Question on understanding the time-bandwidth of a signal I read from a paper (Ozaktas, H.M., et al. (1996) Digital Computation of the Fractional Fourier Transform) that says
The time-bandwidth product can be crudely defined as the product of the temporal extent of the signal and its (double-sided) bandwidth. It is equal to the number of degrees of freedom and the number of
complex numbers required to uniquely characterize the signal among others of the same time-bandwidth product.
I understand the first statement, but the second statement is hard to interpretate.
First, why is the time-bandwidth product equal to the number of degrees of freedom and what is the number of degrees of freedom of a signal? Does it hold for both the deterministic and random signal?
Second, why is there a (minimum) number of complex numbers required to characterize a signal? What if the number is not reached?
Hope you can help clarify these confusions.
This is not a rigorous answer.

First, why is the time-bandwidth product equal to the number of
degrees of freedom and what is the number of degrees of freedom of a
signal? Does it hold for both the deterministic and random signal?

The degree of freedom is the number of samples needed to represent a complex continuous-time signal. The Nyquist Shannon sampling theorem states that for any signal confined within $\left[-\frac{\Delta f}{2},+\frac{\Delta f}{2}\right]$, we need at least $\Delta f$ sample per time unit to perfectly reconstruct it.
If in the time domain, the signal can be approximated by its duration $\left[-\frac{\Delta t}{2},+\frac{\Delta t}{2}\right]$, the number of samples needed is thus $\Delta f \Delta t$, which is the time-bandwidth product defined in [Ozaktas et al.].

Second, why is there a (minimum) number of complex numbers required to
characterize a signal? What if the number is not reached?

If the number is not reached, the discrete-time signal represented by the insufficient number of samples cannot be used to reconstruct the continuous-time signal. An aliasing will be observed, for example.