Since the spectrum of a continuous time
signal is no longer periodic, it becomes now interesting to look at the effective
support in frequency of said spectrum. So for continuous time signals, there is
new concept called bandlimitedness and a bandlimited function is a function
whose spectrum has a finite support. Or in other words,
there exists a frequency Fs. So that the Fourier transform is 0 for f outside of the minus Fs
over 2 to Fs over 2 interval. The total support, therefore, of the function is Fs hertz. What is the prototypical
bandlimited function then? Well, the simplest spectrum that we
can imagine is something that is equal to some constant over the finite
support of with Fs and 0 everywhere else. We have already seen this kind of spectrum
when we studied the ideal low-pass. And if we do the exercise again, this time
in continuous time, we take the inverse Fourier transform of this spectral shape
which is of course, a rect of with Fs. And we find that in the time domain, the expression for
the bandlimited function will be a sinc. We can still play with the constant
G's to the things turn out pretty. So let's see,
we want a total bandwidth of Fs hertz. We defined Ts as 1 over Fs. This would be the period
associated to the frequency Fs. And if we choose the normalization
constant G equal to one of our Fs or equal to Ts, we obtain the following pair. In the frequency domain,
we have that the prototypical bandlimited function will be a rect of with Fs and
amplitude 1 over Fs. And in the time domain, we will have a sinc with time
constant Ts equal to 1 over Fs. How do these things look graphically? So this is our prototypical
bandlimited function in the frequency domain with support Fs. And in the time domain,
we have a sinc function that intersects the horizontal axis at every multiple
of Ts where Ts is 1 over Fs. When Ts is equal to 1, this formula
has become particularly simple. So in the frequency domain,
we have rect(f). In the time domain, we have sinc(t). Graphically, we have
a support of one hertz. And in the time domain,
the sinc will be equal to 0 every second.