General solution that have the rectangular function as amplitude of the Fourier transform?

Question

The sinc function has as Fourier transform the rectangular function. However, this is only one function that produces the rectangular function as amplitude response. Supposing that you are only interested in the amplitude (absolute value) component of the Fourier transform being equal to the rectangular function (so the phase can be anything), what would be a general solution?

So, in equation:

$$ |h(\omega)| = \left|\int_{-\infty}^{\infty} f(t)e^{-2{\pi}i{\omega}t}\, dt\right|=A \cdot rect\left(\frac{\omega}{B}\right) $$

With A and B real scale and rate parameters, (real) $ \omega $ angular frequency, and $ \left| g \right| $ the absolute value of complex value $ g $.

Does this have a general solution in closed form? Or, alternatively, a way to approach the problem numerically?

PS: this question was first asked on the mathematics stack exchange, but as was pointed out, may fit better on the DSP stack exchange.

Answer 1

No matter what phase function you choose, you can always represent the final transfer function as the cascade of a real rectangle and an allpass filter.

There are quite a few different types of allpass filters around and many different choices of a phase function. See picture below for a few example: pure sinc, random phase and the application of a few Schroeder Allpass filters (which are often used in reverb algorithms). The time domain signals look completely different but the magnitude response is identical for all.

Does this have a general solution in closed form? Or, alternatively, a way to approach the problem numerically?

Numerically it's easy enough: apply your phase and calculate the inverse Fourier Transform. I don't think you can do a closed form without constraining the phase function in some way.

Answer 2

The most general expression for the impulse response of an ideal lowpass filter with cut-off frequency $\omega_0$ and with phase $\phi(\omega)$ is

$$h(t)=\frac{1}{2\pi}\int_{-\omega_0}^{\omega_0}e^{j[\phi(\omega)+\omega t]}d\omega\tag{1}$$

If we restrict ourselves to real-valued lowpass filters, for which the phase must satisfy $\phi(\omega)=-\phi(-\omega)$, Eq. $(1)$ can be written as

$$h(t)=\frac{1}{\pi}\int_{0}^{\omega_0}\cos[\phi(\omega)+\omega t]d\omega\tag{2}$$

From $(1)$ and $(2)$ it is clear that in general there is no closed-form solution for $h(t)$. There might be a closed-form solution for a few special cases, such as a linear phase $\phi(\omega)=-\omega\tau$, in which case Eq. $(2)$ produces the well-known result

$$h(t)=\frac{\sin[\omega_0(t-\tau)]}{\pi(t-\tau)}\tag{3}$$

For more general phase functions, we could resort to numerical integration techniques to obtain an approximate solution of $(2)$.