Reconstructing Signal From Its Cyclic Autocorrelation Can a signal be reconstructed from its cyclic autocorrelation? Specifically, if we know

$$
R^{\alpha}(\tau) = \int{x(t)x^{\ast}(t-\tau)e^{-j2\pi\alpha t}\mathrm{d}t},
$$

can we reconstruct $x(t)\in\mathbb{C}^{N}$? It seems like the answer should be yes, but it's not obvious to me how to do it. I know there has been work on reconstructing a signal from its autocorrelation, and these algorithms of course require some other knowledge about the signal (e.g., Discrete signal reconstruction from its autocorrelation function and one sample). I guess I am hoping knowing the cyclic autocorrelation will at least relax the required assumptions. Moreover, how does the problem change if we also had the conjugate version:

$$
\bar{R}^{\alpha}(\tau) = \int{x(t)x(t-\tau)e^{-j2\pi\alpha t}\mathrm{d}t}
$$

Edit: Let me try rewording the question. I understand that autocorrelation is not an isomorphism. Is this also true of cyclic autocorrelations? Specifically, can we recover $x\in\mathbb{C}^N$ from

$$
R(\alpha,m)=\sum_{n=0}^{N-1}x(n)x^{\ast}(n-m)e^{-j2\pi n \alpha}.
$$

If we only have knowledge for $\alpha=0$, then we have just the regular autocorrelation and the answer is in general no. However, does knowing $R$ for $\alpha=0,1/N,2/N,\dots$ enable one to recover $x$ or at least reduce the required assumptions necessary for doing so? If not, what if the conjugate version is also available? Can we recover $x$ then?
No, you cannot reconstruct the original signal from the cyclic autocorrelation. The fundamental reason is that it results from an averaging operation. Like the autocorrelation and PSD, the cyclic autocorrelation is a many-to-one functional (you already noted that I think). For example, all BPSK signals with identical timing parameters (symbol clock phase and carrier phase), identical pulse functions, and IID symbols have the exact same cyclic autocorrelation functions. Yet there are an infinite number of such signals obtained by all the different sequences of transmitted bits. In general, there is no single unique signal that gives rise to a particular cyclic autocorrelation.

I note that none of the equations you wrote are the cyclic autocorrelation. In general, it is the infinite time average of the second-order lag product of the signal multiplied by a complex sine wave. Aside from the missing scale factor $1/N$, your third equation is an estimate of the cyclic autocorrelation for discrete-time signals.

There are special cases, though, such as periodic signals. Such signals are non-random. The time-varying autocorrelation function $R_s(t, \tau)$ for such signals is identically equal to the second-order lag product itself $s(t+\tau/2)s^*(t-\tau/2)$. Since the time-varying autocorrelation is composed of the sum of sine-wave-weighted cyclic autocorrelations, one might recover $s(t)$ from the set of cyclic autocorrelations, to within a scale factor of $\pm 1$.

The cyclic autocorrelation is sensitive to phase in the sense that the cyclic autocorrelation for $s(t)$ differs from that for $s(t-D)$ in general. This is a basis for synchronization algorithms.

I'm curious why it is obvious to you that there "should be a tremendous amount of redundancy in the cyclic autocorrelation." Redundant what?