I learned about the inverse system. INV

Suppose I have an impulse response $h(t)$ which is a box function. ($A = 1, T = 2$) BOX

If I take the Fourier transform and referring the transform pairs, $$H(j\omega) = 2\operatorname{sinc}(\omega), \\ H(j\omega)H_1(j\omega) = 1, \\ H_1(j\omega) = \frac{1}{H(j\omega)} = \frac{1}{2\operatorname{sinc}(\omega)}$$ where $\operatorname{sinc}(u) \triangleq \frac{\sin(u)}{u}$ , right?

Then, what is the reverse of this Fourier Transform $\frac{1}{2\operatorname{sinc}(\omega)}$?

Or instead of using the Fourier Transform, should I use a different approach to get the inverse system of the box function?

  • 3
    $\begingroup$ Note that not all systems are invertible. For your example, consider a cosine with period $T$ as an input signal $x(t)$. Then your box function will yield zero, i.e., $y(t)=0$. Therefore, no matter how you design $h_1(t)$, you can never recover $x(t)$. More generally, almost any system will filter out some frequencies (consider e.g. lowpass filters), which are lost and cannot be recovered afterwards. $\endgroup$ – Florian May 28 '19 at 8:35
  • 3
    $\begingroup$ A general inverse does not exist. $H(j \omega)$ is zero for some values of $\omega$ and inverting this would result in "divide by zero". You can still create "approximate" inverse, but that's more complicated and depends a lot on the specific application. $\endgroup$ – Hilmar May 28 '19 at 12:41

As pointed out in the comments, the inverse system is not realizable. However, it is possible to write down an expression for the impulse response of the inverse system. We want a system that results in a Dirac impulse when convolved with the given rectangular pulse $h(t)$. This can be done as follows. We delay the rectangular impulse by $T/2$, so it starts at $t=0$. Then we add an infinite number of delayed versions of it such that the result becomes a step function $u(t)$. Each delay is an integer multiple of $T$:


The system that achieves this has an impulse response given by


Now we have a step function, from which we can generate a Dirac impulse by taking the derivative. So the inverse system is given by the concatenation of the system with impulse response $f(t)$ given by $(2)$ and an ideal differentiator. The total impulse response of the inverse system is then


where $\delta'(t)$ is the distributional derivative of the Dirac impulse.

Clearly, the system with impulse response $g(t)$ given by $(3)$ is not realizable, but as a theoretical concept it is interesting, just like other ideal systems (e.g., ideal frequency selective filters, differentiators, Hilbert transformers, etc.).


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