Let's look at what is generally known:
Time domain is generally inversely related to frequency domain in as much as if you have a narrow width in the time domain the frequency domain spectrum will be wider. and vice versa. When you have narrower time domain pulses you need higher frequencies to represent it in the frequency domain therefore more bandwidth -> wider plot.
Looking at your sinc function to find when the function will go to zero you simply need to look at the sin aspect of it. At what values does sin = \$0\$ ? Answer \$ 0, \pi , 2\pi , 3\pi \$ ... That means the value \$ {\tau}f = 0, \pi.2\pi,3\pi ... \$
There is one exception here, as \$ {\tau}f\$ -> \$0\$ sinc ->\$1\$ becasue the sin and the \$\tau\$ cancel in the limit. so you only need to look at \$\pi,2\pi,3\pi, ...\$
Lets pick \$\tau=1\$ then the rect function will run from \$ -\frac{\tau}{2}\$ to \$ \frac{\tau}{2}\$ and the first zero in the frequency domain will be at f=1.
Lets pick \$\tau=2\$ then the rect function will run from \$ -\frac{\tau}{1}\$ to \$ \frac{\tau}{1}\$ and the first zero in the frequency domain will be at f=\$\frac{1}{2}\$.
a wider time pulse means a narrower frequency band.
Lets pick \$\tau=\frac{1}{2}\$ then the rect function will run from \$ -\frac{\tau}{4}\$ to \$ \frac{\tau}{4}\$ and the first zero in the frequency domain will be at f=\$\frac{2}{1}= 2\$.
a narrower time pulse means a wider frequency band.
How do you calculate the power and energy of a signal given only the frequency domain form of the signal function?
For power you do it in exactly the same way that you would in the time domain;
Time domain: \$ P(t)=V(t)I(t) \$
S-Domain: \$ \tilde{P}(\omega)=\tilde{V}(\omega)\tilde{I}(\omega) \$
If you don't know \$ \tilde{V}(\omega) \$ and \$ \tilde{I}(\omega) \$ Then the situation is the same as if you were to try and calculate the power in the time domain without knowing \$ V(t) \$ and \$ I(t) \$, it is not possible.
For energy you pick any number you like..
The s-domain does not care about time. If we assume that you have a signal at 1kHz with a power of 1W. if you observe this signal for 1s in the time domain then the energy is 1J, is you observe it for 1000s in the time domain then the energy is 1kJ.
Best Answer
Many people mistake necessary and sufficient reasons. Or if they don't mistake them, they don't spell out the difference, thinking it's obvious.
A time-limited signal will be unlimited in the frequency domain.
It's necessary that a signal be time unlimited, for it to be able to have a limited band in the frequency domain, but that's still not sufficient.
We can construct the spectrum of the rectangular pulse train using the convolution rule. As the rectangular pulse train is a pulse, convolved with an infinite series of impulses, its spectrum will be the spectrum of the pulse, multiplied by the spectrum of an infinite series of impulses.
As the spectrum of an infinite series of impulses is itself an infinite series of impulses, the total spectrum will still go out to infinity.
If instead you had a time signal that was a rectangular pulse convolved with (say) a Gaussian pulse (which is limited in both time and in its spectrum), then the spectrum of that would indeed be frequency limited.