, A_{2}=0, A_{3}= -\omega_{0}^{3}, \ldots$). Therefore, we see that the free oscillations are a response to discontinuities.

Let us now assume that a right-sided quasi-sinusoidal signal f(t) is the input of the filter L which has very short impulse characteristics l(t). For times comparable with the duration of l(t), the filter ``does not know" that f(t) differs from $\sin (\omega_{0} t)$.Therefore one can expect that the response will contain two parts, the first part is y₀(t), the reaction to the beginning of the signal (that is, to the discontinuity!) with the dominating frequency of the l(t). The second part represents the response to the signal itself with the dominating frequency of the signal f(t). A corresponding theory was proposed by S. Katz (1966) (see also my book (1974)).