Inverse system of a box function

I learned about the inverse system. Suppose I have an impulse response $$h(t)$$ which is a box function. ($$A = 1, T = 2$$) 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?

• 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. – Florian May 28 at 8:35
• 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. – Hilmar May 28 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 $$T$$:

$$u(t)=\sum_{n=0}^{\infty}h(t-T/2-nT)\tag{1}$$

The system that achieves this has an impulse response given by

$$f(t)=\sum_{n=0}^{\infty}\delta(t-T/2-nT)\tag{2}$$

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

$$g(t)=(f\star\delta')(t)=\sum_{n=0}^{\infty}\delta'(t-T/2-nT)\tag{3}$$

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.).